Lectures on Algebraic Geometry

G. Harder

Table of contents

Table of contents I. Categories, products, projective and inductive limits

I.1. The notion of a category and examples  S. 1
I.2. Functors  S. 3
I.3. Products, projective limits and direct limits in a category  S. 4

II. Basic concepts of homological algebra

II.1. The category Mod_{Γ} of Γ-modules:  S. 12
II.2. More functors  S. 14
II.3. The derived functors  S. 20
II.4. The functors Ext and Tor  S. 29

III. Presheaves and sheaves

III.1. Presheaves  S. 36
III.2. Manifolds as locally ringed spaces  S. 40
III.3. Stalks and sheafification  S. 43
III.4. The functors f_{∗} and f^{∗}  S. 46
III.5. Products  S. 48
IV. Cohomology and sheaves  S. 53
IV.1 Examples  S. 53
1.1 Sheaves on Riemann surfaces  S. 53
1.2 Cohomology of the circle  S. 55
IV.2 The derived functor  S. 58
2.1 Introduction  S. 58
2.2 A direct definition of H^{1}  S. 59
VI.3 Fiber bundles and non abelian H^{1}  S. 62
3.1 Fibrations  S.62
3.2 The non abelian H^{1}  S.64
3.3 The reduction of the structure group  S.65
IV.4 Fundamental properties of the cohomology of sheaves.  S. 69
4.1 Introduction .  S. 68

4.2 The derived functor to f_{∗}  S. 68
4.3 Functorial properties of the cohomology.  S. 70
4.4 Paracompact spaces.  S. 72
IV.5 Čech cohomology of sheaves  S. 79
5.1 The Čech complex  S.79
5.2 The Čech resolution of a sheaf.  S.83
IV.6 The spectral sequence  S. 86
6.1 Introduction  S. 86
6.2 The vertical filtration  S. 89
6.3 The horizontal filtration  S. 94
6.4 The derived category  S. 97
6.5 The spectral sequence of a fibration  S. 102
6.6 Čech complexes and the spectral sequence.  S. 104
6.7 Products  S. 111
6.8 An excursion into homotopy theory  S. 116
IV.7 Cohomology with compact supports  S. 120
7.1 The definition  S. 120
7.2 An example  S. 121
7.3 The fundamental class  S. 123
IV.8 Cohomology of manifolds  S.124
8.1 Local systems  S.124
8.2 The Čech resolution of local systems  S.125
8.3 Poincaré-Duality.  S.130
8.4 The cohomology in top degree and the homology.  S.133
8.5 The fundamental class of a submanifold  S.139
8.6 Cup product and intersections  S.140
8.7 Compact oriented surfaces  S.141
8.8 The cohomology rimg of ℙ^{n}(ℂ)  S.142
IV.9 The de-Rham and the Dobeault isomorphism  S. 143
9.1. Real manifolds  S. 143

9.2. Cohomology of holomorphic bundles on complex manifolds  S. 147
9.3. Chern classes  S. 152
IV.10 Hodge theory  S. 155
10.1. Real manifolds.  S. 155
10.2 Complex manifolds.  S. 160
10.3 Hodge Theory on Tori.  S. 165
V.Compact Riemann surfaces and Abelian Varieties  S. 167
V.1 Compact Riemann Surfaces  S. 167
1.1 The Hodge structure on H^{1}(S, ℂ)  S.168
1.2. Cohomology of holomorphic bundles  S. 173
1.3. The theorem of Riemann-Roch  S. 180
1.4. The algebraicity of Riemann-Roch surfaces  S. 185
1.5 The Jacobian of a compact Riemann surface  S. 199
1.6 The classical version of Abels theorem  S. 200
1.7. Riemann period relations  S. 202
V.2 Line bundles on complex tori  S. 207
2.1. The construction of line bundles  S. 204
2.2 Homomorphisms between complex tori  S. 210
2.3. The self duality of the Jacobian  S. 214
2.4. Ample line bundles and the algebraicity of the Jacobian  S. 215
V.3 Towards an algebraic theory  S. 236
3.1 Introduction  S. 233
3.2 The structure of End(J)  S. 238
3.3 An algebraic substitute for the cohomology  S. 246
VI. Basic notions in algebraic geometry  S. 252
VI.1 Affine schemes  S. 252
1. Localization  S. 123
2. The spectrum of a ring A  S. 124
3. The Zariski topology on Spec(A) = X  S. 128
4. The structure sheaf on Spec(A)  S. 130

5. Quasicoherent sheaves  S. 132
6. Schemes as locally ringed spaces  s. 133
VI.2 Schemes  S. 265
1. Construction of quasicoherent sheaves  S. 137
2. Fibered products  S. 137
3. Points, T -valued points and geometric points  S. 143
4. Closed points and geometric points on schemes of finite type  S. 145
5. Morphism, T -valued points and geometric points  s. 146
6. Flat morphisms  S. 147
7. Formal schemes and infinitesimal schemes  S. 163
VII. Some commutative algebra  S. 294
VII.1 .....  S. 165
1. Finite A-algebras  S. 165
2. Dimension theory for finitely generated k algebras  S. 167
3. Theorem  S. 168
4. Rings with finiteness conditions  S. 169
VII.2 Minimal prime ideals and decomposition into irreducibles  S. 172
1. Decomposition into irreducibles for noetherian schemes  S. 172
2. Affine schemes over k and change of scalars  S. 172
3. Local irreducibility  S. 176
4. Low dimensional rings  S. 178
5. Finiteness properties of Tor  S. 182
6. Dominant morphisms  S. 183
VII.3 Smooth points  S. 185
1. Theorem  S. 187
2. The Jacobi criterion  S. 189
3. Theorem: Generic smoothness  S. 191
4. Relative differentials  S. 194
VIII. Projective Schemes  S. 202

VIII.1....  S. 331
1. The projective space ℙ_{A}^{n}  S. 202
2. Locally free sheaves on ℙ^{n}  S. 208
3. Ample and very ample sheaves  S. 214
VIII.2 Cohomology of coherenct sheaves, the finiteness theorems  S. 216
1. .....  S. 216
2. Base change  S. 227
3. Semicontinuity  S. 236
IX. Curves and the theorem of Riemann Roch  S. 379
IX.1 Basic notions and results  S. 238
1. The structure of O_{C,𝔭}  S. 240
2. Curves and their function fields  S. 242
3. Line bundles and divisors  S. 245
IX.2 The theorem of Riemann-Roch  S. 248
1. Differentials and residues  S. 249
2. The “different”sheaf  S. 258
3. Vector bundles on curves  S. 261
4. The degree  S. 263
IX.3 ....  S. 268
1. Applications of the Riemann-Roch theorem  S. 268
2. A special case of the Grothendieck-Riemann-Roch theorem  S. 272
3. Hodge index theorem  S. 280
4. Curves over finite fields  S. 280
X. The Picard functor  S. 444
1.Generalities on the Picard functor  S. 334
2. The infinitesimal properties of the Picard functor  S. 343
3. Symmetric powers  S. 348
4 The algebraic construction of the Jacobian of a curve  S. 353
5. Properties of the Scheme Pic(C)  S. 367

6. The reprensentability of Pic(X)  S. 377
7. Descend to the ground field  S.382
8 The endomorphism ring again  S. 382
9. Should come earlier- See remark above  S. 392
10. The Weil pairing  S. 393
11 The Neron-Severi group ( and the ring of correspondences ?)  S. 396
4. Etale cohomology and the Weil conjecture  S. 400
5. The cyclotomic character  S. 400
6. The geometric étale cohomology groups  S. 405

vii

Introduction

Before I describing the contents of this book I have to say a few words about its history. My original plan was to write a book on the cohomology of arithmetic group and I still have this intention. Actually there exists a first version of this book in German. They resulted from a series of lectures I gave on this subject. This subject requires a lot of background from different aereas of mathematics for intance linear algebraic groups, arithmetic groups, number theory. For a while it is possible to keep this input on an elementary level because one can work with special examples like Sl_{2}, the action of Sl_{2}(ℤ) on the upper half plane is not so difficult to undestand. But then the basic concepts of homological algebra and sheaf cohomology enter.

Therefore I wrote an exposition of sheaf cohomology and an introduction into homological algebra. Then it happened that I gave a course on algebraic geometry and again the same concepts were fundamental. So I used this first part as an introduction into algebraic geometry and I completed it by writing up some notes for the students which covered some algebraic geometry.

At a certain point I decided write an introduction into algebraic geometry which also included the basic facts of sheaf theory and homological algebra. Again after a while I realized that this volume would cover more than 500 pages and so I decided to split this volume again in two parts so that my plan is now to deliver three volumes.

The present volume I starts with a very informal introduction into category theory. It continues with an introduction into homological algebra. In view of the content of the third volume Chapter II is an introduction into homological algebra based on the example of cohomology of groups. Chapter III introduces into the theory of sheaves. The role of sheaves is twofold: They allow to formulate the concepts af manifolds is locally ringed spaces (C_{∞}-manifolds, complex manifolds, algebraic manifolds ) this is discussed in III.2 and we have the cohomology of sheaves which is covered in Chapter IV. Up to here the book may serve as an introduction into algebraic topology but with a strong focus on applications in algebraic geometry and in the cohomology of arithmetic groups. The discussion of singular homology is rather short.

My original notes where very informal but after a while I felt the desire to give a rather self contained account and so it happened that the introduction into sheaf cohomology became rather complete up to a certain level. I included Poincare duality, spectral sequences, the cup product and Hodge theory. Only in the section on Hodge theory I refer to some analytical results which are not proved in this book.

The last chapter V we apply sheaf cohomology to the theory of compact Riemann surfaces. In the first section of Chapter V we prove the theorem of Riemann-Roch, we show that Riemann surfaces are smooth projective algebraic curves. We prove Abels theorem and we discuss the theory of Jacobians over the field C. In the second section we discuss the theory of line bundles on these Jacobians and more general on abelian varieties.

This last chapter goes beyond homological algebra and algebraic topolgy but it shows the enormous usefulness of these concepts. The Chapter V can also be seen as a preparation for the second volume which is an introduction into algebraic geometry.

Chapter I

Categories, products, projective and inductive limits

I.1. The notion of a category and examples I want to give a very informal introduction to the theory of categories. The main problem for a beginner is to get some acquaintance with the language and to get used to the abstractness of the subject. A category C is

(i) a collection of objects Ob(C).

We do not insist that this collection is a set. For me this means that we do not have the notion of equality of two objects. If we write N ∈ Ob(C) then we mean that N is an object in the category C.

(ii) To any two objects N, M ∈ Ob(C) is attached a set

HomC(N, M)

which is called the set of morphisms between these two objects. Usually we denote a morphism ϕ ∈ HomC(N, M) by an arrow ϕ : N → M.

(iii) For any three objects N, M, P we have the composition of morphisms

HomC(N, M) × HomC(M, P) → HomC(N, P)
(ϕ, ψ) ↦ ψ ∘ ϕ.

If a morphism η is a composition of ϕ and ψ then we denote this by a commutative diagram (or commutative triangle)

A \underrightarrow{ϕ} B
η ↘ ↙ ψ
P

We require that this composition is associative in the obvious sense (if we have four objects…). The reader should verify that this associativity can be formulated in terms of a tetrahedron all of whose for sides are commutative triangles. Here we use that the morphisms between objects form a set. In a set we know what equaliy between elements means.

(iv) For any object N ∈ Ob(C) we have a distinguished element Id_{N} ∈ HomC(N, N) which is an identity on both sides under the composition.

1.1 Examples: Everybody has seen categories:

a) The category Ens of all sets where the arrows are arbitrary maps.

b) The category Vect_{k} of vector spaces over a given field k where the sets of morphisms are the k-linear maps.

c) the category Mod_{A} of modules over a ring A where the maps are A-linear maps. We also have the category of abelian groups Ab, the category Groups of all groups where the morphisms are the homomorphisms of groups.

d) the category Top of topological spaces where the maps are the continuous maps.

I said in the beginning that we do not have the notion of equality in a category. But we we can say that two objects N, M ∈ Ob(C) are isomorphic. This means that we can find two arrows ϕ : N → M and ψ : M → N such that Id_{N} = ψ ∘ ϕ, Id_{M} = ϕ ∘ ψ. But in general it may be possible to find many such isomorphisms between the objects and hence we have many choices to identify them. Then it is better to refrain from considering them as equal. For instance we can consider the category of finite dimensional vector spaces over a field k. Of course two such vector spaces are isomorphic if they have the same dimension. Since we may have many of these isomorphisms, we do not know how to identify them and the notion of equality does not make sense. But if we consider the category of framed finite dimensional k− vector spaces, i.e. vector spaces V equipped with a basis which is indexed by the numbers 1, 2, …, n = dim(V) and morphisms which are linear maps which send basis elements to basis elements and which respect the ordering, then the situation is different. In this case we can say the objects form a set: If two such objects are isomorphic then the isomorphism is unique. It is important to accept the following fact: The axioms give us a lot of flexibility, at no point we require that the elements in HomC(N, M) are actual maps between sets (with some additional structure). Insofar all the above examples are somewhat misleading. The simplest example of a situation where the arrows are not maps is the following one:

e) We may start from a an ordered set I = (I, ≤) and we consider its elements as the objects of a category. For any pair i, j ∈ I we say that HomI(i, j) consists of one single element ϕ_{i,j} if i ≤ j and is empty otherwise. The composition is the obvious one obtained from the transitivity of the order relation. The reader may say that this is not a good example, because the ϕ_{i,j} can be considered as maps between the two sets {i}, {j} but that is the wrong point of view. To make this clear we can also construct a slightly different category J from our ordered set. We may define the sets of morphisms as:

HomJ (i, j) are finite sequences {i_{0}, i_{1}, …, i_{n}} with i_{ν} ≤ i_{ν+1} and i = i_{0}, j = i_{n}.

We leave it to the reader to verify that we have a composition and an identity. Now we may have many arrows between two objects {i}, {j} which are sets consisting of one element. We may also do the following which may look strange at the first glance. If we have a category C we may revert the arrows and form the so called opposite category C^{opp} which has the same objects but where

HomC^{opp}(N, M) = HomC(M, N).

I.2. Functors: We need the notion of a functor F from one category C to another category C′. A functor is a rule that transforms an object N ∈ Ob(C) into an object F (N) ∈ Ob(C′) and for any two objects N, M ∈ Ob(C) it provides maps

F_{N,M} : HomC(N, M) ⟶ Hom_{C′}(F (N), F (M)).

In other words: For any ϕ : N → M the functor produces an arrow

F_{N,M}(ϕ) = F (ϕ) : F (N) → F (M)

and this production should satisfy the obvious consistency conditions namely respect identity elements and composition. Such an F together with the collection of map between the sets of morphisms is called a covariant functor because direction of the arrows is preserved. We also have the notion of a contravariant functor which turns the arrows backwards or what amounts to the same: it is a functor from the opposite category C^{opp} to C′. Any object X of a category defines functors from this category to the category Ens: We attach to it the covariant functor

h_{X}(Z) = HomC(X, Z).

If we have two objects Z, Z′ and ψ : Z → Z′ then the composition produces h_{X}(ψ) : HomC(X, Z) → HomC(X, Z′) which sends ϕ : X → Z to ψ ∘ ψ. We may also put X into the second free place in the HomC( , ) and consider h_{X}^{∘}(Z) = HomC(Z, X). This gives us a contravariant functor.

2.1 Examples:

a) We have a contravariant functor from the category of vector spaces into itself: We send a vector space V ∈ Ob(Vect_{k}) to its dual space V^{∨} = Hom_{k}(V, k).

b) A very clever example of a functor is the homology of topological space (see Chap. IV). To any topological space X (i.e an object in the category Top) we may attach the homology groups H_{0}(X, ℤ), H_{1}(X, ℤ), …, H_{i}(X, ℤ), … the index set runs over all positive integers. These homology groups are abelian groups which depend functorially on the space X: A continuous map

f : X ⟶ Y

between spaces induces a homomorphism between their homology groups

f_{i} : H_{i}(X, ℤ) ⟶ H_{i}(Y, ℤ) for all indices i.

This functor transforms a very complicated object - a toplogical space- into a simpler but not too simple object namely a family of abelian group. This can be used to prove that ℝ^{n} is not homeomorphic to ℝ^{m} if n ≠ m. To see this we remove the origin from ℝ^{n} and from ℝ^{m} and we will see that the resulting spaces will have non-isomorphic homology groups if n ≠ m.(Chap IV, 7.3). On the other hand if we had a homeomorphism between the two spaces we could arrange that it maps the origin to the origin. Hence we would get a homeomorphism between the modified spaces which then must induce isomorphisms on the homology groups and this is impossible. If I am right these homology groups are historically the first examples where the concept of functors has been used. We will see many more interesting functors in the chapter on homological algebra.

I.3. Products, projective limits and direct limits in a category. Let us assume that we have a category C and an ordered set I = (I, ≤). Furthermore we assume that to any i ∈ I we have attached an object X_{i} ∈ Ob (C) and for any pair i ≤ j of indices we have an arrow

ϕ_{ij} ∈ HomC(X_{j}, X_{i})

We assume that always ϕ_{ii} = Id_{X_{i}} and for any triple i ≤ j ≤ j′ we have

ϕ_{ij} ∘ ϕ_{jj′} = ϕ_{ij′}.

We have seen in example e) that we may consider our ordered set (I, ≤) as a category I. Then we can summarize our assumptions by saying that i → X_{i} is a contravariant functor from the category I to the category C. The family ({X_{i}}_{i∈I}, ϕ_{ij}) is called a projective or sometimes inverse system of objects in C. For any object Z ∈ Ob (C) we define a set

HomC(Z, ({X_{i}}_{i∈I}, ϕ_{ij}))

which consists of families {ϕ_{i}}_{{i∈I}} of morphisms

ϕ_{i} : Z → X_{i}

such that for any pair i ≤ j the diagram

Z \underrightarrow{ϕ_{j}} X_{j}
ϕ_{i} ↘ ↙ ϕ_{ij}

X_{i} commutes. It is clear that

Z → HomC(Z, ({X_{i}}_{i∈I}, ϕ_{ij}))

is a contravariant functor from C to Ens because a morphism ϕ : Z′ → Z induces a map HomC(Z, ({X_{i}}_{i∈I}, ϕ_{ij})) → HomC(Z′, ({X_{i}}_{i∈I}, ϕ_{ij})) which is induced by the composition. We should think of ({X_{i}}_{i∈I}, ϕ_{ij}) as a huge diagram

⟶ X_{i} ⟶ X_{j}
↘
↘  X_{c} ⟶
↗
X_{a}
↗
⟶ X_{b}  ↘
↘
X_{r} ⟶ X_{u} ⟶

where we did not draw the compositions because they are redundant and make the picture complicated. Then an element ϕ ∈ HomC(Z, ({X_{i}}_{i∈I}, ϕ_{ij})) is a system of arrows {φ_{ν} : Z → X_{ν}}_{ν∈I} into this diagram:

⟶ X_{i} ⟶ X_{j}
↘
↗ ↘ X_{c} ⟶
↗
Z  X_{a}
↘  ↗
⟶ X_{b}  ↘
↘
X_{r} ⟶ X_{u} ⟶

so that every diagram induced by a i ≤ j commutes. Again we suppressed the compositions. ( In the special diagram, are the two arrows from Z to X_{i} and X_{b} arbitrary or is there a constraint?). An object P ∈ C together with an element

Φ ∈ Hom_{C}(P, ({X_{i}}_{i∈I}, ϕ_{ij}))

is called projective limit of the system ({X_{i}}_{i∈I}, ϕ_{ij})) if doe any Z ∈ Ob(C) the map

\begin{matrix}HomC(Z, P) & ⟶ & HomC(Z, (\{X_{i}\}_{i∈I}, ϕ_{ij})) \\ ψ & ⟶ & \{Φ_{i} ∘ ψ\}_{\{i∈I\}} \end{matrix}

is a bijection. This is the so called universal property of (P, Φ)

In terms of our above diagrams this means that a projective limit P is an object that is squeezed between Z and the diagram i.e. we have given our Φ ∈ Hom_{C}(P, ({X_{i}}_{i∈I}, ϕ_{ij})) and any ϕ from any Z into the diagram is obtained by first giving an arrow Z → P and then composing with the universal arrow Φ. Such a projective limit may not exist in our category. But if it exists then this gives us a first example of a representable functor: Starting from the functor Z → HomC(Z, ({X_{i}}_{i∈I}, ϕ_{ij})) we find a P such our functor is equivalent to the functor h_{P}^{∘} which we attached to P. More precisely we have an universal element Φ ∈ Hom_{C}(P, ({X_{i}}_{i∈I}, ϕ_{ij})) such that the equivalence of the functors is given by the universal property above.

3.1 The Yoneda lemma: We have a simple categorical argument which is called the Yoneda lemma which shows that such a (P, Φ) is unique up to a canonical isomorphism. If we have a second pair (P′, Φ′) then we get from the universal property that Φ′ is obtained from a uniquely defined morphism ψ′ : P′ → P composed with Φ and conversely we get Φ from Φ′ by composing with a unique ψ : P → P′. Finally the universal property yields that the composition ψ′ ∘ ψ and ψ ∘ ψ′ must be the identities. So we can conclude: If a projective limit exists it is unique up to a canonical isomorphism and is denoted by

P = lim_{\overleftarrow{i∈I}} X_{i}

This limit is also called the inverse limit because the arrow points backwards. We also should remember the the arrows in our system {X_{i}} point from objects with a larger index to objects with smaller index. The universal map u is sometimes suppressed in the notation. I will discuss some examples of projective limits which belong to the general education of anybody working in algebra or topology.

3.2 Examples.

(1) We consider the case where C = Ens and the order relation on I is trivial, i.e. i ≤ j if and only if i = j. Then we may take the product of these sets P = ∏_{i∈I} X_{i} and the Φ_{i} : P → X_{i} are the usual set theoretic projections. Then {P, Φ_{i}}_{i∈I} is also the product in the categorical sense.

(2i) We take the set of positive natural numbers ℕ_{+} and we define as order relation the divisibility relation, i.e. n ≤ m ⇔ n | m. For any m we can define the quotient rings ℤ/mℤ and if m | m′ then we have the projection

ϕ_{m,m′} : ℤ/m′ℤ → ℤ/mℤ,

and φ_{m,m′}(x_{m′}) = x_{m} means that x_{m′} ≡ x_{m} mod m. We can define a ring

ℤ̂ = {(…, x_{n}, …)_{n∈ℕ_{+}} | x_{n} ∈ ℤ/nℤ, x_{n′} = x_{n} mod n′ if n′|n}

where addition and multiplication are taken componentwise, and we have the projection map

ℤ̂ \underrightarrow{ϕ_{n}} ℤ/nℤ

which is the projection to the n-th component. Then (ℤ̂, φ_{n})_{n∈ℕ^{+}} is the projective limit in the category of rings.

2ii) We may also look at the ordered set {p^{n}}_{{n=1,2,...}} where p is a prime. Then we get the projective system

ℤ/p^{n}ℤ → ℤ/p^{m}ℤ →

and the projective limit

ℤ_{p} = {(…, x_{n}, …) | x_{m} ≡ x_{n} mod p^{n} if n ≤ m}.

Each component x_{n} determines completely the x_{m} with m ≤ n but if we go backwards we get more and more refined information. We can put a topology onto ℤ_{p} where a basis of open sets is given by the elements of the form y + p^{k}ℤ_{p}. The ring ℤ_{p} contains ℤ as a dense subring, it is a local ring without zero divisors, the unique maximal ideal is pℤ_{p} = (p). Its quotient field is the field ℚ_{p} of p-adic numbers.

It follows from elementary number theory ( The Chinese remainder theorem) that

ℤ̂ = ∏_{p} ℤ_{p}.

This ring ℤ̂ is not integral, it has zero divisors.

(3) It is not too difficult to see that in Ens projective limits exist. One simply forms the product

∏_{i∈I} X_{i}

and takes the subset of those elements

x = (…, x_{i}, …)_{i∈I}

which satisfy ϕ_{ij}(x_{j}) = x_{i}. This implies that also in such categories like the category of rings, the category of modules over a given ring products and projective limits exist. But in the category of fields we even cannot form the product of two fields, because we cannot avoid zero divisors.

(4) A very important example of a projective limit is the Galois group of a field k. We assume that we have constructed an algebraic closure k̄ of k, this is a field with the followind two properties (i) Every α ∈ k̄ is algebraic over k, i.e. satisfies a nontrivial equation

α^{n} + a_{1}α^{n−1} + … a_{n} = 0

with a_{i} ∈ k.

(ii) The field k̄ is algebraically closed. (Such a field can always be constructed if we use the axiom of choice). We have the set of finite normal extensions

k ⊂ K ⊂ k̄,

this is an ordered set by inclusion. For any normal extension k ⊂ k_{1} ⊂ k̄ let Gal(k_{1}/k) be the group of automorphisms of k_{1} whose restriction k induces the identity. For a tower of finite normal extensions

k ⊂ K ⊂ L

we have a surjective map

Gal(L/k) → Gal(K/k)

which is simply given by restriction. We can form the projective limit

lim_{\overleftarrow{K/k}} Gal(K/k)

of this system. It exists by the above remark. The resriction defines an isomorphism

Gal(k̄/k) → lim_{\overleftarrow{K/k}} Gal(K/k).

This is clear if we know that every automorphism σ : K → K over k can be extended to an automorphism of the algebraic closure.

(5) It is of course obvious that in the category Ensfin of finite sets we cannot have infinite products. But if we have a family ({X_{i}}_{i∈I}, ϕ_{ij})} of finite sets we can form the product in Ens and we define a topology on this product. This should be the coarsest topology such that the projections

p_{i} : ∏_{j∈I} X_{j} → X_{i}

become continuous. (On X_{i} we take the discrete topology, every subset is open). Hence we get a basis for the topology if we take finite intersections

⋂_{i∈E} p_{i}^{−1}({x_{i}})

where E is finite and x_{i} ∈ X_{i} a point. It is not too difficult to prove that the product endowed with this topology becomes a compact space. The same holds if we take projective limits of finite sets (groups, rings,.....), these limits are compact topological spaces (groups, rings, ...). The resulting projective limits are called profinite sets (groups, rings,.....). For instance the ring

ℤ̂ = lim_{\overleftarrow{m}} ℤ/mℤ

is such a profinite ring. The Galois group Gal(\overline{k}/k) of a field k is a profinite group. The topology of this groups is called the Krull topology.

3.3 Representable functors I want to say a few words about representable functors. We discussed the example of projective limits. But the notion of representability for a functor is much more general. It may be applied to any contravariant or covariant functor which takes values in the category of sets. If we have a contravariant functor

F : C → Ens

we may ask whether we can find an object X and an element u ∈ F (X) such that for any Z ∈ Ob(C) we get a bijection

Hom_{C}(Z, X)→^{∼}F (Z)

which is given by the universal rule

ϕ → F (ϕ)(u)?

If such an object X together with u ∈ F (X) exists then the so-called Yoneda lemma asserts that it is unique up to a canonical isomorphism. This means that the data provide a distinguished isomorphism between two solutions of the problem. (The proof is basically the same as in the case of projective limits: If we have two such objects X, X′ we have Hom_{C}(X′, X)→^{∼}F (X′). Now the u′ ∈ F (X′) provides a morphism in HomC(X′, X). Interchanging the two arguments gives us a morphism in the opposite direction. The compositions must be the identities.)

3.4. Direct limits: I begin with the simplest example. If we have a family {X_{i}}_{i∈I} of sets then we can form the disjoint union

⨆_{i∈I} X_{i}.

This construction satisfies

Hom_{Ens}(⨆_{i∈I} X_{i}, Z) = ∏_{i∈I} HomC(X_{i}, Z).

Here is becomes clear that the formation of a disjoint union and a product are dual to each other this means that the arrows are turned backwards. We formulate a principle:

3.4.1. The product is constructed so that we know what the arrows into it are, the disjoint union so that we know what the arrows from it are.

To describe inductive (or direct) limits we start again from an ordered set (I, ≤). Now we consider a covariant functor which attaches to any i an X_{i} ∈ Ob(C) and an element

ϕ_{ij} ∈ Hom_{C}(X_{i}, X_{j})

whenever i ≤ j. So in contrast to the case of projective limits the arrows point from objects with a smaller index to objects with a larger index. Such a system (or functor) is called an inductive system. This time we look at

Hom_{C}(({X_{i}}_{i∈I}, ϕ_{ij}), Z),

these are now maps from the diagram to objects in C. We say that an object L together with a map

Ψ ∈ Hom_{C}(({X_{i}}_{i∈I}, ϕ_{ij}), L)

is a direct limit of ({X_{i}}_{i∈I}, ϕ_{ij}) if

Hom_{C}(L, Z) ⟶^{∼} Hom_{C}(({X_{i}}_{i∈I}, ϕ_{ij}), Z)

and where the bijection is given by composition

ϕ_{i} = ϕ ∘ Ψ_{i}

and ϕ_{i} ∈ Hom_{C}(X_{i}, Z), ϕ_{i} ∈ Hom_{C}(L, Z). If such a limit exists we write

L = lim_{\overrightarrow{i∈I}} X_{i}.

It is clear that in the category Ens direct limits exist: Starting from an inductive system of sets ({X_{i}}_{i∈I}, ϕ_{ij}) we form the disjoint union ⨆_{i∈I} X_{i}.. We introduce an equivalence relation ∼ on this disjoint union. This equivalence relation will satisfy x_{i} ∼ x_{j} whenever ϕ_{ij}(x_{i}) = x_{j}. This is not necessarily an equivalence relation, but we simply take the equivalence relation generated by the relation. Then it is not hard to see that the quotient of the disjoint union by this relation is a direct limit.

3.5 An ordered set (I, ≤) is called directed if for any two i, j ∈ I we can find an element l ∈ I such that i < l, j < l. If we have inductive system of sets ({X_{i}}_{i∈I}, ϕ_{ij}) over a directed set, then the equivalence relation in our construction above can be described directly

x_{i} ∼ x_{j} ↔ ∃ l ∈ I, i ≤ l, j ≤ l and ϕ_{il}(x_{i}) = ϕ_{jl}(x_{j}).

We may also look at the opposite case where the ordering relation on the set I is trivial, i.e. we have i ≤ j if and only if i = j. If we have an inductive system ({X_{i}}_{i∈I}, ϕ_{ij}) over such a set then the inductive limit should be called a disjoint union. More examples of such direct limits will be constructed in Chapter III where we shall see that stalks of sheaves are direct limits. Generally we had projective limits as subsets of products, direct limits will be quotients of disjoint unions. By the way in some sense this discussion of direct limits is superfluous because if we pass to the opposite category the direct limits become projective limits.

3.6 Exercise:

1) Do we have disjoint unions in the category Vect_{k}? If so how does the disjoint union of two vector spaces look like.

2) We may ask the same question for the category Rings of rings, for the category of commutative rings and for the category of groups. 2b) In any category we can consider diagrams of the form

\begin{aligned}& & B \\ & ↗ \\ A \\ & ↘ \\ & & C \end{aligned}

we can interprete this as an inductive system and we can ask whether the limit exists. If our category is the category of groups then the limit does exist and it is given by the almaganated product.

3) Let me assume we have an index set (I, ≤) and a projective system ({X_{i}}_{i∈I}, ϕ_{ij}) on it. Let us assume that the indexing set contains a maximal element m, i.e. m ≥ i for all elements i ∈ I. I claim that the projective limit exists. How does it look like? Can you formulate an analogous assertion for injective limits.

4) Let us assume that we have a directed set (I, ≤) We assume that we have inductive system of rings {R_{i}}_{i∈I}. Does the direct limit exist? Hint: Forget the ring structure and consider the R_{i} as sets. Form the limit in the category of sets. Now you can reintroduce the ring structure on this limit by observing that any pair (or even finite set ) of elements can be represented by elements in a suitable member R_{i} of the family.

5) We have seen that we may interpret an ordered set (A, ≤) as a category. What does it mean for such a category that the product of two elements exists?

Chapter II

Basic Concepts of Homological Algebra

In this chapter I want to explain the fundamental concepts of homological algebra. I will do this for the specific case group (co-) homology.

This example will become important to us in the third volume of this book where we discuss the cohomology of arithmetic groups. But since in this particular case the basic principles of homological become very clear I have chosen this example as introduction into the subject. The cohomology of sheaves, which can serve as a second example, will be discussed in Chapter IV.

II.1. The category of Γ-modules: In the following Γ will always be a group. A Γ-module is an abelian group M together with an action of Γ: This means we have a map Γ × M ↦ M, (γ, m) → γm, which satisfies 1_{Γ}m = m, (γ_{1}γ_{2})m = γ_{1}(γ_{2}m) and γ(m_{1} + m_{2}) = γm_{1} + γm_{2}. These Γ-modules are the objects of the category of Γ-modules: If we write M ∈ Ob(Mod_{Γ}), then this means that M is a Γ-module. If M_{1}, M_{2} ∈ Ob(Mod_{Γ}), then we may consider the set

Hom_{Mod_{Γ}}(M_{1}, M_{2}) = Hom_{Γ}(M_{1}, M_{2}) =
\{φ : M_{1} → M_{2} | \begin{matrix}φ homomorphism of abelian groups \\ φ(γm_{1}) = γφ(m_{1}) for all γ, m_{1} \end{matrix} \}

On Hom_{Γ}(M_{1}, M_{2}) we have a natural structure of an abelian group: For any two elements φ, ψ ∈ Hom_{Γ}(M_{1}, M_{2}) we put (φ + ψ)(m_{1}) = φ(m_{1}) + ψ(m_{1}). Here we have another typical example of a category: We have a collection of objects — this collection is not a set in general — and for any two such objects we have a set of morphisms. (In our special case these sets of morphisms are abelian groups.) A certain bunch of axioms has to be satisfied: We have the identity Id_{M} ∈ Hom_{Γ}(M, M), we have a composition Hom_{Γ}(M_{1}, M_{2}) × Hom_{Γ}(M_{2}, M_{3}) → Hom_{Γ}(M_{1}, M_{3}) and Id_{M} is neutral with respect to this composition. (See the introduction in Chap. I) In our special case this composition is bilinear. The special category Mod_{Γ} has some extra features: Given φ : M → N we can form the kernel and the image

ker(φ) = {m | φ(m) = 0}, Im(φ) = {φ(m) | m ∈ M} ,

clearly these are also Γ-modules. If N ⊂ M is a Γ-submodule of M, then we may form the quotient module

M/N = MmodN,

this is again a Γ-module. Finally, we have direct sums and direct products

⨁_{i∈I} M_{i} = {(… m_{i} …)_{i∈I} | m_{i} ∈ M_{i}, almost all m_{i} = 0}
∏_{i∈I} M_{i} = {(… m_{i} …)_{i∈I} | m_{i} ∈ M_{i}}

where the addition and the action of Γ are defined componentwise. All these properties of imply that Mod_{Γ} is an abelian category. The notion of abelian categories can be axiomatized (see [Go], 1.8.).

1.1 Complexes If we have a sequence of maps between Γ-modules

⟶ M_{ν+1} ⟶^{d_{ν+1}} M_{ν} ⟶^{d_{ν}} M_{ν−1} ⟶

then this is called a (homological) complex if d_{ν} ∘ d_{ν+1} = 0 for all indices ν, i.e. if always

Im(d_{ν+1}) ⊂ ker(d_{ν}).

The maps d_{ν} are the differentials of the complex. The complex is called exact if we have Im(d_{ν+1}) = ker(d_{ν}) for all indices ν. We define the homology groups of such a complex as

H_{ν}(M^{•}) = \frac{ker(d_{ν} : M_{ν} → M_{ν−1})}{Im(d_{ν+1} : M_{ν+1} → M_{ν})}.

The complex is exact if and only if its homology groups are trivial. We can also consider complexes where the differentials raise the index by one then we write the indices ν as superscripts

⟶ M^{ν−1} ⟶^{d^{ν−1}} M^{ν} ⟶^{d^{ν}} M^{ν+1} ⟶,

then this is a cohomological complex. Very often we abbreviate and simply write M^{•} or (M^{•}, d) for a (co)homological complex. We define the cohomology groups of a cohomological complex by

H^{ν}(M^{•}) = \frac{ker(M^{ν} ⟶^{d^{ν}} M^{ν+1})}{im(M^{ν−1} ⟶ M^{ν})}.

Again we abbreviate H^{•}(M^{•}) and this is the graded direct sum over all cohomology groups. The elements in the kernel Z^{ν}(M) = ker(M^{ν} ⟶^{d^{ν}} M^{ν+1}) are called the cocycles in degree ν and the elements in B^{ν}(M) = im(M^{ν−1} ⟶ M^{ν}) are the coboundaries. Hence the cohomology is the group of cocycles modulo the coboundaries.

Actually we may also view these cohomology groups as a complex, the differentials are zero. Again is clear that a complex is exact, if and only if all its cohomology groups vanish. A map between two complexes

φ^{•} : M^{•} → N^{•}

is a sequence of maps ϕ^{ν} : M^{ν} → N^{ν} which commutes with the differentials. It is clear that such a map induces a map between the cohomology groups H^{•}(φ^{•}) : H^{•}(M^{•}) → H^{•}(N^{•}). A short exact sequence is an exact complex

0 ⟶ M′ ⟶^{i} M ⟶^{p} M′′ ⟶ 0,

i.e. i is injective, Im(i) = ker(p) and p is surjective, i.e. M′′ is isomorphic to the quotient of M by the submodule i(M′) ≃ M′.

II.2. More Functors As I explained already in the first chapter a functor is a rule that produces in a functorial way an object in a target category from an object in the source category. If for instance the source category is Mod_{Γ} and the target category is the category Ab of abelian groups, then a functor

F : Mod_{Γ} ⟶ Ab

associates to any Γ-module M ∈ Ob(Mod_{Γ}) an abelian group F (M). Functoriality means that for any M_{1}, M_{2} ∈ Ob(Mod_{Γ}) we have a map

F_{M_{1},M_{2}} : Hom_{Mod_{Γ}}(M_{1}, M_{2}) ⟶ Hom_{Ab}(F (M_{1}), F (M_{2}))

which sends Id_{M} to Id_{F (M)} and compositions into compositions. If we require that this map is a homomorphism F_{M_{1},M_{2}} between the abelian groups, then this functor is an additive functor between abelian categories. There are two very simple functors between the category Mod_{Γ} and the category Ab of the abelian group

Forget : Mod_{Γ} ⟶ Ab
Trivial : Ab ⟶ Mod_{Γ}

where the first factor “forgets” the Γ-module structure on the abelian group M and the second introduces the trivial Γ-action on an abelian group A, i.e. every element γ ∈ Γ induces the identity on A. These two functors are exact functors, this means that they transform exact sequences into exact sequences. Homological algebra owes its existence to the fact that many important additive functors are not exact.

Here comes the first example. If M is a Γ-module, we define the module M^{Γ} = {m | γm = m for all γ ∈ Γ} of invariants. It is an abelian group and hence we defined a functor

Invariants : Mod_{Γ} ⟶ Ab from the category of Γ-modules to the category of abelian groups. If A is a trivial Γ-module, then

Hom_{Mod_{Γ}}(A, M) = Hom_{Ab}(A, M^{Γ}),

and this property also characterizes the submodule M^{Γ} in M. The module M_{Γ} of coinvariants is defined as a quotient

M ⟶ M_{Γ}

where M_{Γ} is a trivial Γ-module and for any Γ-module with trivial action by Γ we have

Hom_{Mod_{Γ}}(M, A) = Hom_{Ab}(M_{Γ}, A).

To give a different description of M_{Γ} we recall the notion of the group ring R = ℤ[Γ] of our group Γ. It consists of all finite linear combinations

∑_{γ∈Γ} n_{γ}γ n_{γ} ∈ ℤ, almost all n_{γ} = 0,

where we add componentwise (i.e. the additive group is the free abelian group over the set), and where we multiply

(∑_{γ} n_{γ}γ) · (∑_{η} m_{η}η) = ∑_{γ,η} n_{γ}m_{η}γη = ∑_{δ} ( ∑_{γη=δ} n_{γ}m_{η}) δ.

This group ring contains the so called augmentation ideal I_{Γ} which is the two sided ideal

I_{Γ} = {∑ n_{γ}γ | ∑ n_{γ} = 0} .

It is clear that this ideal is generated by elements of the form 1 − γ (as ℤ-module). For any Γ-module M the module I_{Γ}M ⊂ M is a Γ-submodule, and it is also an easy exercise that

M_{Γ} = M/I_{Γ}M = H_{0}(Γ, M)

has the desired property the module of coinvariants should have.

The following fact is the starting point of homological algebra:

In In general the functors M → M^{Γ} and M → M_{Γ} are not exact.

To be more precise: If we start from a short exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

of Γ-modules, then the sequence

0 ⟶ (M′)^{Γ} ⟶ M^{Γ} ⟶ (M′′)^{Γ}

is exact, but the last arrow is not surjective (in general). A similar assertion holds for M_{Γ}. We only get an exact sequence

M′_{Γ} ⟶ M_{Γ} ⟶ M′′_{Γ} ⟶ 0.

We say that H^{0} is a left exact functor and H_{0} is a right exact functor. The goal is to construct the so called derived functors which measure the deviation from exactness. We motivate this by an example:

2.1 The first cohomology group I want to explain why the functor M → M^{Γ} is not exact and to show how this more or less automatically leads to the definition of the derived functor. Let us start from an exact sequence of Γ-modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0.

We get an exact sequence of abelian groups

0 ⟶ (M′)^{Γ} ⟶ M^{Γ} ⟶ (M′′)^{Γ}.

We pick an element m′′ ∈ (M′′)^{Γ}, and we want to understand, why this is not necessarily in the image of M^{Γ}. Of course we can find an element m ∈ M which maps to m′′. But there is no reason why this element should be invariant under Γ, the only thing we know is that for all γ ∈ Γ the difference

m′_{γ} = m − γm ∈ M′.

We get a map

\begin{matrix}Γ & ⟶ & M′ \\ γ & ⟼ & m′_{γ} \end{matrix} ,

and this map satisfies

m′_{γ_{1}} + γ_{1}m′_{γ_{2}} = m′_{γ_{1}γ_{2}}.

A map Γ → M′ satisfying this relation is called a 1-cocycle. On the set of all 1-cocycles we get a structure of an abelian group if we add the values and we denote by

Z^{1}(Γ, M′),

the abelian group of 1-cocycles. Our element m is in M^{Γ} if and only if the cocycle m′_{γ} = m − γm = 0.

We notice that the choice of m is not unique, we may change m → m + m′ with m′ ∈ M′. This is the only possible modification. Then we also modify the cocycle defined by m into γ ↦ m′_{γ} + m′ − γm′. This leads to the definition of the group B^{1}(Γ, M′) of 1-coboundaries. It is the group of those cocycles γ ↦ b_{γ} for which we can find a m′ ∈ M′ such that b_{γ} = m′ − γm′ for all γ. Hence we see: The element m′′ ∈ (M′′)^{Γ} defines an element in Z^{1}(Γ, M′) which is well defined up to a coboundary. We introduce the first cohomology group (provisorial definition)

Ȟ^{1}(Γ, M′) = Z^{1}(Γ, M′)/B^{1}(Γ, M′),

and we have seen that any m′′ ∈ (M′′)^{Γ} defines a class

δ(m′′) ∈ Ȟ^{1}(Γ, M′)

which is zero if and only if m′′ is in the image of M^{Γ} → (M′′)^{Γ}. It is clear that δ is a homomorphism, and that we have extended our exact sequence one step further

0 ⟶ (M′)^{Γ} ⟶ M^{Γ} ⟶ (M′′)^{Γ} ⟶^{δ} Ȟ^{1}(Γ, M′).

The next thing that can be checked easily is the functoriality of M′ → Ȟ^{1}(Γ, M′). If we have a φ ∈ Hom_{Mod_{Γ}}(M′, N) then this induces a map φ̌^{(1)} : Ȟ^{1}(Γ, M′) → Ȟ^{1}(Γ, N), and our above considerations also show that we get an even longer exact sequence

0 → (M′)^{Γ} → M^{Γ} → (M′′)^{Γ} →^{δ} Ȟ^{1}(Γ, M′) → Ȟ^{1}(Γ, M) → Ȟ^{1}(Γ, M′′),

the verification of exactness is left to the reader. But at the end it stops again; the last map needs not to be surjective. We also see that this longer exact sequence depends functorially on the short exact sequence we started from. If we have a map between two exact sequences of Γ-modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0
↓  ↓  ↓
0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0

then this induces a map between the two resulting exact sequences (in the sense of maps between complexes, i.e. all diagrams commute). In principle we can try to extend our sequence beyond Ȟ^{1}(Γ, M′′). We pick an element in Ȟ^{1}(Γ, M′′) and try to lift it to an element in Ȟ^{1}(Γ, M), and then we will see what the obstruction to this lifting will be. This will suggest a definition of a cohomology group Ȟ^{2}(Γ, M′). But actually there is a much more elegant way to define the cohomology functor which is also universal in the sense that it applies to many other cases. This will be done in section II.3.

2.2 At this point we introduce some new notation, instead of M^{Γ} we also write H^{0}(Γ, M) and H_{0}(Γ, M) will be the same as M_{Γ}. This is a very suggestive notation if we use it for our exact sequence above. Of course all this does not yet prove that M → M^{Γ} is not exact in general. For instance, it could be that Ȟ^{1}(Γ, M) = 0 for all Γ and all M, or it could also be that Ȟ^{1}(Γ, M′) → Ȟ^{1}(Γ, M) is always injective. We will show in exercise 2.6. that for Γ ≠ {1} these functors are not trivial.

2.3 Exercise:

If A is a trivial Γ-module, then

Ȟ^{1}(Γ, A) = Hom(Γ, A)

where the last Hom is the Hom in the category of groups.

This shows that for suitable A the module Ȟ^{1}(Γ, A) ≠ 0 if Γ is not equal to its commutator group [Γ, Γ].

Let us now assume that Γ′ ⊂ Γ is a subgroup. We have the important functor from the category of Γ′-modules to the category of Γ-modules which is called induction. For any Γ′-module Y we define an abelian group

Ind_{Γ′}^{Γ} Y = {f : Γ → Y | f(γ′γ) = γ′f(γ) for all γ′ ∈ Γ′, γ ∈ Γ} ,

and we define the action of Γ on Ind_{Γ′}^{Γ} Y by

(γf)(γ_{1}) = f(γ_{1}γ).

This is the induced Γ-module from the Γ′-module Y. It is very easy to check that for any Γ-module X we have an isomorphism (Frobenius reciprocity )

Hom_{Γ}(X, Ind_{Γ′}^{Γ} Y) ˜ ⟶ Hom_{Γ′}(X, Y)

which is given by

φ ⟼ {x ⟼ φ(x)(1)}.

2.4 Exercise:

We have a canonical (this means functorial in Y) isomorphism

Ȟ^{1}(Γ, Ind_{Γ′}^{Γ} Y) ˜ ⟶Ȟ^{1}(Γ′, Y).

This isomorphism is obtained from the following map on the level of cocycles: For any 1-cocycle

{γ ⟼ f_{γ}} ∈ Z^{1}(Γ, Ind_{Γ′}^{Γ} Y)

we define the 1-cocycle

\{γ′ ⟼ \overline{f}_{γ′}\} ∈ Z^{1}(Γ′, Y)

by \overline{f}_{γ′} = f_{γ′}(1). Show that this map sends coboundaries into coboundaries and induces an isomorphism on cohomology. Hint: We have to combine several little observations:

i) We consider an 1-cocycle

{γ ⟼ f_{γ}} ∈ Z^{1}(Γ, Ind_{Γ′}^{Γ} Y),

and we take into account that f_{γ} is actually a Y -valued function on Γ. Then the cocycle relation reads

f_{γ_{1}γ_{2}}(x) = f_{γ_{1}}(x) + (γf_{γ_{2}})(x) = f_{γ_{1}}(x) + f_{γ_{2}}(xγ_{1}).

If we evaluate at x = 1 we get

f_{γ_{2}}(γ_{1}) = f_{γ_{1}γ_{2}}(1) − f_{γ_{1}}(1),

and this relation tells us that we only need to know the values f_{γ}(1). Then the cocycle relation gives us the values of the f_{γ} at any x ∈ Γ. ii) If we have any function

\begin{aligned}h & : & Γ & ⟶ & Y \\ h & : & γ & ⟶ & h_{γ}, \end{aligned}

we may put (think of h_{γ} as being f_{γ}(1))

H_{γ}(x) = h_{xγ} − h_{x},

then H_{γ} is a function on Γ with values in Y. If γ → h_{γ} satisfies

h_{γ′xγ} − h_{γ′x} = γ′(h_{xγ} − h_{x}),

then H_{γ} ∈ Ind_{Γ′}^{Γ} Y and γ → H_{γ} it is a 1-cocycle.

iii) If we have a 1-cocycle γ → f_{γ} in Z^{1}(Γ, Ind_{Γ′}^{Γ} Y), then

γ′ ⟶ f_{γ′}(1) γ′ ∈ Γ′

is a one-cocycle in Z^{1}(Γ′, Y). Hence we have a map

Z^{1}(Γ, Ind_{Γ′}^{Γ} Y) ⟶ Z^{1}(Γ′, Y),

and it is clear that this map sends coboundaries into coboundaries. iv) If we have a 1-cocycle

γ′ ⟶ \overline{f}_{γ′}

in Z^{1}(Γ′, Y), then we want to construct a 1-cocycle γ → f_{γ} so that \overline{f}_{γ′} = f_{γ′}(1). To do this we choose a system γ_{i} of representatives of Γ′\Γ where we choose the identity for the class Γ′. For γ = γ′γ_{i} we put

f_{γ}(1) = f_{γ′}(1)

and apply (ii). The cocycle relation for γ′ → f_{γ′}(1) provides the decisive relation in (ii).This proves the surjectivity of our map between 1-cocycles in (iii).

v) Finally, we have to check that γ → f_{γ} is a boundary if γ′ → \overline{f}γ′ is a boundary. We can write

f_{γ′}(1) = y − γ′y

with y ∈ Y and for all γ′ ∈ Γ′. If we want to write γ → f_{γ} as a boundary, i.e.

f_{γ} = c − γc,

then this reads f_{γ}(x) = c(x) − c(xγ), and evaluation at 1 yields

f_{γ}(1) = c(1) − c(γ).

Hence we choose c(1) = y and put

c(x) = y − f_{x}(1)

and verify that this c bounds f_{γ}.

2.5 Exercise:

Use the previous exercise and prove that for any group Γ ≠ {1} there is a Γ-modules M s.t.

Ȟ^{1}(Γ, M) ≠ 0.

2.6 Exercise:

The group ring ℤ[Γ] is also a Γ-module by multiplication from the left. We get an exact sequence of Γ-modules

0 ⟶ I_{Γ} ⟶ ℤ[Γ] ⟶ ℤ ⟶ 0.

If we apply the functor H_{0} to this sequence, we find

I_{Γ}/I_{Γ}I_{Γ} ⟶ ℤ[Γ]/I_{Γ} ˜ ⟶ ℤ ⟶ 0
∥
ℤ  ,

and hence the sequence is exact if and only if I_{Γ} = I_{Γ}I_{Γ}. It is not difficult to prove that

\begin{matrix}Γ & ⟶ & I_{Γ}/I_{Γ}I_{Γ} \\ γ & ⟶ & 1 − γ \end{matrix}

induces an isomorphism

Γ/[Γ, Γ] = Γ_{ab} ˜ ⟶I_{Γ}/I_{Γ}I_{Γ}

([Γ, Γ] is the commutator subgroup).

II.3. The derived functors After these motivating considerations we explain the fundamental problem to be solved in homological algebra. We have the functor

M ⟶ M^{Γ} = H^{0}(Γ, M)

which is only left exact. We want to construct the right derived functor: This is a collection of functors

M ⟶ H^{i}(Γ, M) i = 0, 1, …,

such that for any short exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

we get a long exact sequence

\begin{aligned}0 ⟶ H^{0}(Γ, M′) & ⟶ H^{0}(Γ, M) ⟶ H^{0}(Γ, M′′) ⟶^{δ} H^{1}(Γ, M′) ⟶ \\ & ⟶ H^{1}(Γ, M) ⟶ H^{1}(Γ, M′′) ⟶^{δ} H^{2}(Γ, M′) ⟶ … \end{aligned}

which depends functorically on the exact sequence in the sense explained before. Finally we want this functor to be minimal (or universal) in the following sense. If we have any other collection of functors

M ⟶ H̃^{i}(Γ, M) i = 0, 1, …

with H^{0}(Γ, M) = H̃^{0}(Γ, M), and the same properties as above, then we find a natural transformation

H^{i}(Γ, M) ⟶ H̃^{i}(Γ, M),

which is compatible with the connecting homomorphisms. We want to indicate the main ideas how to construct these derived functors. The verification that the new construction of the H^{1} gives the same result as our previous Ȟ^{1} will be done in the exercises 4.4.

I want to explain a very simple principle that governs to the construction of these functors.

A simple observation: If we have an exact sequence of Γ-modules

0 ⟶ M′ ⟶^{i} M ⟶^{p} M′′ ⟶ 0,

we say that the sequence splits if one of the following assertions holds:

(i) We have a section to p. This is a Γ-modules homomorphism

s : M′′ ⟶ M

for which p ∘ s = id_{M′′}. (ii) The modules M splits, i.e. we have a Γ-submodules M̃′′ such that

\begin{matrix}M′ ⊕ M̃′′ & ˜ ⟶ & M \\ (m′, m̃′′) & ⟼ & i(m′) + m̃′′. \end{matrix}

(iii) We have a Γ-modules homomorphism

j : M ⟶ M′

s.t. j ∘ i = id_{M′}. Now our observation is that our functors H^{0},H_{0} will transform split exact sequences into split exact sequences, in other words if we restrict them to split exact sequences then they are still exact.

3.1 The simple principle: It is based on the assumption that we have already constructed a derived functor {M → H̃^{i}(Γ, M)}. Let us assume we have a class of C of Γ-modules which are acyclic for this functor, this means that for any X ∈ C we have H̃^{i}(Γ, X) = 0 for all i > 0. An acyclic resolution of M ∈ Ob(Mod_{Γ}) by objects in C is an exact sequence of Γ-modules

0 ⟶ M ⟶ X^{0} ⟶ X^{1} ⟶ X^{2} ⟶

where the X^{ν} ∈ C.

Lemma 3.2.: If C is a class of acyclic objects for the derived functor {M → H̃^{i}(Γ, M)}, and if

0 ⟶ M ⟶ X^{0} ⟶ X^{1} ⟶

is an acyclic resolution of M by objects in C. Then we have an isomorphism

H̃^{i}(Γ, M) ≃ H^{i}((X^{•})^{Γ}).

Proof: By induction on i. For i = 0 we get the exact sequence

0 ⟶ M^{Γ} ⟶ (X^{0})^{Γ} ⟶ (X^{1})^{Γ} ⟶

and

M^{Γ} ˜ ⟶ ker((X^{0})^{Γ} ⟶ (X^{1})^{Γ}) = H^{0}((X^{•})^{Γ}).

Now we cut the resolution into pieces. We write

0 ⟶ M ⟶ X^{0} ⟶ X^{0}/M ⟶ 0,

and we have a resolution by objects in C

0 ⟶ X^{0}/M ⟶ Y^{0} ⟶ Y^{1} ⟶

where Y^{ν−1} = X^{ν}. The first sequence yields a long exact sequence which is interrupted by many zeroes which come from the H̃^{•}-acyclicity of the X^{ν}.

0 ⟶ M^{Γ} ⟶ (X^{0})^{Γ} ⟶ (X^{0}/M)^{Γ} ⟶ H̃^{1}(Γ, M) ⟶ 0
⟶ H̃^{1}(Γ, X^{0}/M) ˜ ⟶ H̃^{2}(Γ, M) ⟶ 0
⋮  ⋮
0 ⟶ H̃^{i−1}(Γ, X^{0}/M) ˜ ⟶ H̃^{i}(Γ, M) ⟶ 0.

We check the case i = 1. Here we find

H̃^{1}(Γ, M) ≃ (X^{0}/M)^{Γ}/(X^{0})^{Γ},

but X^{0}/M ⊂ X^{1} is the kernel of X^{1} → X^{2}, and (X^{0}/M)^{Γ} = ker((X^{1})^{Γ} → (X^{2})^{Γ}), and hence

H̃^{1}(Γ, M) ≃ \frac{ker((X^{1})^{Γ} → (X^{2})^{Γ})}{im((X^{0})^{Γ} → (X^{1})^{Γ})} = H^{1}((X^{•})^{Γ}).

Hence we proved our assertion for i = 1 and then induction is clear. We want to apply this principle to construct the derived functors. But in some sense we are trapped: If we have not yet defined the derived functor, how can we know that certain objects are acyclic? This difficulty is resolved by the notion of injective modules. A Γ-module I is called injective if it has the following property: Whenever we have a diagram of Γ-modules

\begin{matrix}A ⟶^{φ} B \\ ↓ψ \\ I \end{matrix}

where ker(φ) ⊂ ker(ψ), then we can extend the diagram to a commutative diagram

\begin{matrix}A & ⟶^{φ} & B \\ ↓ψ & ↙ λ \\ I \end{matrix}

Our assumption on φ, ψ is valid if φ is injective. If we want to check the injectivity of a module it clearly suffices to check diagrams with φ = 0.

Injective modules have a very important property: Whenever we have a short exact sequence

0 ⟶ I ⟶ M ⟶ M′ ⟶ 0,

and the module I is injective then the sequence splits. We simply apply the defining property of injective modules to

0 ⟶I ⟶ M
↓
I

where the vertical arrow is the identity. Our simple observation above implies that we get exact sequences

0 ⟶ H^{0}(Γ, I) ⟶ H^{0}(Γ, M) ⟶ H^{0}(Γ, M′) ⟶ 0.
0 ⟶ H_{0}(Γ, I) ⟶ H^{0}(Γ, M) ⟶ H^{0}(Γ, M′) ⟶ 0.

whenever the module I on the left is injective. Since the require that the cohomology modules should measure the deviation from exactness and that they should be minimal in this respect, we expect them to vanish for injective modules, in other words we expect that injecive modules should be acyclic. In view of our simple principle above we try to define them by injective resolutions. The following lemma is the starting point:

Lemma 3.3.: Every Γ-modules M can be embedded into an injective module I. Sketch of the proof: First we consider the category Ab of abelian groups. This is the case Γ = {Id}. One proves that the abelian group ℚ/ℤ is injective (this requires Zorn’s lemma), then we see that every abelian group A can be embedded into a suitable product

A ⟶ ∏_{α} ℚ/ℤ.

If we have an Γ-module M we forget the Γ-module structure and embed it into an injective abelian group, i.e. M ↪ J. Now we get Ind_{{1}}^{Γ} M ↪ Ind_{{1}}^{Γ} J, and the module Ind_{{1}}^{Γ} J is injective in the category of Γ-modules. This follows from Frobenius reciprocity. Then we have achieved our goal since we have

M ⟶ Ind_{{1}}^{Γ} M ⟶ Ind_{{1}}^{Γ} J = I.

Now the actual construction of the cohomology functor (the universal derived functor) becomes clear. We noticed that injective modules should be acyclic, i.e. H^{r}(Γ, I) = 0 for r > 1. But our earlier lemmas tell us that we can find an injective resolution of M, i.e.

0 ⟶ M ⟶ I^{0} ⟶ I^{1} ⟶,

in short 0 → M → I^{•}. Then our lemma tells us

H^{ν}(Γ, M) ≃ H^{ν}((I^{•})^{Γ}) = \frac{ker((I^{ν})^{Γ} → (I^{ν+1})^{Γ})}{im((I^{ν−1})^{Γ} → (I^{ν})^{Γ})},

and in some sense that has to be explained and justified. We take this as definition of the cohomology. Of course we have to investigate how these cohomology groups depend on the injective resolution and we have to show functoriality.

3.4 Functoriality If we have two Γ-modules M, N and a φ ∈ Hom_{Mod_{Γ}}(M, N), and if in additon we have chosen two injective resolutions 0 → M → I^{•} and 0 → N → J^{•}, then I claim that we can extend the map φ to a map between the complexes

0 → M ⟶^{i} I^{0} → I^{1} → …
↓φ  ↓  ↓
0 → N ⟶^{j} J^{0} → J^{1} → …

The existence of this extension is proved by induction on the degree. To get the first arrow φ^{0} : I^{0} → J^{0} we apply the defining property of injective modules to get the arrow φ^{0} in the diagram

\begin{matrix}M & ⟶^{i} & I^{0} \\ j ∘ φ↓ & ↙ φ^{0} \\ J & 0 \end{matrix} .

Then we construct φ^{1} by the same principle and it is quite clear that at any step the existence of the vertical arrow follows directly from the defining property of injective modules (we only need that the J^{ν} are injective). This extension φ^{•} : I^{•} → J^{•} induces of course a map between the cohomology group

H^{•}(φ^{•}) : H^{•}((I^{•})^{Γ}) ⟶ H^{•}((J^{•})^{Γ}).

Now we have to worry what happens if we take two different extensions φ^{•}, φ̃^{•} of our map φ. I want to show that these two extensions induce the same map on the cohomology. To see this we can easily reduce to the case where φ = 0, and where φ^{•} is an arbitrary extension of φ = 0. Then I have to show that φ^{•} induces the zero map on the cohomology. I prove this by showing that under this assumption the map

φ^{•} : I^{•} → J^{•}

is actually homotopic to zero. This means that we can construct maps h^{ν} : I^{ν} → J^{ν−1} (h^{0} = 0) such that

φ^{ν} = d ∘ h^{ν} + h^{ν+1} ∘ d

To construct h^{1} we observe that our assumption φ = 0 implies that the kernel of I^{0} → I^{1} is contained in the kernel of the vertical arrow I^{0} → J^{0}. Since J^{0} is injective we can construct h^{1} : I^{1} → J^{0} which produces a commutative diagram

\begin{matrix}I^{0} & → & I^{1} \\ ↓ & ↙ h^{1} \\ J & 0 \end{matrix} .

Now we modify the given vertical arrow I^{1} → J^{1} by subtracting the composition of h^{1} and the horizontal arrow I^{0} → J^{1}. To this modified arrow we can apply the previous argument and it becomes clear how to construct these h^{ν} by induction. Again the existence of such an h^{ν} in any degree follows from the injectivity of the J^{ν−1} and the construction of the previous ones. But if we now apply our functor (invariants under Γ) we get

0 → (I^{0})^{Γ} → (I^{1})^{Γ} → (I^{2})^{Γ} → …
h^{0} ↙ ↓φ^{0} h^{1} ↙ ↓φ^{1} h^{2} ↙ ↓φ^{2}
0 → (J^{0})^{Γ} → (J^{1})^{Γ} → (J^{2})^{Γ} → …

(We should have written φ^{•}^{Γ}, h^{•}^{Γ} to be absolutely correct.) But now it is clear that φ^{•} induces zero in the cohomology. If we have a cycle c_{ν} ∈ (I^{ν})^{Γ} representing a given cohomology class then φ^{ν}(c_{ν}) = d ∘ h(c_{ν}) and hence it represents the trivial class. If we apply this to a module M, the identity and two different resolutions, then we find that we get a unique isomorphism between the resulting cohomology groups. In this sense the cohomology groups do not depend on the chosen resolution. We also get functoriality of the cohomology groups by the same argument.

3.5 Other resolutions If we start from an arbitrary resolution of our module M say 0 → X^{0} → X^{1} → … and if we also choose an injective resolution as above then our considerations in 1.3.4. show that we can construct a morphism of complexes of Γ-modules

0 → M ⟶^{i} X^{0} → X^{1} → …
↓φ  ↓  ↓
0 → N ⟶^{j} J^{0} → J^{1} → …

because we only need the injectivity of the J^{•}. Therefore we get a canonical homomorphism

H^{•}((X^{•})^{Γ}) → H^{•}((J^{•})^{Γ}) = H^{•}(Γ, M)

Our starting principle in 3.1. says that this homomorphism will be an isomorphism if the modules X^{ν} are acyclic.

Now we want to show that we get a long exact sequence in the derived functors if we start from a short exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0.

3.6 Injective resolutions of short exact sequences:

We write our short exact sequence vertically and choose injective resolutions of the two modules M′, M′′ which we write horizontically. Imagine we have done this. Then we can write the direct sum in the middle and we get short vertical exact sequences. It will look like this:

0  0  0
↓  ↓  ↓
0 → M′ →  I^{′0}  →  I^{′1}  → …
↓  ↓  ↓
0 → M  I^{′0} ⊕ I^{′′0}  I′ ⊕ I^{′′1}
↓  ↓  ↓
0 → M′′ →  I^{′′0}  →  I^{′′1}  → …
↓  ↓  ↓
0  0  0

The horizontal arrows in the middle are still missing. Now the injectivity of I^{′0} allows an arrow from M to I^{′0} which yields a commutative diagram

0

↓

0 ⟶ M′ ⟶ I^{′0}
↓ ↗

M

This and the map from M′′ → I^{′′0} gives us an injection

0 ⟶ M ⟶ I^{′0} ⊕ I^{′′0},

which we can fill into the diagram above.

This yields a diagram

0  0  0
↓  ↓  ↓
0 → M′ →  I^{′0}  → U → 0
↓  ↓  ↓
0 → M → I^{′0} ⊕ I^{′′0} → V → 0
↓  ↓  ↓
0 → M′′ →  I^{′′0}  → W → 0
↓  ↓  ↓
0  0  0

We have U ↪ I^{′1} and W ↪ I^{′′1} and again we construct as before an arrow

V ↪ I^{′1} ⊕ I^{′′1}.

This goes on forever and we get a resolution of the exact sequence 3.6.1. Now one has to be aware that in general the homomorphisms of

I^{′ν} ⊕ I^{′′ν} ⟶ I^{′ν+1} ⊕ I^{′′ν+1}

are not the direct sum of the two homomorphisms which are already given by the resolution of the extreme modules. The construction gives that we have to add a homomorphism

Ψ : I^{′′ν} ⟶ I^{′ν+1}

to this direct sum. Is we apply the functor H^{0}(Γ) to these complexes. We get the diagram

0  0  0
↓  ↓  ↓
0 ⟶  (I^{′,0})^{Γ}  ⟶  (I^{′,1})^{Γ}  ⟶ …
↓  ↓
0 ⟶ (I^{′,0} ⊕ I^{′′,0})^{Γ} ⟶ (I^{′,1} ⊕ I^{′′,1})^{Γ} ⟶ …
↓  ↓
0 ⟶ (I^{′′,0})^{Γ} ⟶ (I^{′′,1})^{Γ} ⟶ …
↓  ↓
0  0

where the horizontal complexes compute the cohomology of M′, M and M′′ respectively and where the vertical sequences are exact. Now it is standard in homological algebra that this leads to a long exact sequence of the cohomology groups. We constructed a derived functor using these injective resolutions. It is universal as one sees easily from the requirement that it vanished on injective modules. Essentially the same strategy works for the left derived functor

M ⟶ H_{i}(Γ, M) i = 1, …,

where H_{0}(Γ, M) = M_{Γ}. The defining property of injective modules implied that it is always a direct summand if it sits in a bigger module. The dual notion is the notion of projective modules. A Γ-modules P is called projective if for any diagram

M ⟶^{P} N ⟶ 0
↑ i
P

where the top sequence is exact we can find a map j : P → M so that p ∘ j = i, i.e. we find

\begin{aligned}M & ⟶^{P} & N & ⟶ & 0 \\ & j ↖ & ↑ i \\ & & P \end{aligned} .

It is easily seen that free Γ-modules, i.e.

⨁_{i∈I} ℤ[Γ]

are projective. Hence we find

i) Every Γ-modules M has a projective resolution

⟶ P_{2} ⟶ P_{1} ⟶ P_{0} ⟶ M ⟶ .

ii) Every projective Γ-module P which is a quotient of a Γ-module X is a direct summand, i.e. the sequence

0 ⟶ Y ⟶ X ⟶ P ⟶ 0

splits.

The assertion ii) implies that the sequence

0 ⟶ Y_{Γ} ⟶ X_{Γ} ⟶ P_{Γ} ⟶ 0

is still exact. Hence we should require

H_{i}(Γ, P) = 0  for  i = 1, ….

Now we may apply the same argument. For a module we choose a projective resolution

P_{•} ⟶ M ⟶ 0

and put

H_{•}(Γ, M) = H_{•}((P_{•})_{Γ}).

The same arguments as before show that this gives a universal left derived functor for the functor

M ⟶ M_{Γ}.

All arrows are reversed.

II.4 The functors Ext and Tor: We may look at our previous constructions from a slightly more general point of view. The category of Γ-modules is the same as the category of R = ℤ[Γ]-modules where R is the group ring. We now consider the category Mod_{R} of modules over an arbitrary ring R. To any pair of modules M, N ∈ Ob(Mod_{R}) we can introduce the abelian group

Hom_{R}(N, M) = \{φ : N → M| \begin{matrix}φ homomorphism of abelian groups \\ φ(rm) = rφ(m) for all r, m, \end{matrix} \}

and if we fix N then this becomes a functor from the category Mod_{R} to the category of abelian groups. But we may also fix M and vary N, then our functor becomes a contravariant functor: A R-module homomorphism

Ψ : N_{1} ⟶ N_{2}

induces a map

Hom_{R}(N_{2}, M) ⟶ Hom_{R}(N_{1}, M)

by composition. (So far all our functors were covariant.) It is quite clear that the covariant functor

M ⟶ Hom_{R}(N, M)

is left exact and that the contravariant functor

N ⟶ Hom_{R}(N, M)

is right exact.

For a given N we may try to construct the right derived functor to M ⟶ Hom_{R}(N, M) and for given M we may try to construct a left derived functor to the functor N ⟶ Hom_{R}(N, M). The same principles as before tell us that the right derived functor should be zero on injective modules (same definition), and the left derived functor should be trivial on projective modules. Hence we choose an injective resolution of

0 ⟶ M ⟶ I^{•}

and define the right derived functor by

R ExtR^{•}(N, M) = H^{•}(Hom_{R}(N, I^{•})).

I say againwhat this means: For a fixed R-module N and any exact sequence of R modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0,

we get a long exact sequence

0 → Hom_{R}(N, M′) → Hom_{R}(N, M) → Hom_{R}(N, M′′) → R Ext_{R}^{1}(N, M′) → R Ext_{R}^{1}(N, M) →

But by construction these groups R Ext_{R}^{i}(N, M) are also functorial in N if we fix M, the functors N → Ext^{i}(N, M) are contravariant. Analogously we choose a projective resolution

P_{•} ⟶ N ⟶ 0

and define

L ExtR^{•}(N, M) = H_{•}(Hom_{R}(P_{•}, M)).

Our previous arguments show that we get a left derived functor which has all functorial properties. It is clear that the functors

M ⟶ Hom_{R}(P, M) , resp. N ⟶ Hom_{R}(N, I)

are exact if P is projective (resp. I )is injective. Hence we have in this case

R Ext_{R}^{i}(P, M) = 0 , L Ext_{R}^{i}(N, I).

We will indicate briefly that this implies that we have a functorial isomorphism

L Ext_{R}^{i}(N, M) ≃ R Ext_{R}^{i}(N, M).

To see this we choose two resolutions:

0 ⟶ M ⟶ I^{•} , P_{•} ⟶ N ⟶ 0,

and we form the double complex

Hom_{R}(P_{•} → N → 0, 0 → M → I^{•})

which in full looks like

↑  ↑
⟶ Hom_{R}(P_{1}, M) ⟶ Hom_{R}(P_{1}, I^{0}) ⟶ Hom_{R}(P_{1}, I^{1}) ⟶
↑  ↑  ↑
0 ⟶ Hom_{R}(P_{0}, M) ⟶ Hom_{R}(P_{0}, I^{0}) ⟶ Hom_{R}(P_{0}, I^{1}) ⟶
↑  ↑  ↑
0 ⟶ Hom_{R}(N, M) ⟶ Hom_{R}(N, I^{0}) ⟶ Hom_{R}(N, I^{1}) ⟶
↑  ↑  ↑
0  0  0

Now the first vertical complex computes the L Ext_{R}^{•}(P, M) and the horizontal complex at the bottom computes R Ext_{R}^{•}(P, M). All other vertical or horizontal complexes are exact. Then a simple diagram chase shows that the cohomology of the bottom horizontal complex and the first vertical complex are isomorphic. We summarize 4.1 We have defined a functor in two variables ExtR^{•}(N, M) which can be be computed from an an injective resolution of M or a projective resolution of N. The higher Ext_{R}^{i}(N, M) for i > 0 vanish if M is injective or if N is projective.

4.2 The derived functor for the tensor product Another functor in two variables is given by the tensor product. Here we have to be a little bit carefull in case that our ring R is not commutative. We consider the categories Mod^{L}_{R}, Mod^{R}_{R} of left and right R-modules. The tensor product of a right R-module N and a left R-module M is an abelian group N ⊗_{R} M together with a map

\begin{matrix}Ψ : N × M & ⟶ N ⊗_{R} M \\ Ψ : (n, m) & ⟼ n ⊗ m \end{matrix}

which has the following properties

(i) It is a biadditive, i.e.

\begin{aligned}Ψ(n_{1} + n_{2}, m) & = (n_{1} + n_{2}) ⊗ m = n_{1} ⊗ m + n_{2} ⊗ m \\ Ψ(n + m_{1} + m_{2}) & = m ⊗ (m_{1} + m_{2}) = n ⊗ m_{2} + n ⊗ m_{2} \end{aligned} .

(ii) For all r ∈ R, n ∈ N, m ∈ M we have

nr ⊗ m = n ⊗ rm.

(This is the moment where we need that N is a right R-module and M is a left R-module). (iii) This map is universal: If we have another

Ψ′ : N × M → X

with an abelian group X which satisfies (i) and (ii) then we can find a φ : N ⊗_{R} M → X such that Ψ′ = φ ∘ Ψ. It is easy to construct N ⊗_{R} M, we form the free abelian group which is generated by pairs (n, m) ∈ N × M and divide by the subgroup generated by elements of the form

\begin{matrix}(n_{1} + n_{2}, m) & − (n_{1}, m) − (n_{2}, m) \\ (n, m_{1} + m_{2}) & − (n, m_{1}) − (n, m_{2}) \end{matrix}

and

(nr, m) − (n, rm).

If our ring R is commutative then we can give N ⊗_{R} M the structure of an R-module: We simply define

r(n ⊗ m) = nr ⊗ m = n ⊗ rm.

In this case of a commutative ring R we can assume that both variables N, M are left R-modules. It is a right exact functor but in general it is not exact. This means that for a short exact sequence 0 → M′ → M → M′′ → 0 the sequence

N ⊗_{R} M′ → N ⊗_{R} M → N ⊗_{R} M′′ → 0

will be exact but the first arrow on the left will not be injective in general. We leave it as an exercise to the reader to verify the right exactness. In the section on flat morphisms of schemes we will discuss some examples which explain these phenomena (AS. xxx). But if the module M′′ is projective then the sequence stays exact if we tensorize by any N because the sequence can be split. This allows us to construct the derived functor. We work with a projective resolution

P_{•} ⟶ M → 0,

to define

R TorR_{•}(N, M) = H_{•}(N ⊗_{R} P_{•}).

This is a universal left derived functor of our functor above, it clear that this is a functor in the two variables N, M. We can also choose a projective resolution Q_{•} → N → 0 define the functor L TorR_{•}(N, M) = H_{•}(Q_{•} ⊗_{R} M). Again it is not so difficult to prove that these two functors are indeed equivalent. To see this we consider the double complex defined by the two resolutions and the vertical and horizontal subcomplexes are acyclic in the ”interior”.

Again we summarize 4.3.1 Hence we can define a functor in two variables TorR_{•}(N, M) which can be computed by a projective resolution of N or a projective resolution of M. The higher Tor_{i}^{R}(N, M) vanish for i > 0 if one of the entries is a projective module. A left R-module M is called flat if the functor N → N ⊗_{R} M is exact. The following is obvious 4.3.2 The left R-module M is flat if and only if Tor_{i}^{R}(N, M) = 0 for all i > 0 and all right R-modules N . The functors cohomology and homology of groups are special cases of Ext^{•} and Tor^{•}. We take for our ring the group ring

ℤ[Γ],

and we observe: If ℤ is the abelian group ℤ with trivial Γ-action then

Hom_{ℤ[Γ]}(ℤ, M) = M^{Γ},

and hence we see

Ext_{ℤ[Γ]}^{i}(ℤ, M) = H^{i}(Γ, M)

and

M ⊗_{ℤ[Γ]} ℤ = M_{Γ},

hence

TorR_{•}(M, ℤ) = H_{•}(Γ, M).

We conclude the section with some extra remarks and some exercises. First we observe that we can compute the cohomology of a group also from a projective resolution

⟶ P_{2} ⟶ P_{1} ⟶ ℤ[Γ] ⟶ ℤ ⟶ 0

”

P_{0}  .

Hence we get

H^{i}(Γ, M) = H^{i}(Hom_{Γ}(P_{•}, ℤ)),

4.4 Exercise:

(i) Apply this to the case of a cyclic group Γ = ℤ/nℤ. Let σ be a generator of the group. We have the exact sequence

0 ⟶ I_{Γ} ⟶ ℤ[Γ] ⟶ ℤ ⟶ 0

and I_{Γ} = ℤ[Γ](1 − σ).

ia) In the case n = 0 (the infinite group) we have that I_{Γ} is a free module. This gives simple formulae for the cohomology and shows H^{ν}(Γ, M) = 0 for ν ≥ 2. ib) In the case of a finite group the map

\begin{matrix}ℤ[Γ] & ⟶ & I_{Γ} \\ r & ⟶ & r(1 − σ) \end{matrix}

has the kernel ℤ[Γ](1 + … + σ^{n−1}). Construct a “periodic” resolution from this and compute the cohomology. (ii) We still have to compare our provisorial cohomology groups Ȟ^{1}(Γ, M) and the new ones. This is not so difficult.

a) We observe that our new cohomology groups obviously satisfy: For a subgroup Γ′ ⊂ Γ and a Γ′-module

H^{i}(Γ, Ind_{Γ′}^{Γ} Y) = H^{i}(Γ′, Y).

(Choose an injective resolution of the Γ′-modules Y and ..... .)

b) Hence we have H^{1}(Γ, Ind_{1}^{Γ} M) = 0. We constructed the sequence

0 ⟶ M ⟶ Ind_{1}^{Γ} M ⟶ (Ind_{1}^{Γ} M)/M ⟶ 0,

and we find

((Ind_{1}^{Γ} M)/M)^{Γ} /(Ind_{1}^{Γ} M)^{Γ} ≃ H^{1}(Γ, M),

but this was also the formula for the Ȟ^{1}. (iii) Let us consider the ring R = k[X]/(X^{2}) where k is any field. Then the category of R-modules is the same as the category of k-vector spaces V together with an k-linear endomorphism α : V → V which satisfies α^{2} = 0. If dim_{k} V = 1, then α must be zero. Compute

Ext_{R}^{1}(k, k).

Does this ring a bell?

Chapter III

Presheaves and Sheaves

III. 1. Presheaves We start from a topological space X and we define the category Off(X) of open sets: The objects are the open sets U, V ⊂ X and the morphisms

Hom_{Off(X)}(V, U) = \{ \begin{matrix}∅ & if V ⊄ U \\ \{i\} & i is the inclusion if V ⊂ U. \end{matrix}

1.1. Definition : A presheaf on X with values in a category C is a contravariant functor from the category Off(X) with values in the category C.

We say again what this means: To any open set U ⊂ X our presheaf F associates an object F(U) ∈ Ob(C). Whenever we have an inclusion

V ⟶^{i} U

we get a so-called restriction morphism

r_{U|V} : F(U) ⟶ F(V).

Of course we have r_{U|U} = Id and for V_{1} ⊂ V_{2} ⊂ U we get a commutative diagram

F(U) \underrightarrow{r_{U|V_{2}}} F(V_{2})
r_{U|V_{1}} ↘  ↙ r_{V_{2}|V_{1}}
F(V_{1})

which can be written awkwardly

r_{U|V_{1}} = r_{V_{2}|V_{1}} ∘ r_{U|V_{2}}.  (Sh0)

If this functor F takes values in the category Ab of abelian groups (, rings, modules over a ring, vector spaces, sets) we call it a presheaf of abelian groups (, rings, modules over a ring, vector spaces, sets). For us the target category will always be of one these simpler categories. This means that the objects F(U) will be sets equipped with some kind of additional structure and the morphisms will be maps which respect this additional structure. Under this assumption we we know what the elements in F(U) are they will be called the sections of F over U.

Sometimes it is a nagging question what F(∅) should be. Usually we can can take for F(∅) a so called final object in the category, this is an object Ω such that for any other object X ∈ Ob(C) we have exactly one morphism from X to Ω. For the category of sets we can take any set consisting of just one element and for the category Mod_{R} we can take the zero module. It is clear that presheaves with values in a given category C on X form a category PS_{X} by themselves. If we have two presheaves F, G a morphism

Ψ ∈ Hom_{PS_{X}} (F, G)

is a collection of morphisms

Ψ_{U} : F(U) ⟶ G(U)

which satisfies the obvious rule of consistency: whenever we have V ⊂ U we get a commutative diagram

Ψ_{U} : F(U) ⟶ G(U)
↓r_{U|V}  ↓r_{U|V}
Ψ_{V} : F(V) ⟶ G(V).

(If we were pedantic, we should also denote the r_{U|V} differently (rF_{U|V} or so).)

The category of presheaves (in a suitable target category C) contains a ( so called “full”) subcategory, this is the category of sheaves. Before I can define sheaves I need:

1.1.1 A remark about products Let us assume we have two indexing sets I, J and two families of objects {X_{i}}_{i∈I}, {Y_{j}}_{j∈J} in a category with products. Assume that we have a map

τ : J ⟶ I

and in addition that for every j ∈ J we have a morphism

f(j) : X_{τ(j)} ⟶ Y_{j}.

Then we get for j ∈ J a composition morphism

f(j) ∘ p(τ(j)) : ∏_{i∈I} X_{i} → Y_{j}

It is the definition of the product that this gives us a unique morphism

f_{τ} : ∏_{i∈I} X_{i} ⟶ ∏_{j∈J} Y_{j}

which produces a commutative diagram

∏_{i∈I} X_{i} →^{f_{τ}}  ∏_{j∈J} Y_{j}
↓  ↓
X_{τ(j)} ⟶^{f(j)} Y_{j}.

Hence morphisms from one product into another product can be obtained from maps between the indexing sets in the opposite direction and morphisms between the objects indexed by indices related by this map. We say that this arrow is induced by the maps between the indexing sets and the maps between the objects.

Now we explain the extra condition a presheaf has to satisfy if it wants to be a sheaf. We need that the target category C has products. For our purpose it is good enough if it is a category of rings or a category of modules. Let F be a C-valued presheaf on our space X. Let U ⊂ X be open, let U = ∪_{α∈A}U_{α} be an open covering. Then we get a diagram of maps

F(U) ⟶^{p_{0}} ∏_{α∈A} F(U_{α}) ⟶^{p_{1}} ; ⟶_{p_{2}} ∏_{(α,β)∈A×A} F(U_{α} ∩ U_{β}),

where the arrows are given as follows: The arrow p_{0} is induced by the maps

F(U) ⟶^{r_{U|Uα}} F(U_{α})

and p_{1}, p_{2} are induced in the above sense by the two projections

\begin{matrix}A × A & \overrightarrow{⟶} A \\ (α, β) & ⟼ α \\ (α, β) & ⟼ β \end{matrix}

and the restriction maps F(U_{α}) → F(U_{α} ∩ U_{β}). If we assume that our target category is the category of sets, (abelian) groups, rings ... where the product is the simple-minded product then we can see what happens to s ∈ F(U): It is mapped to

(…, r_{U|U_{α}}(s), …)_{α∈A.}

A section

(… s_{α} …)_{α∈A} ∈ ∏_{α∈A} F(U_{α})

is mapped to

(…, r_{U_{α}|U_{α}∩U_{β}}(s_{α}) …)_{(α,β)∈A×A}

under p_{1} and to

(…, r_{U_{β}|U_{α}∩U_{β}}(s_{β}), …)_{(α,β)∈A×A}

under p_{2}. In any case it is clear from (Sh0) that the first arrow “equalizes” the two arrows p_{1}, p_{2}. This means that p_{1} ∘ p = p_{2} ∘ p. Now we are ready to state the condition a sheaf has to satisfy. For simplicity we assume that our target category is one of the simple ones above.

1.2 Definition A presheaf F is a sheaf if and only if (Sh1) The arrow p_{0} is injective. (Sh2) The image of p_{0} is exactly the set of those elements where p_{1}, p_{2} take the same values. We summarize the two conditions into

F(U) ⟶^{∼}( ∏_{α∈A} F(U_{α}) ⟶^{p_{1}} ; ⟶_{p_{2}} ∏_{(α,β)∈A×A} F(U_{α} ∩ U_{β}))[p_{1} = p_{2}].

We will say that the above sequence is an exact sequence of sets. Comment: In the case of an abstract target category C we would have to explain what injectivity of p_{0} means and how we define the object [p_{1} = p_{2}] for a pair of morphisms A ⟶^{p_{1}} ; ⟶_{p_{2}} B. This is actually not so difficult.

Now we fix a target category C. The sheaves with values in C form a “full” subcategory S_{X} of the category of presheaves with values in C. This means that each sheaf is also a presheaf and for any two sheaves F, G on X we have that the sets of morphisms in the category of sheaves and in the category of presheaves are the same, i.e.

Hom_{PS_{X}} (F, G) = Hom_{S_{X}} (F, G)

1.3. Examples: On any topological space X we have the sheaf C_{0} of continuous ℝ or ℂ-valued functions: For any open set U ⊂ X we put C_{0}(U) = ring of real or complex valued continuous functions on U. The restriction maps are given by the restriction of functions. The properties (Sh1), (Sh2) are obvious because the continuity of a function can be checked locally. We can also define the sheaf

U → ℤ_{X}(U) = locally constant ℤ -valued functions on U.

Note that U → constant ℤ − valued functions on U

would only define a presheaf because (Sh2) will not be satisfied in general.

This makes it clear what the general rule is: whenever we have a class of functions defined by certain properties then they provide a sheaf if these properties can be checked locally. Of course we can replace ℤ by any abelian group A and define the sheaf

U ⟶ A_{X}(U) = locally constant A-valued functions on U.

We may look at these sheaves from a different point of view. We can put the discrete topology on A, and then we see that A_{X}(U) is simply the abelian group of continuous functions on U with values in A. Sometimes we will write \underline{A} instead of A_{X}. If we have a point p ∈ X then we can define the ring of germs of continuous functions in this point p. A germ of a continous function at p is a continuous function f : U_{p} → ℂ defined in an open neighborhood U_{p} of p modulo the following equivalence relation:

(f : U_{p} → ℂ) ∼ (g : V_{p} → ℂ)

if and only if there is a neighborhood W_{p} ⊂ U_{p} ∩ V_{p} of p such that and f|W_{p} = g|W_{p}.

It is clear that the germs form a ring which is called C_{0,X,p}. It is clear that this ring is the direct limit

lim_{\overrightarrow{U∋p}} C_{0}(U) = C_{0,X,p}

(See I.3.4,3.5). This ring is a local ring which means that it has a unique maximal ideal. This maximal ideal 𝔪_{p} is the kernel of the evaluation at p. To see this one has to observe that a germ f which does not vanish at p also does not vanish in a small neighborhood of p and on this neighborhood we can define the continuous function 1/f. This means that f is invertible in C_{0,X,p} and it follows that any ideal in C_{0,X,p} which is not contained in 𝔪_{p} is the entire ring. Of course such a ring of germs is pretty big. If we do the same thing with our sheaf ℤ_{X} then it is clear that a germ at p is determined by its value at p. Hence in this case the ring of germs is simply ℤ_{X,p} = ℤ. This is not a local ring.

III.2. Manifolds as locally ringed spaces At this point I want to explain that the concept of sheaves gives us a better way to think of topological (C^{0}), differentiable (C^{∞}) or complex manifolds. I hope the explanation will also be helpful for the understanding of the concept of sheaves. A topological manifold X is a Hausdorff space such that for each point p ∈ X we can find an open neighborhood U_{p} of p which is homeomorphic to an open set in ℝ^{n}:

U_{p} ⟶^{∼} U ⊂ ℝ^{n}.

This is also called a C^{0}-manifold, on this space we can define the sheaf C_{0,X} of germs of continuous function with values in ℝ or ℂ.

A non-trivial theorem in algebraic topology asserts that two non-empty open sets U ⊂ ℝ^{n} and V ⊂ ℝ^{m} can only be homeomorphic if n = m. (See Chap IV.7.3) This allows us to speak of the dimension of the topological manifold provided it is connected. A C^{∞}-manifold of dimension n is a topological manifold X together with a C^{∞}-atlas: This is a family

{V_{α}, u_{α}}_{α∈A}

of open subsets such that

(i) X = ⋃_{α∈A} V_{α}

(ii) The u_{α} are homeomorphisms

u_{α} : V_{α} ⟶^{∼} V′_{α}

where the V′_{α} are open subsets in ℝ^{n}. (iii) If V_{α} ∩ V_{β} ≠ ∅ then we get a diagram

V_{α} ∩ V_{β} u_{α}(V_{α} ∩ V_{β}) ⊂ V′_{α} ; {u_{α} \atop ↗} ; u_{αβ} ↓ ↑ u_{βα} ; {u_{β} \atop ↘} ; u_{β}(V_{α} ∩ V_{β}) ⊂ V′_{β}

and we demand that u_{αβ}, u_{βα} are C^{∞}-maps. The maps u_{α} : V_{α} ⟶^{∼} V′_{α} are called the local charts of the atlas. In this case it is easier to see that the dimension is well defined.

We may define a complex manifold of dimension n in the same way if we demand that the V′_{α} are open in ℂ^{n} and the u_{αβ}, u_{βα} are holomorphic maps. Of course it is clear that a complex manifold of dimension n also carries a structure of a C^{∞} manifold of dimsion 2n. Once we have the notion of C^{∞}-manifold (resp. complex manifold) we may define the sheaves of germs of C^{∞}− (resp. holomorphic) functions: For U ⊂ X and f : U → ℂ, we say that f is C^{∞} (resp. holomorphic) if for any p ∈ U and any V_{α} with p ∈ V_{α} the map

f̃_{α} : f ∘ u_{α}^{−1} : u_{α}^{−1}((V_{α} ∩ U)) ⟶ ℂ

is C^{∞} (resp. holomorphic). Let us denote these sheaves by C_{X}^{∞} and O_{X} respectively. After defining a C^{∞} (resp. complex) manifold in the way we did it there is still a lot of talking about how to compare different atlases, what are equivalence classes of atlases what are maximal atlases and so on. I want to explain that these concepts of manifolds become much clearer if we follow Grothendieck and introduce the concept of locally ringed spaces.

With our definition we know what it means that a map

h : X → Y

between two C^{∞} (resp. complex) manifolds is a C^{∞} (resp. holomorphic) map. Such a map should be continuous and then we use the atlases to formulate what else should be true, namely that the maps induced by the local charts should be C^{∞} (resp. holomorphic). But we see that there is a different way of formulating that h is C^{∞} (resp. holomorphic): Whenever we have open sets U ⊂ X, V ⊂ Y such that h(U) ⊂ V, i.e.

h : U → V

and a section f ∈ C_{X}^{∞}(V) (resp. f ∈ O_{X}(V)) then the composite f ∘ h is certainly continuous. It is not hard to check, that our map is C^{∞} (resp. holomorphic) if and only if for any such pair U, V and any f the composite map f ∘h is again C^{∞} (resp. holomorphic), i.e. we get a map

∘h : C_{Y}^{∞}(V) ⟶ C_{X}^{∞}(U)

resp.

∘h : O_{Y} (V) ⟶ O_{X}(U).

A better formulation is obtained if we introduce the sheaf (see the following sections on f_{∗}, f^{∗} and the adjointness formula) h^{∗}(C_{Y}^{∞}) on X: For any open subset U ⊂ X the space of section

h^{∗}(C_{Y}^{∞})(U)

consists functions f : U → ℂ which have the following property: For any point p ∈ U we can find a neighborhood U_{p} of p and an open set V_{h(p)} ⊂ Y such that

h(U_{p}) ⊂ V_{h(p)}

and we can find a section f̃ ∈ C_{X}^{∞}(V_{h(p)}) so that

f = f̃∘ h.

Then we can say that a map h : X → Y is C^{∞} (resp. holomorphic) if it is continuous and induces a map

∘h : h^{∗}(C_{Y}^{∞}) → C_{X}^{∞}

resp.

∘h : h^{∗}(O_{Y}) → O_{X}.

Of course the composition with h always induces a map

h^{∗}(C_{Y}^{0} ) → C_{X}^{0}

between the sheaves of continuous functions. A C^{∞} or holomorphic map h has to respect the distinguished subsheaves which have been defined using the atlases.

Now we come to the decisive point, we turn the whole thing around and say:

A C^{∞}- (resp. complex) manifold is a topological space X together with a subsheaf C_{X}^{∞} (resp. O_{X}) in the sheaf of continuous functions such that for any point p ∈ X we have a neighborhood U_{p} of p and a homeomorphism h between U_{p} and an open subset U′ of ℝ^{n} (resp. ℂ^{n}) such that

(U_{p}, C_{X}^{∞}) ≃ (U′_{p}, C_{U′_{p}}^{∞})

resp.

(U_{p}, O_{X}) ≃ (U′_{p}, O_{U′_{p}})

where U_{p} is open in ℝ^{n} (resp. ℂ^{n}) and the sheaves are the C^{∞} (resp. holomorphic) germs on U′_{p} and where ≃ means that the composition ∘h induces an isomorphism between the subsheaves. In very simple words: A so and so manifold is a topological manifold on which we have a sheaf of function which locally looks like the sheaf of it so and so functions on some model. In our examples the stalks are local rings hence we get a examples of so called locally ringed spaces. It is not only so that we get a much clearer concept of C^{∞} or complex manifolds. It turns out that this concept allows generalizations to cases where we cannot work with atlases anymore. (See example 4) below and Chapter VI.) It is possible to define the category of locally ringed spaces. These are topological spaces X together with a sheaf of rings whose stalks are local rings. We will encounter these objects in Chapter VI. I want to discuss a couple of examples and exercises.

2.1 Examples 1a): We define the structure of a complex space on the one dimensional projective space ℙ^{1}(ℂ). As a topological space this is the space of lines in ℂ^{2} passing through the origin. This is also the space of all pairs (z_{0}, z_{1}) ≠ (0, 0) of complex numbers divided by the equivalence relation

(z_{0}, z_{1}) ∼ (λz_{0}, λz_{1}) λ ∈ ℂ^{∗}.

We have the two open subsets U_{0} (respectively U_{1}) where the coordinate z_{0} ≠ 0 (respectively z_{1} ≠ 0.) On these open subsets we can normalize the non zero coordinate to one and get bijections

\begin{matrix}U_{0} ⟶^{∼} ℂ, & U_{1} ⟶^{∼} ℂ, \\ (1, z) ↦ z, & (u, 1) ↦ u \end{matrix}

Now we define the sheaf O_{ℙ^{1}} : For any open subset U ⊂ ℙ^{1}(ℂ) the sections of O_{ℙ^{1}}(U) consist of those ℂ-valued functions whose restriction U_{0} ∩ U resp. U_{1} ∩ U are holomorphic. 1b): Of course we can define the n-dimensional projective space ℙ^{n}(ℂ). Again it is the space of lines in ℂ^{n+1} passing through the origin. We can identify this to the space

{(z_{0}, ⋯, z_{n}) ∈ ℂ^{n+1} | not all z_{i} = 0}/ℂ^{∗}

where ℂ^{∗} acts diagonally. We define the subset

U_{i} = {(z_{0}, ⋯, z_{n}) ∈ ℂ^{n+1} | z_{i} ≠ 0}/ℂ^{∗}

and identify U_{i} ˜ ⟶ℂ^{n} by the map (z_{0}, ⋯, z_{i}, ⋯, z_{n}) ⟼ (\frac{z_{0}}{z_{i}} , ⋯, \frac{z_{n}}{z_{i}} ) . The sheaf of holomorphic functions on ℙ^{n}(ℂ) is the sheaf of those functions whose restriction to the U_{i} is holomorphic. 2): We choose a lattice

Ω = {n_{1}ω_{1} + n_{2}ω_{2} | n_{1}, n_{2} ∈ ℤ}

in ℂ, where ω_{1}, ω_{2} are linearly independent over ℝ. This lattice operates by translations in ℂ, we form the quotient ℂ/Ω as a topological space, the projection

π : ℂ → ℂ/Ω.

is locally a homeomorphism. We define

O_{ℂ/Ω}(U) = {f : U → ℂ | π^{−1}(U) ⟶^{f∘π} ℂ is holomorphic}.

Then it is clear that this gives ℂ/Ω the structure of a complex manifold. 3): Let us assume that we have a holomorphic function f : U → ℂ where U ⊂ ℂ is open and contains the origin. We assume f(0) = 0. We consider f as a germ and we assume that its n-th iteration is the identity, i.e. f(f(… f(z)) …) = z. We assume the f is of exact order n, i. e. no earlier iteration gives the identity. Of course f(z) = ζz + a_{1}z^{2} … where ζ = e^{\frac{2πik}{m}} and (k, m) = 1. We can find a smaller open set D ⊂ U such that f(D) = D. This defines a holomorphic action of the cyclic group G =< f^{ν} > of order n on D and we can form the quotient under this action. This is the space D/G = B. Let π : D → B the projection map. We define a sheaf O_{B} on B : For any open set V ⊂ B we define O_{B}(V) as the ring of holomorphic functions on the inverse image π^{−1}(V) ⊂ D which are invariant under the action of the cyclic group G.

2.1.3 Exercise

Prove that this sheaf defines a structure of a one dimensional complex space on B. Hint: Consider the special case where U = ℂ and f(z) = ζz first. Of course the problem arises only in a neighborhood of the origin 0. There the stalk of the sheaf O_{B} is ring of power series in w = z^{n} which have a strictly positive radius of convergence. Then return to the general case and prove that you can find a germ of a function g(z) = z + b_{2}z^{2} + b_{3}z^{3} … such that f(g(z)) = g(ζz) and show that this reduces the problem to the first case.

4) Let us consider ℂ^{2} and consider the following action of our cyclic group:

f : (z_{1}, z_{2}) ↦ (ζz_{1}, ζ^{−1}z_{2})

If we form the quotient π : ℂ^{2} → ℂ^{2}/G = B we can try to play the same game. Again we get the structure of a two dimensional complex variety except at the point π(0) = 0. Here we see that the germ of our sheaf O_{B} becomes a power series ring in u = z_{1}^{n}, v = z_{2}^{n}, w = z_{1}z_{2} and we have uv = w^{n}. This means u, v, w are not independent variables anymore. At 0 our space is singular and not locally isomorphic to (ℂ^{2}, O_{ℂ^{2}}). But our concepts of locally ringed spaces are strong enough to deal with this and define the more general structure of a complex space which may have singular points.

5a) I want to discuss an example which is much more subtle, and where I need some difficult theorems from local complex analysis. We assume that U ⊂ ℂ^{n} is an open subset and f_{1}(z_{1}, ⋯, z_{n}), ⋯, f_{r}(z_{1}, ⋯, z_{n}) are holomorphic functions on U. Then we can consider the ideal I ⊂ O_{ℂ^{n}}(U) which is generated by these functions. We can look at the subset Y of common zeroes of the f_{i}, i.e.

Y = \{\underline{z} = (z_{1}, ⋯, z_{n}) | f_{i}(\underline{z}) = 0 for all i = 1 ⋯ r\}

and this is of course also the set of common zeroes of all the f ∈ I. For any open subset V ⊂ Y we can look at the open sets U′ ⊂ U with U′ ∩ Y = V, and then we can form the quotient

O_{ℂ^{n}}(U′)/(f_{1}, ⋯, f_{r})

where (f_{1}, ⋯, f_{r}) is the O_{ℂ^{n}}(U′)-ideal generated by the f_{i}. We put

O_{Y} (V) = lim_{\overrightarrow{U′:U′∩Y =V}} O_{ℂ^{n}}(U′)/(f_{1}, ⋯, f_{r}).

Now it is a theorem in local complex analysis that V → O_{Y} (V) is in fact a sheaf (see ????). One checks that the stalk O_{Y,y} = lim_{\overrightarrow{V :y∈V}} O_{Y} (V) is a local ring and the pair

(Y, O_{Y})

is in fact a locally ringed space. I want to point out that we cannot interprete the rings O_{Y} (V) as rings of holomorphic functions on Y. We may for instance consider the case that U = ℂ, and we take the single function f(z) = z^{2}. Then Y = {0} and the local ring is ℂ[z]/(z^{2}). It contains nilpotent elements and cannot be interpreted as ring of holomorphic functions. But still our space (Y, O_{Y}) is called a complex space.

5b) We say that our system of equations satisfies the Jacobi criterion in a point y ∈ Y if the Jacobian matrix

(\frac{∂f_{i}}{∂z_{j}} )_{i,j} (y) i = 1 ⋯ r, j = 1 ⋯ n

has maximal rank r. Then this is still true in a small open neighborhood of y. The implicite function theorem says that in a small neighborhood U_{1} ⊂ ℂ^{n} of y we can perform a change of coordinates

u_{i} = g_{i}(z_{1} ⋯ z_{n}) i = 1 ⋯ n

such that in the new coordinates our functions become f_{1}(u_{1} ⋯ u_{n}) = u_{1}, ⋯, f_{r}(u_{1} ⋯ u_{n}) = u_{r}. Hence we see that in this neighborhood

Y ∩ U_{1} = {(0, ⋯, 0, u_{r+1}, ⋯, u_{n}) | u_{i} suff. small},

and then (Y, O_{Y}) is clearly an n − r-dimensional complex manifold in the neighborhood of y ∈ Y. In this case we do not have to invoke the above mentioned theorem. We can turn this around and say that a subset Y ⊂ U is a d dimensional submanifold of U if can describe it locally as the common set of zeroes of n − d holomorphic functions which satisfy the Jacobi criterion. We come back to the situation in 5a). We say that the ideal I defines a (smooth) submanifold of dimension d if the set of common zeroes Y is a submanifold of dimension d and if in addition at any point y ∈ Y we can find g_{1}, …, g_{n−d} ∈ I which satisfy the Jacobi criterion at the point y. In this situation the argument in 5b) shows that these g_{1}, …, g_{n−d} generate the ideal I we we restrict it to a small neighborhood of y.

5c) A closed subset Y ⊂ ℙ^{n}(ℂ) is a d dimensional complex projective manifold if for any index i the intersection Y ∩ U_{i} is a d-dimensional complex submanifold of U_{i}.

5d) A homogenous polynomial of degree k is a polynomial

f(z_{0}, ⋯, z_{n}) = ∑ a_{ν_{0}···ν_{n}}z_{0}^{ν_{0}} ⋯ z_{n}^{ν_{n}}

where a_{ν_{0}···ν_{n}} = 0 unless ∑ ν_{i} = k. We cannot consider such a polynomial as a function on ℙ^{n}(ℂ). But of course it makes sense to speak of the zeroes of this polynomial on ℙ^{n}(ℂ). Therefore we may consider an ideal I = {f_{1}, ⋯, f_{s}} which is generated by s homogenous polynomials. We can look at the common set of zeroes

Y = \{\underline{z} = (z_{0} ⋯ z_{n}) | \underline{z} ≠ 0, f_{i}(\underline{z}) = 0\}/ℂ^{∗}.

Such a set Y is called an algebraic subset of ℙ^{n}(ℂ). If we restrict a homogenous polynomial f to one of the open sets U_{i} above, then we can interprete it as a function on U_{i} because we can normalize the i-th coordinate of a point to one. Hence our ideal I defines an ideal I_{i} of holomorphic functions on each of the U_{i}. Such a subset Y ⊂ ℙ^{n}(ℂ) is called a smooth projective (algebraic) variety of dimension d if each of the ideals I_{i} defines a smooth submanifold of dimension d in the sense of 5b). This definition is not yet very satisfactory because it needs input from analysis ( the implicit function theorem), for a definition in purely algebraic terms I refer to VII.3. It happens the we need more than n − d equations to describe a smooth projective variety of dimension d. Locally at a point y we can choose n−d equations from our set of equations to describe Y but this subset may vary if the point moves around. If we have such a complex d -dimensional submanifold Y ⊂ ℂ^{n} then the coordinate functions z_{1}, …, z_{n} are of course holomorphic functions on ℂ^{n} and therefore after restriction also on M. If we have a point y ∈ Y we may consider the functions

z̃_{i} = z_{i} − z_{i}(y) for i = 1, …, n

as holomorphic functions on Y. Then it follows from 5b) that we can pick d functions from this set - let us assume that these are z̃_{1}, …, z̃_{d}- such that locally the remaining functions can be written as convergent power series in these, i.e.

z̃_{d+j} = h_{j}(z̃_{1}, …, z_{d}) j = 1 … n − d

Then the

z̃_{i} = z_{i} − z_{i}(y) for i = 1, …, n

are called a system of local parameters at y.

III.3. Stalks and sheafification In our examples above we had the notion of a germ of a function at a point p. This notion can be extended to the more general classes sheaves. Let us assume that we consider the category of (pre-) sheaves on X with values in some nice category (abelian groups, rings or sets). If we have a point p ∈ X then we consider the set 𝔘_{p} of open sets containing our point p. We define an ordering on this set

V ≥ U if and only if V ⊂ U.  (sic!)

Then this is an inductive system which is also directed: to any U_{1}, U_{2} we find a V with U_{1} ≤ V, U_{2} ≤ V. If we have a (pre-)sheaf F on X we define the stalk in p by

F_{p} = lim_{\overrightarrow{U∈𝔘p}} F(U),

and this limit is simply the (abelian group, ring, set) of germs of sections. It inherits the structure of an (abelian group, ring, set); this follows from this directedness and is discussed in the Exercise 4) in Chapter I. An element s_{p} ∈ F_{p} is called a germ of a section. By definition it can always be represented by a section s_{U} ∈ F(U) where U ∈ 𝔘_{p}. If this is so we write s_{U}|_{p} = s_{p} and we say that s_{p} is the restriction of s_{U} to the stalk at p.

3.1 Let s be a section over the open set U. If we have s_{p} = 0 at p ∈ U then we find an open neighborhood V of p such that s restricted to this neighborhood is zero. Hence we can define the support of s: It is the closed subset of U where s_{p} ≠ 0. These stalks help to clarify the difference of the notion of sheaves and presheaves. For any presheaf we can consider the map

F(U) ⟶ ∏_{p∈U} F_{p},

which is given by restricting the sections to the stalks. Then we know:

(i) This map is injective, if and only if our presheaf satisfies (Sh1). (ii) If a presheaf F satisfies (Sh1) then it is a sheaf if and only if the following holds: A collection of germs (… s_{p} …)_{p∈U} is the restriction of a section over U if for any p we find a U_{p} ∈ 𝔘_{p} and a section s̃_{p} ∈ F(U_{p}) such that s̃_{p}|_{q} = s_{q} for all q ∈ U_{p}. We leave the verification of this fact to the reader.

3.2. The process of sheafification of a presheaf: To any presheaf G on a space X we can construct a sheaf G^{#} together with a map j : G → G^{#} (in the category of presheaves) such that for any sheaf F we have

Hom_{PS_{X}} (G, F) = Hom_{S_{X}} (G^{#}, F).

This can also be seen as another example of a representable functor. Our presheaf G defines a functor from the category S of sheaves (with values in the category of rings, abelian groups, sets) into the category of sets namely the functor F → Hom_{PS_{X}} (G, F). our sheaf G^{#} is the representing this functor. Hence by the Yoneda-lemma it is unique up to isomorphism. Using the stalks it is possible to define G^{#} quite directly. We define

G^{#}(U) = \{(… s_{p} …) ∈ ∏_{p∈U} G_{p} | \begin{matrix}For any point p ∈ U ∃ open U_{p} \\ p ∈ U_{p} ⊂ U and s̃_{p} ∈ G(U_{p}), s. t. \\ s̃_{p} | q = s_{q} for all q ∈ U \end{matrix} \} .

The reader should verify, that this defines indeed a sheaf, this sheaf has the same stalks as our original presheaf, we have a map G → G^{#} and it has the required property. There exist some more abstract notions of sheaves on so called Grothendieck topologies, these are in some sense “spaces” which sometimes do not have points anymore. In such a case it is not possible to use the stalks, but still it is possible to construct G^{#}. Therefore I will give here another construction of G^{#} which does not use stalks. We consider coverings 𝔘 = {U_{i}}_{i∈I}, U = ⋃_{i∈I} U_{i} of an open set U. We introduce the category of coverings. An arrow from a covering 𝔙 = {V_{α}}_{α∈A} to the covering 𝔘 = {U_{i}}_{i∈I} is a map

τ : A ⟶ I

s. t. ⋃_{α∈τ^{−1}(i)} V_{α} = U_{i}. We write

τ : 𝔙 → 𝔘

for such a morphism. In general the arrow τ is not determined by the two coverings, but many constructions using this arrow will give results not depending on it. We will say that such an arrow defines a refinement of 𝔘 by 𝔙. Sometimes we will say that 𝔙 is a refinement of 𝔘 if there is an arrow from 𝔙 to 𝔘.

The arrow τ defines a map between diagrams (see the general remark about maps between products at the beginning of this section)

\begin{matrix}G(U) & ⟶^{p_{0}} & ∏_{i∈I} G(U_{i}) & ⟶^{p_{1}} ; ⟶_{p_{2}} & ∏_{(i,j)∈I×I} G(U_{i} ∩ U_{j}) \\ ↓ & & ↓ & & ↓ \\ G(U) & ⟶ & ∏_{α∈A} G(V_{α}) & \overrightarrow{⟶} & ∏_{(α,β)∈A×A} G(V_{α} ∩ V_{β}) \end{matrix} .

For any covering 𝔘 = {U_{i}}_{i∈I} of U we define

G^{𝔘}(U)[p_{1} = p_{2}] := {s ∈ ∏_{i∈I} G(U_{i}) | p_{1}(s) = p_{2}(s)} .

If 𝔙 is a refinement of 𝔘 then our map τ defines a map

\begin{matrix}G(U) & ⟶ & G^{𝔘}(U)[p_{1} = p_{2}] \\ ∥ & & ↓ \\ G(U) & ⟶ & G^{𝔙}(U)[p_{1} = p_{2}] \end{matrix} .

It is not difficult to see that the vertical arrow does not depend on the choice of τ. Now we need the courage to believe that we can extend the notion of direct limit to this situation where we do not have an indexing set but a category which is directed because two coverings have always a common refinement. We put

G^{+}(U) = lim_{\overrightarrow{𝔘}} G^{𝔘}(U)[p_{1} = p_{2}].

We check that G^{+} is again a presheaf, and it satisfies condition (Sh1). Moreover if the original presheaf G satisfies already (Sh1) then G^{+} satisfies even (Sh2). Hence we see that G^{++} = G^{#} is always a sheaf. We have

i : G ⟶ G^{#},

and G^{#} has the required universal property.

III.4. The functors f_{∗} and f^{∗}: Given two topological spaces X, Y and a continuous map f : X → Y, we construct two functors f_{∗}, f^{∗} which transport sheaves on X to sheaves on Y and sheaves on Y to sheaves on X respectively. Let us denote by S_{X} resp. S_{X} the category of sheaves resp. presheaves on X with values in the category of abelian groups, rings or sets. If we have a sheaf F on X we define the sheaf f_{∗}(F) on Y by

f_{∗}(F)(V) = F(f^{−1}(V))

for all open subsets V ⊂ Y. It is clear that f_{∗}(F) is a sheaf on Y, it is called the direct image of F. The functor f^{∗} transforms sheaves on Y into sheaves on X. The idea is that the stalk of f^{∗}(G) in a point x ∈ X is equal to the stalk of the original sheaf G in the point y = f(x), i.e. f^{∗}(G)_{x} = G_{f(x)}. The actual construction is a little bit complicated. At first we define a presheaf f′(G): For U ⊂ X we put

f′(G)(U) = lim_{\overrightarrow{V ⊃f(U)}} G(V).

It is not difficult to verify that this is a presheaf and that for any covering U = ∪_{i∈I}U_{i} we get an injective map

f′(G)(U) ⟶ ∏_{i∈I} f′(G)(U_{i}).

It satisfies (Sh1) but not necessarily (Sh2). We define

f^{∗}(G) = f′(G)^{#}.

We recall that the stalks of the sheafification of a presheaf are equal to the stalks of the presheaf hence we get

f^{∗}(G)_{x} = lim_{\overrightarrow{x∈U}} lim_{\overrightarrow{V ⊃f(U)}} G(V) = lim_{\overrightarrow{V :f(x)∈V}} G(V) = G_{f(x)}.

Since the exactness of sequences of sheaves can be checked stalkwise, it is clear that f^{∗} is an exact functor.

4.1. The adjunction formula: The functors f_{∗}, f^{∗} are adjoint functors. To be more precise: The functor f^{∗} is left adjoint to f_{∗}. This means that we have a functorial isomorphism

Hom_{S_{X}} (f^{∗}(G), F) = Hom_{S_{Y}} (G, f_{∗}(F)).

Here functorial means that from morphisms u : G′ → G and v : F → F′, we get the obvious commutative diagrams. It is not very difficult to verify the adjointness formula. From the construction of the sheafification we have Hom_{PS_{X}} (f′G, F) = Hom_{S_{X}} (f^{∗}G, F). Hence a morphism ψ in Hom_{PS_{X}} (f′G, F) is a collection of ψ_{U} : f′G(U) → F(U). It follows from the definition f′G(U) and the properties of the direct limit that this is nothing else than a collection of maps

ψ_{U,V} : G(V) → F(U)

where U, V run over all open sets in X, Y which satisfy f(U) ⊂ V, and where the maps in this collection satisfy the obvious compatibilities. We will call ψ_{U,V} the evaluation of ψ on U, V. Now we are allowed to evaluate on U = f^{−1}(V) and we get a collection ψ_{f^{−1}(V),V} = ϕ_{V} : G(V) → f_{∗}(F)(V), i.e. an element in Hom_{S_{Y}} (G, f_{∗}(F)). The other direction is also clear. Remark: It is always confusing and hard to memorize which functor is a left (right) adjoint of which. The question is whether f^{∗} has to be placed into the source or the target of the Hom(,). Here is a simple rule that helps. We have to remember that f_{∗} gives directly a sheaf while the construction of f^{∗}G involves the process of sheafification and this uses direct limits. But as I explained in the chapter on categories direct limits are made so that we know what the maps from them are. Hence the free place on the left in Hom( , ) is the place where f^{∗}G belongs. 4.2 Extensions and restrictions: We can consider the special case of an open subset U ⊂ X and let A = X \ U be its complement. Then we have the two inclusions i : A⟶X, j : U ⟶X. For a sheaf F on X the sheaf j^{∗}(F) is very easy to understand since for an open set V ⊂ U we have j^{∗}(F)(V) = F(V). This is called the restriction of F to U. The operation i^{∗}(F) is much more delicate and will cause us some trouble (See IV 1.2.2.1). If we have a sheaf G on U then j_{∗}(G) is a delicate functor since it depends on the local topology in the neighborhood of boundary points ( See IV ???). But for a sheaf G on A the i_{∗}(G) is easy to understand. Its stalks are zero outside of A and equal to the stalks of G on A. It is called the extension by zero.

III.5. Products If we have a family of sheaves {F_{α}}_{α∈A} the we can define the product: For any open set U ⊂ X we put

( ∏_{α∈A} F_{α})(U) := ∏_{α∈A} F_{α}(U)

and it is easy to verify that this is again a sheaf. If our sheaves have values in the category of rings, modules, abelian groups etc. the product is again a sheaf with values in that category. If the F_{α} are abelian groups or modules we might be tempted to take the direct sum of sheaves. But this does not work in general. The naive definition gives only a presheaf because (Sh2) may be violated if the indexing set A is infinite.

5.1. Perhaps here is the right place to explain that the sheaves on X with values in the category of abelian groups form an abelian category . First of all this says that for two such sheaves the set

Ψ ∈ Hom_{S_{X}} (F, G)

are abelian groups: If we have two morphisms Ψ = {Ψ_{U}, Φ_{U}}_{U}, Φ = {Φ_{U}, Φ_{U}}_{U} then Ψ + Φ = {Ψ_{U} + Φ_{U}}_{U}. This group structure is bilinear with respect to composition. If we have a morphism Ψ : F → G then we can define the kernel ker(Ψ) as the subsheaf U → ker(Ψ_{U}). This kernel has a categorial interpretation: For any other sheaf A

Hom_{S_{X}} (A, ker(Ψ)) = {ϕ ∈ Hom_{S_{X}} (F, G)|Ψ ∘ ϕ = 0}.

Now we may consider the preasheaf

K(U) = F(U)/ ker(Ψ)(U).

It is fundamental that this preasheaf is not neccesarily a sheaf and this will be explained in detail in the next Chapter. It is not hard to verify the first sheaf condition (Sh1) but in general it does not satisfy the second condition (Sh2). Of course we can sheafify the preasheaf K and we get the quotient sheaf

F/ ker(Ψ) = K^{#}

This quotient has again a categorial interpretation and it is called the coimage of Ψ. We can also define the image of Ψ as a subsheaf of G. It is simply im(Ψ)(U) = Ψ_{U}(K^{#}(U)) and by construction it is isomorphic to the coimage. These to objects namely the coimage and image can be defined in a categorial context and it is one of the axioms for an abelian category that they should be canonically isomorphic ( See [ Mac Lane ]). In an abelian category we can define the notion of exact sequences but this will be discussed in the following chapter.

Chapter IV

Cohomology of Sheaves

We consider sheaves with values in abelian groups. We can define the notion of an exact sequence of sheaves. A sequence of sheaves on a space X

0 ⟶ F′ ⟶ F ⟶ F′′ ⟶ 0

is exact if for all points x ∈ X the sequence of stalks is exact. It is easy to see that this is equivalent to

(i) For all open sets U ⊂ X the sequence

0 ⟶ F′(U) ⟶ F(U) ⟶ F′′(U)

is exact and

(ii) For any s′′ ∈ F′′(U) we can find a covering U = ⋃_{j} U_{j} by open sets and s_{j} ∈ F(U_{j})

such that s_{j} ↦ s′′|U_{j}

It is the decisive point that the exactness does not imply that F(U) ⟶ F′′(U) is surjective. We can only find local liftings for an s′′ ∈ F′′(U). Applied to U = X this tells us that the functor F → F(X) will not be exact in general. Hence we have to construct a right derived functor to it. As in Chapter II we introduce the notation H^{0}(X, F) for F(X) and we want construct cohomology groups H^{1}(X, F), H^{2}(X, F) which have functorial properties and such that any short exact sequence yields a long exact sequence

0 ⟶ F′(X) ⟶ F(X) ⟶ F′′(X) → H^{1}(X, F′) →

as in the previous chapter. The following two examples are absolutely fundamental. In a nutshell we see everything that makes sheaf cohomology work. I also want to stress the almost perfect analogy between these two examples which will be explained in (1.2.1.).

IV.1 Examples

1.1 Sheaves on Riemann surfaces In the previous section we introduced the notion of a complex manifold (See 2.1). Here I want to consider a compact Riemann surface (X, O_{X}). This means that X is a compact connected complex manifold of dimension 1. For any P ∈ X we find an open neighborhood U_{P} of P such that (See III.2 and III.2.1)

(U_{P} , O_{X}|U_{P}) ≃ (B, O_{B}),

where B = {z ∈ ℂ | |z| < 1} is the open unit disc and where O_{B} is the sheaf of germs of holomorphic functions on B. We assume that the homeomorphism between the spaces maps P to the origin 0 in the disc. The element z ∈ O_{B}(B) yields via the isomorphism an element z_{P} ∈ O_{X}(U_{P}). This element z_{P} vanishes at P and it generates the maximal ideal 𝔪_{P} of the stalk O_{X,P}. Such an element is called a uniformizer or uniformizing element at P. Any power series

u_{P} = f(z_{P}) = az_{P} + bz_{P}^{2} , …

which has a positive radius of convergence and with a ≠ 0 can serve as an uniformizer as well. A complex function

g : U_{P} \{P } → ℂ

is called meromorphic on U_{P} if it is holomorphic and if we can find an integer n such that z_{P}^{n} ·g = h extends to a holomorphic function on U_{P}. We say that g has a pole of order n at P if n is the smallest value for such an integer. We write ord_{P} (g) = −n and by definition g ∈ z_{P}^{−n}O_{X,P}, but g ∉ z_{P}^{−n+1}O_{X,P} . If T is a finite subset of S and if f : S \ T → ℂ is a holomorphic function then we say that f is meromorphic if its singularities at the points of T are at most poles ( and not essential singularities). For any point P ∈ T we have defined ord_{P} (f) and we define the polar divisor

Div_{∞}(f) = ∑_{P ∈T,ord_{P} (f)>0} ord_{P} (f)P

which we consider as an an element in the divisor group Div(S), the free abelian group generated by the points of S. Since S is compact it follows from the principle of analytic continuation that f can only have a finite number of zeroes on U = S \ T and this implies that 1/f is also holomorphic on some open set U′ = S \ T′ where T′ is finite and then 1/f is also meromorphic. We may also define the zero divisor of f as

Div_{0}(f) = −Div_{∞}(1/f)

and the divisor of f as

Div(f) = Div_{0}(f) + Div_{∞}(f).

We have a homomorphism called the degree which is given by

deg : Div(S) → ℤ

which is given by deg : D = ∑ n_{P} P ↦ ∑ n_{P}. A divisor D = ∑_{P} n_{P} P which is the divisor of a meromorphic function will be called a principal divisor. We will see (See V.1.3.2 ) that for a principal divisor the degree deg(D) = ∑ n_{P} = 0.

To any divisor D = ∑_{P} n_{P} P we attach the sheaf O_{X}(D) which which is defined by

O_{X}(D)(U) = {f meromorphic on U | ord_{P} (f) ≥ −n_{P} for all P ∈ U}.

We could also say that f ∈ z_{P}^{−n_{P}} O_{X,P} for all P. A divisor D = ∑ n_{P} P is called effective if all n_{P} ≥ 0, we could also call this a positive divisor and write D ≥ 0. The definition of O_{X}(D)(U) can be reformulated: It consists of those meromorphic functions f on U for which Div(f) + D | U ≥ 0. If D is an effective divisor we have an inclusion of sheaves O_{X} ⊂ O_{X}(D). We form the quotient sheaf 𝕃_{D} = O_{X}(D)/O_{X}. It is clear that the stalk at P is z_{P}^{−n_{P}} O_{X,P} /O_{X,P} . For any point P

z_{P}^{−n}O_{X,P} /O_{X,P} = 𝕃_{P}^{(n)}

is the finite dimensional vector space of Laurent expansions, an element ℓ ∈ L_{P} can be written as

ℓ = \frac{a_{n}}{z_{P}^{n}} + \frac{a_{n−1}}{z_{P}^{n−1}} + … \frac{a_{1}}{z_{P}} mod O_{X,P} .

If a_{n} ≠ 0, we say that ℓ has a pole of order n. So the stalk of this sheaf at a point P ∈ X is the vector space of all Laurent expansions of pole order ≤ n_{P} . Especially the stalk is zero at points where n_{P} = 0 and therefore the sheaf 𝕃_{D} has only a finite number of non-zero stalks. It is called a skyscraper sheaf. We have the exact sequence of sheaves

0 → O_{X} → O_{X}(D) → 𝕃_{D} → 0.

It is clear that the space of sections H^{0}(X, 𝕃_{D}) is simply the direct sum of the stalks in the points P with n_{P} > 0. There is no interaction between the different points. The question whether the sequence

0 → H^{0}(X, O_{X}) → H^{0}(X, O_{X}(D)) → H^{0}(X, 𝕃_{D}) → 0

is exact amounts to whether a given collection of Laurent expansions at the finitely many points P with n_{P} > 0 can be realized by a meromorphic function on X. In general the answer is no and the discrepancy is controlled by the first cohomology group H^{1}(X, O_{X}) which we will define later. To be more precise we will construct a map

δ : H^{0}(X, 𝕃_{D}) → H^{1}(X, O_{X})

such that the extended sequence

0 → H^{0}(X, O_{X}) → H^{0}(X, O_{X}(D)) → H^{0}(X, 𝕃_{D}) → H^{1}(X, O_{X})

becomes exact. The computation of H^{1}(X, O_{X}) is more or less equivalent to the Riemann-Roch Theorem which we will discuss in the chapters on curves and Riemann surfaces.

1.1.1. Exercise:

(1) Prove that in the case X = ℙ^{1}(ℂ) the above sequence is always exact.

(2) Prove that in the case X = ℂ/Ω the above sequence is not always exact.

1.2 Cohomology of the circle We consider the circle S^{1} and the sheaf \underline{ℤ} which is defined by

\underline{ℤ}(V) = \{f : V → ℤ | f is locally constant \}.

We pick a point P ∈ S^{1} and let U = S^{1}\{P }. We define a sheaf ℤ^{(P)} by

\underline{ℤ}^{(P)}(V) = \underline{ℤ}(U ∩ V).

If i : U → S^{1} is the inclusion then this is the sheaf i_{∗}(ℤ) (see III.4.2). Clearly we have an inclusion \underline{ℤ} ⊂ \underline{ℤ}^{(P)} and for all Q ≠ P we have the equality of stalks

\underline{ℤ}_{Q} = \underline{ℤ}_{Q}^{P} = ℤ.

But in the point P we have

ℤ = \underline{ℤ}_{P} ↪ \underline{ℤ}_{P}^{(P)} = ℤ ⊕ ℤ

because on a little interval I_{ϵ} containing P we have \underline{ℤ}(I_{ϵ}) = ℤ but \underline{ℤ}^{(P)}(I_{ϵ}) = \underline{ℤ}(I_{ϵ} ∩ U) = ℤ ⊕ ℤ. Hence we get an exact sequence of sheaves

0 → \underline{ℤ} → \underline{ℤ}^{(P)} → S_{P} → 0

where S_{P} is the skyscraper sheaf whose stalk at P is ℤ and zero elsewhere. We get the sequence of global sections

0 → H^{0}(S^{1}, \underline{ℤ}) →^{∼} H^{0}(S^{1}, \underline{ℤ}^{(P)}) → H^{0}(S^{1}, S_{P})
∥  ∥  ∥
0 →  ℤ  →^{∼}  ℤ  →  ℤ

and we see that the last arrow is not surjective. Again we need a non-zero H^{1}(S^{1}, ℤ) to control the discrepancy. We even can have an idea what this group H^{1}(S^{1}, ℤ) should be. Intuitively we should think that the sheaf ℤ^{(P)} doubles the point P, so our circle becomes an intervall I and it is at least plausible that

H^{1}(I, ℤ) = H^{1}(S^{1}, ℤ^{(P)}).

But the intervall is contractible (see 4.4.11), and we will see then H^{1}(I, ℤ) = 0. Hence we should expect (and we will prove this later) that

H^{0}(S^{1}, S_{P})→^{∼}H^{1}(S^{1}, ℤ) ≃ ℤ.

I want to stress another important point. We can ask whether the isomorphism H^{1}(S^{1}, ℤ) ≃ ℤ is canonical. The answer is no! This becomes clear if we recall that

S^{P} = (ℤ ⊕ ℤ)/ℤ

where ℤ is embedded diagonally. There is no way to distinguish between the two possibilities to identify H^{0}(S^{1}, S^{P}) to ℤ. But we can choose an orientation on S^{1} (see 3.3.1), this eans tha we choose a direction ( a non zero tangent vector ) at each point which varies continously with the point. Then we have distinction between the two intervalls in the intersection

I_{ε} ∩ U = U_{ε}^{+} ∪ U_{ε}^{−}

we say that U_{ε}^{+} is the intervall which the positive tangent vector at P points to. Then

\underline{ℤ}(I_{ε} ∩ U) = \underline{ℤ}(U_{ε}^{+}) ⊕ \underline{ℤ}(U_{ε}^{−}) = ℤ ⊕ ℤ,

and we now have a canonical identification

H^{1}(S, ℤ) = ℤ

where we send

(a, b) mod ℤ ⟶ a.

1.2.1. I want to stress the analogy between the two examples: The sheaves O_{X} and \underline{ℤ} have a property in common: They are very rigid. This eans that any section over a connected open subset U is determined by its restriction to an arbitrarily small non empty open subset V ⊂ U. The analogy goes even further. If we consider the sheaf \underline{ℝ} on a manifold M, then we can characterize \underline{ℝ} as a subsheaf in the sheaf C_{∞}(M): It is the subsheaf of functions with zero derivatives. An analogous statement is true for O_{X}. We can characterize O_{X} as the subsheaf in the sheaf of C_{∞} function annihilated by the Cauchy-Riemann operators.

IV.2 The derived functor.

2.1 Introduction. We want to define a universal derived functor to the functor F → H^{0}(X, F). To do this we use the same ideas as in Chapter II. We define the notion of an injective sheaf: A sheaf I is injective if in any diagram

A ⟶^{φ} B
↓ψ
I

with ker(φ) ⊂ ker(ψ) we can find a map η : B → I which make this diagram commutative. It is rather easy to see that every sheaf F can be embedded into an injective sheaf. The following construction has been invented by Godement (See [Go],???). For any point x ∈ X we embed the stalk F_{x} into an injective abelian group I_{x}. We define the sheaf I by

I(U) = ∏_{x∈U} I_{x}

and the restriction maps ∏_{x∈U} I_{x} → ∏_{x∈V} I_{x} are induced by the inclusion V ⊂ U. To prove the injectivity of I we consider our diagram above stalk by stalk and choose for each x ∈ X an η_{x} such that the diagram commutes

A_{x}  ⟶^{φ_{x}} B_{x}
↓ψ_{x} ↙ η_{x}
I_{x}

commutes. By construction this collection provides an embedding η : F → I, for any open set U ⊂ X the homomorphism η_{U} : F(U) → I(U) is induced by the maps F(U) → F_{x} → I_{x}. Now it is obvious that we can find an injective resolution for any sheaf F:

0 → F → I^{0} → I^{1} → .

Consequently we define

H^{•}(X, F) = H^{•}(I^{•}(X)).

The same arguments as in the previous section show that this defines a universal right derived functor. The reader might or should even be scared: How can we ever compute the cohomology of a sheaf if we use such ”huge and bizarre” sheaves to define it. Our strategy will be to exhibit classes of ”smaller” sheaves which have the property that they are acyclic. One possibility to construct such sheaves is discussed in the following exercise.

2.1.1 Exercise.

Let us assume that we have a sheaf of commutative rings R on X, the rings should have an identity, especially we have 1 ∈ R(X). Let us assume that we have a so called partition of 1: For any covering X = ⋃_{i∈I} U_{i} we can find elements h_{i} ∈ R(X) such that Supp(h_{i}) ⊂ U_{i}, for any point x ∈ X we have only finitely many indices such that h_{ix} ≠ 0 and finally 1 = ∑_{i} h_{i}. Show that sheaves F of R modules are acyclic. Hint: Assume we have a short exact sequence

0 → F′ → F → F′′ → 0

of R-modules. Use the partition of unity to show that F(X) → F′′(X) is surjective.

We will see that on a C^{∞} -manifold M the sheaves of rings of C^{∞} functions have a partition of unity. This will imply that for any C^{∞} vector bundle ( See 3.2 ) E and the sheaf C^{∞}(E) of C^{∞} -sections in it

H^{i}(M, C^{∞}(E)) = 0 for all i > 0

2.2 A direct definition of H^{1}: We want to indicate briefly how we could approach the problem to define a right derived functor for H^{0}(X, F) more directly. The reader should notices the analogy between this approach and the one used to define the first cohomology group in group cohomology (See II.2.1.) Let us assume we have an exact sequence of sheaves

0 ⟶ F′ ⟶ F ⟶ F′′ ⟶ 0.

We look at

F(X) ⟶ F′′(X)

and pick a section s′′ ∈ F′′(X). Locally we can lift this section to a section in F. This means we can find a covering X = ∪U_{α} and sections s_{α} ∈ F(U_{α}) which map to s′′ | U_{α}. But the s_{α} do not necessarily match: The difference

s′_{α,β} = s_{α} − s_{β} | U_{α} ∩ U_{β}

is a section in F′(U_{α} ∩ U_{β}) because it goes to zero in F′′. The collection {s′_{α,β}}_{(α,β)∈A×A} satisfies the cocycle relation, i.e. we have

s′_{α,β} − s′_{β,γ} + s′_{γ,α} | U_{α} ∩ U_{β} ∩ U_{γ} = 0.

This suggests the definition of the group of 1-cocycles with respect to a covering 𝔘 = {U_{α}}_{α∈A}: These 1-cocycles are collections

(…, t_{α,β}, …) ∈ ∏_{(α,β)∈A×A} F′(U_{α} ∩ U_{β})

which satisfy the relation above. They form a group which will be denoted by Z^{1}(𝔘, F′). We may also define the group of coboundaries: An element (… t_{α,β} …) is a coboundary if we can find s′_{α} ∈ F′(U_{α}) s.t. t′_{α,β} = s′_{α} − s′_{β}.

We define H^{1}(X, 𝔘, F′) to be the quotient

H^{1}(X, 𝔘, F′) = Z^{1}(𝔘, F′)/B^{1}(𝔘, F′).

Now it is clear that s′′ defines an element

δ(s′′) ∈ H^{1}(S, 𝔘, F′),

and it is clear that s′′ is in the image if and only if δ(s′′) = 0. If we start from a different covering 𝔘′ then 𝔘 and 𝔘′ have a common refinement (See III.3.1) τ : 𝔚 → 𝔘, τ′ : 𝔚 → 𝔘′. We get maps

H^{1}(X, 𝔘, F′)  H^{1}(X, 𝔘′, F′)
↘  ↙
H^{1}(X, 𝔚, F′).

It is not difficult to see that these maps do not depend on the choice of of the arrows. It is clear that these maps are compatible with δ and hence we get a boundary operator

δ : F′′(X) ⟶ lim_{\overrightarrow{𝔘}} H^{1}(X, 𝔘, F′) = H̆^{1}(X, F′).

Now it is not hard to see that we have a structure of an abelian group on the limit, the boundary operator is a homomorphism and the sequence

0 → H^{0}(X, F′) → H^{0}(X, F) → H^{0}(X, F′′) →^{δ} H̆^{1}(X, F′) → H̆^{1}(X, F) → H̆^{1}(X, F′′)

is exact. Of course we need to compare this construction of cohomology groups with the other one using injective resolutions, this comes next.

2.2.1 A sheaf F on a space X is called flabby if for any open set U ⊂ X the restriction map F(X) → F(U) is surjective. This is a very strange property of a sheaf. For instance the continuous functions on a space almost never have this property.

2.2.2 We want to show that injective sheaves are flabby. To do this we consider an open subset U ⊂ X we denote its inclusion by j : U → X, let A = X \ U let us denote the inclusion of the closed set by i : A → X. For any sheaf F we can take its restriction to A and extend this restriction again to X by using i_{∗}. (Extension by zero: See III 4.2.)

We have a surjective homomorphism of sheaves F → i_{∗}i^{∗}(F) and this gives us an exact sequence of sheaves

0 → j_{!}(F) → F → i_{∗}i^{∗}(F) → 0

where of course j_{!}(F) is just the kernel. A short disgression: We may give a direct definition of this kernel and call it again the extension of F|U to X by zero. To give this direct definition we recall the notion of the support of a section (see III.3.1.) and notice that for any open set V ⊂ X we have more or less by definition

j_{!}(F)(V) = {s ∈ F(V) | the support of s does not meet V ∩ A}.

This means that this sheaf is a little bit delicate. By construction we have an inclusion j_{!}(F)(V)⟶F. In a sense the sheaf j_{!}(F) ”knows” the boundary points of U. Now we come back to our original problem, we wanted to show that injective sheaves are flabby. We have an inclusion

j_{!}I ⟶ j_{∗}I
↓
I

and since I is injective we find a homomorphism ϕ : j_{∗}I → I which makes this diagram commute. If we have a section s ∈ F(U) then this is by definition the same as a section s ∈ i_{∗}(F)(X) and then ϕ(s) ∈ F(X). It is clear from the diagram that ϕ(s) restricted to U is s. Moreover we see that our section ϕ(s) has support contained in the closure Ū, the best we can expect.

2.2.3 Exercise

a) Show that for a flabby sheaf F we have H̆^{1}(X, F) = 0

b) Show that H̆^{1}(X, I) = 0 for an injective sheaf. Show that this implies that for any sheaf F

H̆^{1}(X, F) = H^{1}(X, F).

c) Show that flabby sheaves are acyclic.

I discussed this construction of the first cohomology groups in detail, because here we can see how natural these constructions are. Here we meet a fundamental principle of homological algebra which is applied again and again:

We want to lift a section s′′ ∈ H^{0}(X, F′′) to a section s ∈ H^{0}(X, F). We localize the problem by choosing a covering where we have local liftings. These are not unique and hence it can happen, that they do not match on the intersections. These differences on the intersections yield a cocycle, and the class of this cocycle yields the obstruction to the global solution of the problem. We have seen how the same principle works in group cohomology (Chap. II.2.1.) There we want to lift a Γ invariant section m′′ ∈ (M′′)^{Γ} to a Γ-invariant section m ∈ M^{Γ}. In this context localizing means that we drop the requirement that m should be Γ−invariant. Then we find a non unique lifting. The comparison of the local sections on the intersections of the open sets in the geometric situation corresponds now to the comparison of m with γm where γ runs through the group. This gives the cocycles γ → m − γm ∈ M′

This construction generalizes to higher cohomology groups. We can define the so called Čech cohomology by means of coverings. The cohomology defined by means of injective resolutions and the Čech cohomology coincide on reasonably spaces. We postpone this discussion. At this point we make a short detour. Since we discussed H^{1} in some detail it may be appropriate to discuss the non abelian H^{1}, this means we discuss sheaves with values in non commutative groups and their first cohomology sets.. This non abelian cohomology plays an important role in the theory of bundles and I want to loose some words on this subject.

VI.3 Fiber bundles and non abelian H^{1}.

3.1. Fibrations. I want to introduce the notion of fibre bundles. We consider maps between topological spaces

π : X ⟶ B.

If we have another such map π′ : X′ → B then a map over B is a continuous map f : X → X′ which commutes with the projections. Now we consider a space F (the fibre) and another space B (the base). A continuous map π : X → B is called a (locally trivial) fibration with fibre F, if we can find a covering B = ⋃_{i∈I} U_{i} such that for any i we can find a homoemorphism Ψ_{i} over the base U_{i}

Ψ_{i} : π^{−1}(U_{i})  ⟶^{∼}  U_{i} × F
↘ ↙ p_{1}
U_{i}.

Locally in the base our space is a product of an open set in the base and the given fibre. We also say that X →^{π} B is a fibre bundle with fibre F. If V ⊂ B is open then a section to π over V is a continous map s : V → X for which π ∘ s = Id_{V}.

It is important to consider fibres F which are not only topological spaces but also carry some extra structure.

3.1.1. Vector bundles For instance we can consider the case that F is a finite dimensional ℝ or ℂ-vector space and where F, ℝ, ℂ are equipped with the standard topology. For convenience we denote by 𝕂 a field which is either ℝ or ℂ. In this case we can make an additional assumption on our trivialization. We assume that we have a covering B = ⋃_{i∈I} U_{i} and

Ψ_{i} : π^{−1}(U_{i}) ⟶^{∼} U_{i} × F

as before. But in addition we assume that for any pair i, j of indices the map

G_{ij} = Ψ_{j} | U_{i} ∩ U_{j} ∘ Ψ_{i}^{−1} | U_{i} ∩ U_{j} : (U_{i} ∩ U_{j}) × F → (U_{i} ∩ U_{j}) × F

has the form

G_{ij}(u, x) = (u, g_{ij}(u)x)

where g_{ij}(u) is a linear automorphism of our vector space F. It is clear that u → g_{ij}(u) must be a continuous map from U_{i} ∩ U_{j} into the general linear group G = Gl_{n}(𝕂). Moreover, it is obvious that we have a cocycle relation for any triplet of three indices i, j, k

g_{ij}(u) · g_{jk}(u) = g_{ik}(u) for all u ∈ U_{i} ∩ U_{j} ∩ U_{k}.

If this assumption is fulfilled, we say that

π : X ⟶ B

is a vector bundles. I find this definition is a little bit unsatisfactory because it needs the covering and the Ψ_{i}. We will give a second definition which I think is better.

Of course our data allow us to introduce the structure of a vector space on each fibre π^{−1}(b) such that the vector space structure “varies continuously with b”. What do we mean by that? Our definition also implies that we can find sections

e_{1}, ⋯, e_{n} : U_{i} ⟶ π^{−1}(U_{i}) = U_{i} × F,

such that in each point u ∈ U_{i} the elements e_{1}(u), ⋯, e_{n}(u) ∈ π^{−1}(u) form a basis of this vector space. Now we can identify

π^{−1}(u) ⟶^{∼} 𝕂^{n}

by sending

∑ a_{ν}e_{ν}(u) ⟼ (a_{1}, ⋯ a_{n}),

and we get a map

π^{−1}(U_{i}) ⟶ U_{i} × 𝕂^{n} ,

and the phrase “the vector space structure varies continuously with b” means that this is a homeomorphism.

This allows us to give a different formulation of the concept of a vector bundle. We can say that

π : X ⟶ B

is a vector bundle if:

(a) For any b ∈ B we have the structure of a finite dimensional 𝕂 -vector space on the fibre π^{−1}(b)

(b) For any b ∈ B we can find a neighborhood V of b and sections

e_{1}, ⋯, e_{n} : V ⟶ π^{−1}(V)

such that these sections evaluated at any point v ∈ V form a basis of π^{−1}(v).

(c) The map

π^{−1}(V) ⟶ V × 𝕂^{n}

sending a point x = ∑ a_{i} e_{i}(v) above v ∈ B to (v, a_{1} ⋯ a_{n}) ∈ V × 𝕂^{n} is a homeomorphism.

If we have such a vector bundle

π : X ⟶ B,

and if we have an open set V ⊂ B together with the section

e_{i} : V ⟶ π^{−1}(V) i = 1 ⋯ n

which form a basis at any point v ∈ V, then we call this a local trivialization of a bundle. We can consider the sheaf of germs of sections into X, let us call it C_{0,X}. This is a locally free module over the sheaf of germs of continous functions C_{0,B}. On the other hand it is rather clear that a locally free module E over C_{0,B} also gives us a vector bundle.

3.2 The non abelian H^{1}. We know of course what it means that two vector bundles X → B are isomorphic. Actually it is obvious that the vector bundles over a given base space form a category: A continuous map

φ : X ⟶ X′
↘ ↗
B

is a morphism of vector bundles, if φ restricted to the fibres is linear.

I want to explain the description of the set of isomorphism classes of vector bundles on B in terms of non abelian sheaf cohomology. Given our vector bundle we select a covering 𝔙 = {V_{i}}_{i∈I} of B and local trivializations

e_{i,ν} : V_{i} ⟶ π^{−1}(V_{i}) ν = 1 ⋯ n.

If we have an ordered pair (i, j) of indices, then we get a continuous map

g_{ij} : V_{i} ∩ V_{j} ⟶ GL(n, 𝕂)

such that

g_{ij}(v) · (e_{i,ν}(v)) = e_{j,ν}(v)

i.e. g_{ij}(v) sends the sections {e_{i,ν}}_{v} into the sections {e_{j,ν}}_{v}.

This is clearly a one-cocycle, this means

g_{ij} · g_{jk} = g_{ik} on V_{i} ∩ V_{j} ∩ V_{k}

and

g_{ii} = Id .

This suggests that we introduce the set of one-cocycles with respect to our covering. We introduce the sheaf of germs of continuous maps form our space B to the group G = GL(n, 𝕂), we denote this sheaf by C_{0}(G). Then we define as before

C^{1}(𝔙, C_{0}(G)) = \{\underline{c} = (⋯ g_{ij} ⋯) ∈ ∏_{i,j} C(G)(V_{i} ∩ V_{j}) | \underline{c} = (⋯ g_{ij} ⋯) is a 1- cocycle\}.

If we modify our local trivialization, then we modify the cocycle into

g′_{ij} = h_{i}g_{ij}h_{j}^{−1}  on  V_{i} ∩ V_{j},

where \underline{h} ∈ ∏ C_{0}(G)(V_{i}). This gives us an equivalence relation on C^{1}(𝔙, C_{0}(G)) and dividing by this relation we get a set

H^{1}(B, 𝔙, C_{0}(G)).

Again we may change the covering, we can pass to common refinements and we end up with

H^{1}(B, C_{0}(G)) = lim_{\overrightarrow{𝔙}} H^{1}(B, 𝔙, C_{0}(G)).

Since our sheaf takes values in the category of non abelian groups, we cannot multiply cocycles and we only get a set. Now it follows from our considerations that:

3.2.1. The elements in H^{1}(B, C_{0}(G)) are in one-to-one correspondence with the set of isomorphism classes of vector bundles.

3.3 The reduction of the structure group. Of course we can start from any topological group G, we can consider te sheaf of G valued functions on B and we can look at the cohomology set H^{1}(B, C_{0}(G)). This set classifies so called principal G-bundles, this are bundles P → B with a left action of G such that G acts simply transitively on the fibres. Then G is called the structure group of P → B. 3.3.1 We may introduce different kinds of additinal structures on the fibres of a vector bundle π : X → B. For instance we may choose a euclidian metric < , > on the fibres which varies differentiably with the point. Then we can choose local trivializations e_{1}, …, e_{n} which are given by orthonormal basis vectors. If we compare two such local trivializations then our functions g_{ij} will be functions with values in the orthogonal group O(n) and therefore it will correspond to an element in H^{1}(B, C_{0}(O(n))). In such a situation we say that the additional structure induces a reduction of the structure group. In this case we hve a reduction from Gl_{n}(ℝ) to the orthogonal group O(n). Another such additional structure is an orientation. If we consider the highest exterior power Λ^{n}(X/B), i.e we take the highest exterior power fibre by fibre, then we get a bundle of one dimensional vector spaces. On this bundle we have an action of the multiplicative group of positive real numbers ℝ_{>0}^{∗}. If we divide the bundle by this action then the quotient is a bundle B̃ → B with fibres consisting of two points. If we can find a global section s : B → B̃, then we say that X → B is orientable. If we choose such a section then we say that X → B is oriented. If we have an orientation on B then we may choose local trivializations e_{1}, …, e_{n} for which the ordered basis is positive with respect to the orientation. If we have done this then our g_{ij} will take values in the subgroup Gl_{n}(ℝ)^{+} of matrices with determinant positive and thus we have another case of the reduction of the structure group. On a C_{∞} manifold M we have the notion of the tangent bundle T_{M}. (See a book ???). Locally on M we have coordinate functions x_{1}, …, x_{n} so that any differentiable function is a differentiable function in the variables x_{1}, …, x_{n}. Then the vector fields ∂/∂x_{1}, …, ∂/∂x_{m}. provide a local trivialization of this tangent bundle. If we have in addition an euclidian metric on this bundle then M is called a Riemannian manifold. If we have chosen an orientation (if possible ) then we call M oriented.

3.3.2 Local systems. If B is a topological space and A an abelian group, then we attached to A the sheaf \underline{A} = A_{B} of locally constant functions with values in A (see III, 1.3.). We want to introduce the notion of local A-systems or local systems of A’s. If A is a sheaf of abelian groups on B, then we call A a local A-system, if for any point b ∈ B we can find an open neighborhood V_{b} such that the restriction of A to V_{b} is isomorphic to A_{V_{b}}. This implies that for any point b ∈ B the stalk A_{b} is isomorphic to A. At this point it is reasonable to assume that our space B is locally connected, i.e. for any point b ∈ B and any open neighborhood V_{b} of b we can find a connected open neighborhood U_{b} ⊂ V_{b} of b.

If we have that A | V_{b} is isomorphic to A_{V_{b}} as above, and if we replace V_{b} by the connected open neighborhood U_{b}, then

A(U_{b}) ≃ A,

and for any point u ∈ U_{b} we get an isomorphism

A(U_{b}) ⟶ A_{u}.

If we now fix a covering B = ∪V_{i}, where the V_{i} are connected and we have isomorphisms

Ψ_{i} : A | V_{i} ˜ ⟶A_{V_{i}},

then we may compare the Ψ_{i} on the intersections and we get

g_{ij} : V_{i} ∩ V_{j} ⟶ Aut(A)

which are locally constant (or continuous if Aut(A) is endowed with the discrete topology). Hence we see that the local A-systems are classified by the elements in

H^{1}(B, \underline{Aut}(A))

where \underline{Aut}(A) is the sheaf of locally constant functions in Aut(A).

3.3.3 We may even consider local systems of vector spaces. In this case we endow the vector space ℝ^{d} with the discrete topology.

Such a local system of vector spaces may also be considered as a vector bundle

π : X ⟶ B

where we have local sections e_{1}, …, e_{n}, which are called ”constant”. If we pass to a different open set V′ and a trivialization of X over V′ by constant sections e′_{1}, ⋯, e′_{n}, then on the intersection

e′_{i} = ∑ a_{ij}e_{j}

where now the a_{ij} are locally constant functions on V_{i} ∩ V_{j}.

Of course we can describe the set of isomorphism classes of local systems of vector spaces in terms of non abelian cohomology. We consider the group G_{d} = GL_{n}(𝕂)_{d} which is the general linear group but endowed with the discrete topology. Then it is clear that the isomorphism classes of local systems of n-dimensional 𝕂-vector spaces are given by

H^{1}(B, GL_{n}(𝕂)_{d}).

These local sysytems of vector spaces are the same kind of objects as bundles with a flat connection. (See also 9.1 and 9.2)

IV.4 Fundamental properties of the cohomology of sheaves.

4.1 Introduction. I will now state some results concerning the cohomology of sheaves. They are not so easy to prove. In some cases I can give a hint to how to prove them in an exercise. If we have any space X and an abelian group A then we may define the sheaf \underline{A}_{X} of germs of locally constant sections. It is called the constant sheaf attached to A . Sometimes we simply write \underline{A}. We define the cohomology of X with coefficients in A as

H^{•}(X, A) := H^{•}(X, \underline{A}_{X}).

If A = ℤ then the cohomology groups

H^{•}(X, ℤ) := H^{•}(X, \underline{ℤ}_{X}).

are equal to the ones defined by singular cochains, if the space X is reasonable. (This is a theorem, we come back to it later). The first important result which we will show is that the cohomology of constant sheaves vanishes on certain contractible spaces. We begin by stating a special case which is also the starting point for the more general results:

1.) If D = {(x_{1}, …, x_{n}) ∈ ℝ^{n} | Σx_{i}^{2} ≤ 1} and D^{∘} the interior of D, then

H^{i}(D, A) = H^{i}(D^{∘}, A) = 0 for i ≥ 1.

The following exercise treats the case n = 1. 4.1.1 Exercise.

Let us consider the following property (E) of a sheaf A on the intervall X = [−1, 1]: For any open interval I ⊂ [−1, 1] the restriction map A([−1, 1]) → A(I) is surjective. (We only require that I is open in [−1, 1], i.e. it may contain the boundary points. Condition (E) does not mean that A is flabby!)

1.) Show that the sheaves \underline{A}_{[−1,1]} and injective sheaves have property (E).

2.) If A has property (E) and if we have an exact sequence

0 ⟶ A ⟶ F ⟶ G ⟶ 0

then F(I) → G(I) is surjective for any open interval in [−1, 1]. 3.) If we have sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0

where A and B have property (E) then C also has property (E). 4.) For any sheaf A which satisfies (E) we have H^{q}([−1, 1]), A) = 0 for all q ≥ 1. Especially

we have H^{q}([−1, 1], \underline{A}) = 0. for any abelian group A.

This is some progress, I think we justified the computation in 1.2. But we need a stronger result and this stronger result is provided by corollary 4.4.9 which says that the cohomology groups are invariant under homotopies. To get to this point we need to investigate a relative situation f : X → Y.

4.2 The derived functor to f_{∗}. Given two spaces X, Y and a continuous map f : X → Y, we constructed the two functors f_{∗}, f^{∗} which transport sheaves on X to sheaves on Y and sheaves on Y to sheaves on X respectively (see III.4). Let us denote by S_{X} the category of sheaves on X with values in the category of abelian groups. If we have a sheaf F on X (with values in the category of abelian groups), then we defined the sheaf f_{∗}(F) on Y by

f_{∗}(F)(V) = F(f^{−1}(V))

for all open subsets V ⊂ Y. It is clear that f_{∗}(F) is a sheaf on Y. The functor f_{∗} is left exact but not exact in general. We get our previous case if we take Y to be just one point, i.e. Y = {pt}. Then the stalk of f_{∗}(F)_{pt} in this point is simply F(X) = H^{0}(X, F).

Again we define a derived functor for f_{∗} by the same method as before. We choose an injective resolution

0 ⟶ F ⟶ I^{0} ⟶ I^{1} ⟶ …

of F, and we get a complex of sheaves on Y by taking the direct image

0 ⟶ f_{∗}(I^{0}) ⟶ f_{∗}(I^{1}) ⟶ ….

This is now a complex of sheaves on the space Y. We define the sheaves ( see III.5.1)

R^{q}f_{∗}(F) = \frac{ker(f_{∗}(I^{q}) → f_{∗}(I^{q+1}))}{Im(f_{∗}(I^{q−1}) → f_{∗}(I^{q}))}.

It is clear that the stalk of R^{q}f_{∗}(F) in a point y is simply the cohomology of the complex of stalks. As before, we show that these sheaves do not depend on the choice of the resolution and that for any morphism

u : F ⟶ G

we get the derived map

R^{q}u : R^{q}f_{∗}(F) ⟶ R^{q}f_{∗}(G).

Finally it is clear that f_{∗}(F) = R^{0}f_{∗}(F), and that any short exact sequence

0 ⟶ F′ ⟶ F ⟶ F′′ ⟶ 0

leads to a long exact sequence

0 → f_{∗}(F′) → f_{∗}(F) → f_{∗}(F′′) → R^{1}f_{∗}(F′) → R^{1}f_{∗}(F) → R^{1}f_{∗}(F′′) → R^{2}f_{∗}(F′).

The intuitive idea — which in some cases is right in some cases wrong — (see 4.4.7) is that the stalk of R^{q}f_{∗}(F)_{y} in a point y should be the cohomology of the fibre f^{−1}(y) ⊂ X with coefficients in the restriction of F to this fibre.

4.2.1. The following special case is very important ( see III 4.2.). Let us assume that A ⊂ X is a closed subspace and let i : A → X be the embedding of A into X. Then i_{∗}(F) is a sheaf on X. It is clear that the stalk of i_{∗}(F) is given by

i_{∗}(F)_{x} = \{ \begin{matrix}F_{x} & if x ∈ A \\ 0 & if x ∉ A \end{matrix} .

It is clear that F → i_{∗}(F) is an exact functor.

4.2.2. We also defined the functor f^{∗}. This functor transforms sheaves on Y into sheaves on X. The idea is that the stalk of f^{∗}(G) in a point x ∈ X is equal to the stalk of the original sheaf G in the point y = f(x), i.e. f^{∗}(G)_{x} = G_{f(x)}. Since the exactness of sequences of sheaves can be checked stalkwise, it is clear that f^{∗} is an exact functor. We know that these two functors are adjoint and I recall the adjointness formula

Hom_{S_{X}} (f^{∗}(G), F) = Hom_{S_{Y}} (G, f_{∗}(F)).

We want to discuss the consequences of existence of f^{∗}, f_{∗} and the adjointness formula for the cohomology and its functorial properties.

4.2.3. Lemma. If f : X → Y is continuous, and if I is an injective sheaf on X, then f_{∗}(I) is injective on Y.

Proof. This follows directly from the adjointness formula and the exactness of f^{∗}.

4.3 Functorial properties of the cohomology. If we start from a sheaf G on the target space Y, and if we take an injective resolution

0 ⟶ G ⟶ J^{0} ⟶ J^{1} ⟶ …,

then we get a resolution

0 ⟶ f^{∗}(G) ⟶ f^{∗}(J^{0}) ⟶ f^{∗}(J^{1}) ⟶ ….

As we have seen earlier (III.3.5), this gives us a map

H^{q}(0 → f^{∗}(J^{0})(X) → f^{∗}(J^{1}) → …) ⟶ H^{q}(X, f^{∗}(G)).

On the other hand we have a map between the complexes

0 ⟶ J^{0}(Y) ⟶ J^{1}(Y) ⟶
↓  ↓
0 ⟶ f^{∗}J^{0}(X) ⟶ f^{∗}J^{1}(X) ⟶

(This follows from the definition of f^{∗}.), and hence a functorial map

H^{q}(Y, G) ⟶ H^{q}(X, f^{∗}G).

There is an especially important case of this: If f : X → Y, and we consider the sheaf \underline{ℤ}_{Y} on Y, then we see easily

f^{∗}(\underline{ℤ}_{Y}) = \underline{ℤ}_{X}.

To see this we construct a homomorphism from f′(\underline{ℤ}_{Y}) to \underline{ℤ}_{X}: For U ⊂ X, U open, we have

f′(ℤ)(U) = lim_{\overrightarrow{V ⊃f(U)}} ℤ(V).

For V ⊂ f(U) we have f^{−1}(V) ⊂ U, and of course, we have maps

\underline{ℤ}_{Y} (V) ⟶ \underline{ℤ}_{X}(f^{−1}(V)) ⟶ \underline{ℤ}_{X}(U),

and this provides a map

f′(\underline{ℤ}_{Y})(U) ⟶ \underline{ℤ}_{X}(U).

This is a map from the praesheaf f′(\underline{ℤ}_{Y}) to the sheaf \underline{ℤ}_{X}, and this provides a unique map

f^{∗}(\underline{ℤ}_{Y}) ⟶ \underline{ℤ}_{X}.

Looking at the stalks we see that this map is an isomorphism. This yields the functoriality of the cohomology groups H^{q}(X, ℤ). For any map f : X → Y we get

f^{q} : H^{q}(Y, ℤ) ⟶ H^{q}(X, ℤ).

There is another case: We always get a map H^{q}(Y, f_{∗}F) → H^{q}(X, f^{∗}f_{∗}F) and the adjointness provides the map f^{∗}f_{∗}F → F which corresponds to the identity f_{∗}F → f_{∗}F. The composition of these two maps yields a map f^{q} : H^{q}(Y, f_{∗}F) → H^{q}(X, F). For this map we have an easy theorem:

4.3.1 Theorem. Let us assume that f : X → Y is continuous and F a sheaf on X. If the higher derived sheaves R^{q}f_{∗}(F) = 0 for q ≥ 1 then we get an isomorphism

f^{q} : H^{q}(X, F) ˜ ⟶H^{q}(Y, f_{∗}(F))

for all q ≥ 1.

This is clear: We start from an injective resolution

0 ⟶ F ⟶ I^{0} ⟶ I^{1} ⟶ ….

Then our assumption says that

0 ⟶ f_{∗}(F) ⟶ f_{∗}(I^{0}) ⟶ f_{∗}(I^{1}) ⟶

is a resolution, and the lemma implies that this resolution is injective. Hence

H^{q}(Y, f_{∗}(F)) = H^{q}(0 → f_{∗}(I^{0})(Y) ⟶ f_{∗}(I^{1})(Y) ⟶ …).

But the complex of sections over Y is equal to the complex

0 ⟶ I^{0}(X) ⟶ I^{1}(X) ⟶ …,

which gives the cohomology H^{q}(X, F). One important consequence of this theorem is the case of an embedding

i : A⟶X

where A is a closed subspace of X. In this case we have seen that i_{∗} is an exact functor from sheaves on A to sheaves on X, hence R^{q}i_{∗}(F) = 0 for q ≥ 1 and

H^{q}(A, F) = H^{q}(X, i_{∗}(F)).

4.3.2 If we want to apply the theorem above we have to understand how to compute the sheaves R^{q}f_{∗}(F). We want to show that under certain assumptions the stalks R^{q}f_{∗}(F)_{y} = H^{q}(f^{−1}(y), i_{y}^{∗}(F)). To get more precise informations which will allow us to compute cohomology groups in certain cases we have to make assumptions on our spaces.

4.4 Paracompact spaces. In general the sheaves R^{q}f_{∗}(F) may be very difficult to compute. One possibility is to relate the stalks R^{q}f_{∗}(F)_{y} to the cohomology groups of the fibre f^{−1}(y). This is possible if our spaces have certain finiteness and seperability properties. A covering X = ⋃_{α∈A} U_{α} is called locally finite if for any point x ∈ X we can find a neighborhood V_{x} of x such that V_{x} meets only finitely many of the U_{α}, i.e. the set of indices α for which V_{x} ∩ U_{α} ≠ ∅ is finite. A space X is called paracompact if it is Hausdorff and if for any open U ⊂ X and any covering U = ⋃_{i∈I} U_{i} we can find a locally finite refinement of the covering. Recall that a refinement of the covering is another covering U = ⋃_{j∈J} W_{j} together with a map τ : J → I

such that for all j ∈ J we have the inclusion W_{j} ⊂ U_{τ(j)}. We call such a refinement a strong refinement if even the closures W̄_{j} are contained in U_{τ(j)}. I claim: If our space X is paracompact and locally compact then any covering of an open set U = ⋃_{i∈I} U_{i} has a strong locally finite refinement.

Since our space is Hausdorff and locally compact we know: For any point x ∈ X and any open neighborhood V_{x} of x we can find an open neighborhood W_{x} such that its closure \overline{W}_{x} is contained in V_{x}. Now it is clear how get a strong locally finite refinement of a covering U = ⋃_{i∈I} U_{i}: We can construct a strong refinement of the covering and after that

we construct a locally finite refinement of this strong refinement.

4.4.1 We have a simple criterion for paracompactness. We say that an open subset U ⊂ X is exhaustible by compact subsets if we we can find a increasing sequence of compact subsets

∅ = K_{0} ⊂ … ⊂ K_{n} ⊂ K_{n+1} ⊂ …

s. t. U = ⋃ K_{n} and for any n the compact set K_{n} is contained in the interior K^{∘}_{n+1} of the next one.

4.4.2 Lemma. A Hausdorff space X for which any open subset is exhaustible by compact subsets is paracompact.

To see this we consider U ⊂ X and a covering U = ⋃_{i∈I} U_{i} by open subsets. We choose

inductively finite coverings of K_{n}, where each covering set is contained in K^{∘}_{n+1} , in one of the U_{i} of the given covering and has empty intersection with K_{n−1}. It is clear that the resulting covering is locally finite: For any point x ∈ U we have a smallest n such that

u ∈ K_{n} then it has an open neighborhood which is contained in some K^{∘}_{n+1} and does not meet K_{n−1} and this neighborhood clearly meets only finitely many of the sets in the covering. We say that our space X is exhaustible by compact sets, if the open subset U = X has this property.

4.4.3 It is not difficult to show that a Hausdorff space is paracompact if it is exhaustible by compact sets and if any open set U can be exhausted by a sequence of sets

W_{n} ⊂W^{∘}_{n+1}⊂ W_{n+1}

where the W_{n} are only closed subsets of X.

To see that this is true we observe that a closed subspace A ⊂ X is exhaustible by compact sets. This implies that any covering of A by open sets has a locally finite refinement. ( Same proof as for Lemma 1.5.2. ) Assume that we have a covering of U by open sets. We proceed as in the proof of Lemma 1.5.2 but now we construct locally finite coverings of the W_{n} ( instead of finite ones ) where we obey the same precautions as before. We come to a very technical lemma which says something about extension of sections.

Assume that we have a closed embedding i : A⟶X . We call this a nice emdedding if the following is true: Let F be a sheaf on X, we consider the sheaf i^{∗}(F) on A. By construction it is the sheafification of the preasheaf V → i′(F)(V) where V is open in A and

i′(F)(V) = lim_{\overrightarrow{U⊃V}} (F(U)).

If s̃ ∈ F(U) and if s is its image in i^{∗}(F)(U ∩ A) then we say that s is the restriction of s̃ to A ∩ U = V. Now the condition for being nice is that if any section s ∈ i^{∗}(F)(V) can be extended into some neighborhood U of V in X, this means it is in the image of F(U) → i′(F)(V) for some U which satisfies U ∩ A ⊃ V. This can be reformulated by saying that i′(F) is already a sheaf.

4.4.4 Lemma. (Extension of sections): If X is paracompact and locally compact then any closed embedding i : A⟶X is nice.

Proof. We start with V ⊂ A and our section s ∈ F(V). We know from the definition of i^{∗}(F) that for any point p ∈ V the image of s in the stalk s_{p} ∈ i^{∗}(F)_{p} is the restriction of a section s̃_{p} ∈ F(U_{p}) where U_{p} is an open neighborhood of p in X. Hence we can find a covering ⋃_{α} U_{α} ⊃ V and sections s̃_{α} ∈ F(U_{α}) such that s̃_{α} maps to the restriction s|U_{α}∩V. We may assume that this covering is locally finite since our space is paracompact. Let {W_{j}}_{j∈J} be a strong locally finite refinement of this covering. As usual we denote the map between the indexing sets by τ : J → I. Let q ∈ V we can find an open neighborhood V_{q} of q in X such that V_{q} meets only finitely many of the W_{j} and the U_{α}.. We choose an open neighborhood D_{q} ⊂ V_{q} which is contained in W_{j} for all those ( finitely many ) j for which q ∈ W_{j} and also in all those finitely many U_{α} with q ∈ U_{α}. We may also choose D_{q} so small that D_{q} ∩ W_{j} = ∅ if q ∉ W̄_{j} because D_{q} meets only finitely many of them anyway. It follows from the defintion of i^{∗} that we can take these D_{q} so small that we have s̃_{α}|D_{q} = s̃_{β}|D_{q} whenever q ∈ U_{α} ∩U_{β}. Let ˜ s̃_{p} ∈ F(D_{p}) be the restriction of any of these s̃_{α}. I claim that these sections ˜ s̃_{p}, ˜ s̃_{q} restrict to the same section over D_{p} ∩ D_{q} for any pair p, q. This is clear if D_{p} ∩ D_{q} = ∅ so we may assume that D_{p} ∩ D_{q} ≠ ∅. If D_{p} ⊂ W_{j} then q ∈ W̄_{j} because otherwise we have D_{q} ∩ W_{j} = ∅ by construction and this implies D_{p} ∩ D_{q} = ∅, a contradiction. We have D_{p} ⊂ W_{j} ⊂ U_{τ(j)}. Since the W_{j} formed a strong refinement of the U_{α} we even know that W̄_{j} ⊂ U_{τ(j)}. Hence q ∈ U_{τ(j)} and then we conclude that D_{q} ⊂ U_{τ(j)} again by construction. Consequently we have that D_{p} and D_{q} are contained in U_{τ(j)} and this implies that the sections ˜ s̃_{p}, ˜ s̃_{q} are restrictions of s̃_{τ(j)}. Hence the ˜ s̃_{p} define a section s̃ over U = ∪D_{q} and this is the element we where looking for.

4.4.5. I want to discuss a variant of this lemma. Let us assume that only the closed subset A is locally compact and paracompact. But in addition we assume that can find an subset W ⊂ X with W ⊃ A and such that W →^{∼}W_{0} × A where W_{0} is a topological space and we assume that the isomorphism sends A→^{∼}{w_{0}} × A where w_{0} is a point in W_{0}. I claim that under these assumptions the embedding is nice. This can be shown by a slight modification of the proof of lemma 4.4.4. We proceed as in the proof but we choose the open sets U_{α} to be of the form U_{α} = V_{α} × W_{α} where V_{α} is open in V and W_{α} is a neighborhood of w_{0} in W_{0}. Then we choose a strong locally finite refinement of the covering V = ⋃_{α∈I} V_{α}, let us denote this refinement by V = ⋃_{β∈J} Y_{β} and

let τ : J → I be the map for which Y_{β} ⊂ V_{τ(β)}. This gives us a covering of V by open set in X: We have V ⊂ ⋃_{β∈J} Y_{β} × W_{τ(β)}. This covering now plays the role of the covering by

the W_{j} in the proof of the Lemma 4.4.4. We proceed essetially in the same way as before. We choose neighborhoods D_{q} which satisfy D_{q} ⊂ Y_{β} × W_{τ(β)} if q = (q, w_{0}) ∈ Y_{β} × W_{τ(β)} and D_{q} ∩ Y_{β} × W_{τ(β)} = ∅ if - here we have a slight modification- q ∉ Ȳ_{β} × W_{τ(β)}. From here on the argument is the same.

4.4.6 Lemma. Let i : A⟶X be a nice embedding. If I is an injective sheaf on X then i^{∗}(I) is flabby and hence acyclic.

Proof: This follows from 2.2.2

These technical considerations will be applied to prove the following difficult theorem:

4.4.7 Theorem ( Base change):Let us assume that X is paracompact, that Y is locally compact and Hausdorff and that

f : X ⟶ Y

is a proper map. Then for any sheaf F on X and any y ∈ Y we have

R^{q}f_{∗}(F)_{y} = H^{q}(f^{−1}(y), i_{y}^{∗}(F)).

Recall that f is called proper if the inverse image of a compact set in Y is again compact.

Proof. We shall need a modification of the theorem therefore we will also discuss to what extend we really need our assumptions. Let i_{y} : f^{−1}(y)⟶X be the inclusion of the (closed and compact ) fibre. Then we know from our assumptions that the embedding is nice (Lemma 4.4.4).

We formulate the following condition on our map f

For any open neighborhood of a fibre U ⊃ f^{−1}(y)we find a V_{0}s.t.f^{−1}(V_{0}) ⊂ U. (Cyl)

We show that (Cyl) is valid under the asssumption of the theorem. We consider the intersection

(X \ U) ∩ f^{−1}(\overline{V})

for all closures \overline{V} of compact open neighborhoods if V. Since f^{−1}(\overline{V}) is compact, the intersection is also compact. Since U is a neighborhood of f^{−1}(y) we know that for x ∈ X \ U we have f(x) ≠ y. We may choose open neighborhoods W_{f(x)}, V_{y} s.t. their closure is compact and f(W_{f(x)}) ∩ V_{y} = ∅, hence f(x) ∉ \overline{V}_{y}. Hence x ∉ (X \ U) ∩ f^{−1}(\overline{V}_{y}) and therefore

⋂_{V ∋y}(X \ U) ∩ f^{−1}(\overline{V}) = ∅.

Now it follows from a standard argument on compact spaces that there must be a V, y ∈ V with f^{−1}(V) ⊂ U. The following considerations prove the assertion of the theorem under the two assumptions

a) for all y the fibre f^{−1}(y) is closed and the embedding is nice

b) (Cyl) holds. By definition we have

f_{∗}(A)_{y} = lim_{\overrightarrow{V :y∈V}} A(f^{−1}(V))

and (Cyl) implies that

lim_{\overrightarrow{V :y∈V}} A(f^{−1}(V)) = lim_{\overrightarrow{U:f^{−1}(y)⊂U}} A(U).

Then the fact that the embedding of the fibre is nice yields

lim_{\overrightarrow{U:f^{−1}(y)⊂U}} A(U) = i_{y}^{∗}(A)

and we conclude

f_{∗}(A)_{y} = i_{y}^{∗}(A)

This proves our the theorem for q = 0. To prove it in general, we start from an injective resolution

0 ⟶ F ⟶ I^{0} ⟶ I^{1} ⟶ …

on X. Then (we sometimes drop the brackets inf_{∗})

0 ⟶ f_{∗}F ⟶ f_{∗}I^{0} ⟶ …

is a complex of injective sheaves. If we pass to the sequence of stalks at a point y ∈ Y, we get a complex of abelian groups

0 ⟶ f_{∗}F_{y} ⟶ f_{∗}I_{y}^{0} ⟶ …,

and the cohomology of this complex is the stalk R^{q}f_{∗}(F)_{y}. But this complex is equal to the complex

0 ⟶ i_{y}^{∗}(F)(f^{−1}(y)) ⟶ i_{y}^{∗}I^{0}(f^{−1}(y)) ⟶ …,

and this is the complex of global sections of the complex of sheaves on f^{−1}(y):

0 ⟶ i_{y}^{∗}F ⟶ i_{y}^{∗}I^{0} ⟶ …

which is a flabby and hence acyclic resolution of i_{y}^{∗}F. Hence the cohomology of the above complex of global sections is

H^{q}(f^{−1}(y), i_{y}^{∗}(F)).

4.4.8 Corollary. If X, Y and f : X → Y are as in the theorem and if

H^{q}(f^{−1}(y), i_{y}^{∗}(F)) = 0 for q ≥ 1 and all y ∈ Y,

then

f^{q} : H^{q}(Y, f_{∗}F) ˜ ⟶H^{q}(X, F)

is an isomorphism.

This is the combination of the difficult and the easy theorem. The following corollary is not a direct consequence of the theorem.

4.4.9 Corollary. If X is a Hausdorff space and if

p : X × [0, 1] ⟶ X,

is the projection to the first factor then this projection induces isomorphisms in cohomology

p^{•} : H^{•}(X, \underline{ℤ}) ˜ ⟶H^{q}(X × [0, 1], \underline{ℤ}).

For any t ∈ [0, 1] the inclusion x ↦ x × {t} induces an isomorphism in cohomology.

This is not a direct consequence of the the theorem as it is stated since we do not make any assumption on X except that it is Hausdorff. But first of all our modifified lemma 4.4.6 implies that for any point x_{0} in X the embedding {x_{0}} × I ⟶X × I is nice. (We need that the fibre is closed so we can get away with the weaker assumtion that points in X are closed). Secondly it is clear that the condition (Cyl) in the proof of the Base change theorem is also fulfilled. This means that the proof is valid for the projection p.

The rest is clear since p_{∗}\underline{ℤ} = \underline{ℤ}, and since

H^{q}(\{x\} × [0, 1], \underline{ℤ}) = 0.

The second assertion follows if we compose the inclusion with the projection.

4.4.10 The homotopy axiom. Let X be a Hausdorff space. If we have two maps

f, g : X ⟶ Y

which are homotopic, i.e. there is a map

F : X × [0, 1] ⟶ Y,

so that F (x, 0) = g(x), F (x, 1) = f(x) then

f^{•} = g^{•} : H^{•}(Y, \underline{ℤ}) ⟶ H^{•}(X, \underline{ℤ}).

Proof. Look at

X ⟶^{top} ; ⟶_{bot} X × [0, 1] ⟶^{F} Y

where the arrows are x → (x, 0), x → (x, 1). If we compose these arrows with F we get f, g. 4.4.11. A space X is called contractible to a point p ∈ X if the two maps f = Id and the map g which maps all the points in X to the point p are homotopic. If we apply the homotopy axiom to this two maps we get For a contractible Hausdorff space X we have

H^{i}(X, ℤ) = 0 for all i > 0

It is clear that the space ℝ^{n} is contractible. The same thing holds for any open ball B^{n}{(x_{1}, …, x_{n})| ∑ x_{i}^{2} < 1} and also its closure.

4.4.12 Application. We have the tools to compute cohomology groups of spheres and other simple spaces. We consider the sphere S^{n} = {(x_{0}, x_{1}, …, x_{n}) ∈ ℝ^{n+1}|x_{0}^{2} + x_{1}^{2} + … + x_{n}^{2} = 1}. We cover it by the two balls D_{±} which are defined by x_{n} ≥ 0 or x_{n} ≤ 0 respectively. We have the two inclusions i_{±} : D_{±} ⟶S^{n}. These balls are contractible, we have the sheaves \underline{ℤ}_{D_{±}} which we extend to the two sheaves \underline{ℤ}_{±} = i_{±∗}(\underline{ℤ}_{D_{±}}). We also have maps \underline{ℤ}_{S^{n}} → \underline{ℤ}_{±} which on open sets V ⊂ S^{n} are defined by the restriction \underline{ℤ}_{S^{n}}(V) → \underline{ℤ}_{±}(V) = \underline{ℤ}_{D_{±}}(V ∩ D_{±}). This gives an inclusion \underline{ℤ}_{S^{n}} ⟶\underline{ℤ}_{+} ⊕ \underline{ℤ}_{−} which is an isomorphism in all the stalks which are not over the intersection of the two balls which is the sphere S^{n−1}. In the points x in the intersection the inclusion is given by the diagonal \underline{ℤ}_{x} = ℤ ⊂ (\underline{ℤ}_{+} ⊕ \underline{ℤ}_{−})_{x} = ℤ ⊕ ℤ From this we get an exact sequence

0 → \underline{ℤ}_{S^{n}} → \underline{ℤ}_{+} ⊕ \underline{ℤ}_{−} → \underline{ℤ}_{S^{n−1}} → 0,

where the map s : (\underline{ℤ}_{+} ⊕ ℤ_{−})_{x} = ℤ ⊕ ℤ → ℤ_{S^{n−1,x}} = ℤ is the difference betweeb the + and − component. The cohomology of the two balls is trivial except in degree zero, hence we get

H^{ν−1}(S^{n−1}, \underline{ℤ})→^{∼}H^{ν}(S^{n}, \underline{ℤ})

if ν − 1 > 0 and in degree zero we find the exact sequence

0 → H^{0}(S^{n}, \underline{ℤ}) → H^{0}(D_{+}, \underline{ℤ}) ⊕ H^{0}(D_{−}, \underline{ℤ}) → H^{0}(S^{n−1}, \underline{ℤ}) → H^{1}(S^{n}, \underline{ℤ}) → 0.

Putting all this information together we can prove rather easily that

H^{0}(S^{0}, \underline{ℤ}) = ℤ ⊕ ℤ

and for n > 0 we have

H^{ν}(S^{n}, \underline{ℤ}) = \{ \begin{matrix}0 & for ν ≠ 0, n \\ ℤ & for ν = 0 or ν = n \end{matrix}

This is of course essentially the same calculation as the one one finds in books on topology. In these books the two essential ingredients are homotopy and the so called Mayer-Vietoris sequence. Here the Mayer-Vietoris sequence is replaced by the construction of suitable exact sequences of sheaves. Of course we have to be aware that the isomorphism H^{n}(S^{n}, ℤ) ˜→ℤ is not canonical ( See also the example 1 (ii) at the beginning of this chapter). It depends on the choice of the homomorphism s above. To make this choice we used the explicit description of the sphere to decide what D_{+} and D_{−} is. The point is that it suffices to choose an orientation on S^{n}, then we can identify H^{n}(S^{n}, ℤ) = ℤ ( See 3.3.1 and 7.3,7.4) . To see this we proceed inductively. If we have covered S^{n} = D_{a} ∪D_{b} and D_{a} ∩D_{b} = S^{n−1}, and if we have choosen an orientation on S^{n−1}, then we can do the following: We choose a positively oriented basis e_{1} ⋯ e_{n−1} of tangent vectors in a point x ∈ S^{n−1}. Now we choose a normal vector v to S^{n−1} at x such that e_{1} ⋯ e_{n−1}, v is positively oriented. This normal vector points into D_{a} or D_{b}. Now we put D_{+} = D_{a} if it points to D_{a} or D_{+} = D_{b} in the other case. Then we have a unique choice for s, and we get a canonical isomorphism

H^{n−1}(S^{n−1}, ℤ) ⟶ H^{n}(S^{n}, ℤ).

Since we selected an orientation on S^{n−1} we have H^{n−1}(S^{n−1}, ℤ) = ℤ.

4.4.13. A two dimensional, compact, oriented manifold is called an oriented surface. The simplest example is the 2-sphere S^{2}. If we have such a surface S we can construct a new one by the following construction: We pick two different points p, q ∈ S and we choose two small neighborhoods D_{p}, D_{q} which are homeomorphic to a two dimensional disc. The boundaries ∂D̄_{p}, ∂D̄_{q} can be identified to the oriented circle S^{1}. We form a cylinder S^{1} × [0, 1]. We remove the interior of the two disks from the surface S and map ∂(S^{1} × [0, 1]) = S^{1} × {0} ∪ S^{1} × {1} by taking the identity on each component to the boundaries of our two discs in S \ D_{p} ∪ D_{q}. Using this map we glue the cylinder to our surface, we add a so called handle. It is a theorem in two dimensional topology that any oriented surface S can be obtained from the sphere by adding a certain number of handles.

4.4.14 Exercise.

Let S be a compact oriented surface which has been obtained from the sphere by adding g handles.

Show that H^{0}(S, \underline{ℤ}) = H^{2}(S, \underline{ℤ}) = ℤ and H^{1}(S, \underline{ℤ})→^{∼}ℤ^{2g}.

Hint: Construct a sequence sheaves on S which is suggested by the process of adding a handle and proceed by induction.

We can also understand the cohomology of our oriented surface without such an explicit construction.This will be discussed in the section on Poincaré duality (see 8.3.).

IV.5 Čech cohomology of sheaves

5.1. The Čech-complex: For any space X, any sheaf F on X with values in the category of abelian groups and any open covering

𝔘 = {U_{i}}_{i∈I}, X = ⋃_{i∈I} U_{i}

of X, we will define the Čech-cohomology groups

Ȟ^{q}(X, 𝔘, F), q = 0, 1, …

To define these cohomology groups we introduce the so-called Čech complex. For any set of indices (i_{0}, …, i_{q}) ∈ I^{q+1} we define

U_{i_{0}...i_{q}} = U_{i_{0}} ∩ U_{i_{1}} ∩ … ∩ U_{i_{q}}.

Then we put

C^{q}(X, 𝔘, F) = ∏_{(i_{0},...,i_{q})∈I^{q+1}} F(U_{i_{0}...i_{q}})

for q = 0, 1, . . .. We define a boundary map

d : C^{q}(X, 𝔘, F) → C^{q+1}(X, 𝔘, F)

by the following formula

(dc)_{i_{0}...i_{q+1}} = ∑_{ν=0}^{q+1}(−1)^{ν} res(c_{i_{0},...,î_{ν},...,i_{q+1}}).

We have to explain why this formula makes sense: An element c ∈ C^{q}(X, 𝔘, F) is an element in a product and has components

c_{j_{0}...j_{q}} ∈ F(U_{j_{0}...j_{q}}).

Hence dc will also have components which are indexed by elements in I^{q+2}. An element (i_{0}, …, i_{q+1}) ∈ I^{q+2} provides q + 2 elements in I^{q+1} which are obtained by suppressing one of the components. By (i_{0}, …, î_{ν}, …, i_{q+1}) we denote the element in in I^{q+1} where we removed i_{ν}. For all these q+2 possibilities we have the restriction associated to U_{i_{0}...i_{q+1}} ⊂ U_{i_{0},...î_{μ}...,i_{q+1}} which we simply denote by

res : F(U_{i_{0},...,î_{μ},...,i_{q+1}}) → F(U_{i_{0},...,i_{q+1}}).

Now it is clear that the formula gives the rule to compute the (i_{0} … i_{q+1})-component of dc. We leave it as an exercise to prove that d ∘ d = 0. Hence (C^{•}(X, 𝔘, F), d) is a complex of abelian groups. Let us look at the beginning of our complex

0 → ∏_{i∈I} F(U_{i}) ⟶^{d} ∏_{(i,j)∈I×I} F(U_{i} ∩ U_{j}) → ….

An element

c = (…, c_{i}, …)

in the first term goes to zero if and only if

c_{i}|U_{i} ∩ U_{j} = c_{j}|U_{i} ∩ U_{j} for all i, j

But since F is a sheaf this implies that this is the case if and only if c comes from a uniquely defined global section s ∈ F(X), i.e. s_{i} = s|U_{i} for all i. We define cycles Z^{q}(X, 𝔘, F) to be the kernel of d and boundaries are the elements b ∈ C^{q}(X, 𝔘, F) of the form b = dc with c ∈ C^{q−1}(X, 𝔘, F). The boundaries form a subgroup B^{q}(X, 𝔘, F) of Z^{q}(X, 𝔘, F) and now we put

Ȟ^{q}(X, 𝔘, F) = Z^{q}(X, 𝔘, F)/B^{q}(X, 𝔘, F).

We just saw

Ȟ^{0}(X, 𝔘, F) = F(X), .

5.1.1 Remark. In general these Čech cohomology groups do depend on 𝔘. Later on we shall see that under certain assumptions on the sheaves and on the space and the nature of the covering they will be independent of the covering. We have the notion of a refinement of a covering (See III.3.1). If τ : 𝔙 → 𝔘 is such a refinement the map τ between the indexing set yields a map between the Čech complexes τ^{•}(C^{•}(X, 𝔘, F), d) → (C^{•}(X, 𝔙, F), d) and we get a map

Ȟ^{•}(X, 𝔘, F) → Ȟ^{•}(X, 𝔙, F).

It is possible to show that this map on the level of cohomology does not depend on τ, but we do not need this fact here. Since the coverings form a category we can define the Čech cohomology groups of a space as direct limit

lim ; ⟶_{𝔘} Ȟ^{•}(X, 𝔘, F) = Ȟ^{•}(X, F).

5.1.2 The alternating complex. We can also look at the so called alternating complex Calt^{•}(X, 𝔘, F). It is defined as the subcomplex where the cochains satisfy

(i)  c_{i_{0},...,x,...,x,...,i_{q}} = 0

and

(ii)  c_{i_{0},...,x,...,y,...,i_{q}} = −c_{i_{0},...,y,...,x,...,i_{q}}.

It is not too difficult to prove that Calt^{•}(X, 𝔘, F) is a subcomplex, i.e. the coboundary operator maps it into itself. It is a little bit more difficult to prove that

Calt^{•}(X, 𝔘, F) → C^{•}(X, 𝔘, F)

induces an isomorphism in cohomology. Sometimes it is easier to do computations using this smaller complex. 5.1.3 Exercise.

a) Prove that

Calt^{•}(X, 𝔘, F) → C^{•}(X, 𝔘, F)

induces an isomorphism in cohomology.

b) Consider the simplex

Δ^{n+1} = {(x_{1}, …, x_{n+2}) ∈ ℝ^{n+2}|x_{i} ≥ 0, ∑ x_{i} = 1}.

Then we get a covering 𝔘 of Δ^{n+1} by the open sets

U_{i} = {(x_{1}, …, x_{n+2}) ∈ Δn + 1|x_{i} > 0}.

Show that the cohomology groups

Ȟ^{m}(Δ^{n+1}, 𝔘, \underline{ℤ}) = \{ \begin{matrix}= 0 & if m > 0 \\ = ℤ & if m = 0. \end{matrix}

c) Now we remove the interior of Δ^{n+1} and we get the n-dimensional sphere

∂Δ = S^{n} = {(x_{1}, …, x_{n+2})| at least one of the x_{i} is zero}.

Our covering of Δ^{n+1} induces a covering 𝔘′ on S^{n}.

U_{i} = {(x_{1}, …, x_{n+2}) ∈ S^{n}|x_{i} > 0}.

Show that the Čech-cohomology groups Ȟ^{•}(S^{n}, 𝔘′, \underline{ℤ}) coincide with cohomology groups computed by injective resolutions.

A rather elegant solution of this exercise can be obtained if we use the following Lemma whose proof I give for later references.

5.1.4 Lemma. Let 𝔘 be a covering of an arbitrary space X and let us assume that in our covering 𝔘 = {U_{i}}_{i∈I} is a member y ∈ I for which U_{y} = X. Then we have H^{q}(X, 𝔘, F) = 0 for all q ≥ 1.

Proof. Let us assume we have a cocycle

c = (… c_{i_{0},...,i_{q}} …)_{(i_{0},...,i_{q})∈I^{q+1}} ∈ Z^{q}(X, 𝔘, F).

We construct a cochain b ∈ C^{q−1}(X, 𝔘, F) by

b_{i_{0},...,i_{q−1}} = c_{y,i_{0},...,1_{q−1}}.

We have to observe that

U_{i_{0},...,i_{q−1}} = U_{y,i_{0},...,i_{q−1}}.

Then

\begin{matrix}(db)_{i_{0},...,i_{q}} = Σ(−1)^{ν}b_{i_{0},...,î_{ν},...,i_{q}} \\ = Σ(−1)^{ν}c_{y,i_{0},...,î_{ν},...,i_{q}} = −(dc)_{y,i_{0},...,i_{q}} + c_{i_{0},...,i_{q}} = c_{i_{0},...,i_{q}}. \end{matrix}

To apply this to the exercise above we can consider the inclusion U_{0} ⟶Δ^{n+1}. The covering of Δ^{n+1} induces a covering of U_{0} and these two coverings yield the same Čech complexes.

5.2 The Čech resolution of a sheaf.

5.2.1 Heuristical remark. Let F be a sheaf on X with coefficients in the category of abelian groups. Let us assume that we have a resolution of F

0 → F → G^{0} → G^{1} → … → G^{n} → …

A resolution of a sheaf may be very useful for the computation of the cohomology of F. In ChapterII we showed: If the resolution above is acyclic then we can use it to compute the cohomology groups of F, we have :

H^{•}(X, F) = H^{•}(G^{•}(X)).

But even if a resolution is not acyclic it may also be helpful. For instance we still have a homomorphism

H^{•}(G^{•}(X)) → H^{•}(X, F)

which in general is neither injective nor surjective. But we have some kind of estimate for the deviation from being an isomorphism and in these estimates the cohomology groups H^{q}(X, G^{p}) will enter. In the case of an acyclic resolution these cohomology groups are zero for q > 0 and we get another explanation of our old result. The precise formulation of the result will be given in 6.2.1.

5.2.2. I want to put to Čech complex into this context. Let 𝔘 = {U_{α}}_{α∈A} be a covering of our space X let F be a sheaf with values in the category of abelian groups. We give the indexing set A a total order and we denote by A_{<}^{q+1} the subset of those sequences \underline{α} = (α_{0}, α_{1}, …, α_{q}) where α_{0} < α_{1} … < α_{q}. Again we put U\underline{α} = U_{α_{0}} ∩ U_{α_{1}} ∩ … U_{α_{q}} and let i\underline{α} : U\underline{α} ⟶X be the inclusion map. We restrict F to U\underline{α} and take the direct image of this restriction. This way we obtain the sheaves F ∗ ; \underline{α} = i_{\underline{α}∗}i∗ ; \underline{α}(F). I recall that these sheaves are defined by the rule i_{\underline{α}∗}i∗ ; \underline{α}(F)(V) = F(V ∩ U\underline{α}). The stalk of this sheaf is equal to F_{x} if x ∈ U\underline{α} it is zero if x ∉ Ū\underline{α} and it depends on the local structure of Ū\underline{α} in the boundary points x ∈ ∂Ū\underline{α}. We have always a homomorphism F_{x} → F_{\underline{α}x}^{∗}. I allow myself to write F_{α}^{∗} for F_{{α}}^{∗}.

Now we construct a resolution of our sheaf F

0 → F → ∏_{α∈A} F_{α}^{∗} → ∏_{(α,β)∈A×A_{<}} F_{(α,β)}^{∗} → … → ∏_{\underline{α}∈A_{<}^{q+1}} F ∗ ; \underline{α} → .

The first map is simply

F_{x} → ∏_{α∈A} F_{αx}^{∗}.

The boundary map

d : ∏_{\underline{α}∈A_{<}^{q+1}} F_{α}^{∗} → ∏_{\underline{β}∈A_{<}^{q+2}} F ∗ ; \underline{β}

is given by the following rule: Let s = (… s\underline{α} …) ∈ (∏ F ∗ ; \underline{α})_{x}

(ds)\underline{β} = ∑_{i=0}^{q+1}(−1)^{i}s_{β_{0}...β̂_{i}...β_{q+1}}

where we interpret s_{β_{0}...β̂_{i}...β_{q+1}} as an element in F_{\underline{β},x}^{∗} It is clear that this is a complex of sheaves.

5.2.3 Exercise.

a) Prove that this complex of sheaves is exact. Hint: We have to check exactness in the stalks. If x ∈ X we know that we can find an element γ ∈ A with x ∈ U_{γ}. Now we are in the same situation as in the above Lemma, except that we have modified the Čech complex since we have ordered the index set. But it is not difficult to adapt the Lemma to this situation here.

b) Let E be a finite totally ordered set, i.e. E = {0, 1, …, n}. Let A be an abelian group, for any r ∈ ℤ we define

C^{r}(A) = ⊕_{I⊂E,|I|=r+1}A,

by definition we have C^{r}(A) = 0 if r ∈ {0, …, n} = E. For a subset I ⊂ E and α ∈ I we define p(α, I) as the position of α in I, i.e. p(α, I) = 0 if α is the smallest element, p(α, I) = |I| − 1 if α is the biggest one. We define (co-)boundary operators

\begin{aligned}d : C^{r}(A) & → C^{r+1}(A) \\ δ : C^{r}(A) & → C^{r−1}(A) \end{aligned}

by

\begin{aligned}(da)_{J} & = ∑_{β∈J}(−1)^{p(βJ)}a_{J\\{β\}} \\ (δa)_{J} & = ∑_{β∉J}(−1)^{p(βJ∪\{β\})}a_{J∪\{β\}} \end{aligned}

where a_{J} is the J-th component of

a = (… a_{J} …) ∈ C^{•}(A).

We get two complexes

\begin{aligned}0 & → C^{0}(A) ⟶^{d} … ⟶^{d} C^{r}(A) ⟶^{d} C^{r+1}(A) ⟶^{d} \\ 0 & → … C^{r}(A) ⟶^{d} C^{r−1}(A) ⟶^{d} … ⟶^{d} C^{0}(A) → 0 \end{aligned}

Show that these two complexes are exact using the ideas of the part a) and the previous exercises

5.2.4. Let me come back to the heuristical remark above. I said that the complexes of sections G^{•}(X) of a resolution contain some information concerning the cohomology of F. Now we see that for the special case of the Čech resolution the resulting complex of global sections is very similar to the Čech complex attached to the covering actually it is the ordered Čech complex and it is known that the ordered Čech complex gives us the same cohomology groups. We see that coverings allow us to construct resolutions of sheaves. We already saw some other constructions providing resolutions of sheaves. If we look back to our computation of the cohomology of the spheres ( See 4.5 ) then we see that our first short exact sequence is a resolution. We could extend it by resolving \underline{ℤ}_{S^{n−1}} and so on. This gives us the general idea that these resolution in some sense provide a kind of cutting a space into simpler pieces. (See Exercise 4.5.2 ).

IV.6 The spectral sequence

6.1. Now we have seen that a covering of our space provides resolutions of sheaves. In general the sheaves in the resolution will not be acyclic but perhaps their cohomology is easier to compute because they are supported on smaller sets namely the closures of the U_{i_{0},...,i_{q}}. The method of the spectral sequence allows us to extract informations on the cohomology of the sheaf F in terms of the cohomology groups of the resolving sheaves. We consider a resolution of a sheaf F:

0 → F → G^{0} → G^{1} → … → G^{n} → …

6.1.1. We break the sequence

0 → F → G^{0} → K → 0

and we have seen (II.3.6) that we can find an injective resolution of this short exact sequence

0 → F →  G^{0}  → K → 0
↓  ↓  ↓
0 → I^{0} → I^{0} ⊕ J^{0} → J^{0} → 0
↓ ↓ ↓
0 → I^{1} → I^{1} ⊕ J^{1} → J^{1} → 0
↓  ↓  ↓

We have the second half of the exact sequence

0 → K → G^{1} → … → G^{n} →

and we can apply the same to this sequence. Proceeding in the same way forever, we get a diagram

0  0  0  0
↓  ↓  ↓  ↓
0 → F → G^{0} → G^{1} → ⋯ → G^{n} →
↓ ↓ ↓ ↓
0 → I^{0} → I^{00} → I^{10} → ⋯ → I^{p0} →
↓ ↓ ↓ ↓
0 → I^{1} → I^{01} → I^{11} → ⋯ → I^{p1} →
↓ ↓ ↓ ↓
0 → I^{2} → I^{02} → I^{12} → ⋯ → I^{p2} →
↓  ↓  ↓  ↓

where all the I^{ν} and I^{pq} are injective, all squares commute. This double complex of sheaves has two properties

a)all horizontal sequences are exact.

b)The vertical complexes I^{ν•} are injective resolutions of G^{ν} and I^{•} is an injective resolution of F.

We apply the functor global sections to this diagram and get the augmented double complex Ĩ^{^{•}^{•}}(X)

0  0  0  0
↓  ↓  ↓  ↓
0 → F(X) → G^{0}(X) → G^{1}(X) → ⋯ → G^{n}(X) →
↓ ↓ ↓ ↓
0 → I^{0}(X) → I^{00}(X) → I^{10}(X) → ⋯ → I^{p0}(X) →
↓ ↓ ↓ ↓
0 → I^{1}(X) → I^{01}(X) → I^{11}(X) → ⋯ → I^{p1}(X) →
↓ ↓ ↓ ↓
0 → I^{2}(X) → I^{02}(X) → I^{12}(X) → ⋯ → I^{p2}(X) →
↓  ↓  ↓  ↓

We replace the vertical complex on the left and the horizontal line on the top by zero and then we get the ( non augmented ) double domplex I^{^{•}^{•}}

0  0  0
↓  ↓  ↓
0 → I^{00}(X) → I^{10}(X) → ⋯ → I^{p0}(X)
↓ ↓ ↓
0 → I^{01}(X) → I^{11}(X) → ⋯ → I^{p1}(X)
↓ ↓ ↓
0 → I^{02}(X) → I^{12}(X) → ⋯ → I^{p2}(X)
↓  ↓  ↓

All squares commute and all vertical and horizontal sequences are complexes. We give a name to the differentials

d′_{pq} : I^{pq}(X) ⟶ I^{p+1,q}(X) horizontal
d′′_{pq} : I^{pq}(X) → I^{p,q+1}(X) vertical.

We get a simple complex Ispl^{•} from I^{^{•}^{•}}: We put

I_{spl}^{n} = ⊕_{p+q=n}I^{pq}

and we define

d^{n} : I_{spl}^{n} ⟶ I_{spl}^{n+1}

as

d^{n} = ∑_{p+q=n} d′_{pq} + (−1)^{p}d′′_{pq}

It is clear that the commuting of the squares implies that

d^{n+1} ∘ d^{n} = 0.

6.1.2. The following facts are more or less obvious from the construction 6.1.1.

a) The vertical complexes (I^{p•}(X), d′′) compute the cohomology of the sheaves G^{p},

i.e.

H^{q}(X, G^{p}) = H^{q}(I^{p•}(X), d′′)

b) The horizontal complexes (I^{•q}, d′) compute the cohomology of I^{q} and hence they are exact except in degree zero

H^{0}(X, I^{q}) = I^{q}(X) = H^{0}(I^{•q}(X), d′)

and

H^{p}(I^{•q}(X), d′) = 0 for p > 0.

c) The inclusion

I^{•}(X) ⟶ Ispl^{•}(X)

given by x_{q} ⟶ (x_{q}, 0, ⋯, 0) induces an isomorphism

H^{•}(I^{•}(X)) ≃ H^{•}(Ispl^{•}(X)

and hence we have

H^{•}(X, F) = H^{•}(Ispl^{•}(X))

The last assertion is not quite so obvious, it requires a little argument using b). Let us look at a class which is represented by the cocycle

x = (x_{0,n}, …, x_{n,0}).

The entries of the array are placed in our complex like that:

\begin{matrix}… & … & … & x_{n,0} \\ … & … & x_{n−1,1} & 0 \\ … & ... & … & … \\ x_{0,n} & … & … & … \end{matrix}

The cocycle condition implies d′_{n0}x_{n0} = 0. Hence we find a b = (0, …, y_{n−1,0}) ∈ I_{spl}^{n−1}(X) such that d′_{n−1,0}(y_{n−1,0}) = x_{n0} and x − d_{spl}^{n−1}(b) represents the same class but has its last component in the upper right corner is equal to zero. Repeating this we get that we can represent our element by a cocycle whose components are zero except the first one in the lower left corner. This implies c). We also have the inclusion of the complex

G^{0}(X) → G^{1}(X) → ⋯ → G^{n}(X) →
↓ ↓ ↓
I_{spl}^{0}(X) → I_{spl}^{1}(X) → ⋯ → I_{spl}^{n}(X) →

and hence we get from this construction a homomorphism

H^{n}(G^{•}(X)) ⟶ H^{n}(Ispl^{•}(X)) = H^{n}(X, F)

This is the so called edge homomorphism. If the sheaves G^{p} are acyclic then II.3.2 tells us that this edge homomorphism is an isomorphism. This can also be seen by looking at the the double complex, the same argument which gave us c) in the assertion above implies that the edge homomorphism is an isomorphism. If the G^{p} are not acyclic then the edge homomorphism is neither injective nor surjective in general. In this general case we can get information on H^{n}(X, F) in terms of the cohomology groups H^{q}(X, G^{p}) in other words the cohomology groups of the sheaves G^{p} enter. Before I discuss this in some detail I recommend to the reader to solve the following exercise. It shows how these mechanisms work and it deals with the computation of H^{1}(Ispl^{•}(X)). The cocycles are the elements (x_{0,1}, x_{1,0}) which satisfy d′x_{1,0} = 0, d′′x_{0,1} = 0 and d′′x_{1,0} + d′x_{0,1} = 0. Now a simple calculation solves the following exercise

6.1.3 Exercise:

a) Show that the edge homomorphism

H^{1}(G^{•}(X)) ⟶ H^{1}(X, F)

is injective. It provides an isomorphism to those classes which can be represented by cocycles with x_{0,1} = 0. b) Sending a class to x_{0,1} induces a a homomorphism

H^{1}(X, F) = H^{1}(Ispl^{•}(X)) ⟶ H^{1}(X, G^{0})

and that the kernel of this maps is the image of the map in a).

c) The image of the map H^{1}(X, F)) ⟶ H^{1}(X, G^{0}) lands in the kernel of H^{1}(X, G^{0}) →

H^{1}(X, G^{1}) and we have a homomorphism ker(H^{1}(X, G^{0}) → H^{1}(X, G^{1})) → H^{2}(G^{•}(X)).

d) Show that we get even on exact sequence

0 → H^{1}(G^{•}(X)) → H^{1}(X, F) → ker(H^{1}(X, G^{0}) → H^{1}(X, G^{1})) → H^{2}(G^{•}(X))

Now I explain what happens in higher degrees. The formulation of the becomes more complicated. I will formulate the main result and give some indications why it is true.

6.2 The vertical filtration. In the following discussion we will never need that the horizontal complexes of sheaves are exact. This means that in 6.1.1 we can drop the condition a) and moreover of condition b) we only need that the vertical sequences I^{p•} are injective resolutions of G^{p}. The sheaf F will not play any role. We have a filtration on the double complex, we define F^{p}(I^{^{•}^{•}}(X)) to be the subcomplex where the entries in the first p − 1 vertical collumns are zero. By we denote F^{p}(Ispl^{•}(X)) we denote the resulting simple complex. The inclusion of complexes

F^{p}(Ispl^{•}(X))⟶Ispl^{•}(X)

induces a homomorphism in cohomology

H^{n}(F^{p}(Ispl^{•}(X))) → H^{n}(Ispl^{•}(X)).

and we define F^{p}(H^{n}(Ispl^{•}(X))) as the image of this homomorphism. This yields a filtration of the cohomology, we have F^{0}(H^{n}(Ispl^{•}(X))) = H^{n}(Ispl^{•}(X)) and F^{p}(H^{n}(Ispl^{•}(X))) = 0 for p > n. Our goal is to get some understanding of the filtration steps

F^{p}(H^{n}(Ispl^{•}(X)))/F^{p+1}(H^{n}(Ispl^{•}(X))).

We have an exact sequence of complexes

0 → F^{p+1}(Ispl^{•}(X)) → F^{p}(Ispl^{•}(X)) → F^{p}(Ispl^{•}(X))/F^{p+1}(Ispl^{•}(X)) → 0.

The complex on the right is simply the vertical complex given by the p-th collumn. Hence we know

H^{n}(F^{p}(Ispl^{•}(X))/F^{p+1}(Ispl^{•}(X)))→^{∼}H^{n−p}(X, G^{p}).

We write the exact sequence in cohomology

→ H^{n}(F^{p+1}(Ispl^{•}(X))) → H^{n}(F^{p}(Ispl^{•}(X))) → H^{n−p}(X, G^{p}) → H^{n+1}(F^{p−1}(Ispl^{•}(X)))

which yields an inclusion

H^{n}(F^{p}(Ispl^{•}(X)))/ Im(H^{n}(F^{p+1}(Ispl^{•}(X))) ⊂ H^{n−p}(X, G^{p})

By definition we have a homomorphism

H^{n}(F^{p}(Ispl^{•}(X)))/ Im(H^{n}(F^{p+1}(Ispl^{•}(X))) → F^{p}(H^{n}(Ispl^{•}(X)))/F^{p+1}(H^{n}(Ispl^{•}(X)))

which gives us E1: The filtration steps F^{p}(H^{n}(Ispl^{•}(X)))/F^{p+1}(H^{n}(Ispl^{•}(X))) are isomorphic to subquotients of H^{n−p}(X, G^{p}). We put n − p = q we write E_{1}^{p,q} = H^{q}(X, G^{p}). In addition the structure of a complex on G^{•} induces a structure of a complex

H^{q}(X, G^{•}) = → H^{q}(X, G^{p−1}) → H^{q}(X, G^{p}) → H^{q}(X, G^{p+1}) → .

We denote these boundary operators by d_{1}^{pq} : E_{1}^{p,q} → E_{1}^{p+1,q} and call the complex

(E_{1}^{pq}, d_{1}^{pq})_{pq}

the E_{1}-term of our double complex. Since this is a complex we have the cocyles and coboundaries in it

B_{1}^{pq} ⊂ Z_{1}^{pq} ⊂ E_{1}^{pq}

The next question is: How can we compute the subquotient of E_{1}^{p,q} which is isomorphic to the subquotient F^{p}(H^{n}(Ispl^{•}(X)))/F^{p+1}(H^{n}(Ispl^{•}(X))) of the cohomology? A subquotient of E_{1}^{pq} is by definition of the form Z_{∞}^{pq}/B_{∞}^{pq} where B_{∞}^{pq} ⊂ Z_{∞}^{pq} ⊂ E_{1}^{pq}, we have to compute these two submodules. First of all it is clear from the definition that Z_{∞}^{pq} consists of those classes ξ_{pq} ∈ H^{q}(X, G^{p}) which have a representative x_{pq} ∈ I^{pq}(X) which is the lower left entry of a cocycle

x = \begin{aligned}& & & x_{p+2,q−2} \\ & & x_{p+1,q−1} \\ & x_{pq} \\ 0 \end{aligned}

i.e. given x_{pq} we can place entries x_{p+1,q−1}, …, x_{p+q,0} such that dx = 0.

It is also clear that ξ_{pq} ∈ B_{∞}^{pq} if and only if we can find an element

y ∈ I_{spl}^{n−1}(X)

i.e. an element

\begin{aligned}& & & y_{n−1,0} \\ y & = \\ & & y_{0,n−1} \end{aligned}

such that x − dy ∈ F^{p+1}(I_{spl}^{n}(X)). Our strategy is to approximate these submodules Z_{∞}^{pq} (resp. B_{∞}^{pq}) by a sequence of decreasing (resp. increasing ) submodules, i.e. we will construct sequences of submodules

Z_{1}^{pq} ⊃ Z_{2}^{pq} ⊃ … Z_{∞}^{pq}

and

B_{1}^{pq} ⊂ B_{2}^{pq} ⊂ … B_{∞}^{pq}

such that for large indices r we have Z_{r}^{pq} = Z_{∞}^{pq}, B_{r}^{pq} = B_{∞}^{pq}. The first step in this sequence is easy to describe. It is of course clear, the first problem is to find an x_{p+1,q−1} ∈ I^{p+1,q−1}(X) for which

d′x_{pq} = d′′x_{p+1,q−1}.

The element x_{pq} represents our class ξ_{pq}, and we can find such an x_{p+1,q−1} if and only if the class goes to zero under

d_{1}^{pq} : H^{q}(X, G^{p}) ⟶ H^{q}(X, G^{p+1})

i.e. it lies in the kernel

Z_{1}^{pq} = ker(d_{1}^{pq})

and this implies Z_{1}^{pq} ⊃ Z_{∞}^{pq}. If on the other hand our element

ξ_{pq} ∈ B_{1}^{pq} = Im(d_{1}^{p−1,q} : H^{q}(X, G^{p−1}) → H^{q}(X, G^{p})),

then this means that we can find an element y_{p−1,q} which represents a class in H^{q}(X, G^{p−1}) and therefore satisfies d′′(y_{p−1,q}) = 0, and which maps to x_{pq} under d′. Then we can choose our element

y = \begin{aligned}& 0 \\ & y_{p−1,q} \\ 0 \end{aligned}

and

dy = \begin{aligned}& 0 \\ & x_{pq} \\ 0 \end{aligned}

which means that

ξ_{pq} ∈ B_{∞}^{pq} ⊂ Z_{∞}^{pq}.

We conclude that B_{1}^{pq} ⊂ B_{∞}^{pq} ⊂ Z_{∞}^{pq} ⊂ Z_{1}^{pq}, and we define

E_{2}^{pq} = Z_{1}^{pq}/B_{1}^{pq} = H^{q}(H^{p}(X, G^{•})).

In other words we define E_{2}^{pq} as the cohomology groups of the complex (E_{1}^{pq}, d_{1}^{pq}). We get E2: The filtration steps F^{p}(H^{n}(Ispl^{•}(X)))/F^{p+1}(H^{n}(Ispl^{•}(X))) are isomorphic to subquotients of E_{2}^{pq}. The decisive point is that we can define

d_{2}^{pq} : E_{2}^{pq} → E_{2}^{p+2,q−1}.

such that we get a complex

E_{2}^{p−2,q+1} ⟶^{d_{2}^{p−2,q+1}}E_{2}^{pq}⟶^{d_{2}^{pq}}E_{2}^{p+2,q−1}

such that the cocyles and coboundaries of this complex satisfy B_{2}^{pq} ⊂ B_{∞}^{pq} ⊂ Z_{∞}^{pq} ⊂ Z_{2}^{pq}. To construct this map we consider an element ξ_{pq} ∈ Z_{1}^{pq} we want to lift it to a cocycle x. Since this element maps to zero under the boundary map d_{1}^{pq} we can find an element x_{p+1,q−1} such that d_{1}^{pq}(x_{pq}) = d′′(x_{p+1,q−1}). If we apply the horizontal boundary operator we get d′(x_{p+1,q−1}) = z_{p+2,q−1} ∈ I^{p+2,q−1}(X). This element z_{p+2,q−1} represents a class in H^{p+2,q−1}(X, G^{q−1}) which is in the kernel Z_{1}^{p+2,q−1} of d_{1}^{p+2,q−1} because it is a boundary under the horizontal boundary operator. We made some choices, but it is obvious that another choice modifies z_{p+2,q−1} by an element in B_{1}^{p+2,q−1} and hence we get a homomorphism

d_{2}^{pq} : E_{2}^{pq} → E_{2}^{p+2,q−1}.

It is alo easy to verify that we even get a complex

E_{2}^{p−2,q+1} ⟶^{d_{2}^{p−2,q+1}}E_{2}^{pq}⟶^{d_{2}^{pq}}E_{2}^{p+2,q−1}

It is clear that we must have d_{2}^{pq}(ξ_{pq}) = 0 if ξ_{pq} ∈ Z_{∞}^{pq} i.e. if this class lifts to a class in H^{n}(F^{p}(Ispl^{•}(X)) Or in other words Z_{2}^{pq} = ker(d_{2}^{pq}) ⊃ Z_{∞}^{pq}.

I claim that a class in E_{2}^{pq} goes to zero in H^{n}(Ispl^{•}(X))/F^{p+1}(H^{n}(Ispl^{•}(X))) if it is in the image of d_{2}^{p−2,q+1}. An element in E_{2}^{pq} can be represented by an element y_{p−2,q+1} ∈ I^{p−2,q+1}(X) which goes to zero under the vertical boundary operator and whose image under the horizontal boundary opertar is of the form d′(y_{p−2,q+1}) = d′′(y_{p−1,q}). Then d′(y_{p−1,q}) = x_{pq} represents an element in the image. Then we can take

y = \begin{aligned}& & & 0 \\ & & y_{p−1,q} \\ & (−1)^{p−2}y_{p−2,q+1} \\ 0 \end{aligned}

and

dy = \begin{aligned}& & 0 \\ & x_{pq} \\ 0 \end{aligned}

Hence we get B_{2}^{pq} = Im(d_{2}^{p−2,q+1}) ⊃ B_{∞}^{pq}. Now we define E_{3}^{pq} as the cohomology of the complex, i.e.

E_{3}^{pq} = Z_{2}^{pq}/B_{2}^{pq}

Now it is clear - and I will not give the formal proof - that this construction can extended by induction to all r and we get 6.2.1. Starting from

E_{1}^{pq} = H^{q}(X, G^{q}) and d_{1}^{pq} : H^{q}(X, G^{p}) → H^{q}(X, G^{p+1})

we can define a sequence of terms

(E_{r}^{pq}, d_{r}) d_{r}^{pq} : E_{r}^{pq} → E_{r}^{p+r,q−r+1} where d_{r}^{p+r,q−r+1} ∘ d_{r}^{pq} = 0

such that at any level

E_{r+1}^{pq} = \frac{ker(d_{r}^{pq} : E_{r}^{pq} → E_{r}^{p+r,q−r+1})}{Im(d_{r}^{p−r,q+r−1} : E_{r}^{p−r,q+r−1} → E_{r}^{p,q})}

and such that for all r the subquotients F^{p}(H^{n}(Ispl^{•}(X)))/F^{p+1}(H^{n}(Ispl^{•}(X))) are isomorphic to subquotients of E_{r}^{pq}. Since we are in the positive quadrant, i.e. p, q ≥ 0 the sequence of modules E_{r}^{pq} becomes stationary after a while. We say that the spectral sequence converges to the target H^{n}(X, F) and write

(E_{1}^{pq}, d_{1}^{pq}) ⇒ H^{n}(X, F)

or if one knows the E_{2}-term

(E_{2}^{pq}, d_{2}^{pq}) ⇒ H^{n}(X, F).

6.3 The horizontal filtration. 6.3.1 The horizontal filtration in the case of a resolution We go back to our starting point where we had a resolution of a sheaf

0 → F → G^{0} → G^{1} → … → G^{n} → …

from which we constructed the double complex. Then conditions a) and .b) in 6.1.1. are valid. We now consider consider the filtration by subcomplexes ′F^{q}(Ispl^{•}(X)) ⊂ Ispl^{•}(X) where the entries in the first q −1 horizontal lines are zero. If we apply the same the same arguments to this horizontal filtration we get something that we have done already. Since the I^{•} are acyclic our arguments yield that

H^{•}(′F^{q}(Ispl^{•}(X))/′F^{q+1}(Ispl^{•}(X))

vanishes except we are in degree zero and

H^{0}(′F^{q}(Ispl^{•}(X))/′F^{q+1}(Ispl^{•}(X)) = H^{q}(X, F).

Hence we see that for this filtration

′E_{1}^{pq} = 0 unless p = 0

and

′E_{1}^{0n} = H^{n}(X, F).

We do not have any non trivial differentials. Hence we see again that the double complex computes the cohomology H^{•}(X, F). This is only true since we assumed a) in 6.1.1. If we do not have this assumption then both filtrations are non trivial (see 6.4.2). We will also encounter a case where the vertical filtration is uninteresting.

6.3.2 The general case Assume that we have a complex G^{•} which starts in degree zero and that we have an adjusted injective resolution G^{•} → I^{^{•}^{•}} ( See 6.4.1). We change the notation and give the index q to the vertical complexes. (We want a certain consistency therefore we arrange things so that p is the index for the filtration F^{p}.) Now the horizontal filtration will not be trivial in general. We can apply to it the same method which we applied for the vertical filtration. Let ′F^{p}(I^{^{•}^{•}}) be the subcomplex where the entries in the first p − 1 lines are zero. Now we use the specific form of the adjusted injective resolution. The horizontal complex ′F^{p}(I^{^{•}^{•}})/′F^{p+1}(I^{^{•}^{•}}) is of the form

I_{B}^{q−1,p} ⊕ I_{H}^{q−1,p} ⊕ I_{B}^{qp} → I_{B}^{qp} ⊕ I_{H}^{qp} ⊕ I_{B}^{q+1,p} → I_{B}^{q+1,p} ⊕ I_{H}^{q+1,p} ⊕ I_{B}^{q+2,p}

where the diffrential is always zero on the first two summands and is the identity isomorphism between the third term in degree q and the first term in degree q + 1. This makes it easy to compute the cohomology even after we applied the global section functor

H^{q}(′F^{p}(I^{^{•}^{•}})(X)/′F^{p+1}(I^{^{•}^{•}})(X) = I_{H}^{qp}(X)→^{∼}′E_{1}^{pq}

and the differential ′d_{1}^{pq} : I_{H}^{qp}(X) → I_{H}^{q,p+1}(X) is the differential which obtained from the differential in the resolution H^{q}(G^{•}) → I_{H}^{q,•} after we take the global sections. Hence we see that the E_{2} term of the spectral sequence whose E_{2}-term is

′E_{2}^{pq}→^{∼}H^{p}(X, H^{q}(G^{•}))

and which converges to H^{•}(Ispl^{•}(X)).

If the sheaves G^{p} are acyclic for the functor global sections then we have R^{•}H_{X}^{0}(G^{•})→^{∼} G^{•}(X) But still we may consider an adjusted injective resolution of G^{•} → I^{^{•}^{•}}. In this case the vertical filtration is kind of uninteresting. Its E_{1}^{pq} term is as always H^{q}(X, G^{p}) but this is zero for q > 0. The differentials are given by d : H^{0}(X, G^{p}) → H^{0}(X, G^{p+1}) and his gives us the E_{2}^{pq} term as

E_{2}^{pq} = \{ \begin{matrix}H^{p}(G^{•}(X)) & for q = 0 \\ 0 & for q > 0 \end{matrix}

The higher differentials are zero.

6.3.3 Applications. This method of the spectral sequence has many applications. We will discuss some of these applications lateron. Here we give some indications how such applications can look like. If we consider again the resolution of our sheaf

0 → F → G^{0} → G^{1} → … → G^{n} → …

One typical applications provide finiteness results. For instance it we can show that E_{1}^{pq} or E_{2}^{pq} are finitely generated abelian groups or finite dimensional vector spaces, then we can conclude that the same is true for the target groups ( vector spaces ) H^{n}(X, F). Another typical application concerns Euler characteristics. If we know that the cohomology groups H^{n}(X, F) are finite dimensional vector spaces over a field k which vanish for n >> 0 then we define the Euler characteristic

χ(X, F) = ∑(−1)^{n}dim_{k}(H^{n}(X, F)).

It is of course clear that

χ(X, F) = ∑ ∑_{p+q=n}(−1)^{p+q}dim_{k}(E_{∞}^{pq}).

But if we have for a certain level r that the total dimension of the E_{r}^{pq} is finite then it follows from simple principles in linear algebra that

∑(−1)^{p+q}dim_{k}(E_{r}^{pq}) = ∑(−1)^{p+q}dim_{k}(E_{r+1}^{pq}).

Then we can conclude

χ(X, F) = ∑_{p,q}(−1)^{p+q}dim_{k}(E_{r}^{pq}).

If already the H^{q}(X, G^{q}) have finite total dimension then

χ(X, F) = ∑_{p,q}(−1)^{p+q}dim_{k}(H^{q}(X, G^{p})).

There are interesting cases where one knows the structure of the groups E_{2}^{pq} and one also knows that the d_{2}^{pq} are zero. Then we have E_{2}^{pq} = E_{3}^{pq} and it can happen that the differentials on this level vanish again and that this goes on forever. Then we say that the spectral sequence degenerates at level E_{2}. In such a case the E_{2}^{pq} are equal to the subquotient in the filtration on the target. If for instance the cohomology groups are finite dimensional vector spaces then we can compute the dimensions of the cohomology dim_{k}(H^{n}(X, F)) = ∑_{p,q} dim_{k}(E_{r}^{pq}). But in some cases the computation of the higher differentials becomes an extremely difficult task.

6.4. The derived category. Instead of considering sheaves we consider complexes of sheaves on X

G^{•} = 0 → … → G^{ν} → G^{v+1} → ⋯ →

where we assume that G^{α} = 0 if α << 0, we say that the complex is bounded to the left.

We introduce the sheaves of cocycles Z(G^{ν}) = ker(G^{ν} → G^{ν+1}) and the sheaf of coboundaries B(G^{ν}) = im(G^{ν−1} → G^{ν}) and the cohomology sheaves

H^{ν}(G^{•}) = \frac{ker(G^{ν} → G^{ν+1})}{im(G^{ν−1} → G^{ν})}.

We may have cohomology in negative degrees. If we have two such complexes we have an obvious notion of a morphism

G1^{•} ⟶^{ψ} G2^{•}

It is clear that ψ induces a morphism of between the cohomology sheaves

H^{ν}(G1^{•}) → H^{ν}(G2^{•})

Now the derived category D(S_{X}) is defined as a “quotient” category of the category of complexes: A morphism

ψ : G1^{•} → G2^{•}

is declared to be an isomorphism if it induces an isomorphism on the cohomology sheaves. This means that the objects are the complexes of sheaves which are bounded to the left but the sets of morphisms are the morphisms between complexes divided by an equivalence relation: Two morphisms

G1^{•} \overrightarrow{→} G2^{•}

become equal if there is third complex G3^{•} and a morphism G2^{•} → G3^{•} which equalizes the two maps and which introduces an isomorphism H^{•}(G2^{•}) → H^{•}(G3^{•}). This categorial construction will not be discussed in further detail here. (Reference ???)

This gives us a new way to speak of resolutions. If we have a sheaf F we can view it as a complex

F[0] : 0 → 0 → ⋯ → F → 0 → 0 → ⋯

where the sheaf sits in degree zero. If we have a resolution

0 → F → G^{0} → G^{1} → ⋯ → G^{n} →

we write this as morphism ψ : F[0] → G^{•}:

0 → ⋯ → F  →  0 → ⋯ →
↓  ↓
0 → G^{0} → ⋯ → G^{n}  →

and the fact that G^{•} is a resolution translates into the fact that ψ is an isomorphism in the derived category.

We can introduce the notion of a resolution of a complex. This is a double complex

0 → 0 → ⋯ → 0 →  0  → ⋯ →
↓  ↓  ↓
0 → G^{ν_{0}} → ⋯ → G^{n} → G^{n+1} → ⋯ →
↓ ↓ ↓
0 → A^{ν_{0}0} → ⋯ → A^{n0} → A^{n+1,0} → ⋯ →
↓ ↓ ↓
0 → A^{ν_{0}1} → ⋯ → A^{n1} → A^{n+1,1} → ⋯ →
↓  ↓  ↓

where the vertical complexes are exact.

We can drop the line on the top and consider the double complex of sheaves A^{^{•}^{•}} and we can pass to the simple complex Aspl^{•}. Of course we have a morphism

r : G^{•} ⟶ Aspl^{•}

and I leave it as an exercise to the reader to prove that this is an isomorphism in the derived category. (See 2.3.)

6.4.1 Injective resolutions of complexes The point is that we can construct injective resolutions of complexes of sheaves by double complexes. We adapt the approach we used when we constructed the double complex for the resolution G^{•} of F in 6.1.1. The only difference is that we do not assume that the complex of sheaves G^{•} is exact, we have to take the cohomology sheaves into account. (The condition 6.1.1 a) is not valid) We start at the left end of our complex and we break it

0 → Z(G^{ν_{0}}) → G^{ν_{0}} → B(G^{ν_{0}+1}) → 0

and resolve this by the standard construction (see Chap. Hom. Alg. ???). Our resolution looks as follows

0 → H^{ν_{0}}(G^{•}) →  G^{ν_{0}}  → B(G^{ν_{0}+1}) → 0
↓  ↓  ↓
0 → I_{H}^{ν_{0}0} → I_{H}^{ν_{0}0} ⊕ I_{B}^{ν_{0}+1,0}  → I_{B}^{ν_{0}+1,0} → 0
↓ ↓ ↓
0 → I_{H}^{ν_{0}1} → I_{H}^{ν_{0}1} ⊕ I_{B}^{ν_{0}+1,1}  → I_{B}^{ν_{0}+1,1} → 0
↓  ↓  ↓

For the following indices we always have the sequences

0 → B(G^{q}) ⟶ Z(G^{q}) → H^{q}(G^{•}) → 0

and

0 → Z(G^{q}) → G^{q} ⟶ B(G^{q+1}) → 0

We always resolve the first sequence by this method and then we use the resolution of the term in the middle for the left term in the second sequence, resolve the term on the right and then proceed by the standard construction to resolve the term in the middle.

This goes on forever and we get an injective resolution of the complex G^{•} .

0 → 0 → ⋯ → 0 →  0  → ⋯ →
↓  ↓  ↓
0 → G^{ν_{0}} → ⋯ → G^{n} → G^{n+1} → ⋯ →
↓ ↓ ↓
0 → I^{ν_{0}0} → ⋯ → I^{n0} → I^{n+1,0} → ⋯ →
↓ ↓ ↓
0 → I^{ν_{0}1} → ⋯ → I^{n1} → I^{n+1,1} → ⋯ →
↓  ↓  ↓

We call such a resolution an adjusted injective resolution. It is also rather clear that this construction is functorial in G^{•}, if we have a morphism between two complexes ψ : G1^{•} → G2^{•} then this extends to a morphism of the injective resolutions

Ψ̃ : I1^{••} ⟶ I2^{••}

and if we pass to the simple complexes

I1,spl^{•} ⟶ I2,spl^{•}

this extension is unique by the definition of the derived category.

Once we have the notion of the derived category we get a new idea of derived functor should be. I explain this in the context of sheaves on spaces and the global section functor but it works in a much more general context. If we have a complex of sheaves on our space X

G1^{•} ∈ Ob(D(S_{X}))

we can take an injective resolution

r : G^{•} ⟶ I^{^{•}^{•}}.

We apply the functor global sections and we get

r_{X} : G^{•}(X) ⟶ I^{^{•}^{•}}(X)

and we may pass to the simple complex on the right hand side. Then

Ispl^{•}(X)

is a complex of abelian groups and can be viewed as an object in the derived category of abelian groups. Our functor F → F(X) = H^{0}(X, F) = H_{X}^{0}(F) is a functor from the category of sheaves on X to the category of abelian groups. Now we constructed the derived functor

R^{•}H_{X}^{0} : D(S_{X}) ⟶ D(Ab)

from the derived category of sheaves on X to the derived category of abelian groups. We apply it to our sheaf F. We view it as a complex F[0] and consider

R^{•}H^{0}(F[0]).

We recover the cohomology groups H^{•}(X, F) as the cohomology groups of the complex R^{•}H^{0}(F[0]).

In general we can say that the derived functor R^{•}H^{0}(F[0]) is a “higher level object” it contains more information then the cohomology groups.

6.4.2. In section II.3.1 I explained that we may - after defining the derived functor by using injective resolutions - compute it from acyclic resolutions. The same is of course true in the context of derived categories. If we have a resolution of our complex G^{•} → A^{^{•}^{•}} as above and if the A^{pq} are acyclic for the functor H^{0}(X, ) then we have

R^{•}H_{X}^{0}(G^{•})→^{∼}H^{0}(X, A^{^{•}^{•}}) = A^{^{•}^{•}}(X).

To see this we choose an injective resolution G^{•} → I^{^{•}^{•}} The definition of injective sheaves allows us to construct a commutative diagram of complexes

\begin{aligned}G^{•} & → & A^{^{•}^{•}} \\ ↓ & & ↓ \\ G^{•} & → & I^{^{•}^{•}} \end{aligned}

which then induces a homomorphism of complexes

A^{•}(X)_{spl} → I^{•}(X)_{spl}

which must be an isomorphism in the derived category. To see this we look at the vertical filtration (see 6.2. ???) and find that we get the same E_{1}^{pq} term namely H^{q}(X, G^{p}) on both sides. Here we used the acyclicity of the A^{^{•}^{•}} resolution. Now the rest follows from a simple argument of functoriality: We get an isomorphism for the E_{∞}^{pq} and hence the homomorphism must be an isomorphism.

6.4.3 The composition rule. The concept of derived categories allows a very elegant formulation of the content of the theory of spectral sequences. I want to explain this in a special case but it will be clear what happens in more general situations. Now we assume that we have a continuous map f : X → Y between two topological spaces. Let F be a sheaf on X with values in the category of abelian groups. We have the functors H^{0}(X, ) = H_{X}^{0}( ) and f_{∗}. It is clear that H_{X}^{0}( ) is the composition of f_{∗} and H_{Y} (0). We want to understand what this composition implies for the derived functors. We introduced the higher direct images R^{•}f_{∗}(F) as the derived functor of the direct image functor f_{∗}. This is just a collection of sheaves on Y which are indexed by degrees. But we defined the derived functor

R^{•}f_{∗} : D(S_{X}) ⟶ D(S_{Y})

which sends a complex of sheaves on X into an object in the derived category of sheaves on Y. The cohomology of this object gives us the derived sheaves R^{•}f_{∗}(F). Now we have a little lemma which makes the whole thing click.

6.4.4 Lemma. The direct image f_{∗}(I) of an injective sheaf I on X is a sheaf which is acyclic for the functor H^{0} on Y.

This is lemma 1.5.1. I stated the Lemma in the weaker form because the acyclicity f_{∗}(I) is what we actually need and it can be applied in more general situations. Now come back to our complex of sheaves. We choose on injective resolution G^{•} → I^{^{•}^{•}}, apply f_{∗} and get

R^{•}f_{∗}(G^{•}) ≃ f_{∗}(Ispl^{•})

Now we apply R^{•}H_{Y}^{0} to these complexes. But since the sheaves in f_{∗}(Ispl^{•}) are acyclic we find

R^{•}H_{Y}^{0} ∘ R^{•}f(G^{•}) = H^{0}(Y, F_{∗}(Ispl^{•})) ≃ H^{0}(X, Ispl^{•}) ≃ R^{•}H_{X}^{0}(G^{•})

Hence we get

H_{X}^{0}(G^{•}) = R^{•}H_{Y}^{0} ∘ R^{•}f_{∗}(G^{•}).

This gives us a general principle: If we pass to the derived category then the derived functor of a composition is the composition of the derived functors provided the above lemma is available. Of course this formulation is very elegant but it may be difficult to extract explicit information from this rule. In our case here we can apply 2.4.2. and we get that we have a spectral sequence with E_{2} term H^{p}(Y, R^{q}f_{∗}(F)) which converges to H^{n}(X, F).

6.5 The spectral sequence of a fibration. This spectral sequence is especially useful if we can apply it in combination with base change. We say that the map

f : X ⟶ Y

is a cohomological fibration for the sheaf F if the sheaves

R^{q}f_{∗}(F)

are local systems on Y (see 3.2.1) whose stalk in y is given by

H^{q}(f^{−1}(y), F).

If our space Y is locally connected this means we have base change, i.e.

R^{q}f_{∗}(F)_{y} ≃ H^{q}(f^{−1}(y), F).

and for any point y we can find a connected neighborhood V_{y} such that R^{q}f_{∗}(F) restricted to V_{y} is isomorphic to the sheaf of locally constant sections into R^{q}f_{∗}(F)_{y}. The intuitive meaning of this notion is that R^{q}f_{∗}(\underline{A}) is “the system of cohomology groups of the fibres”.

To produce examples of such cohomological fibrations we consider maps f : X → Y which are locally trivial fibrations with some fibre F (See 3.1). Furthermore we assume that for any local trivialization

Ψ_{i} : f^{−1}(U_{i})  ⟶^{∼}  U_{i} × F
↘ ↙ p_{1}
U_{i},

the restriction of F to f^{−1}(U_{i}) is isomorphic to a pullback with respect to the projection to F of a sheaf on the fibre F. Finally we assume that X, Y are Hausdorff and Y should be locally contractible, i.e. each point y ∈ Y has arbitrarily small contractible (see 4.4.11) neighborhoods. Then I claim Under these assumptions f : X → Y is a cohomological fibration. If we assume in addition that our space Y is pathwise connected and if we pick a base point y_{0} ∈ Y, then we will also show (See 8.1) that our local system is basically the same object as a representation of the fundamental group

π_{1}(Y, y_{0})

on H^{q}(f^{−1}(y_{0}), \underline{A}). Especially for a simply connected base space Y we even have

R^{q}f_{∗}(\underline{A}) = H^{q}\underline{(f^{−1}(y), \underline{A})}.

I want to discuss some special cases where this spectral sequence for a fibration becomes very useful.

Let us consider a fibre space

π : X ⟶ Y

(see ???) where the fibre F is homeomorphic to a sphere S^{n−1}. Then we have the E_{2}-term in the spectral sequence

E_{2}^{pq} = H^{p}(Y, R^{q}π_{∗}(ℤ)),

and it is clear that R^{0}π_{∗}(ℤ) = \underline{ℤ}. We have R^{q}π_{∗}(ℤ) for q ≠ 0, n − 1 and R^{n−1}π_{∗}(ℤ) is a local system where the stalcks are isomorphic to ℤ. We say that this fibration by spheres is orientable if the local system is trivial, and we say that the fibration is oriented if we fix an isomorphism

R^{n−1}π_{∗}(ℤ) ≃ \underline{ℤ}.

Now we consider the E_{2}-term of the spectral sequence. It looks like

H^{0}(Y, R^{n−1}π_{∗}(ℤ)) H^{1}(Y, R^{n−1}π_{∗}(ℤ)) ⋯ H^{p}(Y, R^{n−1}π_{∗}(ℤ))
0  0  0
⋮  ⋮  ⋮
0  0  0
H^{0}(Y, ℤ)  H^{1}(Y, ℤ)  ⋯  H^{p}(Y, ℤ),

and the differential operator d_{2} is given by an arrow that points 2 steps to the right and one step down. Thus it is zero (unless we have n − 1 = 1) and stays zero for a while. So the terms E_{2}^{pq} = E_{3}^{pq} ⋯ for a while until we come to the differential d_{n} which sends

d_{n} : H^{p}(Y, R^{n−1}π_{∗}(ℤ)) ⟶ H^{p+n}(Y, R^{0}π_{∗}(ℤ)),

and now the E_{n+1}^{p,q} may be different from E_{n}^{pq}. After that the spectral sequence degenerates. Therefore we get an exact sequence

⟶ H^{p+n−1}(Y, ℤ) ⟶ H^{p+n−1}(X, ℤ) ⟶ H^{p}(Y, R^{n−1}π_{∗}(ℤ)) ⟶ H^{p+n}(Y, R^{0}π_{∗}(ℤ))

which is the so called Gysin sequence. It contains relevant information concerning the fibration. If for instance, one of the differentials is non zero, then the map

H^{•}(Y, ℤ) ⟶ H^{•}(X, ℤ)

is not injective. From this we can conclude that under this assumption the fibration cannot have a section

s : Y ⟶ X

to π.

If the bundle is oriented, then R^{n−1}π_{∗}(ℤ) = ℤ, and we have a canonical generator e ∈ R^{n−1}π_{∗}(ℤ). This gives a class in H^{0}(Y, R^{n−1}π_{∗}(ℤ)) which is mapped to a class

e = d_{n}^{0,n−1}(e_{0}) ∈ H^{n}(Y, ℤ),

and this class is the Euler class of the fibration. If it is non zero, then the bundle has no section.

6.6 Čech complexes and the spectral sequence. I return to the Čech resolutions constructed from coverings 𝔘 = {U_{α}}_{α∈A} (See 5.2.2.):

F𝔘^{•} = 0 → F → Π_{α∈A}F_{α} → Π_{(α,β)∈A_{<}^{2}}F_{α_{β}} → ⋯ →

In view of our previous discussion this means that we have an isomorphism in the derived category

0 →  F  →  0  →  0
↓  ↓  ↓
0 → Π_{α∈A}F_{α} → Π_{(α,β)∈A_{<}^{2}}F_{α,β} → Π_{(α,β,γ)∈A_{<}^{3}}F_{α,β,γ} → ⋯ →

and hence these two complexes have isomorphic derived functors. The sheaves F\underline{α} are concentrated on the closed subsets \overline{U}\underline{α} = \overline{U_{α_{0}} ∩ … ∩ U_{α_{p}}} and our resolution is acyclic if and only if the sheaves F\underline{α} on Ū_{α} are acyclic. In this case we say that the covering 𝔘 provides a F-acyclic resolution.

We consider the vertical filtration ( See 6.2 and 6.3). We get for our E_{1} term

∏_{\underline{α}∈A_{<}^{p+1}} H^{q}(X, F\underline{α}) = ∏_{\underline{α}∈A_{<}^{p+1}} H^{q}(\overline{U}\underline{α}, F\underline{α}) = H^{q}(X, F𝔘^{•})

The E_{2}^{p,0} term is the Čech cohomology and the edge homomorphism yields a homomorphism

Ȟ^{p}(X, 𝔘, F) → H^{p}(X, F).

It is clear

6.6.1. If the covering provides a F acyclic resolution then the edge homomorphism is an isomorphism or in other words the Čech complex computes the cohomology of F. In general this edge homomorphism needs not to be injective because we may have a non trivial differential

d_{2} : E_{2}^{p−2,1} → E_{2}^{p0} = Ȟ^{p}(X, 𝔘, F).

But for p = 1 this differential is zero and it follows that edge homomorphism

Ȟ^{1}(X, 𝔘, F) → H^{1}(X, F)

is injective. (Exercise 6.1.3) But of course it may be non surjective, its image is the kernel of

H^{1}(X, F) → ∏_{α∈A} H^{1}(X, F_{α}) = ∏_{α∈A} H^{1}(\overline{U}_{α}, F_{α})

I want to consider a special case. We cover our space X by two open sets X = U ∪ V then our resolution becomes very short:

0 → F → F_{U} ⊕ F_{V} → F_{U∩V} → 0

where F_{U} = i_{U∗}i_{U}^{∗}(F) and so on. Then our spectral sequence has only two collums: We have as E_{1} term

H^{q}(U, F) ⊕ H^{q}(V, F) → H^{q}(U ∩ V, F) → 0
⋮  →  ⋮  → 0
\underbrace{H^{0}(U, F) ⊕ H^{0}(V, F)} → \underbrace{H^{0}(U ∩ V, F)} → 0
p = 0  p = 1

and the horizontal boundary operator is taking the difference of the restriction maps. Then we se that the spectral sequence degenerates on E_{2} level and we find that we have a long exact sequence

H^{q−1}(U ∩ V, F) → H^{q}(X, F) → H^{q}(U, F) ⊕ H^{q}(V, F) → H^{q}(U ∩ V, F) →

which is called the Mayer-Vietoris sequence. It is of course nothing else that the long exact sequence obtained from the short exact sequence which is given by the resolution. With a slight modification we used this Mayer-Vietoris already when we computed the cohomology of spheres ( See 4.5). 6.6.1.1 A CW-complex is a space which is obtained by succesive attachment of cells. This construction is easy to explain. We start with a point, this is the simplest CW-complex. If Y is already a CW-complex, and if

f : S^{n−1} ⟶ Y

is a continuous map, then we construct a new space X = D^{n} ∪_{f} Y which is again a CW-complex. To construct his new space X we consider S^{n−1} as boundary of D^{n} and X = D^{n} ∪_{f} Y is obtained from the disjoint union D^{n} ⊔ Y by identifying x ∈ S^{n−1} to f(x) ∈ Y. This process is called “attaching a n-cell” to Y. We can relate the cohomology of the spaces X and Y. If we consider a tubular neighborhood T of the boundary sphere (i.e. {(x_{1}, ⋯, x_{n}) | 1 − ε < ∑ x_{i}^{2} ≤ 1}) then V = T ∪_{f} Y is open in X and clearly the

inclusion Y ↪ V is a homotopy equivalence. The open ball D^{n}^{∘}= U is also open in X and we have a covering

X = U ∪ V.

The open set U is acyclic and U ∩ V is homotopy equivalent to S^{n−1}. Thus our spectral sequence yields for q ≥ 1

⟶ H^{q−1}(S^{n−1}, ℤ) ⟶ H^{q}(X, ℤ) ⟶ H^{q}(Y, ℤ) ⟶ H^{q}(S^{n−1}, ℤ) ⟶ .

This tells us that we have some control how the cohomology of Y changes if we attach a n-cell. More precisely we can say that we can compute the cohomology of X if we know already the cohomolgy of Y and if we understand the boundary operator on the E_{1} term:

H^{n−1}(Y, ℤ) ⟶ H^{n−1}(S^{n−1}, ℤ).

There is a very prominent example where this method of computing the cohomology is especially successful. We consider the n-dimensional complex projective space ℙ^{n}(ℂ) (III.2 Examples).

Exercise: a) Show that the topological space ℙ^{n}(ℂ) is obtained from ℙ^{n−1}(ℂ) by attaching a 2n-cell.

b) Show that

H^{•}(ℙ^{n}(ℂ), ℤ) = ⨁_{i=0}^{n} ℤ e_{i}

where e_{i} ∈ H^{2i}(ℙ^{n}(ℂ), ℤ) is a free generator.

6.6.2 A criterion for degeneration. Let us assume that our complex of sheaves has the following property: For any index q we can construct a splitting of the sequence

0 ⟶ Z(G^{q}) ⟶ G^{q} ⟶ B(G^{q+1}) ⟶ 0.

If we now construct our adjusted resolution, then we can achieve that the vertical differentials

I_{Z}^{qp} ⊕ I_{B}^{q+1,p}
↓
I_{Z}^{q,p+1} ⊕ I_{B}^{q+1,p+1}

are the direct sum of the differentials of the resolutions of Z(G^{q}) and B(G^{q+1}) (see II, 3.6.1). This means for the adjusted resolution of the complex we get

I^{qp}  =  I_{B}^{qp} ⊕ I_{H}^{qp} ⊕ I_{B}^{q+1,p}  = I_{Z}^{qp} ⊕ I_{B}^{q+1,p}
↓  ↓  ↓
I^{q,p+1}  = I_{B}^{q,p+1} ⊕ I_{H}^{q,p+1} ⊕ I_{B}^{q+1,p+1} = I_{Z}^{q,p+1} ⊕ I_{B}^{q+1,p}

and the rightmost vertical arrow can be taken as the direct sum of the arrows in the resolution of Z(G^{p}) and B(G^{p+1}).

I claim that this implies that the two spectral sequences obtained from the vertical and the horizontal filtration degenerate on E_{2}-level.

We consider the horizontal filtration. We just saw that the E_{2}-term is given by H^{p}(X, H^{q}(G^{•})). An element in this group is represented by the element

ξ_{qp} ∈ I_{H}^{qp}(X)

which is mapped to zero under the vertical boundary map

\begin{matrix}I_{H}^{qp}(X) \\ ↓ \\ I_{H}^{q,p+1}(X). \end{matrix}

But if we view it as an element in I^{qp}(X), then it is mapped to an element

η_{q,p+1} ∈ I_{B}^{q,p+1}(X) ⊂ I_{B}^{q,p+1}(X) ⊕ I_{H}^{q,p+1}(X) ⊕ I_{B}^{q+1,p+1}(X).

We look at the boundary map

I^{q−1,p+1}(X)  →  I^{q,p+1}(X)
∥  ∥
I_{B}^{q−1,p+1}(X) ⊕ I_{H}^{q−1,p+1}(X) ⊕ I_{B}^{q,p+1}(X) → I_{B}^{q,p+1}(X) ⊕ I_{H}^{q,p+1}(X) ⊕ I_{B}^{q+1,p+1}(X),

and we see that our element η_{q,p+1} is the image of the element

η̃_{q−1,p+1} = (0, 0, η_{q,p+1}) ∈ I^{q−1,p+1}(X)

under this boundary map. Now our assumption implies that this element goes to zero under the vertical differential, because this vertical differential respects the decomposition

I^{q−1,p+1}(X) = I_{Z}^{q−1,p+1}(X) ⊕ I_{B}^{q,p+1}(X).

But then the element

ξ̃_{qp} = ξ_{qp} + (−1)^{p−1}η̃_{q−1,p+1}

is a cocycle. This implies that

E_{∞}^{p,q} = E_{2}^{p,q},

and this is the degeneration of the spectral sequence.

6.6.3. We even get more. We know that the E_{2} term is a step in the filtration and hence

H^{p}(X, H^{q}(G^{•})) ˜ ⟶′F^{p}H^{n}(X, G^{•})/′F^{p+1}H^{n}(X, G^{•})

but we just constructed a homomorphism

i_{pq} : H^{p}(X, H^{q}(G^{•})) ⟶ H^{n}(X, G^{•})

because to any class ξ_{qp} we constructed a cocycle ξ̃_{qp} in I_{spl}^{n}(X). Hence we even get a splitting

H^{n}(X, G^{•}) = ⨁_{p+q=n} H^{p}(X, H^{q}(G^{•})).

This splitting is not canonical because it may depend on the choice of the splitting G^{p} = Z(G^{p}) ⊕ B(G^{p+1}) since this choice influences the correction term η. But the images of the H^{p}(X, H^{q}(G^{•})) are well defined modulo the horizontal filtration.

6.6.4 An application. We consider a space Z = X × Y, and we assume that Y has a finite covering Y = ⋃_{α∈I} U_{α} by open sets which is \underline{ℤ} acyclic (See 6.4.5). Now we consider the Čech resolution of the sheaf \underline{ℤ} on Z:

0 ⟶ \underline{ℤ} ⟶ ∏_{α} \underline{ℤ}_{α} ⟶ ∏_{(α,β)} \underline{ℤ}_{\{α,β\}} ⟶

as in 5.2.2. We abbreviate the notation and denote the Čech complex simply by \underline{A}^{•}. Then

0 ↓

(\underline{ℤ}) / ↓

0 ⟶ \underline{A}^{0} ⟶ \underline{A}^{1} ⟶ ⋯

is an isomorphism in the derived category and get

H^{•}(Z, \underline{ℤ}) ≃ H^{•}(Z, A^{•}).

Now it follows from our assumptions on the covering that the sheaves \underline{A}^{p} are acyclic for the projection map p_{1} to the factor X. We have that

p_{1,∗}(\underline{A}^{p}) = \underline{∏_{\underline{α}} \underline{ℤ}\underline{α}(X)} = \underline{∏_{\underline{α}} ℤ(U_{α_{0}} ∩ … ∩ U_{α_{r}})}

i.e. it is the complex of locally constant sheaves on X associated to the abelian groups ∏_{α} ℤ(U_{α_{0}} ∩ ⋯ U_{α_{r}}). This is a complex of finite generated free \underline{ℤ} modules, we denote it by B^{•}. Then we know

H^{•}(Z, \underline{ℤ}) = H^{•}(X, \underline{B}^{•}).

We apply the previous observation. Since the complex B^{•} is a complex of finitely generated free ℤ modules, we can conclude that the quotient A^{p}/Z(A^{p}) is also free and hence we know that we can split off the boundaries. Hence we know that the spectral sequence degenerates, and we get an isomorphism

K : ⨁_{p+q} H^{p}(X, H^{q}(Y, \underline{ℤ}))→^{∼}H^{n}(Z, \underline{ℤ}).

This isomorphism may depend on the splitting because this spl;itting influences the choice of the correction term η above. Under our assumptions the modules H^{q}(Y, ℤ) are finitely generated abelian groups. This allows us to write these groups as quotient of two finitely generated free abelian groups, i.e. we have an exact sequence

0 ⟶ M_{1} ⟶ M_{2} ⟶ H^{q}(Y, ℤ) ⟶ 0

where M_{1}, M_{2} are finitely generated and free. Now we have obviously

H^{p}(X, M_{i}) = H^{p}(X, ℤ) ⊗ M_{i},

and hence we get an exact sequence

\begin{matrix}H^{p}(X, ℤ) ⊗ M_{1} ⟶ & H^{p}(X, ℤ) ⊗ M_{2} ⟶ H^{p}(X, H^{q}(X, ℤ)) ⟶ \\ ⟶ H^{p+1}(X, ℤ) ⊗ M_{1} ⟶ H^{p+1}(X, ℤ) ⊗ M_{2}. \end{matrix}

This yields a short exact sequence. We observe that the first arrow on the left yields a kokernel

H^{q}(X, ℤ) ⊗ M_{2}/M_{1} = H^{q}(X, ℤ) ⊗ H^{p}(Y, ℤ),

and the last arrow on the right has the kernel Tor_{ℤ}^{1}(H^{q+1}(X, ℤ), H^{p}(Y, ℤ)) (see Chap II.4.3), and hence our short exact sequence will be

0 → H^{q}(X, ℤ) ⊗ H^{p}(Y, ℤ) → H^{q}(X, H^{p}(Y, ℤ)) → Tor_{ℤ}^{1}(H^{q+1}(X, ℤ), H^{p}(Y, ℤ)) → 0.

If we make the further assumption that H^{•}(X, ℤ) is finitely generated the module on the right is finite. Then the restriction of K to the tensor products gives us a homomorphism

⨁_{p+q=n} H^{p}(X, ℤ) ⊗ H^{q}(Y, ℤ) ⟶ H^{n}(X × Y, ℤ)

which is injective and has a finite cokernel. This is the so called Künneth homomorphism. This homomorphism does not depend on the choice of the splitting. To see that this is the case we assume that our space X has a finite ℤ-acyclic covering {V_{β}}_{β∈B} by open sets. In this case we can consider our locally constant sheaves \underline{A}^{p} on X and take their Čech resolution provided by {V_{β}}_{β∈B}. Taking sections we get a double complex in which the (p, q) component is

∏_{\underline{β}∈B_{<}^{p+1}} ∏_{α∈A_{<}^{q+1}} ℤ((X\underline{β} × Y) ∩ (X × U\underline{α}),

and where the vertical and horizontal boundary operators are induced from the boundary operators in the Čech complezes. But then it is clear: If we have cocycles

ξ^{p} ∈ ∏_{\underline{β}∈B_{<}^{p+1}} ℤ(V\underline{β}), η^{q} ∈ ∏_{\underline{α}∈A<^{q+1}} ℤ(U\underline{α}),

then we can define

ξ^{p}η^{q} = (⋯ ξp ; \underline{β}ηq ; \underline{α} ⋯),

and this is a cocycle for the resulting simple complex which computes H^{•}(Z, ℤ). Hence we see that we do not need the correction in 6.6.4 which shows that the class does not depend on the splitting. In the next section where I discuss products in a more general context and then we will see that K does not depend on the choice of the covering.

We apply the same reasoning to the vertical filtration. A slightly different argument gives us another construction of the canonical homomorphism

K : ⨁_{p+q=n} H^{p}(X, ℤ) ⊗ H^{q}(Y, ℤ) ⟶ H^{n}(X × Y, ℤ).

. We may interchange the role of X, Y this means we study the spectral sequence attached to the map p_{2} : X × Y → Y. Now we assume that X also has a ℤ-acyclic covering by open sets. Then the E_{2} term is H^{q}(X, H^{p}(Y, ℤ)) and we get homomorphisms If we compute the cohomology starting from Čech coverings, and if we interchange the two spaces, then the two simple complexes resulting from the double complexes are actually isomorphic. We simply have to reflect along the diagonal. But we have to observe the sign convention in the definition of the differentials. This forces us to put signs. This eventually results in the formula: If we look at the two product maps

H^{p}(X, ℤ) ⊗ H^{q}(Y, ℤ)
↘^{i_{1}}
H^{n}(X × Y, ℤ),
↗_{i_{2}}
H^{q}(Y, ℤ) ⊗ H^{p}(X, ℤ)

then we have

i_{1}(α ⊗ β) = (−1)^{pq}i_{2}(β ⊗ α).

6.7 Products We want to discuss products in a more general context. We start with a commutative ring R with identity and we consider sheaves of R-modules on topological spaces. If we have two such sheaves F, G on a space X, then we can consider the tensor product sheaf F ⊗_{R} G on X. It is plausible that this should be defined as the sheaf attached to the presheaf

U ⟶ F(U) ⊗_{R} G(U)

(see III.3.1.), and it is really not too hard to show that the stalk of this sheaf is given by

(F ⊗_{R} G)_{x} = F_{x} ⊗_{R} G_{x}

for all points x ∈ X. Now we consider two spaces X, Y amd the two projections p_{1}, p_{2} from X × Y to X and Y respectively. If now F and G are sheaves of R-modules on X and Y respectively, then we can define the exterior tensor product

F ˜⊗_{R}G = p_{1}^{∗}(F) ⊗_{R} p_{2}^{∗}(G)

as a sheaf on X × Y. We want to construct a R-module homomorphism

m : H^{i}(X, F) ⊗ H^{j}(Y, G) ⟶ H^{i+j}(X × Y, F ˜⊗_{R}G).

It is not so entirely obvious how this can be done because if we take injective resolutions

0 ⟶ F ⟶ 0
↓
0 ⟶ I^{0} ⟶ I^{1} ⟶

and

0 ⟶ G ⟶ 0
↓  ↓
0 ⟶ J^{0} ⟶ J^{1} ⟶

then the resulting morphism of complexes

F ˜⊗_{R}G ⟶ (I^{•} ˜⊗_{R}J^{•}) spl

needs not to be an isomorphism in the derived category. In other words, the simple complex of sheaves on the right hand side is not necessarily exact because the tensor product is not exact. Therefore it seems to be reasonable to assume that one of the sheaves is flat and admits a flat acyclic resolution, say

0 ↓ G ↓

0 ⟶ A^{0} ⟶ A^{1} ⟶ …

where flat means of course that the stalks A_{x}^{i} are flat R-modules. Then we find that the double complex

F ˜⊗_{R}G ⟶ F ˜⊗_{R}A^{0} ⟶ F ˜⊗_{R}A^{1} ⟶
↓ ↓ ↓
I^{0} ˜⊗_{R}G ⟶ I^{0} ˜⊗_{R}A^{0} ⟶ I^{0} ˜⊗_{R}A^{1} ⟶
↓ ↓ ↓
I^{1} ˜⊗_{R}G ⟶ I^{0} ˜⊗_{R}A^{0} ⟶ I^{1} ˜⊗_{R}A^{1} ⟶
↓  ↓  ↓

has exact rows and exact columns and hence we get a resolution of F ˜⊗_{R}G by the simple complex (I^{•} ˜⊗_{R}Aspl^{•}) which we write down

0 ⟶ F ˜⊗_{R}G ⟶ I^{0} ˜⊗_{R}A^{0} ⟶ I^{1} ˜⊗_{R}A^{0} ⊕ I^{0} ˜⊗_{R}A^{1} ⟶ ….

Hence we get a map

H^{n}((I^{•} ˜⊗_{R}A^{•})spl(X × Y) ⟶ H^{n}(X × Y, F ˜⊗_{R}G).

We get a map

H^{n}((I^{•} ˜⊗_{R}A^{•})spl(X × Y)) ⟶ H^{n}(X × Y, F ˜⊗_{R}G).

By construction we have a map

m′ : H^{i}(I^{•}(X)) ⊗ H^{j}(A^{•}(Y)) → H^{n}((I^{•} ˜⊗_{R}A^{•})(X × Y))

if i + j = n the composition of these two is the map which we want to construct. It is not clear that this product is independent of the resolution, so it may not be canonical. But if we assume in addition that also F is flat and has a acyclic resolution by flat sheaves, then it is clear that this product does not depend on the resolution. To see that this is so we first consider exact sequences

0 ⟶ F_{1} ⟶ F_{2} ⟶ F_{3} ⟶ 0

of sheaves on X. This provides without any assumption on the F_{i} an exact sequence

0 ⟶ F_{1} ˜⊗_{R}G ⟶ F_{2} ˜⊗_{R}G ⟶ F_{3} ˜⊗_{R}G ⟶ 0,

and we get two exact sequences

⟶ H^{i−1}(X, F_{2}) ⟶ H^{i−1}(F_{3}) ⟶^{δ} H^{i}(X, F_{1}) ⟶,

and

H^{i−1+j}(X × Y, F_{2} ˜⊗_{R}G) ⟶^{δ} H^{i−1+j}(X × Y, F_{3} ˜⊗_{R}G) ⟶ H^{i+j}(X × Y, F_{1} ˜⊗_{R}G).

Now the formula

m(δ(ξ) ⊗ η) = δ(m(ξ ⊗ η))

for ξ ∈ H^{i−1}(X, F_{3}) and η ∈ H^{j}(Y, G) is obvious by construction. This reduces the problem of the uniqueness of the map m to the assertion that

m : H^{0}(X, F) ⊗ H^{j}(Y, G) ⟶ H^{j}(X × Y, F ˜⊗_{R}G)

is independent of the resolution. But this is obvious because in this case m is the following map: Any element s ∈ H^{0}(X, F) induces a morphism

m(s) : p_{2}^{∗}(G) → p_{1}^{∗}(F) ⊗_{R} p_{2}^{∗}(G)

which is given by multiplication and clearly

m(s ⊗ ξ) = m(s)^{j}(ξ)

for all ξ ∈ H^{j}(X, G). Now it is clear that the general considerations fit into the context of our earlier discussion of the Künneth-formula and the cup product. If we consider spaces X, Y which have a nice acyclic covering, then the acyclic resolutions

0 ⟶ ℤ ⟶ ∏_{α} \underline{ℤ}_{\{α\}} ⟶ ∏_{(α,β)} \underline{ℤ}_{\{α,β\}} ⟶ …

are resolutions by free ℤ-modules and therefore they are also flat. Since we have

\underline{ℤ} ˜⊗_{ℤ}\underline{ℤ} = \underline{ℤ},

we see that the above considerations generalize the previous ones.

6.7.1 The Cupproduct. We may take X = Y, and we consider the product

H^{p}(X, ℤ) ⊗ H^{q}(X, ℤ) ⟶ H^{p+q}(X × X, ℤ).

Now we consider the diagonal X → X × X, and we can consider the restriction

Δ^{∗}i(α ⊗ β) = α ∪ β,

and this is the cupproduct of the two classes.

Now we have seen – at least for reasonable spaces – that the cohomology groups

H^{•}(X, \underline{ℤ}) = ⨁_{p} H^{p}(X, \underline{ℤ})

carry the additional structure of a graded anticommutative algebra. We want to determine the structure of this algebra in some special cases.

6.7.2 An example. Let us consider a n-dimensional vector space V over ℝ and let Γ ⊂ V be a lattice, i.e. a free submodule of rank n such that V/Γ becomes a compact space. We can choose a basis e_{1}, ⋯, e_{n} of Γ, this is also a basis for V and we get an isomorphism

V/Γ ≃ (ℝ/ℤ)^{n} = (S^{1})^{n}.

The Künneth formula yields

H^{•}((S^{1})^{n}, ℤ) = H^{•}(S^{1}, ℤ) ⊗ ⋯ ⊗ H^{•}(S^{1}, ℤ),

and we see that the cohomology

H^{•}((S^{1})^{n}, ℤ) = ⨁_{ν} H^{ν}((S^{1})^{n}, ℤ)

is a free module of rank 2^{n} over ℤ. It remains to determine the structure as a graded algebra.

First of all we notice that H^{1}(ℝ/ℤ, \underline{ℤ}) = ℤ (see ....). Now we consider the cohomology in degree p. If we have a class ξ ∈ H^{p}(V/Γ, \underline{ℤ}), then we can attach to it an alternating p-linear map

φ_{ξ} ∈ Hom_{alt}^{p}(Γ, ℤ).

To define this element we have to give the value φ_{ξ}(γ_{1}, ⋯, γ_{p}) for any p-tuple \underline{γ} = (γ_{1}, ⋯, γ_{p}) of elements in Γ. We take these elements and construct a homomorphism

α\underline{γ} : ℝ^{p}/ℤ^{p} ⟶ V/Γ

which is given by

α\underline{γ}(x_{1}, ⋯, x_{p}) = x_{1}γ_{1} + ⋯ + x_{p}γ_{p}.

Then we can restrict our class ξ to ℝ^{p}/ℤ^{p}. We consider α∗ ; \underline{γ}(ξ) ∈ H^{p}(ℝ^{p}/ℤ^{p}, \underline{ℤ}) = ℤ, and this is our definition

φ_{ξ}(γ_{1}, ⋯, γ_{p}) = α∗ ; \underline{γ}(\underline{ξ}).

We see rightaway that this value is zero if γ_{1} ⋯ γ_{p} are linearly dependent because then the image of α\underline{γ} is a (S^{1})^{p′} with p′ ⊂ p.

We have to show that the map φ_{ξ} is p-linear. This is easily reduced to the following special case: We consider ℝ^{p+1}/ℤ^{p+1}, and we consider the three inclusions i_{1}, i_{2}, Δ : ℝ^{p}/ℤ^{p} → ℝ^{p+1}/ℤ^{p+1} given by

\begin{aligned}I_{1} & : & (x_{1}. · · · , x_{p}) & ↦ & (x_{1}, 0, x_{2}, ⋯ x_{p}) \\ i_{2} & : & (x_{1}, ⋯ x_{p}) & ↦ & (0, x_{1}, x_{2}, ⋯ x_{p}) \\ Δ & : & (x_{1}, ⋯, x_{p}) & → & (x_{1}, x_{1}, x_{2} ⋯ x_{p}) \end{aligned}

and for a class ξ ∈ H^{p+1}(ℝ^{p+1}/ℤ^{p+1}, \underline{ℤ}) we have to show that

i_{1}^{∗}(ξ) + i_{2}^{∗}(ξ) = Δ^{∗}(ξ).

Both sides are linear in ξ and hence we have to check this equality for classes

ξ_{1} ∈ H^{1}(ℝ/ℤ, \underline{ℤ}) ⊗ H^{0}(ℝ/ℤ, ℤ) ⊗ H^{p−1}(ℝ^{p−1}/ℤ^{p−1}, \underline{ℤ})

and

ξ_{2} ∈ H^{0}(ℝ/ℤ, \underline{ℤ}) ⊗ H^{1}(ℝ/ℤ, \underline{ℤ}) ⊗ H^{p−1}(ℝ^{p−1}/ℤ^{p−1}, \underline{ℤ}),

and then it is obviously true. This gives us a homomorphism of graded modules

α : H^{•}(V/Γ, \underline{ℤ}) ⟶ Homalt^{•}(Γ, ℤ).

It is a well known elementary fact that the right hand side has the structure of an anticommutative graded algebra where the product is given by

(φ ∧ ψ)(γ_{1} ⋯ γ_{m}) = ∑_{t}(−1)^{ε(t)}φ(γ_{i_{1}} ⋯ γ_{i_{p}}) · ψ(γ_{j_{1}} ⋯ γ_{j_{q}})

where φ is a p-form, ψ is a q-form m = p + q and so on.

Perhaps it is not so much of a surprise that:

The homomorphism

α : H^{•}(V/Γ, \underline{ℤ}) ⟶ Homalt^{•}(Γ, ℤ)

is an isomorphism of graded algebras. To verify this we wrtie V/Γ = (ℝ/ℤ)^{n}, and we have the following basis for the cohomology: We look at ordered subsets i_{1} < i_{2} ⋯ < i_{p} of {1, ⋯ n} and form

1 ⊗ ⋯ ⊗ 1 ⊗ e_{i_{1}} ⊗ ⋯ e_{i_{2}} ⋯ e_{i_{p}} ⊗ 1 ⋯ ⊗ 1 = ξ\underline{i}

where e_{i_{ν}} ∈ H^{1}(ℝ/ℤ, \underline{ℤ}) is the canonical generator. The e_{i} can be viewed as basis elements for Γ at the same time, then

φ_{ξ\underline{i}}(e_{j_{1}}, ⋯, e_{j_{p}}) = \{ \begin{matrix}1 & if \underline{i} and i_{1} = j_{1} ⋯ i_{p} = j_{p} \\ 0 & else \end{matrix}

and clearly

ξ\underline{i} ∪ ξ\underline{j}′ = \{ \begin{matrix}0 & if \underline{i} and \underline{i}′ are not disjoint \\ (−1)^{ε(\underline{i},\underline{i}′)}ξ_{\underline{i}∪\underline{i}′} & else \end{matrix}

This proves the assertion.

6.8 An excursion into homotopy theory. We want to discuss briefly an application of the spectral sequence which is not directly related to the goals of this book but which is certainly important and beautiful.

If we consider a pathwise connected space X together with a base point x_{0} we can define the homotopy groups

π_{n}(X, x_{0}).

On the other hand we have the singular homology groups H_{i}(X, ℤ) which are not discussed here (exept in the chapter on cohomology of manifolds), and we always have the so called Hurewič homomorphism

π_{n}(X, x_{0}) ⟶ H_{n}(X, ℤ).

A famous theorem of W. Hurewič asserts:

Let X be pathwise connected with base point x_{0}. Let n > 0 be an integer and let us assume that

H_{i}(X, ℤ) = 0  for  0 < i < n.

Then the Hurewič homomorphism

π_{n}(X, x_{0}) ⟶ H_{n}(X, ℤ)

is an isomorphism if n > 1 and for n = 1 we get an isomorphism

π_{1}(X, x_{0})_{ab} = π_{1}(X, x_{0})/[π_{1}(X, x_{0}), π_{1}(X, x_{0})] ⟶^{∼} H_{1}(X, ℤ).

Here π_{1}(X, x_{0})_{ab} is the abelianized fundamental group, i.e. the maximal abelian quotient.

We cannot prove this theorem here, since we did not define the homology groups. But we can also define the singular cohomology group H_{sing}^{n}(X, A) and for reasonable spaces we have

H_{sing}^{n}(X, A) ≃ H^{n}(X, A)

i.e. the singular cohomology with coefficients in A is isomorphic to sheaf cohomology. Under these circumstances we can show

Hom(H_{i}(X, ℤ), ℚ/ℤ) ≃ H^{i}(X, \underline{ℚ/ℤ}),

and hence we can reformulate the Hurewič theorem:

An element [φ] ∈ π_{n}(X, x_{0}) is represented by a homotopy class of maps of pointed spaces ( the basepoints are pt and x_{0})

ϕ : (S^{n}, pt) → (X, x_{0})

and hence it defines a map

φ^{∗} : H^{n}(X, \underline{ℚ/ℤ}) ⟶ H^{n}(S^{n}, \underline{ℚ/ℤ}) = ℚ/ℤ.

Now the map ξ ↦ φ^{∗}(ξ) defines a homomorphism

H^{n}(X, \underline{ℚ/ℤ}) ⟶ Hom(π_{n}(X, x_{0}), ℚ/ℤ)

(the dual of the Hurewič map), and this map is an isomorphism is H^{i}(X, \underline{ℚ/ℤ}) = 0 for 0 < i < n.

Indication of proof: We introduce the space (ΣX, x_{0}) of continuous path starting at x_{0}, i.e. the space of all σ

\begin{matrix}σ : [0, 1] & ⟶ & X \\ σ(0) & = & x_{0}. \end{matrix}

Then this space is contractible and we have a map

\begin{aligned}e & : & (ΣX, x_{0}) & ⟶ & X \\ e & : & σ & ⟶ & σ(1). \end{aligned}

This map is a fibration, the fibre over x_{0} is the loop space Ω(X, x_{0}). Hence we can say

H^{•}(ΣX, \underline{ℚ/ℤ}) = H^{•}(X, H^{•}(Ω(X, x_{0}))

where H^{•}(Ω(X, x_{0})) is the local system of cohomology groups of the loop space.

We prove the Hurewič theorem by induction on n. The key is the observation that H^{i}(ΣX, \underline{ℚ/ℤ}) = 0 for i > 0 since ΣX is contractible.

If n = 1, then we consider the E_{2}-term in our spectral sequence in degree one

\begin{aligned}H^{0}(X, H^{1}(Ω(X, x_{0})) \\ & H^{1}(X, H^{0}(Ω(X, x_{0})) \end{aligned}

It follows that H^{1}(X, H^{0}(Ω(X, x_{0})) = 0 and

H^{0}(X, H^{1}(Ω(X, x_{0})) ⟶ H^{2}(X, H^{0}(Ω(X, x_{0}))

is an isomorphism.

The local system H^{0}(Ω(X, x_{0})) is easy to compute, it is a module under the fundamental group Γ = π_{1}(X, x_{0}). We have a continuous map

Ω(X, x_{0}) ⟶ X

where X̃ is the universal convering and the fibres of this map are exactly the connected components of Ω(X, x_{0}). Then it is easy to see that H^{0}(Ω(X, x_{0})) is the local system given by the Γ module

C(Γ, ℚ/ℤ)

of all ℚ/ℤ-valued functions on Γ where Γ acts by translations. This module contains the constant functions and hence we get an exact sequence

0 → \underline{ℚ/ℤ} → H^{0}(Ω(X, x_{0})) → M → 0,

where M is the quotient sheaf. We get a long exact sequence in cohomology

0 → H^{0}(X, \underline{ℚ/ℤ}) → H^{0} (X, H^{0}(Ω(X, x_{0}))) → H^{0}(X, M) → H^{1}(X, ℚ/ℤ) → 0.

For the local systems the sections H^{0}( ) are simply the invariants under Γ, it is easy to see that we get an exact sequence

(ℚ/ℤ)^{Γ} ⟶^{∼} (C(Γ, ℚ/ℤ))^{Γ} → M^{Γ} ⟶^{∼} H^{1}(X, ℚ/ℤ) → 0

Now it follows from our results on the cohomology of groups that

M^{Γ} ≃ H^{1}(Γ, ℚ/ℤ) = Hom(Γ, ℚ/ℤ),

and hence we proved the result for n = 1.

For n > 1 we apply the same method. Now we know that Ω(X, x_{0}) is connected and hence we see that

H^{0}(Ω(X, x_{0})) ≃ \underline{ℚ/ℤ}.

Then we find many zeroes in the bottom row of the spectral sequence and the system of cohomology groups H^{i}(Ω(X, x_{0})) will be trivial. This shows that the E_{2}-term in the spectral sequence looks as follows

H^{0}(X, H^{n−1}(Ω(X, x_{0})))
⋮
H^{0} (X, H^{1}(Ω(X, x_{0})))
H^{0} (X, H^{0}(Ω(X, x_{0}))) 0 ⋯ 0 H^{n} (X, H^{0}(Ω(X, x_{0}))) .

We conclude that H^{i}(Ω(X, x_{0})) = 0 and that the differential

d : H^{0} (X, H^{n−1}(Ω(X, x_{0}))) ⟶ H^{n} (X, H^{0}(Ω(X, x_{0})))

must be an isomorphism.

This implies that

H^{n−1}(Ω(X, x_{0}))_{x_{0}} ≃ H^{n}(X, \underline{ℚ/ℤ})

i.e.

H^{n−1}(Ω(X, x_{0}), \underline{ℚ/ℤ}) ⟶^{∼} H^{n}(X, \underline{ℚ/ℤ}).

Now we have the exakt sequence for homotopy groups which say that

π_{i−1}(Ω(X, x_{0})) ≃ π_{i}(X, x_{0}),

and the Hurewič theorem follows.

It is quite amusing to consider the special case of X = S^{n}. In this case we find

H^{i}(ΩS^{n}, ℚ/ℤ) = \{ \begin{matrix}ℚ/ℤ & for i = k(n − 1) \\ 0 & else \end{matrix}

IV.7 Cohomology with compact supports

7.1. The definition Let X be a locally compact space and F a sheaf of abelian groups. If we have a section s ∈ H^{0}(X, F) then its support is the set of x ∈ X with s_{x} ≠ 0. It is always closed. We can define the submodule H_{c}^{0}(X, F) of sections with compact support. This yields a left exact functor and we define the cohomology with compact supports as the right derived functor of H_{c}^{0}(X, F). In accordance with our general principles we choose an injective resolution

0 → F → I^{0} → I^{1} →

of F and define

H_{c}^{i}(X, F) = H^{i}(H_{c}^{0}(X, I^{•})).

The cohomology with compact supports has different properties from the ordinary cohomology, it does not satisfy the homotopy axiom. We will see in the section on the cohomology of manifolds that it is dual to the ordinary cohomology. Of course on a compact space X we have Hc^{•}(X, F) = H^{•}(X, F). If we have open sets U ⊂ V ⊂ X then we have natural map

H_{c}^{i}(U, F) → H_{c}^{i}(V, F) → H_{c}^{i}(X, F)

here we see that the restriction maps which we had in the theory of sheaves are turned backwards. On the other hand if we have a map f : X → Y then we will not be able to define a map from H_{c}^{i}(Y, \underline{ℤ}) to H_{c}^{i}(X, \underline{ℤ}) unless the map is proper.

Let us assume that U ↪^{i} X is an open subset of our space let us assume that its closure is compact, then its boundary ∂Ū \ U is also compact. Let F be a sheaf on U. We defined two new sheaves on X: The direct image i_{∗}(F) where

i_{∗}(F)(V) = F(U ∩ V)

and the the extension by zero

i_{!}(F)(V) = {s ∈ F(V) | |s| does not meet V ∩ ∂Ū}

One checks easily that i_{!}(F) has the stalks

\begin{aligned}i_{!}(F)_{x} & = F_{x} if x ∈ U \\ i_{!}(F)_{y} & = 0 if y ∉ U \end{aligned} .

We have a morphism of sheaves i_{!}(F) → i_{∗}(F) which is an isomorphism in all stalks except the ones on the boundary ∂Ū.

7.1.1 Proposition. If X is a space and i : U ↪ X an open subset with compact closure and if F is a sheaf of abelian groups on U then

Hc^{•}(U, F) = H^{•}(X, i_{!}(F)).

Proof. This is almost clear from the definition. We choose an injective resolution of the sheaf F on U

0 → F → I^{0} → I^{1} → …

and we notice that

i_{!}(I^{•})(X) = H_{c}^{0}(U, I^{•}).

7.2. An example Now we consider the sheaf \underline{ℤ} on the open ball D^{n} ⊂ ℝ^{n}. We want to compute Hc^{•}(D^{n}, \underline{ℤ}). To do this we embed D^{n} ↪^{i} D̄^{n}. On D̄^{n} we have an exact sequence

0 → i_{!}(\underline{ℤ}) → i_{∗}(\underline{ℤ}) → i_{∗}(\underline{ℤ})/i_{!}(\underline{ℤ}) → 0.

The sheaf i_{∗}(\underline{ℤ}) is \underline{ℤ}D̄^{n} and the sheaf i_{∗}(\underline{ℤ})/i_{!}(\underline{ℤ}) is concentrated on D̄ \ D = S^{n−1} and on this space it is simply \underline{ℤ}_{S^{n−1}}. We write the long exact sequence in cohomology, exploit our proposition and get

→ H_{c}^{ν}(D, \underline{ℤ}) → H^{ν}(D̄, \underline{ℤ}) → H^{ν}(S^{n−1}, \underline{ℤ}) →→ H_{c}^{ν+1}(D, \underline{ℤ}).

We have H^{ν}(D̄, \underline{ℤ}) = 0 for ν > 0 and hence we get for ν = 0

0 → H^{0}(D̄^{n}, \underline{ℤ}) → H^{0}(S^{n−1}, \underline{ℤ}) → H_{c}^{1}(D^{n}, \underline{ℤ}) → 0
∥
ℤ

and for ν ≥ 0

H^{ν}(S^{n−1}, \underline{ℤ}) ⟶^{∼} H_{c}^{ν+1}(D^{n}, \underline{ℤ}).

Our computation of the cohomology groups of spheres yields

H_{c}^{ν}(D^{n}, \underline{ℤ})→^{∼} \{ \begin{matrix}ℤ for ν = n \\ 0 for ν ≠ n \end{matrix}

If we have a homeomorphism f : D_{n}→^{∼}D_{n} then it induces necessarily an isomorphism f^{∗} : H_{c}^{d}(D^{n}, \underline{ℤ})→^{∼}H_{c}^{d}(D^{n}, \underline{ℤ}) which can only be multiplication by ±1. If we take for instance the homoeomorphism that sends (x_{1}, x_{2}, …, x_{n}) ↦ (x_{1}, x_{2}, …, −x_{n}), i.e. we change the sign of the last coordinate then we get multiplication by −1 on H_{c}^{d}(D^{n}, \underline{ℤ}). We say that f preserves the orientation, if it induces the identity on H_{c}^{d}(D^{n}, \underline{ℤ}). (???? Noch was murmeln)

7.2.1. We want to consider a relative situation. Let us assume that we have a diagram

X  ↪^{i} \overline{X}
π ↓  ↙ π̄
Y,

and we want to assume this is some kind of fibration by n-dimensional balls. By this I means that locally in Y we can trivialize our diagram

V × D^{n}  ↪^{i} V × D̄^{n}
↓  ↙
V.

If we choose a covering Y = ∪V_{α} such that we have trivializations over the V_{α}, then we get identifications (see 3.1.1.). For v ∈ V_{α} ∩ V_{β} we have homeomorphisms

g_{α,β}(v) : (D^{n}, D̄^{n}) ⟶ (D^{n}, D̄^{n})

which means g_{α,β}(v) is a homeomorphism of D̄^{n} which maps the interior to the interior and the boundary to the boundary. We call this fibration orientable if the g_{α,β}(v) preserve the orientation, and if we selected a consistent orientation on the fibres . We consider the sheaf ℤ on X and its extension i_{!}(ℤ) to X̄. We want to compute the cohomology H^{•}(X̄, i_{!}(ℤ)). We apply the spectral sequence for a fibration (see 6.6.2), and we have the E_{2}-term H^{p}(Y, R^{q}π_{∗}(i_{!}(ℤ)). Our computation in the previous section yields

R^{q}π_{∗}(i_{!}(ℤ)) = \{ \begin{matrix}0 & q ≠ n \\ \underline{ℤ} & q = n \end{matrix}

(remember that we have the orientation) and consequently the spectral sequence degenerates and

H^{p+n}(X̄, i_{!}(ℤ)) = H^{p}(Y, R^{n}π_{∗}(i_{!}ℤ)) = H^{p}(Y, ℤ).

7.2.2 Formulae for cup products : We want to explain some formulae for prducts and for this purpose it is convenient to replace the open (resp. closed) ball D^{n} (resp. D̄^{n}) by the open (resp. closed) box

B^{n} = {(x_{1}, ⋯, x_{n}) | |x_{i}| < 1} ⊂ B̄^{n} = {(x_{1}, ⋯, x_{n}) | |x_{i}| ≤ 1}

because the pairs (D^{n}, D̄^{n}) and (B^{n}, B̄^{n}) are homeomorphic. Let us assume that we have two numbers d_{1}, d_{2} with d_{1} + d_{2} ≥ n. We consider products

B̄^{d_{1}} × B^{n−d_{1}}  and  B̄^{d_{2}} × B^{n−d_{2}},

and we consider embeddings

\begin{aligned}i_{1} & : & B̄^{d_{1}} × B^{n−d_{1}} ⟶ B̄^{n} \\ i_{1} & : & ((x_{1}, ⋯, x_{d_{1}}), (y_{1}, ⋯, y_{n−d_{1}})) ⟼ (x_{1}, ⋯, x_{d_{1}}, y_{1}, ⋯, y_{n−d_{1}}) \end{aligned}

and

\begin{aligned}i_{2} & : & B^{n−d_{2}} × B̄^{d_{2}} ⟶ B̄^{n} \\ i_{2} & : & ((y_{1}, ⋯, y_{n−d_{2}}), (x_{1}, ⋯, x_{d_{2}})) ↦ (y_{1}, ⋯, y_{n−d_{2}}, x_{1}, ⋯, x_{d_{2}}). \end{aligned}

We get the sheaves i_{1,!}(ℤ) and i_{2,!}(ℤ) on B̄^{n}, and clearly

\begin{matrix}H^{n−d_{1}}(B̄^{n}, i_{1,!}(ℤ)) & ˜ ⟶ & ℤ \\ H^{n−d_{2}}(B̄^{n}, i_{2,!}(ℤ)) & ˜ ⟶ & ℤ \end{matrix}

where we chose the orientations as given by the ordering of the y-coordinates. We want to consider the cup product

H^{n−d_{1}}(B̄^{n}, i_{1,!}(ℤ)) × H^{n−d_{2}}(B̄^{n}, i_{2,!}(ℤ)) ⟶ H^{2n−d_{1}−d_{2}}(B̄^{n}, i_{1,!}(ℤ) ⊗ i_{2,!}(ℤ)).

We must understand the tensor product of the two sheaves. We observe that we have an embedding

i_{1,2} : B̄^{d_{1}+d_{2}−n} × B^{2n−d_{1}−d_{2}} ⟶ B̄^{n}
i_{12} : ((x_{1}, ⋯, x_{d_{1}+d_{2}−n}), (y_{1}, ⋯, y_{2n−d_{1}−d_{2}})) ⟶
(y_{1}, ⋯, y_{n−d_{2}}, x_{1}, ⋯, x_{d_{1}+d_{2}−n}, y_{n−d_{2}+1}, ⋯, y_{n})

and an isomorphism provided by the multiplication on the stalks

i_{1,!}(ℤ) ⊗ i_{2,!}(ℤ) ˜ ⟶i_{12,!}(ℤ)

We choose the orientation on B^{2n−d_{1}−d_{2}} which is given by the ordering of the coordinates. Then all the cohomology groups in

H^{n−d_{1}}(B̄^{n}, i_{1,!}(ℤ))) × H^{n−d_{2}}(B̄^{n}, i_{2,!}(ℤ)) ⟶ H^{2n−d_{1}−d_{2}}(B̄^{n}, i_{12,!}(ℤ))

are identified to ℤ. Now I claim: Under these identifications the cup product is given by the multiplication. I have the feeling that the proof can be left as an exercise to the reader. First of all we can restrict to the case d_{1} + d_{2} = n, and now we have enough flexibility to reduce to the case d_{1} = n − 1, d_{2} = 1. Then we recall that

H_{c}^{1}(B^{1}, ℤ) = H^{1}(B̄^{1}, i_{!}(ℤ))

can be computed from the exact sequence

0 → i_{!}(ℤ) ⟶ \underline{ℤ} ⟶ \underline{ℤ}/i_{!}(ℤ) ⟶ 0

which reduces it to a cup product with an H^{0} in it.

7.3 The fundamental class. Let M be a connected C_{0}-manifold of dimension n. If we have a point p ∈ M then we can find a neighborhood V_{p} of p which is homeomorphic to an open ball D ⊂ ℝ^{n}. Then we have H_{c}^{d}(V_{p}, \underline{ℤ})→^{∼}ℤ but this isomorphism is not canonical. If we have two points p, q and two small open neighborhoods V_{p}, V_{q} of these points, then we have no consistent way to identify H_{c}^{d}(V_{p}, \underline{ℤ}) and H_{c}^{d}(V_{q}, \underline{ℤ}). But if we choose a path γ : [0, 1] → M which starts at p and ends at q, then we get an identification along the path (see the discussion of this argument in the section on local systems). We say that M is orientable if for any two points this identification does not depend on the path. If M is orientable then we can choose a generator in H_{c}^{d}(V_{p}, \underline{ℤ}) for all p which is consistent with the above identifiction along paths. Once we have choosen such generators we call the manifold oriented. If our manifold has a differentiable structure, then we have another notion of orientation on M. (see ....). In this case it is easy to see that the two concepts of being oriented coincide. In the next chapter we will prove that for a connected and oriented C^{∞}- manifold M of dimension n a point p ∈ M and any open ball D_{p} ⊂ M containing p the map H_{c}^{d}(D_{p}, \underline{ℤ}) → H_{c}^{d}(M, \underline{ℤ}) is an isomorphism. This is the starting point to get Poincaré duality. The image of the generator in H_{c}^{d}(D_{p}, \underline{ℤ}) is the so called fundamental class.

IV.8 Cohomology of manifolds

8.1 Local systems. I want to study the cohomology of C^{∞}-manifolds with coefficients in local systems See 3.3.2). We will know that the stalks at two different points x, y are always isomorphic but in general we do not have the possibility to identify them in a consistent way. The following argument is the same that everybody has seen during the discussion of the principle of analytic continuation in theory of complex functions. Since M is connected, we can choose a path γ : [0, 1] → M with γ(0) = x and γ(1) = y. We cover the path by finitely many sufficiently small open sets U_{i}, on which V is trivial. This gives us a subdivision 0 = t_{0} < t_{1} < … < t_{n} = 1 such that the γ[t_{i}, t_{i+1}] are entirely in one of the covering sets U_{ν} and hence we can identify V_{γ(t_{i})} = V_{γ(t_{i+1})} = V(U_{ν}). This sequence of identifications yields an identification

Ψ_{γ} : V_{x} ⟶^{∼} V_{y}.

This identification depends on the path, but it is not difficult to see that it only depends on the homotopy class [γ] of the path. If we choose a base point x_{0} ∈ M and consider closed paths which start and end at x_{0}, then we get a representation of the fundamental group

\begin{matrix}ρ : π_{1}(M, x_{0}) & → Aut(V_{x_{0}}) \\ ρ : [γ] & ↦ (Ψ_{[γ]} : V_{x_{0}} → V_{x_{0}}) \end{matrix} .

It is not hard to see that the local system can be reconstructed from this representation: We consider the set of pairs ([γ], v) where [γ] is a homotopy class of path from x_{0} to x and v ∈ V_{x_{0}}. The stalk at a point x ∈ M will be this set divided by the equivalence relation

([γ], v) ∼ ([γ_{1}], v_{1})

if and only if

ρ([γ_{1}]^{−1} ∘ [γ])(v_{1}) = v.

We can express this by saying that we have an equivalence of categories: Abelian groups V together with an action of π_{1}(M, x_{0}) and local systems V whose stalk at x_{0} is isomorphic to V. If we have a local system V on our manifold M then under certain assumptions we can construct a dual local system V^{∨}. We want to study

H^{i}(M, V) and H_{c}^{i}(M, V^{∨})

and we will again under certain assumptions construct a duality between H^{i}(M, V) and H_{c}^{d−i}(M, V^{∨}) where d is the dimension of M. This will be the Poincaré duality.

8.2 Čech resolutions of local systems. If M is a C^{∞}-manifold then we can find a covering M = ⋃_{α∈A} U_{α} by open sets which has the following two properties

(1) The covering is locally finite, i.e. to any point p ∈ M we can find an open neighborhood V_{p} containing p such that we have only finitely many α ∈ A such that U_{α} ∩ V_{p} ≠ ∅.

(2) For any finite set α_{0}, …, α_{q} ∈ A the intersection U_{α_{0}} ∩ … ∩ U_{α_{q}} is diffeomorphic to an open ball of dimension n and the intersection Ū_{α_{0}} ∩ … ∩ Ū_{α_{q}} is contractible. I want to explain briefly why we can find such a covering It is part of the definition of a manifold, that it should be paracompact (See 4.4) . This means that any covering admits a locally finite refinement. This follows if we assume that M is countable at infinity, i.e. we can find an increasing sequence of relatively compact open sets

W_{n} ⊂ W_{n+1}

where W̄_{n} ⊂ W_{n+1} for all n and which exhausts the manifold M, i.e. ⋃ W_{n} = M. We can use the paracompactness to introduce a Riemannian metric on M. To do this we construct a partition of unity on M. This is a family {h_{i}}_{i∈I} of positive C^{∞} functions which has the following properties.

(i) The support of such an h_{i} is small so that we can find an open set U_{i} ⊂ M which is C^{∞}-isomorphic to an open ball D ⊂ ℝ^{d} and Supp(h_{i}) ⊂ U_{i}.

(ii) For any point x ∈ M there are only finitely many indices j ∈ I with h_{j}(x) ≠ 0. (iii) We have

∑_{i∈I} h_{i} = 1.

The construction of such a partition of unity is standard and quite easy.

We can construct a Riemannian g_{i} metric on each of the U_{i} simply by transporting the standard metric on the ball by means of the diffeomorphism. We multiply this metric by h_{i} then it extends to a positive quadratic form on the tangent bundle of M which is positive definite on the support of h_{i}. Adding up gives the desired metric. Now we invest some differential geometry. Any point x ∈ M has an open neighborhood V_{x} which is diffeomorph to a ball and which has the property that it is convex: Any two points y, z ∈ V_{x} can be joined by a unique geodesic. We may simply take a small ball B(x, ϵ) = V_{x}, these are all those points which have distance < ϵ from x. The closure of such a ball is diffeomorphic to a closed ball in ℝ^{d}, the boundary \overline{∂B(x, ϵ)} is a sphere. It is a smooth hypersurface in M. Now I come back to the construction of a covering with the required properties. I assume that M is countable at infinity. We can exhaust it by a sequence of relatively compact open sets W_{n} which in addition have the property that W̄_{n} ⊂ W_{n+1}. We start at an index n and cover W̄_{n} by a finite family of such small balls as above. We require that these balls are contained in W_{n+1}. Now we proceed with W̄_{n+1} but we require in addition that these balls have empty intersection with W̄_{n−1}. Then it is clear that the union of all these families provides a covering {U_{α}}_{α∈A} of M. The intersections U_{α_{0}} ∩ … ∩ U_{α_{q}} are diffeomorphic to open balls in ℝ^{n}. If we have a point in the boundary

x ∈ \overline{U_{α_{0}} ∩ … ∩ U_{α_{q}}} \ U_{α_{0}} ∩ … ∩ U_{α_{q}} = ∂(U_{α_{0}} ∩ … ∩ U_{α_{q}})

then the intersection B(x, ϵ)∩\overline{U_{α_{0}} ∩ … ∩ U_{α_{q}}} is contractible: each point in the intersection is joined to x by a unique geodesic in this intersection. We give the indexing set A a total order, in other word we identify it to {0, 1, …, n} or ℕ. We consider q + 1-tuples \underline{α} = (α_{0}, …, α_{q}) ∈ A^{q+1} where the indices are increasing. Let us denote this subset of indices by A_{<}^{q+1}. Then we have for such an \underline{α} = (α_{0}, …, α_{q}) the inclusion

i\underline{α} : U_{α_{0}} ∩ … ∩ U_{q} ↪ M

and as in 2.2.we form the sheaf

V∗ ; \underline{α} = i_{\underline{α}∗}i∗ ; \underline{α}(V).

This sheaf has non-zero stalks only in the points x ∈ \overline{U_{α_{0}} ∩ … ∩ U_{q}} and in such a point the stalk is equal to V_{x}. Here we need that for any point x ∈ \overline{U_{α_{0}} ∩ … ∩ U_{q}} and a small ball B(x, ϵ) that

V(B(x, ϵ)) = i_{α}^{∗}(V)(B(x, ϵ) ∩ \overline{U_{α_{0}} ∩ … ∩ U_{α_{q}})}.

Outside of \overline{U_{α_{0}} ∩ … ∩ U_{q}} the sheaf has been mowed. Now we consider the Čech a resolution of our sheaf V (see 5.2 and 6.5 ):

0 → V → ∏_{α∈A} V_{α}^{∗} → ∏_{(α,β)∈A×A_{<}} V_{(α,β)}^{∗} → … → ∏_{\underline{α}∈A_{<}^{q+1}} V_{α}^{∗} → .

This is now an acyclic resolution since all the sheaves V_{α}^{∗} are acyclic by the homotopy axiom.

Hence we see that the cohomology groups H^{ν}(M, V) can be computed from the complex of global sections (See 6.5.1)

\begin{matrix}∏_{α∈A} V_{α}^{∗}(M) → & ∏_{(αβ)∈A×A_{<}} V_{(α,β)}^{∗}(M) → … → & ∏_{α∈A_{<}^{q+1}} V∗ ; \underline{α}(M) → \\ ∥ & ∥ & ∥ \\ ∏_{α∈A} V(U_{α}) → & ∏_{(α,β)∈A×A_{<}} V(U_{α} ∩ U_{β}) → … → & ∏_{α∈A_{<}^{q+1}} V(U_{α_{0}} ∩ … ∩ U_{α_{q}}) → \end{matrix}

which is the Čech-complex attached to the resolution. We introduce the Čech coresolution. To do this we consider the sheaves

i_{α!}i∗ ; \underline{α}(V) = V ! ; \underline{α}.

These sheaves are zero outside of the open sets U_{α_{0}} ∩ … ∩ U_{α_{q}} and on these sets they coincide with V. On these open sets our sheaves V are isomorphic to a constant sheaf. We computed the cohomology with compact supports ( Prop. 7.2)

H^{ν}(M, V_{α}^{!}) = \{ \begin{matrix}0 & for ν ≠ d \\ V(U_{α}) & for ν = d \end{matrix} .

Now we can define a complex of sheaves (???)

→ ∏_{\underline{α}∈A_{<}^{q+1}} V! ; \underline{α} → … → ∏_{(α,β)∈A×A_{<}} V_{α,β}^{!} → ∏_{α∈A} V_{α}^{!} → V → 0

where the boundary operator is given by

(ds_{x})_{α_{0},...,α_{q}} = ∑_{β}(−1)^{ϵ(β,\underline{α})}s_{x,α_{0},...,β,...,α_{q}}

where β runs over those indices which do not occur in \underline{α} where ϵ(β, \underline{α}) gives us the position of β with respect to the ordering, where s_{x,α_{0},...,β,...,α_{q}} is an element in the stalk V_{α_{0},...,β,...,α_{q},x}^{!}. The last homomorphism on the right is simply summation ∑ s_{x,α}. Again it is clear that this is an exact complex of sheaves.(See ???).

8.2.1 Comment. In a sense we will see that this is an acyclic resolution for the right exact functor

V → H_{c}^{d}(M, V)

but one has to define a suitable category of sheaves for which this makes sense. For a local system on a manifold the H_{c}^{d}(M, V) plays the same role as the global sections for the ordinary cohomology. Except that in addition all the arrows point in opposite directions and this is duality.

Now I claim that we have H_{c}^{ν}(M, ∏_{\underline{α}∈A_{<}^{q+1}} V_{α}^{!}) = 0 if ν ≠ d and

H_{c}^{d}(M, ∏_{\underline{α}∈A_{<}^{q+1}} V_{α}^{!}) = ⨁_{\underline{α}∈A_{<}^{q+1}} H^{d}(M, V_{α}^{!}).

To see this we take the injective resolution constructed by Godement for the V! ; \underline{α} (See 2.1) Then the product of the sheaves in the resolution gives a resolution of the product:

0 → ∏_{\underline{α}∈A_{<}^{q+1}} V! ; \underline{α} → ∏_{\underline{α}∈A_{<}^{q+1}} I0 ; \underline{α} → ∏_{\underline{α}∈A_{<}^{q+1}} I1 ; \underline{α} →

and to compute the cohomology with compact support we look at the resulting complex of global sections with compact support. But since any compact set meets only finitely many of the open sets U_{α} we see that

H_{c}^{0}(M, ∏_{\underline{α}∈A_{<}^{q+1}} Iq ; \underline{α}) = ⨁_{\underline{α}∈A_{<}^{q+1}} H_{c}^{0}(M, Iq ; \underline{α}).

To see this we have to take into account that the stalks of the sheaves Iq ; \underline{α} are zero ouside U|\underline{α} which is clear from the construction. Then

H_{c}^{ν}(M, ∏_{\underline{α}∈A_{<}^{q+1}} V! ; \underline{α}) = ⨁_{\underline{α}∈A_{<}^{q+1}} H^{ν}(M, V! ; \underline{α})

We apply the functor H_{c}^{d} applied to the complex yields a complex

→ ⨁_{α∈A_{<}^{q+1}} H_{c}^{d}(M, V_{α}^{!}) → … → ⨁_{α∈A} H_{c}^{d}(M, V_{α}^{!}) → 0.

I want to make the additional assumption that the number of indices α for which U_{α} contains a given x is not only finite but even bounded independently of x. This has the consequence that our complex of sheaves is even finite. We numerate the complex so that the boundary operator has degree -1 and the complex starts at zero at the right hand side, i.e. it looks like

→ Y_{m} → … Y_{1} → Y_{0} → 0.

8.2.2 Theorem.The cohomology of this complex is the cohomology with compact supports

H^{d−i}(M, V) = H^{i}(→ ⨁_{α∈A_{<}^{q+1}} H^{d}(M, V_{α}^{!}) → … → ⨁_{α∈A} H^{d}(M, V_{α}^{!}) → 0.)

Proof. We used the same arguments which we used when we proved that we can compute cohomology groups by acyclic resolutions. We break the complex of sheaves into pieces

→ ⨁_{\underline{α}∈A_{<}^{q+1}} V ! ; \underline{α} → … → ⨁_{α,β∈A_{<}^{2}} V_{(α,β)}^{!} → G → 0

and

0 → G → ⨁_{α∈A} V_{α}^{!} → V → 0.

The second complex gives us a long exact sequence if we apply the cohomology with compact supports. Since the sheaf in the middle has only cohomology with compact supports in degree d we get

H_{c}^{i−1}(M, V) ≃ H_{c}^{i}(M, G) for i ≠ d − 1, d

and

0 → H_{c}^{d−1}(M, V) → H_{c}^{d}(M, G) → ⨁_{α∈A} H^{d}(M, V_{α}^{!}) → H_{c}^{d}(M, V) → H_{c}^{d+1}(M, G) → 0.

At first we want to conclude that H_{c}^{m}(M, V) = 0 for m > n. If not we would have H_{c}^{m}(M, G) ≠ 0 for some m > n. But G sits in a short exact sequence

0 → G_{1} → ⨁_{α,β∈A_{<}^{2}} V_{α,β}^{1} → G → 0

and G_{1} is the end of the complex

→ ⨁_{(α,β,γ)∈A_{<}^{3}} V_{α,β,γ}^{1} → G_{1} → 0.

We would get H_{c}^{m}(M, G_{1}) ≠ 0 for some m > n and applying the same procedure again and again we get a contradiction because the complex is finite to the left. Hence we get in degree n

0 → H_{c}^{d−1}(M, V) → H_{c}^{d}(M, G) → ⨁_{α∈A} H^{d}(M, V_{α}^{!}) → H_{c}^{d}(M, V) → 0.

Induction on the length of the complex gives us that the complex

0 → … → ⨁_{\underline{α}∈A_{<}^{q+1}} H^{d}(M, V_{α}^{!}) → … H^{d}(M, V_{(α,β)}^{!}) → 0

computes the cohomology H_{c}^{d−•}(M, G) and the theorem follows.

8.3 Poincaré-Duality. We assume that we have a local system V on M which has values in the category of Rmodules where R is a commutative ring with identity. We can also consider the dual local system V^{∨} = Hom_{R}(V, R). We compute the cohomology H^{•}(M, V) and the cohomology with compact support Hc^{•}(M, V^{∨}) by means of the two complexes which we obtain from a suitable covering. We write the complexes

0 → ∏_{α∈A} V(U_{α}) → ∏_{(α,β)∈A_{<}^{2}} V(U_{α} ∩ U_{β}) → … → ∏_{α∈A_{<}^{q+1}} →
∥  ∥  ∥
X^{0} ⟶^{d}  X^{1} → …  ⟶^{d} X^{q}  →

and

⨁_{\underline{α}∈A_{<}^{q+1}} V^{∨}(U\underline{α}) … → ⨁_{(α.β)∈A_{<}^{2}} V^{∨}(U_{α} ∩ U_{β}) → ⨁_{α∈A} V^{∨}(U_{α}) → 0
∥  ∥  ∥
Y^{q} ⟶^{δ}  ⟶^{δ} Y^{1}  ⟶^{δ} Y^{0}  →

where we made the identification V^{∨}(U\underline{α}) = H_{c}^{d}(M, V^{∨,!}). We have a pairing

<, >: Y^{q} × X^{q} → R

which is given by the formula

\begin{aligned}s & = (… s\underline{α} …) ∈ ∏ V(U\underline{α}) \\ t & = (… t\underline{α} …) ∈ ⨁ V^{∨}(U\underline{α}) \\ < s, t > & = ∑_{\underline{α}} s\underline{α} · t\underline{α} \end{aligned}

where s\underline{α}·t\underline{α} is the pairing induced by the pairing on the coefficient systems. The expression makes sense because t has only finitely many non-zero entries. We have

< ds, t >=< s, δt >

for s ∈ X^{q}, t ∈ Y^{q+1}. From this we get a pairing

H^{q}(M, V) × H_{c}^{d−q}(M, V^{∨}) → R.

We want to discuss the properties of this pairing.

8.3.1 Theorem: Poincaré-duality. We assume that V is a local system of finite dimensional k-vector spaces. We also assume that we can find a convex covering such that the function x → #{α|x ∈ U_{α}} is bounded. Furthermore we assume that

dim_{k}H^{i}(M, V) and dim_{k}H_{c}^{i}(M, V)

are finite for all i. Then the pairing

H^{q}(M, V) × H_{c}^{d−q}(M, V^{∨}) → k

is non-degenerate for all q. If R is a discrete valuation ring and if V is a local system of free R-modules of finite rank and if M admits a finite acyclic covering then the pairing

H^{q}(M, V)/Tors × H_{c}^{d−q}(M, V^{∨})/Tors → R

is non-degenerate.

Proof. We start from the computation of the cohomology by means of the two complexes

\begin{aligned}→ ∏_{\underline{α}∈A_{<}^{q}} H^{0}(M, V∗ ; \underline{α}) & → … & → ∏_{α∈A} H^{0}(M, V_{α}^{∗}) & → \\ → ⨁_{\underline{α}∈A_{<}^{q}} H^{d}(M, V_{α}^{∨!}) & → … & → ⨁_{α∈A} H^{d}(M, V_{α}^{∨!}) & → \end{aligned} .

We abbreviate these complexes by X^{•} and Y^{•} respectively. We have the pairing

<, >: Y^{q} × X^{q} → k.

These spaces may be of infinite dimension. We say that a linear form λ : X^{q} = ∏_{α∈A_{<}^{q}} H^{0}(M, V_{α}^{∗}) → R is continuous if it factors over a quotient ∏_{α∈E_{λ}} H^{0}(M, V_{α}^{∗}) where E_{λ} is a finite subset of A_{<}^{q}. Then it is clear that Y^{q} is the space of continuous linear forms on X^{q}. It is also clear that X^{q} is the space of all linear forms on Y^{q}. In X^{q} (resp. Y^{q}) we have the subspaces of cocycles of coboundaries

\begin{aligned}B^{q}(X^{•}) & ⊂ Z^{q}(X^{•}) ⊂ X^{q} \\ B^{q}(Y^{•}) & ⊂ Z^{q}(Y^{•}) ⊂ Y^{q} \end{aligned} .

Since B^{q}(X^{•}) = d(X^{q−1}) by definition we find that

Z^{q}(Y^{•}) = {y ∈ Y^{q}|δy = 0} = B^{q}(X^{•})^{⊥} = {y ∈ Y^{q}| < B^{q}(X^{•}), y >= 0}

and by the same argument we find that

Z^{q}(X^{•}) = B^{q}(Y^{•})^{⊥}.

The spaces X^{q}, Y^{q} are in perfect duality. If they were finite dimensional we could conclude, that for any subspace W of one of them we have (W^{⊥})^{⊥} = W. This is also true in our situation here for subspaces W ⊂ Y^{q} because X^{q} is the space of all linear forms on Y^{q} and for an element y ∉ W we can find an x ∈ X^{q} with < W, x >= 0 and < y, x >≠ 0. (Zorn’s Lemma).

But for subspaces W ⊂ X^{q} the same argument is only true for closed subspaces, which means

W = {x ∈ X^{q} | λ(x) = 0 for continuous linear forms λ which vanish on W }.

Now we use a little Lemma. The space of cocycles Z^{q}(X^{•}) is closed. If

dim_{k}Z^{q}(X^{•})/B^{q}(X^{•}) < ∞

then B^{q}(X^{•}) is also closed.

Proof. Exercise

Since we assumed that the various cohomology groups should be finite dimensional, we conclude that all the B^{q}(X^{•}) are closed and hence

\begin{aligned}B^{q}(X^{•})^{⊥⊥} & = B^{q}(X^{•}) \\ B^{q}(Y^{•})^{⊥⊥} & = B^{q}(Y^{•}) \end{aligned} .

But then it is obvious that the pairing

H^{q}(M, V) × H_{c}^{d−q}(M, V^{∨}) → k

is non-degenerate. Now we come to the second half of the theorem, we assume that V is a local system of free R-modules of finite rank where R is a discrete valuation ring. We also assume that we have an acyclic covering which is finite. In this case we have that the modules X^{q}, Y^{q} are free of finite rank and the conclusion

\begin{aligned}Z^{q}(X^{•}) & = B^{q}(Y^{•})^{⊥} \\ Z^{q}(Y^{•}) & = B^{q}(X^{•})^{⊥} \end{aligned}

is still valid. But we cannot conclude that B^{q}(Y^{•})^{⊥⊥} = B^{q}(Y^{•}) and B^{q}(X^{•})^{⊥⊥} = B^{q}(X). But Z^{q}(X^{•}), Z^{q}(Y^{•}) are direct summands in X^{q} resp. Y^{q} and therefore

X^{q}/Z^{q}(X^{•}) and Y^{q}/Z^{q}(Y^{•})

are torsion free. I claim that B^{q}(X^{•})^{⊥⊥}/B^{q}(X^{•}) and B^{q}(Y^{•})^{⊥⊥}/B^{q}(Y^{•}) must be torsion. Let K is field of fractions of R. The claim follows because we know that B^{q}(X^{•})^{⊥⊥}⊗_{R}K = B^{q}(X^{•}⊗_{R}K)^{⊥⊥} It follows that

\begin{matrix}= B^{q}(X^{•})^{⊥⊥} ⊂ Z^{q}(X^{•}) \\ = B^{q}(Y^{•})^{⊥⊥} ⊂ Z^{q}(Y^{•}) \end{matrix}

and if we look at

Z^{q}(X^{•})/B^{q}(X^{•}) = H^{q}(X^{•}) → Z^{q}(X^{•})/B^{q}(X^{•})^{⊥⊥}

then the kernel of this map is the torsion subgroup. The same holds for Y^{•}. Now it is an easy exercise to prove that the pairing

Z^{q}(X^{•})/B^{q}(X^{•})^{⊥⊥} × Z^{q}(Y^{•})/B^{q}(Y^{•})^{⊥⊥} → R

is non-degenerate.

8.4 The cohomology in top degree and the homology. We start from a local system V and we assume that we obtained it form an action of the fundamental group π = π_{1}(M, x_{0}) on an abelian group (or R-module) V. We have V_{x_{0}} = V. We want to compute the cohomology with compact support in top degree. We will see that this can be expressed completely in terms of the action of π on V. Let I_{π} be the augmentation ideal, we introduced the module of coinvariants V/I_{π} = V_{π}. We will see that H_{c}^{d}(M, V)→^{∼}V_{π}. This makes it clear that the cohomology with compact supports behaves like homology.

We start from our complex

⨁_{(α,β)∈A_{<}^{2}} H^{d}(M, V_{αβ}^{!}) → ⨁_{α∈A} H^{d}(M, V_{α}) → H_{c}^{d}(M, V) → 0.

Let α_{0} be an index such that x_{0} ∈ U_{α_{0}}. I claim that the map

H^{d}(M, V_{α_{0}}^{!}) ≃ V → H_{c}^{d}(M, V)

is surjective and induces an isomorphism

V/I_{π}V ⟶^{∼} H_{c}^{d}(M, V).

Let α be any other index. We can choose a sequence α_{0}, α_{1}, …, α_{r} = α of indices such that U_{α_{i}∩U_{α_{i+1}}} ≠ ∅ for all i. For any pair of consecutive indices α_{i}, α_{i+1} we restrict the boundary operator

δ : ⨁_{(α,β)∈A_{<}^{2}} H^{d}(M, V_{αβ}^{!}) → ⨁_{γ∈A} H^{d}(M, V_{γ}^{!})

to the direct summand

H^{d}(M, V_{α_{i}α_{i+1}}^{!})

(we assume α_{i} < α_{i+1} otherwise we interchange the indices). It is clear that the image of this restriction in the target module lies in the submodule

H^{d}(M, V_{α_{i}}^{!}) ⨁ H^{d}(M, V_{α_{i+1}}^{!}).

We have a natural isomorphism

Ψ_{α_{i},α_{i+1}} : H^{d}(M, V_{α_{i+1}}^{!}).

which is the composition of the isomorphisms

\begin{aligned}H^{d}(M, V_{α_{i}α_{i+1}}^{!}) & ⟶^{∼} H^{d}(M, V_{α_{i}}^{!}) \\ H^{d}(M, V_{α_{i}α_{i+1}}^{!}) & ⟶^{∼} H^{d}(M, V_{α_{i+1}}^{!}) \end{aligned}

which are induced by the inclusions U_{α_{i}} ∩ U_{α_{i+1}} ↪ U_{α_{i}} and U_{α_{i}} ∩ U_{α_{i+1}} ↪ U_{α_{i+1}}. It is clear from the definition that the image of δ restricted to H^{d}(M, V_{α_{i}α_{i+1}}^{!}) is the submodule

V_{i}, −Ψ_{α_{i}α_{i+1}}(V_{i})),

and hence we see that this submodule is in the kernel of

H^{d}(M, V_{α_{i}}^{!}) ⨁ H^{d}(M, V_{α_{i+1}}^{!}) → H_{c}^{d}(M, V).

Now our claim of indices gives us by composition an isomorphism

Ψ_{α_{0},α_{1}} …, α_{r} = α : H^{d}(M, V_{α_{0}}^{!}) → H^{d}(M, V_{α}^{!})

and it is clear that the elements

(V, −Ψ_{α_{0},α_{1}...,α_{v}=α}(V))

lie in the kernel of

H^{d}(M, V_{α_{0}}^{!}) ⨁ H^{d}(M, V_{α}^{!}) → H_{c}^{d}(M, V).

From this it follows that the summand H^{d}(M, V_{α_{0}}^{!}) ≃ V is mapped surjectively to H_{c}^{d}(M, V^{)}. We can take α = α_{0} then we can construct a path

γ : [0, 1] → M

with γ(0) = γ(1) = x_{0} which is obtained by joining x_{0} inside of U_{α_{0}} to a point in U_{α_{0}} ∩U_{α_{1}}, this point to a point in U_{α_{1}} ∩ U_{α_{2}} and so on and finally joining the point in U_{α_{r−1}} ∩ U_{α_{αr}} to x_{0}. The homotopy class of this path is uniquely determined by the chain of indices. Then it is clear from the construction of the local system from the action ρ : π → Aut(V) that Ψ_{α_{0},α_{1}} …, α_{r} = α = ρ([γ]). Hence we see that all elements of the form (Id −ρ([γ])V lie in the kernel of

H^{d}(M, V_{α_{0}}^{!}) → H_{c}^{d}(M, V)

and the surjective map factors

\begin{aligned}V & ⟶ & H_{c}^{d}(M, V) \\ ↘ & & ↗ \\ & V/I_{π}V \end{aligned} .

But now it follows that

V/I_{π}V ⟶^{∼} H_{c}^{d}(M, V^{)}

must be an isomorphism, because the group

⨁_{(α,β)∈A_{<}^{2}} H_{c}^{d}(M, V_{αβ}^{!})

is generated by its direct summands. We can also define the singular homology groups H_{i}(M, V). To do this we consider map

σ : Δ_{1} → M

where Δ_{q} = {(t_{0}, …, t_{q}) ∈ ℝ_{+}^{q+1}| ∑ t_{i} = 1} is the q-dimensional standard simplex. We can consider the pull back σ^{∗}(V) of our local system and since Δ_{q} is contractible, we have

σ^{∗}(V)(Δ_{q}) = σ^{∗}(V)_{p}

where p is any point in our simplex. We form linear combinations

∑ m_{σ} · σ

where m_{σ} ∈ σ^{∗}(V)(Δ_{q}). These linear combinations form an abelian group

C_{q}(M, V).

We define a boundary operator

∂_{q} : C_{q}(M, V) → C_{q−1}(M, V).

To do this we observe that we have face maps

\begin{matrix}τ_{i} : & Δ_{q−1} → Δ_{q} \\ τ_{i} : & (t_{0} … t_{q}) → (t_{0}, …, t_{i−1}, t_{i}, … t_{q}) \end{matrix}

and we put

∂_{1}(m_{σ}σ) = ∑(−1)^{i}m_{σ} · σ ∘ τ_{i}

where we use the fact that

(σ ∘ τ_{i})^{∗}(V)(Δ_{q−1}) = σ^{∗}(V)(Δ_{q}).

An easy computation yields ∂_{q−1} ∘ ∂_{q} = 0 hence we get the chain complex with coefficients in V

→ C_{q}(M, V) → C_{q−1}(M, V) → ⋯ → C_{0}(M, V) → 0

and the homology groups of this complex are the homology groups of M with coefficients in V:

H_{q}(M, V) = H_{q}(C_{•}(M, V)).

It is clear what H_{0}(M, V) is: we see that C_{0}(M, V) is the group given by linear combinations

∑_{x∈M} m_{x} · x

where m_{x} ∈ V_{x}. Of course we see that

m_{x}x − m_{y}y is a boundary

if we can find a path γ : [0, 1] → M with γ(0) = x, γ(1) = y and [γ]m_{x} = m_{y}. Hence it is clear that

H_{0}(M, V) ≃ H_{c}^{d}(M, V) = V/I_{π}V.

This suggests that we have

H_{i}(M, V) ≃ H_{c}^{d−i}(M, V)

and this is actually very easy to prove.

8.4.1. If we want to understand the cohomology Hc^{•}(M, V) it is sometimes very useful to embed M into a compact space. Let us consider an open embedding

i : M ↪ \overline{M},

where \overline{M} is compact. Then we can consider the sheaf i_{!}(V), and we know

Hc^{•}(M, V) = H^{•}(\overline{M}, i_{!}(V)).

We may also consider the direct image i_{∗}(V). Here we have to be careful because the functor i_{∗} is not exact in general. But if we assume that our local system is acyclic with respect to i_{∗}, then we know that

H^{•}(M, V) = H^{•}(\overline{M}, i_{∗}V).

Especially we may have the situation that \overline{M} is an oriented manifold with boundary and M is the interior of \overline{M}. Then it is clear that M ↪ \overline{M} is a homotopy equivalence and a local system V on M extends to a local system on \overline{M}, which we denote by \overline{V}. Under these circumstances we have

H^{•}(M, V) = H^{•}(\overline{M}, i_{∗}V) = H^{•}(M̄, \overline{V}).

If now V is a local system of free R-modules of finite rank and V^{∨} = Hom(V, R) the dual system, then we have the Poincaré pairing

H_{c}^{q}(M, V) × H^{m−q}(M, V^{∨}) ⟶ H_{c}^{m}(M, R) ≃ R

which we may also write as

H^{q}(M̄, i_{!}(V)) × H^{m−q}(M̄, i_{∗}(V^{∨})) ⟶ R.

It should not be too much of a surprise that this pairing can also be expressed in terms of the cup product. We start with the observation that both sheaves i_{!}(V) and i_{∗}(V^{∨}) have flat acyclic resolutions. In this situation we defined the product (see 6.6.1)

H^{q}(\overline{M}, i_{!}(V)) × H^{m−q}(\overline{M}, i_{∗}(V^{∨})) ⟶ H^{m}(\overline{M}, i_{!}(V) ˜⊗i_{∗}(V^{∨})),

and we have the pairing

i_{!}(V) ˜⊗i_{∗}(V^{∨}) ⟶ i_{!}(R).

Now the cup product provides a pairing

H^{q}(\overline{M}, i_{!}(V)) × H^{m−q}(\overline{M}, i_{∗}(V^{∨})) ⟶ H^{m}(\overline{M}, i_{!}(R)) = H_{c}^{m}(M, R) = R.

We want to convince ourselves that this pairing must be the Poincaré pairing. To see that this is the case we compute the cohomology groups from Čech resolutions. Our situation is a little bit different from the previous one since now our manifold has a boundary. But we may put a Riemannian metric as before and at first we cover a neighborhood of ∂M by small open “half-balls” with center on the boundary. Then the complement of the union of these balls is compact, and we cover it by small balls whose closure does not hit the boundary. Let us denote this covering by {U_{α}}_{α∈A}.

For any α_{0} ⋯ α_{q} we consider U_{α_{0}} ∩ ⋯ ∩ U_{α_{q}} = U\underline{α}, and we remove the boundary points

from it (if there are any) and call the result U^{∘} \underline{α}. Then

i\underline{α} :U^{∘}⟶ \overline{M}

is the inclusion, and we define

i_{\underline{α},!}i∗ ; \underline{α}(V) = V_{α}^{!}.

Now we compute our cohomology groups from the Čech resolution and the coresolution as before. We have

⟶ ⨁_{α,β} V_{{α,β}}^{v,!} ⟶ ⨁ V_{α}^{∨,!} ⟶ i_{!}(V^{∨}) ⟶ 0,

and

0 ⟶ V ⟶ ∏_{α} V_{α}^{∗} ⟶ ∏_{(α,β)} V_{(α,β)}^{∗} ⟶ · · · .

Of course it is clear that the Poincaré pairing

H^{0}(\overline{M}, i_{∗}(V)) × H^{m}(\overline{M}, i_{!}(V^{∨})) ⟶ R

is given by the cup product. Then we proceed by induction on the degree. We breack the two resolutions

0 → G → ⨁ V_{α}^{∨,!} → i_{!}(V^{∨}) → 0

0 → V → ∏_{α} V_{α}^{∗} → H → 0

and we get the folowing pieces of long exact sequences

0 ⟶ H^{n−1}(\overline{M}, i_{!}(V^{∨})) ⟶ H^{n}(\overline{M}, G) ⟶ ⋯

and

H^{0}(\overline{M}, H) ⟶ H^{1}(\overline{M}, i_{∗}(V)) ⟶ 0.

The pairing i_{!}(V^{∨}) × i_{∗}(V) → i_{!}(R) induces a pairing

(⨁ V_{α}^{v,!}) × ∏_{α} V_{α}^{∗} ⟶ i_{!}(R)

(see 8.3), and this induces a pairing

G × H ⟶ i_{!}(R).

If we look at the definition of the Poincaré pairing of two classes ξ ∈ H^{n−1}(\overline{M}, i_{!}(V^{∨})) and η = δ_{ψ} ∈ H^{1}(\overline{M}, i_{∗}(V)), then we have

⟨ξ, η⟩ = ⟨δξ, ψ⟩.

But the right hand side is also the cup product of the classes δξ ∈ H^{0}(\overline{M}, G) and ψ ∈ H^{n}(\overline{M}, H). But we have to see that also the cup product satisfies the rule

ξ ∪ η = ξ ∪ dψ = δξ ∪ ψ,

and we have seen that the cup product and the Poincaré pairing coincide on H_{c}^{n−1} × H^{1}. The general case follows by the same argument inductively.

8.5 The fundamental class of a submanifold : The homology groups can be defined for any space X and they provide a covariant functor from spaces to abelian groups: if we have a continuous map f : X → Y, then we get a homomorphism

f_{∗,i} : H_{i}(X, ℤ) ⟶ H_{i}(Y, ℤ)

for all degrees i. This suggests that we should also have this kind of functoriality for the cohomology with compact supports on an oriented manifold M. I want to discuss a special case where we see this functoriality. We consider a manifold M and a oriented submanifold N ⊂ M, let m, n be the dimensions of M and N respectively. I allow that N has several connected components. We assume that we have a relative orientation, this means that we have an orientation of the normal bundle T_{M/N}. Let V be a local system on M, let V′ be its restriction to N. If we consider the homology groups then we get get - directly from the definition - a homomorphism

H_{i}(N, V′) ⟶ H_{i}(M, V).

On our manifolds the homology groups are isomorphic to cohomology groups with compact support and hence we get a homomorphism

H_{c}^{j}(N, V′) ⟶ H_{c}^{m−n+j}(M, V),

and I want to construct this homomorphism directly. We choose an auxiliary Riemannian metric. This Riemannian metric splits the tangent bundle of M along N into T_{M} = T_{N} ⊕ T_{M/N}. Using the exponential map we can construct a tubular neighborhood i_{N} : T N ⊂ M ( See [ ]). We have the projection π : T N → N where the fibres π^{−1}(b) can be identified to small open balls in T_{M/N,b}. By \overline{T N} we denote the closure of T N. This gives us a fibration by open and closed balls as in 7.3.1. Let V_{N} be the restriction our local system V to the open subset T N, then we put V_{N}^{!} = i_{N,!}(V_{N}) we have an inclusion V_{N}^{!} ⟶V^{!} and therefore a homomorphism

Hc^{•}(T N, V_{N}) = H^{•}(M, V_{N}^{!}) → H^{•}(M, V^{!})

I claim that we have a canonical iomorphism

H^{j}(N, V′)→^{∼}H_{c}^{j+m−n}(T N, V_{N}) = H^{j+m−n}(\overline{T N}, V_{N}^{!})

We apply the same argument as in 7.3.1 and get

H^{j+m−n}(\overline{T N}, V_{N}^{!})→^{∼}H^{j}(N, R^{m−n}π_{∗}(V))

and then it is obvious that V′ = R^{m−n}π_{∗}(V). We assume in addition that M and N are oriented and the relative orientation is compatible with these orientations. Finally we assume that V = ℤ is the trivial local system.

Especially we have the map

ℤ = H^{0}(N, ℤ) ⟶ H_{c}^{m−n}(M, ℤ),

and the image of 1 under this map is a class [N] ∈ H_{c}^{m−n}(M, ℤ). It is called the fundamental class of N in M. Let ω be cohomology class in the complementary dimension n = dim N, then we can consider the cup product

[N] ∪ ω ∈ H^{m}(M, ℤ) = ℤ

or we can restrict ω to N and get a class i^{∗}(ω) ∈ H^{n}(N, ℤ) = ℤ. It follows directly from the definition and the compatibility of the Poincaré-pairing and the cup product that

i^{∗}(ω) = [N] ∪ ω.  8.5.1

8.6 Cup product and intersections Let us assume we have two oriented compact submanifolds N_{1}, N_{2} of codimension d_{1}, d_{2} in our oriented manifold M. We get two classes [N_{1}], [N_{2}] in the cohomology with compact support, they sit in degrees d_{1}, d_{2}. We want to understand the cup product of these two classes. We assume that our two submanifolds intersect transversally. This means that in any point p of N_{1} ∩ N_{2} the intersection of the two tangent spaces T_{N_{1},p} ∩ T_{N_{2},p} has dimension dim M −d_{1} −d_{2}. This implies that the intersection N_{1} ∩N_{2} is again a compact submanifold of codimension d_{1} + d_{2}. It may have several connected components. We write

N_{1} ∩ N_{2} = ⋃ C_{j}

where the C_{j} are the connected components. For any point p ∈ C_{j} we get an exact sequence of tangent spaces

0 → T_{C_{j},p} → T_{N_{1},p} ⊕ T_{N_{2},p} → T_{M,p} → 0

where the arrow from the direct sum to the tangent space of M is given by : first component minus second component. This gives us an isomorphism

Λ^{c}(T_{C_{j},p}) ⊗ Λ^{m}(T_{M,p})→^{∼}Λ^{n_{1}}(T_{N_{1},p}) ⊗ Λ^{n_{1}}(T_{N_{2},p})

and this puts an orientation O_{j} on C_{j} for all j. Now I claim

∑[C_{j}] = [N_{1}] ∪ [N_{2}]  8.6.1

We can look at the special case where d_{1} + d_{2} = n. In this case the cup product lands in H_{c}^{m}(M, ℤ) and hence it is a number. If we keep the assumption of transversality then the intersection is a finite number of points. Now the tangent space of a point has always a canonical orientation. If now c ∈ N_{1} ∩ N_{2} then we define

m(c) = \{ \begin{matrix}1 & if the orientatation O_{c} is canonical \\ −1 & if not \end{matrix}

Our formula becomes

[N_{1}] ∪ [N_{2}] = ∑_{c∈N_{1}∩N_{2}} m(c)

It is purely local problem to verify these formalae. According to (8.5.1) we have to restrict the class [N_{2}] to N_{1}. We recall the construction of [N_{2}], this class was the image of a class in H_{c}^{m−d_{2}}(T N_{2}, ℤ) = H^{m−d_{2}}(M, ℤ_{N_{2}}^{!}). If we restrict this class we get a class in H^{m−d_{2}}(N_{1}, ℤ ⊗ ℤ_{N_{2}}^{!}) the rest is clear but has to be written.

8.7 Compact oriented surfaces:

Let S be a compact oriented 2 dimensional manifold.We have seen that for any ring R

H^{0}(S, R) = R and H^{2}(S, R) = R

and the only unknown cohomology sits in degree one. For any prime p we have the exact sequence of sheaves

0 → \underline{ℤ} → \underline{ℤ} → 𝔽_{p} → 0

and in the resulting long exact sequence we find the piece

0 → H^{1}(S, ℤ) → H^{1}(S, ℤ) → H^{1}(S, 𝔽_{p}) → 0

we have zeroes at both ends because H^{0}(S, ℤ) → H^{0}(S, 𝔽_{p})( resp. H^{2}(S, ℤ) → H^{2}(S, ℤ)) is surjective (resp. injective). This implies that H^{1}(S, ℤ) is torsion free. Since we also know that these cohomology groups are finitely generated we conclude that H^{1}(S, ℤ) is free of some rank.

Now we have the Poincare duality pairing

H^{1}(S, ℤ) × H^{1}(S, ℤ) → ℤ

which is non degenerate and alternating. A well known result from elementary algebra tells us that we can find a basis e_{1}, …, e_{g}, f_{1}, …, f_{g} of H^{1}(S, ℤ) such that the duality pairing is given by

e_{i} ∪ f_{j} = δ_{ij}, e_{i} ∪ e_{j} = 0 , f_{i} ∪ f_{j} = 0

The number g is called the genus of the surface. For a curve of genus 3 we can draw the following picture We three pairs of 1-cycles which form a basis in homology. But we also may view these cycles as submanifolds isomorphic to S^{1} which are oriented by the arrows. These submanifolds have fundamental classes e_{1}, f_{1}, e_{2}, f_{2}, e_{3}, f_{3} in H^{1}(S, ℤ) and if we numerate them in the right way we have the above values of the intersection pairing.

8.8 The cohomology ring of ℙ^{n}(ℂ) We are now able to determine the structure of the cohomology ring H^{•}(ℙ^{n}(ℂ), ℤ) (See 6.5.1.1 Exercise). The fundamental class of the hyperplane ℙ^{n−1}(ℂ) ⊂ ℙ^{n}(ℂ) gives us a multiple of the generator ae_{1} ∈ H^{2}(ℙ^{n}(ℂ), ℤ) (Since we are dealing with complex manifolds, all manifolds have a canonical orientation.) Now we can put n such hyperlanes in general position such that they intersect transversally and L_{1} ∩ ⋯ ∩ L_{n} is a point. The fundamental class of a point is the generator in H^{2n}(ℙ^{n}(ℂ), ℤ). We conclude that a^{n}e_{1} ∪ e_{1} … e_{1} is this generator, it follows that a = 1, e_{1} is the fundamental class of the hyperplane and

H^{•}(ℙ^{n}(ℂ), ℤ) = ℤ[e_{1}]/(e_{1}^{n+1}).

IV.9. The de-Rham and the Dolbeault Isomorphism

9.1 Real manifolds. Let M be a C^{∞}-manifold and let V be a local system consisting of finite dimensional ℝ or ℂ-vector spaces. (See 3.3.2). Locally on small connected open subsets U ⊂ M we have a trivialization of V by constant sections e_{1}, ⋯, e_{n} and

V(U) = {∑_{i=1}^{n} a_{i}e_{i} | a_{i} ∈ 𝕂} .

We define

V_{∞}(U) = {∑_{i=1}^{n} f_{i}e_{i} | f_{i} ∈ C^{∞}(U)} ,

and this gives us the sheaf of C^{∞}-sections in V, we forget the discrete topology on the fibres. Let Ω_{M,∞}^{p} be the sheaf of C^{∞} − p-forms on M (3.3.2). We can define a differential

d : V_{∞}(U) ⟶ V_{∞}(U) ⊗ Ω_{M,∞}^{1}(U)

by

d : ∑_{i=1}^{n} f_{i}e_{i} ⟼ ∑_{i=1}^{n} e_{i} ⊗ df_{i}.

If we pass to another open set U′ and if we choose a trivialization e′_{1}, ⋯, e′_{n} over U′ then we get expressions

e_{i} = ∑ a_{ij}e′_{j}

over U ∩ U′ where the a_{ij} are locally constant. Thus we see that we can define a global differential

d : V_{∞} ⟶ V_{∞} ⊗ Ω_{M,∞}^{1}.

It is clear from the definition that for any open set U_{1} ⊂ M

V(U_{1}) = {s ∈ V_{∞}(U_{1}) | ds = 0}.

We can extend our differential to forms of higher degree

s : V_{∞} ⊗ Ω_{M,∞}^{p} ⟶ V_{∞} ⊗ Ω_{M,∞}^{p+1}

by

d(∑ s_{i} ⊗ ω_{i}) = ∑ s_{i} ⊗ dω_{i} + ∑ ds_{i} ∧ dω_{i}

where ds_{i} = ∑ g_{ij} ⊗ ω′_{j} and ds_{i} ∧ dω_{i} = ∑_{i} g_{ij} ⊗ ω′_{j} ∧ ω_{i}. It is well known that dd = 0, and we get the so called de-Rham complex of sheaves

0 → V → C^{∞}(V) → C^{∞}(V) ⊗ Ω_{M,∞}^{1} → … → C^{∞}(V) ⊗ Ω_{M,∞}^{d} → 0.

A form ω ∈ C^{∞}(V)(U) ⊗ Ω_{M,∞}^{d}(V) is called closed if dω = 0.

9.1.1 Remark. If we have a C^{∞}-vector bundle E over M (see 3.1.1, here the g_{ij} have to be C^{∞}-functions), then we may consider differentials

d : C^{∞}(E) ⟶ C^{∞}(E) ⊗ Ω_{M,∞}^{1}

which satisfy

\begin{aligned}d(s_{1} + s_{2}) & = & ds_{1} + ds_{2} \\ d(fs_{1}) & = & fds_{1} + s_{1} ⊗ df \end{aligned}

for local sections s_{1}, s_{2} and local C^{∞}-functions f. Such differentials are called connections on E. Such a connection is called a flat connection if

d(ds_{1}) = d(∑ s_{i} ⊗ ω_{j}) = ∑ s_{j} ⊗ dω_{j} + ds_{i} ∧ ω_{j} = 0.

We saw that starting from a local coefficient system V we have a canonical flat connection on V_{∞}. But in turn, if we have a flat connection, then we can attach a local system E_{0} to E by defining

E_{0}(U) = {s ∈ C^{∞}(E)(U) | ds = 0}.

It is of course clear that the flatness of the connection is necessary for the construction of the de-Rham complex.

It is not very hard to see that the de-Rham complex is exact. This follows from the wellknown Lemma of Poincaré which says that a closed p-form ω on a convex open set U in ℝ^{d} can be written as dψ = ω with ψ ∈ Ω_{∞}^{p−1}(V)(U) . Hence the de-Rham complex gives us a resolution of the sheaf V. We want to introduce the notation Ω_{∞}^{p}(V) for V_{∞} ⊗ Ω_{M,∞}^{p}. I claim that this resolution is also acyclic, we have

H^{i}(M, Ω_{∞}^{p}(V)) = 0 for i ≥ 1 and all p ≥ 0.

To see that this is the case we apply the Exercise 2.1.1. We have seen (Seee 8.2) that we have a partition of unity for the sheaf C^{∞}(M). If we have any C^{∞}(M)− vector bundle E on M then the sheaf C^{∞}(E) is a sheaf of C^{∞}(M) modules. Then our exercise 2.1.1 yields that the higher cohomology groups C^{∞}(E) vanish. We apply the functor global sections and then the resulting complex of global differential forms computes the cohomology (See II.3.1). We get the de-Rham-Isomorphism:

H^{i}(M, V) = H^{i}(Ω∞^{•}(V)(M)) = \frac{\{ω ∈ Ω_{∞}^{p}(V)(M)|dω = 0\}}{\{dψ|ψ ∈ Ω_{∞}^{p−1}(V)(M)\}}.

The argument in Exercise 2.1.1 also applies to the cohomolgy with compact supports and get by the same token

H_{c}^{i}(M, V) = H^{i}(H_{c}^{0}(M, Ω∞^{•}(V)).

If for instance we take the trivial system ℝ on M = ℝ^{d} then a closed form ω of degree p > 0 on ℝ^{d} can be written as dψ with ψ ∈ Ω_{∞}^{p−1}(ℝ^{d}). If p = 0 then a closed form is a constant function f = c thus we get H^{0}(ℝ^{n}, ℝ) = ℝ and H^{i}(ℝ^{n}, ℝ) = 0 for i > 0. (See 4.4.11) If we consider the cohomology with compact supports then a closed form in degree zero which has compact support must vanish. Hence we get H_{c}^{0}(ℝ^{d}, ℝ) = 0. But if we have a form ω with compact support on ℝ^{d} which is of degree d then we may not be able to write it as ω = dψ where also ψ has compact support. If we could do so we would have

∫_{ℝ^{d}} ω = ∫_{D} ω = ∫_{∂D} ψ = 0

where D is a big closed ball which contains the support of ω and ψ. Hence we get a surjective linear form

\begin{matrix}H_{c}^{d}(ℝ^{n}, ℝ) & → ℝ \\ [ω] & ↦ ∫_{ℝ^{d}} ω \end{matrix} .

Since it is easy to see that a form ω with compact support for which in addition ∫_{ℝ^{d}} ω = 0 can be written as ω = dψ with ψ ∈ Ω_{c}^{d−1}(ℝ^{d}) we get that the above map is an isomorphism. 9.1.2: We want to discuss the product structure of the cohomology in the context of the de-Rham isomorphism. If we have two manifolds M and N, then the resolution of the sheaf \underline{ℝ} by the two de-Rham complexes are flat ( Comp. the discussion in 6.7.). If we consider the product M ×N and the two projections p_{1}, p_{2}, then we have a homomorphism of complexes

p_{1}^{∗}(ΩM,∞^{•}) ⊗_{ℝ} p_{2}^{∗}(ΩN,∞^{•}) = ΩM,∞^{•} ˜⊗_{R}ΩN,∞^{•} ⟶ ΩM×N,∞^{•}

which is given by the exterior multiplication of the differential forms. Hence it is clear that the product

H^{p}(M, ℝ) × H^{q}(N, ℝ) ⟶ H^{p+q}(M × N, ℝ)
(α, β) ⟶ α ˜⊗_{R}β

is induced by the exterior multiplication of the differential forms which represent the classes α, β. Especially it becomes clear that the cup product on H^{•}(M, ℝ) is induced by the structure of exterior algebra on the differential forms. Finally it is clear that the Poincaé duality pairing on an open oriented manifold M

H_{c}^{i}(M, ℝ) × H^{m−i}(M, ℝ) ⟶ H_{c}^{m}(M, ℝ) ⟶ ℝ

is given by

[ω] × [η] ⟶ ∫_{m} ω ∧ η

if the classes [ω] ∈ H_{c}^{i}(M, ℝ) and [η] ∈ H^{m−i}(M, ℝ) are represented by ω and η. 9.1.3: The de-Rham isomorphism also provides a different way looking at the notion of the fundamental class and the formulae for the cup product (see 8.5, 8.6). Let us consider an open ball D^{n} ⊂ M in our manifold of dimension n. We assume it to be oriented. If we remove the origin from D, then we have a diffeomorphism

D^{n} \ {0} ≃ (0, 1) × S^{n−1}

which is given by

(x_{1}, ⋯, x_{n}) ⟶ (\sqrt{x_{1}^{2} + ⋯ + x_{n}^{2}}, (\frac{x_{1}}{\sqrt{x_{1}^{2} + ⋯ + x_{n}^{2}}} , ⋯, \frac{x_{n}}{\sqrt{x_{1}^{2} + ⋯ + x_{n}^{2}}} )) = (r, φ).

On S^{n−1} we have a unique differential form ω in degree n − 1 which is invariant under the orthogonal group SO(n), and which satisfies

∫_{S^{n−1}} ω = 1.

Now we choose a C^{∞}-function h(r) which is identically equal to 1 if r is close to 0 and identically equal to zero if r is close to one and this provides the differential form

h(r)ω = ψ

on D \ {0}. If we take its exterior derivative

dψ = \frac{∂h(r)}{∂r} · dr ∧ ω = ω,

then ω is a form on D^{n} \ {0} which vanishes identically in a small open ball around zero and near the boundary of D^{n}. Therefore we can extend it to a differential form on M and clearly we have

∫_{M} ω = 1.

Thus we constructed a form which represents the canonical generator in H^{n}(M, ℤ). Let us assume that M is an oriented manifold and N ⊂ M is an oriented submanifold. We construct a tubular neighborhood T N of N such that we have the projection

π : T N ⟶ N

and such that locally in N we have

π^{−1}(V) ˜ ⟶ V × D^{m−n}
↓
V.

On V × D^{m−n} we construct a m − n-form ω_{V} which is the pullback of a form ω on D^{m−n} which is constructed as above. Now we choose a covering N = ⋃_{i∈I} V_{i} which is locally finite and which trivializes π : T N → N, and we choose a portion of unity 1 = ∑ h_{i} with supp(h_{0}) ⊂ V_{i}. On each π^{−1}(V_{i}) we construct ω_{i} and we put

ω = ∑ h_{i}ω_{i}.

For any point in x ∈ T N we have

(dω)_{x} = ∑(dh_{i})_{x} ∧ ω_{i} − d(Σh_{i})_{x} ∧ ω_{i} = 0,

and we see that ω_{N,∞} ∈ ω^{m−n}(M) is a closed form. It is clear that this form represents the fundamental class

[N] ∈ H^{m−n}(M, ℝ).

If now N_{1}, N_{2} are two oriented submanifolds in M, and if we assume that one of them is compact, then we have the two classes

[N_{1}] = [ω_{N_{1}}], [N_{2}] = [ω_{N_{2}}]

where one of the forms has compact support. We just saw that

[N_{1}] ∪ [N_{2}] = ω_{N_{1}} ∧ ω_{N_{2}}.

If now these two submanifolds are if complementary dimension, and if they intersect transversally, then it is easy to see that

∫_{M,∞} ω_{N_{1}} ∧ ω_{N_{2}} = ∑_{c∈N_{1}∩N_{2}} ∫_{D(c)} ω_{N_{1}} ∧ ω_{N_{2}}

where D(c) is a small ball containing the local support of ω_{N_{1}} ∧ ω_{N_{2}}. Now it is easy to verify that these contributions from the points are equal to m(c) (see 8.6.1)

9.2. Cohomology of holomorphic bundles on complex manifolds

Let M be a complex manifold. From our discussion in IV.3.2 it is rather clear what a homolorphic vector bundle E on M is. This is of course a bundle π : E → M of ℂ− vector spaces for which the transition functions g_{ij} : V_{i} ∩ V_{j} → GL(n, ℂ) are holomorphic. This allows us to define the sheaf of holomorphic sections which also denote by E. It follows from our general principles in IV.3.3 that the holomorphic vector bundles are classified by H^{1}(M, GL_{n}(O_{M,∞})) where GL_{n}(O_{M,∞}) is the sheaf of holomorphic functions from M to GL_{n}(ℂ). To such a holomorphic vector bundle E we have the sheaf of germs of holomorphic sections which will be denoted by the same letter. This sheaf will be a locally free O_{M}-module and a locally free O_{M} gives in turn a holomorphic vector bundle. We also can define the sheaf E_{∞} of C^{∞}- sections in the bundle, we have the inclusions of sheaves E ⟶E_{∞}. The following discussion will show that this concept is completely analogous to the concept of local systems. (See 9.2.4) 9.2.1 The tangent bundle: We pick a point p ∈ M and an open neighborhood U_{p} of p such that

(U_{p}, O_{M|U_{p}}) ≃ (D_{p}, O_{D_{p}})

where D_{p} is an open ball in ℂ^{d} whose center is p = (0, …, 0). The tangent T_{M} is of course a holomorphic bundle which over U_{p} can be trivialized by the derivations \frac{∂}{∂z_{1}} , …, \frac{∂}{∂z_{d}}. We write the complex coordinates by their real and imaginary parts

(z_{1}, …, z_{d}) = (x_{1} + iy − 1, …, x_{d} + iy_{d}).

Then the tangent bundle of the C^{∞}-manifold M ( we forget the complex structure) has a basis locally at p which is given by

\frac{∂}{∂x_{1}} , \frac{∂}{∂y_{1}} , …, \frac{∂}{∂x_{d}} , \frac{∂}{∂y_{d}} .

These section are only sections in T_{M,∞}. This bundle of 2d-dimensional real vector spaces has the structure of a bundle of d dimensional complex vector spaces where locally the multiplication by i is given by

I : \frac{∂}{∂x_{1}} ↦ \frac{∂}{∂y_{1}} ; \frac{∂}{∂y_{1}} ↦ −\frac{∂}{∂x_{1}} ; ⋮ ; I : \frac{∂}{∂x_{d}} ↦ \frac{∂}{∂y_{d}}; \frac{∂}{∂y_{d}} ↦ −\frac{∂}{∂x_{d}} .

We have a priviledged orientation on the underlying C^{∞}-manifold which determined by requiring that dx_{1} ∧ dy_{1} ∧ … dx_{d} ∧ dy_{d} is positive. We can take the tensor product

T_{M} ⊗_{ℝ} ℂ = T_{M,ℂ}

and get a bundle of 2d-dimensional complex vector spaces. On this bundle of complex 2d-dimensional vector spaces we still have the linear transformation I above and T_{M,ℂ} decompose into two eigenspaces which are the eigenspaces with eigenvalues i and −i for I:

T_{M,ℂ} = T_{M,ℂ}^{10} ⊕ T_{M,ℂ}^{01}

where T_{M,ℂ}^{10} is the eigenspace for the eigenvalue i for I and T_{M,ℂ}^{01} is the eigenspace for the eigenvalue −i for I. It is easy to see that locally on M we T_{M,ℂ}^{10} has the basis (fibre by fibre)

\frac{∂}{∂z_{1}} = \frac{∂}{∂x_{1}} − i ⊗ \frac{∂}{∂y_{1}} ; ⋮ ; \frac{∂}{∂z_{d}} = \frac{∂}{∂x_{d}} − i ⊗ \frac{∂}{∂y_{d}} .

This provides a structure of a holomorphic vector bundle on T_{M,ℂ}^{10} the local trivialization is given by the above basis. The composition map T_{M} → T_{M} ⊗ ℂ → T_{M,ℂ}^{10} induces an isomorphism of complex vector bundles. Hence we can also view T_{M,∞} as a holomorphic bundle. The composition T_{M} → T_{M} ⊗ ℂ → T_{M,ℂ}^{01} is antilinear.

9.2.2 Here we apply some very simple principles of linear algebra which on the other hand can be confusing and their application requires some care.

If we have a ℂ-vector space V, we may define the complex conjugate space V̄. Its underlying abelian group is V but the scalar multiplication

ℂ × V̄ → V̄

is given by

(z, v) ⟼ z̄ · v,

where the dot on the right hand side denotes the scalar multiplication of v ∈ V by z̄ ∈ ℂ. Hence we see that the identity map Id : V → V̄ is antilinear.

If we consider our complex vector space V over ℂ as a real vector space together with a linear transformation I with I^{2} = −Id, then we can extend I to a linear transformation on V ⊗ ℂ and decompose into the eigenspace V_{ℂ}^{10} (resp. V^{01}) with eigenvalues i (resp. −i).

The vector spaces V, V̄ considered as real vector spaces are isomorphic by the identity map. In the following diagram the composition of the horizontal maps

V ⟶ V ⊗_{ℝ} ℂ ⟶^{pr_{10}} V_{ℂ}^{10}
↓ Id
V̄ ⟶ V ⊗_{ℝ} ℂ ⟶^{pr_{01}} V_{ℂ}^{01}

are isomorphisms of ℂ-vector spaces.

The thing that may cause confusion is the following fact: On V ⊗_{ℝ} ℂ we have the complex conjugation on the coefficients which may also be denoted by v → v̄. Then we get obviously

\overline{V_{ℂ}^{10}} = V_{ℂ}^{01},

but now putting a bar on V_{ℂ}^{10} has a different meaning, we get a different underlying set in contrast to our convention above.

On the other hand we can say that we constructed canonical isomorphisms

\begin{aligned}V & ⟶^{∼} & V_{ℂ}^{10} \\ V̄ & ⟶^{∼} & V_{ℂ}^{01} \end{aligned}

which allow us to identify V to V_{ℂ}^{10} and V̄ to V_{ℂ}^{01}. If we insert the map given by complex conjugation on the right end of our diagram above, then we ge a commutative diagram and the inconsistency in notation dissolves. 9.2.3 We can form the dual bundle Ω_{M}^{1}. Attached to this bundle we have the sheaf of C^{∞} sections in this bundle which is also denoted by Ω_{M,∞}^{1}. We have a decomposition

Ω_{M,ℂ}^{1} = Ω_{M}^{1} ⊗ ℂ ⟶^{∼} Ω_{M}^{10} ⊕ Ω_{M}^{01}.

The sheaf Ω_{M}^{1} is locally generated by dz_{1} … dz_{d}, we have

\begin{aligned}< dz_{ν}, \frac{∂}{∂x_{μ}} − i ⊗ \frac{∂}{∂y_{μ}} > & = 2δ_{νμ} \\ < dz_{ν}, \frac{∂}{∂x_{μ}} + i ⊗ \frac{∂}{∂y_{μ}} > & = 0 \end{aligned} .

We can define the fibres of Ω_{M,ℂ}^{1} at a point p simply as

Ω_{M,ℂ,p}^{1} = Hom_{ℝ}(T_{M,p}, ℂ)

and then Ω_{M,p}^{10} = {ω|ω(It_{p}) = iω(t_{p})} for all tangent vectors t_{p} ∈ T_{M,p}, in other words

Ω_{M,p}^{10} = Hom_{ℂ}(T_{M,p}, ℂ).

Analogously we have that Ω_{M,p}^{01} are the antilinear 1-forms. If we have a local section ω ∈ Ω_{M}^{10}(U) then the complex conjugate ω̄ is given by

ω̄(t_{p}) = \overline{ω(t_{p})},

where t_{p} ∈ T_{M,p} is a tangent vector at the point p ∈ U. Again we can form the complex of differential forms

Ω_{M,ℂ,∞}^{n−1} → Ω_{M,ℂ,∞}^{n} → Ω_{M,ℂ,∞}^{n+1}

but this sheaf (vector bundle) of n-forms decomposes

Ω_{M,ℂ,∞}^{n} = ⨁_{p+q=n} Ω_{M,∞}^{p,q}

where

Ω_{M,∞}^{p,q} = Λ^{p}Ω_{∞}^{10} ⊗ Λ^{q}Ω_{∞}^{01}.

( Once we have a superscript p, q we know that we are dealing with complex vector spaces, we omoit the subscript ℂ) Locally we get that a (p, q) form can be written as

ω = ∑_{α,β} f_{α,β}dz_{α_{1}} ∧ … ∧ dz_{α_{p}} ∧ dz̄_{β_{1}} ∧ … ∧ dz̄_{β_{q}}

where f_{α,β} is a complex valued C^{∞} function on U (the open set where we have these local coordinates). We get a decomposition of the operator

d : Ω_{M,∞}^{n} → Ω_{M,∞}^{n+1}

as d = \frac{1}{2}(d′ + d′′) where

d′ω = ∑_{γ} \frac{∂f_{α,β}}{∂z_{γ}} dz_{γ} ∧ … ∧ dz_{α_{p}} ∧ dz̄_{β_{1}} ∧ … ∧ dz̄_{β_{0}}

and

d′′ω = (−1)^{p} ∑_{δ} \frac{∂f_{α,β}}{∂z̄_{f}} dz_{α_{1}} ∧ … ∧ dz_{α_{p}} ∧ dz̄_{δ} ∧ … dz̄_{β_{q}}

where the factor \frac{1}{2} is explained by the fact that \frac{∂}{∂z_{v}} , \frac{∂}{∂,z̄_{μ}} and dz_{v}, dz̄_{μ} are not exactly dual bases of each other. (Nochmal nachrechnen). We have

\begin{matrix}d′ : & Ω_{M,∞}^{p,q} → Ω_{M,∞}^{p+1,q} \\ d′′ : & Ω_{M,∞}^{p,q} → Ω_{M,∞}^{p,q+1} \end{matrix} .

9.2.4 Now we come back to our holomorphic vector bundle E. We can embed the sheaf E of holomorphic sections into the sheaf of C^{∞}-sections, we write

0 → E → E_{∞} = Ω_{∞}^{0}(E).

As in the case of local systems we can characterize the subsheaf of holomorphic sections by a differential equation. We observe that we can define the operator

d′′ : C^{∞}(E) → Ω_{M,∞}^{01}(E_{∞}) = E_{∞} ⊗ Ω_{M,∞}^{01}.

To do this we write a local section on U in C^{∞}(E) in the form

s = ∑ f_{i}s_{i}

where the f_{i} are C^{∞}-functions and the s_{i} form a basis of holomorphic sections. Then we put

d′′s = ∑ \frac{∂f_{i}}{∂z̄_{v}} s_{i} ⊗ dz̄_{v}.

This is well-defined because just as in the case of local systems where the corresponding s_{i} were constant we put d′′s_{i} = 0 and this is consistent with the change of trivializations because holomorphic functions f are characterized by \frac{∂f}{∂z̄_{v}} = 0. As in the case of local systems we get a complex of sheaves

0 → E → Ω_{∞}^{0}(E) → Ω_{∞}^{0}(E) → Ω_{∞}^{01}(E) → Ω_{∞}^{0d}(E) → 0.

We need an analogon of the lemma of Poincaré, this is the lemma of Dolbeault: 9.2.5 This complex of sheaves is exact Hier muss ich noch was zu sagen. Combined with our previous observation, namely that the sheaves Ω_{∞}^{pq}(E) are acyclic, gives us an acyclic resolution of the sheaf E. From our general principles we get the Dolbeault Isomorphism:

H^{i}(M, E) = H^{i}(Ω_{∞}^{0•}(E)(M)).

9.3 Chern classes : We consider holomorphic line bundles L on our compact complex manifold M. The isomorphism classes of these line bundles form a group under the tensor product and this group is the first cohomology H^{1}(M, O_{M,∞}^{∗}) ( IV.3.3). We have a homomorphism from the sheaf of holomorphic functions O_{M} to O_{M}^{∗} which is given by the exponential function

\begin{matrix}O_{M}(U) & ⟶ & O_{M}^{∗}(U) \\ f & ↦ & e^{2πif}, \end{matrix}

and this is a surjective homomorphism of sheaves. The kernel is the sheaf of locally constant functions in ℤ, thus we get an exact sequence of sheaves

0 ⟶ ℤ ⟶ O_{M} ⟶ O_{M}^{∗} ⟶ 1

which leads to the exact sequence in cohomology

⟶ H^{1}(M, O_{M}) ⟶ H^{1}(M, O_{M}^{∗}) ⟶^{δ} H^{2}(M, ℤ) ⟶ .

If we have a line bundle L, and its isomorphism class corresponds to [L] ∈ H^{1}(M, O_{M}^{∗}), then the image under δ is called the (first)-Chern class of L, i.e.

δ([L]) = c_{1}(L) ∈ H^{2}(M, ℤ).

We want to give a geometric interpretation of this class. We assume that our holomorphic bundle has a non zero section

s ∈ H^{1}(M, L)

which has an additional property, namely it defines a “smooth divisor”. By this I mean the following: for any open set U ⊂ M over which our bundle becomes trivial and where we selected a nowhere vanishing section 1_{U} ∈ H^{0}(U, L). Our section s can be written as

s = f_{U} · 1_{U}

where f_{U} is a holomorphic function. Now we require that the differential df_{U} is non zero in all the points where f_{U} – and therefore s – is zero. The implicit function theorem implies that the set of zeroes of s is a submanifold Y ⊂ M which is of complex codimension one. This is our smooth divisor. Since we are in the complex case, we know that M and Y have natural orientations, and this also defines a relative orientation (see 8.5). In this situation we attached a fundamental class

[Y] ∈ H^{2}(M, ℤ)

to Y.

Proposition 9.3.1 Under these conditions we have the equality

[Y] = c_{1}(L).

Proof. We choose a Riemannian metric in M and a tubular neighborhood T Y of Y as in 8.5. We shrink this tubular neighborhood slightly by making the discs a little bit smaller. We cover the shrinked neighborhood

T_{ε}Y = ∪ Y_{α} × D_{α} = U_{α}

where Y_{α} is open in Y and D_{α} is a disc. The indices run over an indexing set A. Now we cover M \ T_{ε}Y by open sets U_{γ} such that \overline{U}_{γ} ∩ Y = ∅ and γ runs through C. We assume that we can trivialize the bundle on each of these open sets and on the U_{γ}, γ ∈ C we may trivialize by using the section s. Then we get our 1-cocycle g_{αβ} ∈ O_{M}^{∗}(U_{α} ∩ U_{β}), and g_{βγ} = 1 if β, γ ∈ C. We assume that the covering sets are convex ( See 8.2) that we can form h_{αβ} = \frac{1}{2πi} log g_{αβ} on U_{α} ∩ U_{β}, and we choose h_{αβ} = 0 if α, β ∈ C. Then we get the \underline{ℤ} valued two cocycle

c_{αβδ} = h_{αβ} − h_{βδ} + h_{αδ} on U_{α} ∩ U_{β} ∩ U_{δ}

and this 2-cocycle represents our class c_{1}(L). But we notice that c_{αβδ} = 0 if all three open sets lie in the complement of Y. This means that c_{αβδ} ≠ 0 implies that at least one of the indices lies in A. Consequently

U_{α} ∩ U_{β} ∩ U_{δ} ⊂ T_{ε}Y.

Now we consider the sheaf i_{!}(ℤ) on M where i_{!} : T Y → M (this is the larger tubular neighborhood), and we see that our cocycle lies in

C^{2}(𝔄, i_{!}ℤ).

We have the inclusion i_{!}(ℤ) → \underline{ℤ} and we conclude that c_{1}(L) is the image of the class

c_{1}^{Y} (L) ∈ H^{2}(M, i_{!}ℤ)

which is the class represented by this cocycle. But we know that

H^{2}(M, i_{!}ℤ) = H^{0}(Y, R^{2}π_{∗}(i_{!}ℤ))

(see 8.5.). Since we have a relative orientation we have R^{2}π_{∗}(i_{!}(ℤ)) = \underline{ℤ} on Y and by definition

[Y] = 1_{Y} = constant ℤ-valued function 1.

We want to show that c_{1}^{Y} (L) = [Y]. This can be checked locally on Y. If we want to do this, we consider a point p ∈ Y and a neighborhood U_{p} of p in M such that we have an isomorphism

(U_{p}, O_{U_{p}}) ≃ (B, O_{B})

where B ⊂ ℂ^{n} is an open polydisc, say

B = {(z_{1}, ⋯, z_{m}) | |z_{i}| < 1}.

We take U_{p} so small that we can trivialize our line bundle L | U_{p} by a section s_{1} ∈ H^{0}(U_{p}, L) which is non zero everywhere. Our global section s ∈ H^{0}(M, L) is of the form

s = fs_{1},

if we restrict it to U_{p}. We assumed that df ≠ 0 at p and therefore we may even assume that f(z_{1}, ⋯, z_{m}) = z_{1} and then Y ∩ U_{p} = {(0, z_{2}, ⋯, z_{m}) | z_{i} ∈ C}. Of course it may be necessary to pass to a smaller neighvorhood and after doing this we can still think that

U_{p} ≃ {(z_{1}, z_{2}, ⋯, z_{m}) | |z_{i}| < 1}

and s = z_{1}s_{1}. We cover the disc D = {z_{1} | |z_{1}| < 1} by open sets. The first one is V_{0} = {z | |z| < r} where r < 1 but close to one. The set

V_{1} = {z_{1} | Re(z_{1}) > ε, z_{1} ∈ D}

where ε > 0 on small. Then

V_{2} = e^{\frac{2πi}{3}} V_{1} , V_{3} = e^{\frac{4πi}{3}} V_{1}.

This yields a covering of U_{p} if we multiply by the disc

D′ = {(z_{2}, …, z_{m}) | |z_{i}| < 1}.

We compute the 1-cocycle by the recipe given in our discussion above. We get

g_{ij} = 1 if 1 ≤ i, j ≤ 3

and

g_{0i} = z for i = 1, 2, 3.

We have to write these g_{αβ} as e^{2πih_{αβ}} with some function h_{αβ} on V_{αβ}. Of course we take h_{ij} = 0 for 1 ≤ i, j ≤ B. To define the h_{0j} we take a path γ from 1 to a point z ∈ V_{0} ∩ V_{j} which goes counterclockwise around zero, and

h_{0j} = \frac{1}{2πi} ∫_{γ} \frac{dζ}{ζ} .

We have to compute the differences

c_{0ij} = h_{0i} − h_{0j} + h_{ij}

and get

c_{012} = c_{123} = 0

but

c_{013} = 1.

Now it is clear that this 2-cocycle with values in i_{!}ℤ yields the positive generator in

H^{2}(U_{p}, i_{!}ℤ) = H^{0}(Y ∩ U_{p}, R^{2}π_{∗}i_{!}ℤ)

and this proves the proposition. 9.3.2 This formula gives us a very nice interpretation for Chern classes. Let us assume that dim M = d and let us assume that L_{1}, …, L_{d} are line bundles. We assume that each of these line bundle has a section s_{i} ∈ H^{0}(M, L) and let us assume furthermore that the zero sets are smooth divisors and Y_{i} = [s_{i} = 0] intersect transversally (See 8.6). This has the consequence that the intersections Y_{1} ∩ Y_{2} … ∩ Y_{k} = Z_{k} are smooth complex submanifolds. Let us consider a point p in the intersection of all the Y_{i} and local trivializations t_{i} ∈ H^{0}(U_{p}, L_{i}) of the line bundles at p. Then locally at p we have s_{i} = f_{i}t_{i} where f_{i} is holomorphic at p and f_{i}(p) = 0. Then our transversality assumption implies that f_{1}, f_{2}, …, f_{d} is a sytem of local coordinates at p. The point is isolated in the intersection. We can invoke our formula 8.6.1. It tells us that the cup product of the Chern classes is a class in H^{2d}(M, ℤ) and hence a number and

c_{1}(L_{1}) ∪ c_{1}(L_{2}) … ∪ c_{1}(L_{d}) = |Y_{1} ∩ Y_{2} ∩ … ∩ Y_{d}|.

Of course we may always form the above cup product of d Chern classes of line bundles and we call the result the intersection number of the line bundles. We may even take one line bundle L and call c_{1}(L)^{d} the d-fold or total selfintersection number of the line bundle. We will discuss later (See V.3.1.2) on projective smooth varieties this cup product can always be interpreted as an intersection number of smooth divisors.

9.3.3 The line bundles O_{ℙ^{n}(ℂ)}(k):

I want to outline the construction of a familiy of line bundles O_{ℙ^{n}(ℂ)}(k) on ℙ^{n}(ℂ). I begin with construction of O_{ℙ^{n}(ℂ)}(1). We consider the coordinate functions z_{i} : ℂ^{n+1} → ℂ as linear forms on ℂ^{n+1}. Starting from these linear forms we construct the bundle O_{ℙ^{n}(ℂ)}(1). This bundle becomes trivial when we restrict it to one of the open subsets U_{i} and over this subset z_{i} is a trivializing section, i.e. it it is nowhere zero. For any pair i, j of indices we have the two trivializing sections z_{i}, z_{j} on U_{i} ∩ U_{j}. They are related by the equation z_{i} = (z_{i}/z_{j})z_{j} and z_{i}/z_{j} = g_{ij} is a holomorphic nowhere vanishing function on U_{i} ∩ U_{j}. These quotients define the transition functions ( See III. 3.2 and 9.2 ) defining the bundle O_{ℙ^{n}(ℂ)}(1). It is clear that z_{i} defines in fact a global section in H^{0}(ℙ^{n}(ℂ), O_{ℙ^{n}(ℂ)}(1)) and this section which defines a smooth divsisor [z_{i} = 0] and this is the hyperplane at infinity for those people who live in U_{i}. Hence we see that the Chern class of the bundle O_{ℙ^{n}(ℂ)}(1) is the fundamental class of an arbitrary hyperplane in ℙ^{n}(ℂ). In view of our considerations in IV.8.8 this means that the Chern class c_{1}(O_{ℙ^{n}(ℂ)}(1)) is a generator in H^{2}(ℙ^{n}(ℂ), ℤ). Now the other bundles are simply the tensor products

O_{ℙ^{n}(ℂ)}(1)(n) = O_{ℙ^{n}(ℂ)}(1)^{⊗n}

and their Chern classes are given by n-times the generator.

IV.10. Hodge Theory

10.1. Real manifolds. In this section I describe some very powerful analytical tools which provide insight into the structure of cohomology groups. They are based on the construction of certain linear elliptic differential operators (Laplace operators) which arise if we try to write down an inverse for the operators d, d′, d′′ in the de-Rham or Dolbeault complexes. We need some results on elliptic linear differential operators which we not prove here. We go back to the situation where we have an oriented manifold M, and a local system of finite dimensional ℝ or ℂ-vector spaces V on M. Let d be the dimension of M. We have the de-Rham complex

0 → V → V_{∞} → V_{∞} ⊗ Ω_{M}^{1} → … → V_{∞} ⊗ Ω_{M}^{d} → 0.

If we take global sections and if we drop the first term the resulting complex computes the cohomology groups H^{ν}(M, V). We have seen that we can construct a Riemannian <, > metric on M and using the same method we construct an euclidean (or hermitian) metric < , >_{h} on V. The metric on the tangent bundle provides a metric on the bundle of differential forms Ω_{M}^{p}. This allows us to define a scalar product on the sections f ∈ V_{∞} ⊗ Ω_{M}^{p}(M): It is clear that the metric on V_{∞} and the Riemannian metric together give us a metric on the tensor product of fibres V_{x} ⊗ Ω_{Mx}^{p} at any point x. Hence we get a function x →< v_{1x} ⊗ ω_{1x}, v_{1x} ⊗ ω_{2x} > on M for any two v_{1}⊗ω_{1}, v_{2}⊗ω_{2} ∈ V_{∞}⊗Ω_{M}^{p}(M). Since our manifold M is oriented and Riemannian we have a unique section ω_{top} ∈ Ω_{M}^{d}(M) which has length one and is positive with respect to the orientation. Hence we can integrate

< v ⊗ ω_{1}, v_{2} ⊗ ω_{2} >= ∫_{M,∞} < v_{1x}, v_{2x} >< ω_{1x}, ω_{2x} > ω_{top}.

Here we have to assume that the integral converges. This is centainly so if M is compact. Otherwise we have to introduce the notion of introduce integrable sections. There is another way to describe this scalar product. We have the ∗-operator on the bundle of forms

∗ : Ω_{M}^{p} → Ω_{M}^{n−p}

which is defined pointwise by the requirement

ω_{1} ∧ ∗ω_{2} =< ω_{1}, ω_{2} > ·ω_{top}.

Hence we get as our scalar product for two sections v_{1} ⊗ ω_{1}, v_{2} ⊗ ω_{2} ∈ V_{∞} ⊗ Ω_{M}^{p}(M) the formula

< v_{1} ⊗ ω_{1}, v_{2} ⊗ ω_{2} >= ∫_{M,∞} < v_{1}, v_{2} >_{h} ω_{1} ∧ ∗ω_{2}.

Now it becomes clear that we can define an adjoint operator

δ : V_{∞} ⊗ Ω_{M}^{p}(M) → V_{∞} ⊗ Ω_{M}^{p−1}(M)

which we simply put

δ = (−1)^{d(p−1)+1} ∗ d ∗ .

We have to verify that for v_{1} ⊗ ω_{1} ∈ V_{∞} ⊗ Ω_{M}^{p−1}(M), v_{2} ⊗ ω_{2} ∈ V_{∞} ⊗ Ω_{M}^{p}(M) we have

< dv_{1} ⊗ ω_{1}, v_{2} ⊗ ω_{2} >=< v_{1} ⊗ ω_{1}, δv_{2} ⊗ ω_{2} > .

To see this we perform a simple calculation

\begin{matrix}< dv_{1} ⊗ ω_{1}, v_{2} ⊗ ω_{2} >= ∫_{M,∞} dv_{1} ⊗ ω_{1} ∧ ∗v_{2} ⊗ ω_{2} = \\ ∫_{M,∞} d · (v_{1} ⊗ ω_{1} ∧ ∗v_{2} ⊗ ω_{2}) − (−1)^{p−1}v_{1} ⊗ ω_{1} ∧ d ∗ v_{2} ⊗ ω_{2}. \end{matrix}

From this moment on we assume that M is compact, then ∫_{M,∞} d(ω_{1} ∧ d ∗ v_{2} ⊗ ω_{2}) = 0 and hence

\begin{matrix}< dω_{1}, v_{2} ⊗ ω_{2} >= (−1)^{p+(d−p+1)(p−1)} ∫_{M,∞} dω_{1} ∧ ∗v_{2} ⊗ ω_{2} = \\ (−1)^{p+1+(d+p)p} ∫_{M,∞} < ω_{1} ∧ ∗ ∗ d ∗ v_{2} ⊗ ω_{2} =< ω_{1}, ∗d ∗ v_{2} ⊗ ω_{2} > \end{matrix} .

We define the Laplace operator

Δ = dδ + δd

which sends p-forms to p-forms. It is clear that this is a linear operator of second order and it is elliptic. [Erkläre was ein ellipt. Operator für beliebige Vektorbündel ist

D : E_{∞}(M) → E_{∞}(M).]

The theory of elliptic operators tells us that:

10.1.1 We have a decomposition into eigenspace

”V_{∞} ⊗ Ω_{M}^{p}(M) = ∑_{λ} V_{∞} ⊗ Ω_{M}^{p}(M)(λ)”

where

V_{∞} ⊗ Ω_{M}^{p}(M)(λ) = {ω ∈ V_{∞} ⊗ Ω_{M}^{p}(M) | Δω = λω}.

The eigenspaces have a finite dimension and the eigenvalues tend to infinity, i.e. on any finite interval [0, T] we have only finitely many eigenvalues λ. The sign ∑_{λ} means that any ω can be written as

ω = ∑_{λ} ω_{λ}

where the convergence is uniform on M and stays uniform if we apply a finite number of derivatives. Once we believe this we can compute the cohomology very easily: The operators d and δ respect the decompositions, they send eigenspaces into eigenspaces to the same eigenvalue. Now we can compute the cohomology. Let ω ∈ V_{∞} ⊗ Ω_{M}^{p}(M)(M) be a closed form. Write

ω = ω_{0} + ω′

where Δω_{0} = 0 is the harmonic component, the one in the eigenspace to λ = 0. Then

dω = dω_{0} + dω′ = 0

and hence dω_{0} = 0 and dω′ = 0. But

ω′ = ∑_{λ≠0} ω′_{λ}

where dω′_{λ} = 0. Hence

ω′_{λ} = \frac{1}{λ}Δω′_{λ} = \frac{1}{λ}(dδ + δd)ω′_{λ} = \frac{1}{λ}δω′_{λ}

and therefore

ω′ = d(∑_{λ≠0} \frac{1}{λ}δω′_{λ}).

This means that ω_{0} represents the same cohomology class as ω and we have the

10.1.2 Theorem:The harmonic forms satisfy dω = δω = 0. Sending a harmonic form to its cohomology class provides an isomorphism

ℍ^{p}(V_{∞} ⊗ Ω_{M}^{p})(M) = {ω ∈ V_{∞} ⊗ Ω_{M}^{p}(M)|Δω = 0} ⟶^{∼} H^{p}(M, V).

We almost proved this theorem. We observe that Δ is a positive operator. We have < Δω, ω >=< dω, dω > + < δω, δω >≥ 0. If Δω = 0 then we conclude

0 =< δω, δω >=< dω, dω >

this implies the first assertion. Since harmonic forms are closed they define cohomology classes. If ω is harmonic and ω = dΨ then < ω, ω >=< ω, dΨ >=< δω, Ψ >= 0 and hence ω = 0. The map from harmonic forms to cohomology is injective. The surjectivity has been shown above. 10.1.3 : We can give some indications how these results can be proved. Since we introduced the scalar product on V_{∞} ⊗Ω_{M}^{p}(M) we may take the completion with respect to this scalar product, and we get the Hilbert space

L^{2}(V) ⊗ Ω_{M,(2)}^{p}(M)

of quadratically integrable differential forms with values on V. If we have a closed form ω ∈ V_{∞} ⊗ Ω_{M}^{p}(M), then we can modify it by a form dψ and we can try to minimize the square of the norm

∥ω + dψ∥^{2} = ∫_{M}⟨ω + dψ, ω + dψ⟩.

We look at the lim inf of all the real numbers ∥ω + dψ∥_{2}^{2} where ψ varies. We can find a sequence

ω + dψ_{n} = ω_{n}

such that ∥ω_{n}∥_{2}^{2} converges to this infimum. Since the unit ball in our Hilbert space is weakly compact, we can find a weakly convergent subsequence, i.e. we may assume that ω_{n} converges weakly to a form

ω_{0} ∈ L^{2}(V) ⊗ Ω_{M,(2)}^{p}(M).

We would like to prove that ω_{0} is a C^{∞}-form, that it is harmonic and that this form represents the given class, i.e. ω_{0} = ω + dψ_{0}. Assume that we know that ω_{0} is a harmonic form. This means that it is C^{∞} and satisfies dω_{0} = δω_{0} = 0. Then we have

\begin{aligned}⟨dω_{0}, η⟩ & = & ⟨ω_{0}, δη⟩ & = & 0 \\ ⟨δω_{0}, ψ⟩ & = & ⟨ω_{0}, dψ⟩ & = & 0 \end{aligned}

for all ψ ∈ V_{∞} ⊗ Ω_{M}^{p−1}(M), η ∈ C_{∞}(V) ⊗ Ω_{M}^{p+1}(M). Of course the equalities

⟨ω_{0}, δη⟩ = 0 ⟨ω_{0}, dψ⟩ = 0

make sense for any square integrable form. I claim that they are true. The first one follows because

⟨ω_{0}, δη⟩ = lim_{n→∞}⟨ω + dψ_{n}, δη⟩ = ⟨dω + ddψ, η⟩ = 0

(this is the definition of weak convergence) and the second one follows from the minimality of the norm ∥ω_{0}∥_{2}^{2}. The really deep input from analysis is that the validity of the two equations

⟨ω_{0}, δη⟩ = ⟨ω_{0}, dψ⟩

for all η, ψ implies that ω_{0} must be indeed C^{∞} and then it follows that ω_{0} must be harmonic. The rest is easy again. We need to know that ω_{0} still represents the given cohomology class. This follows from Poincaré-duality. We consider the dual local system V^{∨}. We have the non degenerate pairing

H^{p}(M, V) × H^{n−p}(M, V^{∨}) ⟶ ℝ

which in terms of differential forms is given by integration over M. Hence we see that for any cohomology class [ω′] ∈ H^{n−p}(M, V^{∨}) which is represented by a C^{∞} − (n − p)-form ω′ that

[ω] ∪ [ω′] = ∫_{M,∞} tr(ω ∧ ω′) = ∫ tr((ω + dψ_{n}) ∧ ω′

and weak convergence gives that this integral is equal to

∫ tr(ω_{0} ∧ ω′) = [ω_{0}] ∪ [ω′].

This theorem has consequences.

(i) We conclude that for a compact oriented C^{∞}-manifold the cohomology H^{p}(M, V_{∞}) has finite dimension for any local system of finite dimensional ℝ or ℂ vector spaces. (ii) If we have such a local system V_{∞} and if V^{∨} is the dual system then ????. Both these consequences are known to us, they even hold for more general local systems. But in the next section where we discuss the analogous situation of holomorphic bundles on complex manifolds and in this case these consequences are only obtained by the analytic methods.

10.2. Complex manifolds:

Now we consider a compact complex manifold M. We introduce a hermitian metric h on the tangent bundle T_{M,∞}.

10.2.1 Some linear algebra. I have to recall some simple facts from linear algebra which concern these metrics. Therefore I start from a complex vector space V of finite dimension d. In the following I view V as a real vector space of dimension 2d which is endowed with a linear transformation I : V → V which satisfies I^{2} = −Id. The structure as a ℂ-vector space is regained if we define scalar multiplication of v ∈ V by i by v → I(v). If we have a hermitian form h on V then we can write

h(v_{1}, v_{2}) = Re h(v_{1}, v_{2}) + i · Im h(v_{1}, v_{2})

and it is clear that

Re h : V × V → ℝ is symmetric Im h : V × V → ℝ is alternating .

Since h(Iv_{1}, Iv_{2}) = h(iv_{1}, iv_{2}) = h(v_{1}, v_{2}) we see that both components satisfy

Re h : (Iv_{1}, Iv_{2}) = Re h(v_{1}, v_{2}) Im h : (Iv_{1}, Iv_{2}) = Im h(v_{1}, v_{2}) ,

in other words I is an isometry for the real part and the imaginary part. But we may also recover h from either part. We simply write

h(v_{1}, Iv_{2}) = Re h(v_{1}, Iv_{2}) + i Im h(v_{1}, Iv_{2})

and since h(v_{1}, Iv_{2}) = −ih(v_{1}, Iv_{2}) this yields

h(v_{1}, v_{2}) = − Im h(v_{1}, Iv_{2}) + i Re h(v_{1}, Iv_{2})

and from this we get

Re h(v_{1}, v_{2}) = − Im h(v_{1}, Iv_{2}) Im h(v_{1}, v_{2}) = Re h(v_{1}, Iv_{2}) .

Hence we see that a sesquilinear form h on V (this is a hermitian form without the requirement that it should be positive definite) is the same thing as a symmetric form or an alternating form

Re h = V × V → ℝ, Im h = V × V → ℝ

for which I is an isometry. The form h is hermitian (positive definite) if and only if Re h is euclidean. We complexify V and extend Re h to a bilinear form

Re h_{ℂ} : V_{ℂ} × V_{ℂ} → ℂ.

We have the decomposition

V_{ℂ} = V_{ℂ}^{10} ⊕ V_{ℂ}^{01}

and it is clear that V_{ℂ}^{10}, V_{ℂ}^{01} are isotropic with respect to Re h_{ℂ}, i.e.

Re h_{ℂ}(V^{10}, V^{10}) = Re h_{ℂ}(V^{01}, V^{01}) = 0.

This follows from the definition of the V^{10}, V^{01} as eigenspaces for I with eigenvalue i, −i. But the pairing

Re h_{ℂ} : V^{10} × V^{01} → ℂ

will be not trivial in general. If for instance the form h is positive definite then this pairing is a perfect duality. We have an isomorphism of complex vector spaces

j : V → V^{10}

which is obtained by the embedding of V into V_{ℂ} followed by the projection. Under this isomorphism we send

j : v → \frac{1}{2}(v − Iv ⊗ i)

and we can recover the hermitian form h from Re h_{ℂ} by the formula

h(v_{1}, v_{2}) = \frac{1}{2} Re h_{ℂ}(j(v_{1}), \overline{j(v_{2})})

where is of course the antilinear isomorphism from V^{10} to V^{01} introduced by complex conjugation on the factor ⊗ℂ. We introduce a so-called Hodge structure on the pair vector space V together with its complex structure I. This is a homomorphism

h_{D} : ℂ^{∗} → GL(V)

and it is defined as

h_{D}(z) = h_{D}(a + bi) = a · Id + b · I.

It is clear that this map is a homomorphism. With respect to the euclidean metric on V it has the property that

< h_{D}(z)v_{1}, h_{D}(z)v_{2} >= zz̄· < v_{1}, v_{2} >,

it is not an isometry but a similitude. If we complexify the space to V_{ℂ} then it is obvious that

\begin{aligned}V^{10} & = \{v|h_{D}(z)v = zv\} \\ V^{01} & = \{v, h_{D}(z)v = z̄v\} \end{aligned} .

The action of ℂ^{∗} commutes with complex conjugation we have \overline{h_{D}(z)v} = h_{D}(z)v̄ on V_{ℂ}. We can extend this action of ℂ∗ to the exterior powers Λ^{n}V and Λ^{n}V_{ℂ} simply by

h_{D}(z)(v_{1} ∧ … ∧ v_{n}) = h_{D}(z)v_{1} ∧ … ∧ h_{D}(z)v_{n}

and it is clear that we can characterize the subspace Λ^{p}V^{10} ⊗ Λ^{q}V^{01} ⊂ Λ^{p+q}V_{ℂ} as

Λ^{p}V^{10} ⊗ Λ^{q}V^{01}{ω ∈ Λ^{p+q}V_{ℂ}|h_{D}(z)φ = z^{p}z̄^{q}φ}.

Of course if we extend Re h_{ℂ} to a bilinear form on Λ^{n}V_{ℂ} by

Re h_{ℂ}(φ, ψ) = Re h_{ℂ}(v_{1} ∧ … ∧ v_{n}, w_{1}, ∧ … ∧ w_{n}) = det(Re h_{ℂ}(v_{1}, w_{j}))

then we have

Re h_{ℂ}(h_{D}(z)φ, h_{D}(z)ψ) = (zz̄)^{n} Re h_{ℂ}(φ, ψ).

This implies that the ∗-operator sends ∗ : Λ^{p}V^{10}⊗Λ^{q}V^{01} → Λ^{d−p}V^{10}⊗Λ^{d−q}V^{01}. This must be so because the product v_{pq}∧∗w_{pq} is in top degree and h_{D}(z)v_{pq}∧∗w_{pq} = (zz̄)^{d}h_{D}(z)v_{pq}∧ ∗w_{pq} If can extend our hermitian form h to a hermitian form on Λ^{p}V^{10} ⊗ Λ^{q}V^{01} by

h(φ, ψ) = Re h_{ℂ}(φ, ψ̄).

10.2.2. Now we come back to our compact complex manifold M of dimension d, we assume that we have introduced a hermitian metric <, >_{h} on T_{M,∞}. This introduces a hermitian metric on Ω_{M,∞}^{1}. We have the decomposition

Ω_{M,ℂ}^{1} = Ω_{M,∞}^{10} ⨁ Ω_{M,∞}^{01}

and

Λ^{n}Ω_{M,ℂ} = ⨁ Λ^{p}Ω_{M,∞}^{10} ⊗ Λ^{q}Ω_{M,∞}^{01} = ⊕_{p+q=n}Ω_{M,∞}^{pq}.

We have the euclidean metric Re h on Ω_{M,∞}^{1} and it introduces a star operator

∗ : Λ^{n}Ω_{M,∞}^{1} → Λ^{2d−n}Ω_{M,∞}^{1}.

We have seen in the above section on linear algebra that we should extend this antilinearily to

∗ : Λ^{n}Ω_{M,ℂ}^{1} → Λ^{2d−n}Ω_{M,ℂ}^{1}

and that this operator sends

∗ : Ω_{M,∞}^{pq} → Ω_{M,∞}^{d−p,d−q}.

We define the scalar product on the sections Ω_{M,∞}^{pq}(M) by

< ω_{1}, ω_{2} >= ∫_{M,∞} ω_{1} ∧ ∗ω_{2}.

Now we are able to define the adjoint operators to d′ and d′′, we put

\begin{aligned}δ′ & = − ∗ d′∗ \\ δ′′ & = − ∗ d′′∗ \end{aligned} .

The sign factor simplifies because our manifold has an even dimension when we consider it as a real manifold. Of course we have to verify the adjointness formular

\begin{aligned}< d′ω_{1}, ω_{2} > & =< ω_{1}, δ′ω_{2} > \\ < d′′ω_{1}, ω_{2} > & =< ω_{1}, δ′′ω_{2} > \end{aligned} .

To see this we observe that it is enough to check the case where ω ∈ Ω^{p−1,q}(M), ω_{2} ∈ Ω^{r,s}(M). Let us consider the first case. We see that both sides are zero unless p = d − r, q = d − s. So we assume that this is the case. Now we perform the same calculation as in the real case where we at certain places we have to replace d′ by d and then again d by d′. We observe that

< d′ω_{1}, ω_{2} >=< dω_{1}, ω_{2} >=< ω_{1}, δω_{2} >=< ω_{1}, δ′ω_{2} > .

This allows us to define the Laplace operator

Δ′ = d′δ′ + δ′d′ and Δ′′ = d′′δ′′ + δ′′d′′.

We want to compare these operators to the real Laplacian. Here we find

\begin{aligned}Δ & = (d′ + d′′)(δ′ + δ′′) + (δ′ + δ′′)(d′ + d′′) = \\ & = Δ′ + Δ′′ + δ′′d′ + d′δ′′ + δ′d′′ + d′′δ′ \end{aligned} .

This is not so good unless we know that the mixed contributions disappear. This is indeed the case if our metric h satisfies a certain condition, which I want to explain. Our metric h on the tangent bundle has its imaginary part

Im h : T_{M,∞} × T_{M,∞} → ℝ.

Hence the imaginary part defines a 2-form ω_{h} on the manifold. If we complexify the tangent bundle and if we observe that I is an isometry for ω_{h} then we see that ω_{h} is a form of type (1,1) because it must be zero on T^{10} ⊗ T^{10} and T^{01} ⊗ T^{01}. This is the so-called Kähler form of the metric. Kähler discovered the following

10.2.3 Theorem: If the form ω_{h} is closed, i.e. dω_{h} = 0 then the two mixed terms are zero and we have

Δ = Δ′ + Δ′′.

I will not prove this theorem here. But in our later discussion of the special case of Riemann surfaces - in this case we have automatically dω_{h} = 0 - I will carry out the necessary calculations. This theorem has the following important consequence:

The Laplace operator Δ respects the decomposition

Ω_{M,ℂ}^{n}(M) = ⊕Ω_{M,∞}^{p,q}(M)

and the harmonic forms

ω ∈ Ω_{M,ℂ}^{n}(M)

are sums of harmonic forms

ω = ∑_{p+q=n} ω_{p,q}

and

Δω = 0 ⟺ Δ′ω = Δ′′ω = 0.

A form ω is harmonic if and only if it satisfies all the equations

d′ω = d′′ω = δ′ω = δ′′ω

This follows by the same positivity argument which we used in the real case. This provides us the famous Hodge decomposition of the cohomology of a compact complex manifold which admits a Kähler metric:

H^{n}(M, ℂ) = ⊕_{p+q=n}H^{p,q}(M, ℂ).

Now we apply the same methods to compute the cohomology of a holomorphic bundle E on M. We choose a hermitian metric on the tangent bundle and on the bundle E itself. We consider the Dolbeault complex

0 → C^{∞}(E)(M) ⟶^{d′′} C^{∞}(E) ⊗ Ω^{01}(M) ⟶^{d′′} C^{∞}(E) ⊗ Ω^{02}(M) → .

Again we introduce a scalar product on the sections by

< s_{1} ⊗ ω_{1}, s_{2} ⊗ ω_{2} >= ∫_{M,∞} < s_{1}, s_{2} > ω_{1} ∧ ∗ω_{2}

where < s_{1}, s_{2} > is the hermitian form on the bundle E. We can construct the adjoint operator

δ′′ : C^{∞}(E) ⊗ Ω_{M,∞}^{0q}(M) → C^{∞}(E) ⊗ Ω_{M,∞}^{0,q−1}(M)

it is given by

δ′′ = (−1)^{?} ∗ d′′∗

and we have the Laplacian

Δ′′ = δ′′d′′ + d′′δ′′.

Again we conclude The cohomology groups

H^{p}(M, E) = ℍ^{p}(C^{∞}(E) ⊗ Ω_{M,∞}^{0p}(M)) = {ω ∈ C^{∞}(E) ⊗ Ω_{M,∞}^{0p}(M)|Δ′′ω = 0}.

Especially we can conclude that these groups are finite dimensional. The finite dimensionality is fundamental and there is no easy way to get it. (In contrast to the case of C^{∞}-manifolds and cohomology with coefficients in local systems). Let us consider the dual bundle E^{∨} = Hom(E, O_{M,∞}). We introduce the dual metric on E^{∨} (the dual basis to an orthonormal basis of a fibre E_{x} should be an orthonormal basis of E_{x}^{v}).

10.3. Hodge theory on tori: We have a special case where the two main theorems of Hodge theory are easy to prove. We consider the lattice Γ ⊂ ℝ^{n} (see 6.5.7), and we consider the C^{∞}-manifold

M = ℝ^{n}/Γ.

For any point p ∈ M we have a canonical identification T_{p}M = ℝ^{n}. If we take the standard euclidian metric < , > on ℝ^{n} then we get a Riemannian metric on M. If x_{1} ⋯ x_{n} are the coordinates on ℝ^{n}, then the differential forms can be written as

ω = ∑ f_{i_{1}···i_{p}} dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{p}}.

I want to consider complex valued differential forms, the f_{i_{1}···i_{p}} are complex valued C^{∞}- functions. A basically simple and straightforward computation yields a formula for the Laplace operator:

Δω = ∑ (− ∑ \frac{∂^{2}f_{i_{1}···i_{p}}}{∂x_{i}^{2}} ) dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{p}}.

Now we consider the dual lattice

Γ^{∨} = {φ ∈ ℝ^{n} | ⟨φ, Γ⟩ ⊂ ℤ}

and for φ ∈ Γ^{∨} the function

e_{φ}(x) = e^{2πi⟨φ,x⟩}

on M is an eigenfunction for the Laplacian

Δe_{φ}(x) = 4π^{2}⟨φ, φ⟩e_{φ}(x).

Now any C^{∞}-function on M has a Fourier expansion

f = ∑_{φ∈Γ^{∨}} a_{φ}e_{φ}(x)

where the absolute values |a_{φ}| tend to zero very rapidly. Consequently any differential form can be written as convergent infinite sum

ω = ∑_{φ} ω_{φ}

where Δω_{φ} = +4π^{2}⟨φ, φ⟩ω_{φ}. This is the “decomposition” in 10.1.1.. It has the required property: There are only finitely many φ which satisfy 4π^{2}⟨φ, φ⟩ ≤ T. We apply our arguments in the “proof” of theorem 10.1.2. . The harmonic forms are the constant forms

ω_{0} = ∑ a_{i_{1}···i_{p}}dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{p}}

where a_{i_{1}···i_{p}} ∈ ℂ. We conclude that the cohomology ring H^{•}(M, ℂ) is the exterior algebra of the complexified dual tangent bundle Hom(T_{p}M, ℂ). This agrees with 6.5.7 but the result overthere is slightly sharper because it gives the structure over ℤ.

If we consider a complex torus

M = ℂ^{n}/Γ

where Γ is a lattice of rank 2n, then M is a complex manifold and the tangent space is the complex vector space ℂ^{n} in any point of M. On this tangent space we introduce the standard hermitian metric

∑_{v=1}^{n} z_{v}\overline{z}_{v} = h(z, z).

Again we perform a simple computation and find

Δ′ = Δ′′ : ω = ∑_{\underline{α},\underline{β}} f_{\underline{α}\underline{β}}dz\underline{α} ∧ d\overline{z}\underline{β} → ∑_{\underline{α},\underline{β}} (− ∑_{ν} \frac{∂^{2}f_{\underline{α}\underline{β}}}{∂z_{ν}∂\overline{z}_{v}} ) dz\underline{α} ∧ d\overline{z}\underline{β}.

Again we can introduce the dual lattice

Γ^{∨} = {φ ∈ ℂ^{n} | Re h(φ, γ) ∈ ℤ for all γ ∈ Γ},

and we can expand C^{∞}-functions

f(z) = ∑ a_{φ}e^{2πi Re h⟨φ,z⟩}.

Now we argue as before. We have the Dolbeault complex

0 ⟶ O_{M,∞} ⟶ C^{∞}(M) ⟶^{d′′} Ω_{∞}^{01}(M) ⟶ ⋯,

and we have the adjoint δ′′. The operator Δ′′ = δ′′d′′ + d′′δ′′ has the form above, we can “decompose” into eigenspaces. If we take global sections, we find that

H^{•}(M, O_{M,∞}) = ℍ^{•}(M, Ω_{∞}^{0,•}(M)) = {ω ∈ H^{0}(M, Ω_{∞}^{0,•}(M)) | Δ′′ω = 0}

and again the harmonic forms are the constants. We conclude that

H^{•}(M, O_{M,∞}) = Hom_{ℂ}^{alt}(\overline{T_{p}(M)}, ℂ),

and this will be used in the next chapter.

Chapter V

Compact Riemann Surfaces and Abelian Varieties

V.1 Compact Riemann Surfaces

1.0 Introduction: A compact Riemann surface is a compact complex manifold of dimension 1. Let S be such a surface. It has a canonical orientation (See IV.9.2) In IV 4.4.13 and IV.8.7 we have seen that the cohomology groups of such a surface are given by

\begin{aligned}H^{0}(S, ℤ) & = ℤ \\ H^{1}(S, ℤ) & = ℤ^{2g} \\ H^{2}(S, ℤ) & = ℤ \end{aligned}

and the Poincaré-Duality gives us in addition an alternating perfect pairing

<, >_{cup}: H^{1}(S, ℤ) × H^{1}(S, ℤ) → ℤ.

The number g is called genus of the surface. The genus g is also a measure fo the complexity of the Riemann surface. We will show that a Riemann surface S of genus g = 0 is isomorphic to the so called Riemann sphere ℙ^{1}(ℂ) (see III.2.1 1a) and 1.3.6 in this chapter). In the exercise in IV. 6.5.1 we showed that H^{1}(ℙ^{1}(ℂ), ℤ) = 0 and therefore ℙ^{1}(ℂ) has genus zero. Of course it is clear that a holomorphic function on a compact Riemann surface S must be constant. We will work very hard to show that on any compact Riemann surface we can find a non constant meromorphic function (see 1.3 ). Once we have a non constant meromorphic function f we can cover S by the two open sets U_{0}, U_{1} where f or f^{−1} are holomorphic. We get a holomorphic map

\begin{aligned}f & : & U_{0} & ⟶ & ℂ \\ & & u & ⟼ & f(w) \end{aligned}

and

\begin{aligned}f^{−1} & : & U_{1} & ⟶ & ℂ \\ & & u & ⟶ & f^{−1}(u) \end{aligned}

and it is clear that these two maps provide a surjective map which is also denoted by

f : S ⟶ ℙ^{1}(ℂ)

(see III, 1.2.1a). It will turn out that this map has finite fibres and the number of points in the fibres ( counted with the right multiplicities) is equal to the degree of the polar divisor ( See IV.1.1.) This will become a decisive tool for the understanding of Riemann surfaces (see 1.4.).

1.1 The Hodge structure on H^{1}(S, ℂ) 1.1.1 : We study the cohomology with coefficients in ℂ. I want to change the notation slightly. On our Riemann surface Ω_{S}^{1} will be the sheaf of holomorphic one forms. The sheaves of C_{∞} differential forms will be denoted by Ω∞^{•}. We consider the de-Rham complex

0 → ℂ → Ω_{∞}^{0} → Ω_{∞}^{1} → Ω_{∞}^{2} → 0.

Then

H^{i}(S, ℂ) = H^{i}(Ω∞^{•}(S)).

We recall our results from IV.9.2. We have a complex structure on the tangent bundle this is a linear transformation I : T_{S} → T_{S} which satisfies I^{2} = −Id. We get a decomposition

T_{S,ℂ} = T_{S}^{10} ⊕ T_{S}^{01}.

This provides a decomposition of the complex of differential forms, which only effects 1-forms:

0 → ℂ → Ω_{∞}^{0} ⟶^{d′+d′′} Ω_{∞}^{10} ⊕ Ω_{∞}^{01} ⟶^{d′+d′′} Ω_{∞}^{11} → 0.

The sheaf Ω_{∞}^{10} contains the sheaf Ω_{S}^{1} of holomorphic 1-forms. (See IV,??). In local coordinates at a point p we have

ω = ω′ + ω′′ = fdz + gdz̄

(see IV.9.2.1 ). For a function f we have

df = d′f + d′′f = \frac{∂f}{∂z} dz + \frac{∂f}{∂z̄}dz̄,

and for a 1-form ω = fdz + gdz̄ = ω′ + ω′′ we get

dω = d′′ω′ + d′ω′′ = (\frac{∂f}{∂z̄} − \frac{∂g}{∂z} )dz ∧ dz̄.

Especially we see that a 1-form ω = fdz = ω′ is holomorphic if and only if d′′ω′ = 0 We introduce a hermitian metric h on the tangent bundle T_{S} (see IV.10.2.1). As I explained in general discussion such a hermitian metric is the same as an euclidean metric Re h = h_{0} on the tangent bundle T_{S} which satisfies h_{0}(x, y) = h_{0}(Ix, Iy) for any two tangent vectors x, y ∈ T_{S,p} and any point p. This induces a metric on the dual bundle T_{S}^{∗} which we will denote by <, >. If we pick a point p ∈ S and a local coordinate z at p then it identifies a neighborhood U_{p} to a disc around zero in ℂ. The differential dz is a generator of the O_{S}(U_{p}) module of holomorphic differentials Ω^{1}(U_{p}). In the neighborhood U_{p} of p our hermitian metric is given by a strictly positive function (see section on linear algebra in section on Hodge theory)

u → Re h_{ℂ} < dz(u), dz̄(u) >

which we simply denote by < dz, dz̄ >. Since we can view U_{p} as an open disc in ℂ we have dz = dx + idy and < dz, dz̄ >=< dx, dx > + < dy, dy >= 2 < dx, dx > because the complex structure I which sends dx to dy is an isometry. The metric and the orientation give us a distinguished form ω_{top} in degree 2 which is positive with respect to he orientation and has length 1 with respect to the metric. It is given by

ω_{top} = i\frac{dz ∧ dz̄}{< dz, dz̄ >} = i\frac{(dx + idy) ∧ (dx − idy)}{< dx, dx > + < dy, dy >} = i(−i)\frac{dx ∧ dy}{< dx, dx >} = \frac{(dx ∧ dy)}{< dx, dx >}.

Of course dω_{top} = 0 and hence we see that our Riemann surface is a Kähler manifold (See IV.10.2.3 ).

Now it is rather easy to check that the ∗ operator does the following

\begin{matrix}∗ : & f → if̄\frac{dz ∧ dz̄}{< dz, dz̄ >} \\ ∗ : & fdz → if̄dz̄ \\ ∗ : & gdz̄ → −iḡdz \\ ∗ : & dz ∧ dz̄ → −i < dz, dz̄ > . \end{matrix}  (1.1.2)

We can introduce the adjoint operators δ′, δ′′ (See IV.10.2.3) and define the Laplacian

Δ = (d′ + d′′)(δ′ + δ′′) =

= d′δ′ + δ′d′ + d′′δ′′ + δ′′d′′ + d′δ′′ + δ′′d′ + d′′δ′ + δ′d′′ = Δ′ + Δ′′ + extra terms.

The extra terms add up to zero because the metric is a Kähler metric. (See IV.10.2.3). Since I stated this result without proof in the general case I will carry out the calculation for special situation. On the forms of degree 0 or 2 this is rather clear. If we consider for instance an f ∈ Ω_{∞}^{0}(S) then

δ′′d′f = δ′′ \frac{∂f}{∂z} dz = ∗d′′ ∗ \frac{∂f}{∂z} dz = −i ∗ d′′ \overline{\frac{∂f}{∂z}} dz̄ = 0

and the same principle works for the other combinations. But for forms of degree one we have to work a little bit. Let us consider ω = fdz then we see easily that for two of the four terms that they must vanish simply by looking at the degree:

δ′′d′ω = d′δ′′ω = 0

For the other two terms we have to compute.

δ′d′′fdz = δ′ \frac{∂f}{∂z̄}dz̄ ∧ dz = − ∗ d′ ∗ \frac{∂f}{∂z̄}dz ∧ dz̄
∗id′ \frac{∂f̄}{∂z} < dz, dz̄ > = ∗i(\frac{∂^{2}f̄}{∂z^{2}} < dz, dz̄ > +\frac{∂f̄}{∂z}\frac{∂}{∂z} < dz, dz̄ >) ∧ dz

= (\frac{∂^{2}f}{∂z̄^{2}} < dz, dz̄ > +\frac{∂f}{∂z̄} \overline{\frac{∂}{∂z} < dz, dz̄ >})dz̄

Since < dz, dz̄ > is positive and therefore real we have

\overline{\frac{∂}{∂z} < dz, dz̄ >} = \frac{∂}{∂z̄} < dz, dz̄ >

Now we treat the second term

d′′δ′fdz = −d′′ ∗ d′ ∗ fdz = −d′′ ∗ (−i)d′f̄dz̄

= −id′′ ∗ \frac{∂f̄}{∂z}dz ∧ dz̄ = −d′′ \frac{∂f}{∂z̄} < dz, dz̄ >=

= −(\frac{∂^{2}f}{∂z̄^{2}} < dz, dz̄ > +\frac{∂f}{∂z̄} \frac{∂}{∂z̄} < dz, dz̄ >)dz̄.

Hence we see that the two terms add up to zero and (δ′d′′ + d′′δ′)fdz = 0. We apply our general theorem in the section on Hodge theory to this case. We are mainly interested the first cohomology group. We get that it is given by the harmonic forms in degree one and these harmonic forms are sums of harmonic forms in the degrees (10), (01). A form ω = fdz ∈ Ω_{∞}^{10}(S) is harmonic if and only if d′ω = d′′ω = δ′ω = δ′′ω = 0. Two of these equations are automatically fulfilled the other two are equivalent to ω being holomorphic. We get the Hodge decomposition

H^{1}(S, ℂ) = H^{0}(S, Ω_{S}^{1}) ⊕ \overline{H^{0}(S, Ω_{S}^{1})}  (1.1.3)

The ℂ vector space H^{1}(S, ℂ) = H^{1}(S, ℝ) ⊗_{ℝ} ℂ has the complex conjugation on it as an antilinear map and \overline{H^{0}(S, Ω_{S}^{1})} is the complex conjugate of H^{0}(S, Ω_{S}^{1}) under this complex conjugation.(See IV. 10.2.1 especially the discussion concerning the formation of the complex conjugate space of a ℂ-vector space) 1.1.4 : I want to give an indication how this consequence of the general Hodge theory can be proved in this special situation. Of course there is no problem in degree zero and degree 2. So we look at the case p = 1 and start from a one form

ω = ω^{10} + ω^{01}

which is closed and represents a cohomology class [ω] ∈ H^{1}(S, ℂ). We have seen that we can construct a weakly convergent sequence ω_{n} = ω + dψ_{n} (see IV.10.1.3) such that the weak limit

ω_{0} ∈ Ω_{(2)}^{1}(S)

satisfies

⟨ω_{0}, dψ⟩ = 0

for all ψ ∈ C_{∞}(S). We also have

⟨ω_{0}, δη⟩ = 0

for all η ∈ Ω′′_{∞}(S) because this is true for alle ω + dψ_{n}. Now we decompose

ω_{0} = ω_{0}^{10} + ω_{0}^{01}.

I claim that even

⟨ω_{0}^{10}, d′ψ⟩ = ⟨ω_{0}^{10}, δ′η⟩ = 0

for all ψ ∈ C_{∞}(S), η ∈ Ω′′_{∞}(S). We have

⟨ω_{0}^{10}, d′ψ⟩ + ⟨ω_{0}^{01}, d′′ψ⟩ = 0

and

⟨ω_{0}^{10}, δ′′η⟩ + ⟨ω_{0}^{01}, δ′η⟩ = 0

for all ψ, η. We take η = ∗\overline{ψ} and then we get from our local formulae ((See 1.1.2))

\begin{aligned}δ′′η & = & −id′ψ \\ δ′η & = & id′′ψ \end{aligned}

and the second line becomes

i⟨ω_{0}^{10}, d′ψ⟩ − i⟨ω_{0}^{01}, d′′ψ⟩ = 0.

We find that in the first line both terms are zero, i.e.

⟨ω_{0}^{10}, d′ψ⟩ = ⟨ω_{0}^{01}, d′′ψ⟩ = 0

for all ψ ∈ Ω_{∞}^{0}(S). We want to conclude that ω_{0}^{10} is itself a holomorphic 1-form. The holomorphicity is a local property of ω_{0}^{10}. We choose a point p and a neighborhood U_{p} such that we can identify (U_{p}, O_{U_{p}}) with the disc (D, O_{D}). Let z be the coordinate function on D. Our differential form can be written

ω_{0}^{10} = f(z)dz

and since the restriction of f to D must be square integrable, we have

∫_{D} |f(z)|^{2}i \frac{dz ∧ dz̄}{⟨dz, dz̄⟩} < ∞.

Since the fuction < dz, dz̄ > is bounded and bonded away from zero the square integrability condition is equivalent to

∫_{D} |f(z)|^{2}i dz ∧ dz̄ < ∞.

Now we exploit the orthogonality relation ⟨ω_{0}^{10}, d′ψ⟩ = 0 for C_{∞}-functions ψ with compact support in D. Our local formulae for the ∗-operator implies that the space of forms ∗d′ψ is the same as the space of forms d′′ψ amd therefore the orthogonality condition is equivalent to

∫_{D} ω_{0}^{10} ∧ d′′ψ = 0 for all compactly supported ψ ∈ Ω_{∞}^{0}(D)

We introduce polar coordinates and write

f(z) = f(r, φ) = ∑_{m∈ℤ} a_{m}(r)e^{imφ},

and we have that a_{m}(r) is square integrable on [0, 1] with respect to rdr and we have

∑_{m} ∫_{0}^{1} |a_{m}(r)|^{2}rdr < ∞.

We can choose our function ψ, and we take

ψ(z) = b(r)e^{−inφ}

where b(r) is C_{∞} on [0, 1) and has compact support in [0, 1). Then an easy computation shows

\frac{∂}{∂z̄}ψ(z) = \frac{1}{2} e^{iφ} (\frac{∂}{∂r} b(r) + \frac{n}{r}b(r)) e^{−inφ} = \frac{1}{2} (\frac{∂}{∂r} b(r) + \frac{n}{r}b(r)) e^{−i(n−1)φ}.

Consequently our assumption implies

∫_{0}^{1} a_{n−1}(r) (\frac{∂}{∂r} b(r) + \frac{n}{r}b(r)) rdr = 0

for all such choices of b(r) and n. This means that

∫_{0}^{1} (−r \frac{∂}{∂r} a_{n−1}(r) + na_{n−1}(r)) b(r)dr = 0

and therefore we can conclude that

a_{n−1}(r) = c_{n−1}r^{n−1}

with some constant c_{n−1}. It follows that a_{n}(r) = 0 for n < 0 because in this case

∫_{0}^{1} r^{2n}rdr = ∞,

and we get

f(z) = ∑_{n=0}^{∞} c_{n}r^{n}e^{inφ}

and

∑ |c_{n}|^{2} · \frac{1}{(2n + 2)^{2}} < ∞.

This is good enough to show that f is holomorphic on the disc. The cup product <, >_{cup}: H^{1}(S, ℤ) × H^{1}(S, ℤ) → H^{2}(S, ℤ) = ℤ extends to a bilinear pairing <, >_{cup}: H^{1}(S, ℂ) × H^{1}(S, ℂ) → H^{2}(S, ℂ) = ℂ and we know (See IV. 9.1) that this pairing is given by

< [ω_{1}], [ω_{2}] >_{cup}= ∫_{S} ω_{1} ∧ ω_{2}

where ω_{1}, ω_{2} are closed forms which represent the classes [ω_{1}], [ω_{2}] in the cohomology.

With respect to this pairing the two subspaces H^{0}(S, Ω_{S}^{1}), \overline{H^{0}(S, Ω_{S}^{1})} are maximal isotropic spaces and hence the cup product induces a perfect bilinear pairing

<, >_{cup}: H^{0}(S, Ω_{S}^{1}) × \overline{H^{0}(S, Ω_{S}^{1})} → ℂ.

1.2. Cohomology of holomorphic bundles 1.2.1 For any holomorphic vector bundle on E on S we consider the Dolbeault complex

0 → E ⟶^{d′′} Ω_{∞}^{0}(E) ⟶^{d′′} Ω_{∞}^{01}(E) → 0.

The cohomology group of E are computed from the complex

0 → Ω_{∞}^{0}(E)(S) ⟶^{d′′} Ω_{∞}^{01}(E)(S) → 0.

Now choose in addition a hermitian metric <, >_{h} the bundle E and on T_{S}. The metrics on E and on T_{S} provide an adjoint operator δ′′ : Ω_{∞}^{01}(E)(S) → Ω^{0}(E)(S) and now Hodge theory implies (See IV.10.2.4 noch einfuegen)

\begin{aligned}H^{0}(S, E) & = \{s ∈ Ω_{∞}^{0}(E)|d′′s = 0\} \\ H^{1}(S, E) & = \{s ∈ Ω_{∞}^{01}(E)|δ′′s = 0\}. \end{aligned}  (1.2.2)

and the cohomology groups are finite dimensional. I emphazise that the proof of this finite dimensionality needs the full strength theory of elliptic operators. Actually for the cohomology in degree zero H^{0}(S, E) the finite dimensionality is not difficult but the H^{1} is by no means easy.

For compact Riemann surfaces these results are much easier to prove and therefore I will outline the proofs here. At first I consider the assertion dim H^{0}(S, E) < ∞. We proceed by induction on the rank of the bundle. Let us assume that we have a non zero section s ∈ H^{0}(S, E). Then we show

1.2.3 Lemma: To this non zero section s we can attach a line subbundle L ⊂ E such that s ∈ H^{0}(S, L).

Proof: The section s provides a map

\begin{matrix}O_{S} & ⟶ & E \\ f & ⟼ & fs \end{matrix}

for any holomorphic function f on some open subset U ⊂ S. This yields indeed a line subbundle L′ but it is not yet the one we want. If we are at a point x ∈ S where s(x) = 0, then we can choose a neighborhood U_{x} and a local trivialization of E by local sections e_{1} ⋯ e_{n} which are nowhere zero on U_{x}. Our section s can be written as

s = ∑_{i=1}^{n} f_{i}e_{i}

with f_{i} holomorphic at x and f_{i}(x) = 0 for all i = 1 ⋯ n. This implies that the set of zeroes of s is a finite subset of S. But since dim S = 1 we have a local uniformizer π_{x} ∈ 𝔪_{x} ⊂ O_{S,x} and f_{i} = π_{x}^{n_{i}}h_{i} where h_{i} ∈ O_{S,x}^{∗}. Let m be the minimum of the n_{i}. Then

π_{x}^{−m}s = ∑ π_{x}^{−m}f_{i}e_{i}

extends to a section in E which is defined iover U_{x}. This section defines subbundle L^{(x)} ⊂ E|_{U_{x}} . But this line subbundle coincides with the above bundle L′ if we restrict to the complement of the point x. Hence we see that we can glue the L′ and the L^{(x)} to a line bundle L on S. We have L ⊂ E, the quotient E/L is a vector bundle of smaller rank and s ∈ H^{0}(S, L). This reduces the proof of the finite dimensionality to the case of line bundles. But if we have a line bundle L and a section s ≠ 0, then we look again at the inclusion O_{S} → L induced by the sectiuon, and we get an exact sequence

0 ⟶ O_{S} ⟶ L ⟶ L/O_{S} ⟶ 0

and now L/O_{S} is a sky scraper sheaf (see IV.1.1 ). Since H^{0}(S, O_{C}) = ℂ and obviously dim H^{0}(S, L/O_{S}) is finite dimensional, we are through. The proof of the second assertion is much more difficult. In a first step we construct an antilinear map H^{1}(S, E) ⟶ H^{0}(S, E^{∨} ⊗ Ω^{1}), where the target space has finite dimension. In a second step we show that this map is injective.

The argument in the first step ia similar to the one in 1.1.3. Of course an section ω ∈ Ω_{∞}^{01}(E)(S) defines a class in H^{1}(S, E). On these sections we have the scalar product

⟨ω, ω_{1}⟩ = ∫_{S} ω ∧ ∗ω_{1}

and again we introduce the Hilbert space of square integrable sections

Ω_{(2)}^{01}(E)(S).

We can modify a given section ω by an element d′′ψ, and we can find a sequence ω = ω + d′′ψ_{n} such that

∥ω_{n}∥_{2}^{2} ⟶ inf_{ψ} ∥ω + d′′ψ∥_{2}^{2}.

Again we choose a subsequence which converges weakly to an element ω̃ ∈ Ω_{(2)}^{01}(E)(S), and then we have again

⟨ω̃, d′′ψ⟩ = 0

for all ψ ∈ Ω^{0}(E)(S). The hermitian metric yields an antilinear map j_{h} = E → E^{∨} and together with the hermitian metric on the tangent bundle we get

∗ = j̃_{h} : Ω_{(2)}^{01}(E)(S) ⟶ Ω_{(2)}^{10}(E^{∨})(S).

We have of course ⟨∗ω̃, δ′′η⟩ = 0 for all η ∈ Ω_{∞}^{11}(E^{∨})(S) and basically the same argument which we used in 1.1.3 implies that ∗ω̃ = j̃_{h}(ω̃) is holomorphic. We see that we get a map

H^{1}(S, E) = Ω_{∞}^{01}(E)(S)/d′′Ω_{∞}^{0}(E)(S) ⟶ H^{0}(S, E^{∨} ⊗ Ω^{1})

which is surjective because the star operator above yields a bijection. We know already that H^{0}(S, E^{∨} ⊗ Ω^{1}) is finite dimensional so it remains to show that this map is injective (our second step). We have the two operators

\begin{aligned}d′′ & : & Ω^{0}(E)(S) ⟶ Ω^{01}(E)(S) \\ δ′′ & : & Ω^{01}(E)(S) ⟶ Ω^{0}(E)(S) \end{aligned}

and we take the two orthogonal decomposions of the two Hilbert spaces

\begin{aligned}Ω_{(2)}^{0}(E)(S) & = & E(S) ⊕ Ω̃_{(2)}^{0}(E)(S) \\ Ω_{(2)}^{01}(E)(S) & = & ker(δ′′) ⊕ Ω̃_{(2)}^{01}(E)(S). \end{aligned}

Since E(S) and ker(δ′′) consist of holomorphic sections, we get the same decomposition for C_{∞}-sections

\begin{aligned}Ω_{∞}^{0}(E)(S) & = & E(S) ⊕ Ω̃_{∞}^{0}(E)(S) \\ Ω_{∞}^{01}(E)(S) & = & ker(δ′′) ⊕ Ω̃_{∞}^{01}(E)(S), \end{aligned}

\begin{aligned}d′′ & : & Ω̃_{∞}^{0}(E)(S) ⟶ Ω̃_{∞}^{01}(E)(S) \\ δ′′ & : & Ω̃_{∞}^{01}(E)(S) ⟶ Ω̃^{0}(E)(S). \end{aligned}

These restrictions of d′′.δ′′ to Ω̃∞^{•} are injective and they have seen that there images are dense. But we need that these restricted operators provide isomorphisms and this follows from the decomposition therem in Hodge theory (See IV.10.1.1 ) In our special situation here there is an argument which proves

d′′ : Ω̃_{∞}^{0}(E)(S) ˜ ⟶Ω̃_{∞}^{01}(E)(S)

more directly. Our argument is based on some simple principles of functional analysis. We consider the local problem, this means we choose a small disc D around a point p, we assume that we identified it to the unit disc and we assume that z is a local parameter. We consider the Hilbert spaces Ω(2)^{•}(E)(D). Our first principle says the following: 1.2.4 Let us assume we have a form ω ∈ Ω_{∞}^{01}(E)(D) and let us assume that this form is square integrable. Then we can find an η ∈ Ω_{∞}^{0}(E)(D) such that d′′η = ω and we can bound the L^{2} norm of η

∥η(z)∥_{2,D} ≤ C∥ω∥_{2,D}

Since our bundle is locally trivial we may asum that its restriction to D is trivial. It is also clear that the validity of the L^{2} estimates does not depend on the hermtian metric. Thesetwo facts together allow us to restrict to the case where D is the unit disc and where E|D→^{∼}O_{D} and the metric is the trivial metric. Then we have to show: If f : D → ℂ is a C_{∞} -function on D which is square integrable, i.e.

∥f∥_{2}^{2} = ∫_{D} |f(z)|^{2}idz ∧ dz̄ < ∞

then we can find C_{∞}-function u on D which satisfies

\frac{∂u}{∂z̄} = f

and

∥u∥_{2} ≤ C∥f∥_{2}.

The point is that we can write down an explicit solution for this diferential equation:

u(z) = \frac{1}{2πi} ∫_{D} \frac{f(ζ)}{ζ − z} dζ ∧ dζ̄.

We leave it as anexercise to the reader to show that this is a solution for our equation. It is quite amusin to show that forf = 1 we get u(z) = z̄ and then the general formula is easy.

It is easy to see that this function is C_{∞}. If z_{0} ∈ D, then we can find a C_{∞}-function χ on D which is one on a small neighborhood of z_{0} and zero on a small neighborhood of the boundary of D. Then

u(z) = \frac{1}{2πi} ∫_{D} \frac{χ(ζ)f(ζ)}{ζ − z}dζ ∧ dζ̄ + \frac{1}{2πi} ∫_{D} \frac{(1 − χ(ζ))f(ζ)}{ζ − z}dζ ∧ dζ̄.

The second summand is holomorphic at z_{0} and the first summand is

\frac{1}{1πi} ∫_{D} \frac{χ(ζ + z)f(ζ + z)}{ζ}dζ ∧ dζ̄,

and now we can differentiate under the integral sign because χ · f has compact support and the singularity disappears if we change to polar coordinates. It is an amusing exercise to show that for f = 1 we have u(z) = z̄ and from this it follows easily that u satisfies the differential equation. I think it is also very easy to see that the integral

\frac{1}{2πi} ∫ \frac{1}{|ζ − z|}dζ ∧ dζ̄

is bounded by a constant not depending on z. We may work with polar coordinates around z. To get the L^{2}-estimate we start from

\begin{aligned}|u(z)|^{2} & = & \frac{1}{4π^{2}} | ∫_{D} \frac{f(ζ)}{ζ−z}dζ ∧ dζ̄|^{2} ≤ \\ & = & \frac{1}{4π^{2}} | ∫_{D} \frac{|f(ζ)|}{|ζ−z|^{1/2}} · \frac{1}{|ζ−z|^{1/2}} dζ ∧ dζ̄|^{2} \end{aligned}

This is the square of the scalar product of two L^{2} functions, and we get by Cauchy’s integral formula that the right hand side

≤ \frac{1}{4π^{2}} (∫_{D} \frac{|f(ζ)|^{2}}{|ζ − z|} dζ ∧ dζ̄) (∫_{D} \frac{1}{|ζ − z|}dζ ∧ dζ̄) .

We mentioned already that second factor is bounded by a constant not depending on z and hence we get

|u(z)|^{2} ≤ C \frac{1}{2πi} ∫_{D} \frac{|f(ζ)|^{2}}{|ζ − z|} dζ ∧ dζ̄.

Consequently we get

\frac{1}{2πi} ∫_{D} |u(z)|^{2}dz ∧ dz̄ ≤ C ∫_{D} ∫_{D} \frac{|f(ζ)|^{2}}{|z − ζ|} dζ ∧ dζ̄ · dz ∧ dz̄,

and if we change integration and use our above estimate a second time, then we get

∥u∥_{2}^{2} ≤ C′ · ∥f∥_{2}^{2}.

Now we come to a second principle. Inside of our Hilbert space Ω_{2,D}^{0}(E)(D) we can consider the holomorphic square integrable functions. The second principle says that this subspace is closed and even better:

1.2.5 Any weakly convergent sequence of holomorhic functions η_{n} with bounded Ω_{2,D}^{0}-norm in Ω_{2,D}^{1}(E)(D) converges locally uniformly to a holomorphic function on D. This is an immedeate consequence of Cauchys integral formula. If we pick a point Q ∈ D and we put three concentric discs around Q:

Q ∈ D_{1} ⊂ D_{2} ⊂ D_{3} ⊂ D

each of them is slightly bigger than the previous one. If we have a circle Γ ⊂ D_{3} \ D_{2} then we get from Cauchys formula for z ∈ D_{2}

η_{n}(z) = \frac{1}{2πi} ∫_{Γ} η_{n}(ζ)\frac{1}{ζ − z} dζ.

Now we integrate over all Γ_{τ} between D_{2} and D_{3} and consider z ∈ D_{1}. We get

η_{n}(z) = \frac{c}{2πi} ∫_{D_{3}\D_{2}} η_{n}(ζ)\frac{1}{ζ − z} dζ ∧ dζ̄,

where c is a consant depending on the width of the annulus. We can read this as a skalar product, since the sequence η_{n} is weakly convergent to η we see that η_{n} converges pointwise to the function

η̃: z ↦ \frac{c}{2πi} ∫_{D_{3}\D_{2}} η(ζ)\frac{1}{ζ − z} dζ ∧ dζ̄

which is holomorphic on D_{1}. But now it the Cauchy-formula also gives us that the η_{n} are equicontinous and then it follows that the convergence η_{n} → η̃ is locally uniform and that η = η̃ We choose an ω ∈ Ω̃_{∞}^{01}(E)(S), and we know that we can find a sequence of functions ψ_{n} ∈ Ω̃_{∞}^{0}(E)(S) such that

∥dψ_{n} − ω∥_{2} ⟶ 0.

I claim that the sequence of L^{2}-norms {∥ψ_{n}∥}_{n∈ℕ} is bounded. We cover S by a finite family of discs

S = ⋃_{α∈A} D_{α}.

We restrict the members of the family {ψ_{n}}_{n∈ℕ} to these discs and call the restrictions ψ_{n}^{(α)}. Now we can decompose the restriction

Ω_{(2)}^{0}(E)(D_{α}) = Ω_{(2)}^{0,′}(E)(D_{α}) ⊕ Ω_{(2)}^{(0,hol)}(E)(D_{α})

where the second component is the Hilbert space of holomorphic square integrable sections on D_{α} and accordingly we have

ψ_{n}^{(α)} = ψ_{n}^{(α,′)} + ψ_{n}^{(α,hol)}.

We get

d′′ψ_{n}^{(α)} = d′′ψ_{n}^{(α,′)}.

Since we have an η_{n}^{α} for which d′′η_{n}^{α} = d′′ψ_{n}^{α} and ∥η_{n}^{α}∥_{2,D_{α}} ≤ C∥d′′ψ_{n}^{α}∥_{2,D_{α}} we can conclude that ∥ψ_{n}^{(α,′)}∥_{2,D_{α}} stays bounded because ∥ψ_{n}^{(α,′)}∥_{2,D_{α}} ≤ ∥η_{n}^{α}∥_{2,D_{α}}.

Hence we see: If ∥ψ_{n}∥_{2} is unbounded, then there must be an α such that ∥ψ_{n}^{(α,hol)}∥_{2,D_{α}} is unbounded and even

∥ψ_{n}^{(α,hol)}∥_{2,D_{α}} ≥ C′∥ψ_{n}∥_{2}

with some constant C′. Now we consider the sequence of functions ψ_{n}/∥ψ_{n}∥_{2}. We can extract a subsequence which is weakly convergent, on any D_{α} this sequence has the same limit as ψ_{n}^{(α,hol)}/∥ψ_{n}∥_{2} hence it converges to a holomorphic function. This function must be zero because our ψ_{n} where chosen from the orthogonal complement of the holomorphic sections. It follows from our second principle that the sequence ψ_{n}/∥ψ_{n}∥ converges uniformly to zero. This cannot be the case because the L^{2}-norm of the members of the sequence is one. We get a contradiction. So we see that the sequence of norms ∥ψ_{n}∥ is bounded. Of course we extract a weakly convergent subsequence. If ψ is the limit of this subsequence we found the element which satisfies dψ = ω. This finishes the proof of 1.2.2. (Genau genug ???) If we write locally s = f ⊗ dz̄, then

δ′′s = − ∗ d′′ ∗ (f ⊗ dz̄) = − ∗ d′′(∗f) ⊗ dz = − ∗ (\frac{∂}{∂z̄} ∗ f)dz̄ ∧ dz.

As I explained in the general section on Hodge theory the ∗-operator provides an antilinear map from the bundle E into its dual bundle.(Nachsehen, ob das da schon steht) Hence we see that ∗f ⊗ dz ∈ C_{∞}(E^{∨}) ⊗ Ω^{10}(S) and the condition \frac{∂}{∂z̄} ∗ f ⊗ dz = 0 means that ∗f ⊗ dz is holomorphic and hence we get an antilinear isomorphism

H^{1}(S, E) → H^{0}(S, E^{∨} ⊗ Ω_{S}^{1}).

This antilinear isomorphism will depend on the metric. But if our line bundle is the structure sheaf O_{S} then we may of course choose the standard metric which gives the value 1 to the constant section 1 at every point. This gives us a canonical identification

\overline{H^{0}(S, Ω_{S}^{1})}→^{∼}H^{1}(S, O_{S}).

Remark: In the context of algebraic geometry we had a duality pairing for line bundles L on curves:

H^{1}(C, L) × H^{0}(C, L^{∨} ⊗ Ω_{S}^{1}) → ℂ.

This is related to the assertion above. But in the analytic context we get on ℂ-antilinear isomorphism instead of a pairing. This is so because we have chosen an hermitian metric on L. We will compare these pairings later after the discussion of the theorem of Riemann-Roch.

1.3. The theorem of Riemann-Roch We consider the group Pic(S) isomorphism classes of line bundles on our Riemann surface. This group is isomorphic to H^{1}(S, O_{S}^{∗}). The exact sequence of sheaves ( see IX. 9.3 ????)

0 → ℤ → O_{S} → O_{S}^{∗} → 1

on our Riemann surface provides a long exact cohomology sequence

\begin{aligned}0 & → ℤ → ℂ → ℂ^{∗} → 0 \\ & → H^{1}(S, ℤ) → H^{1}(S, O_{S}) → H^{1}(S, O_{S}^{∗}) → H^{2}(S, ℤ) → 0. \end{aligned}

(We have seen that H^{2}(S, O_{S}) = 0 because the Dolbeault complex stops in degree 1). In IV.1.2 we attached a sheaf O_{S}(D) to any divisor D, it is obvious that O_{S}(D) is a line bundle. It is also clear that O_{S}(D) ⊗ O_{S}(D_{1})→^{∼}O_{S}(D + D_{1}) and that O_{S}(D)→^{∼}O_{S} if and only if D is principal and hence we get

Div(S)/ principal divisors →^{∼}Pic(S)

. For a line bundle L, which we can view as an element [L] in H^{1}(S, O_{S}^{∗}) we define the degree deg(L) = δ([L]) ∈ H^{2}(S, ℤ) = ℤ. (We have a canonical orientation on S). This degree has various properties.

1.3.1 Lemma : If we have a line bundle L on S and a point P then

deg(L(P)) = deg(L) + 1

Proof: This is a special case of proposition IV.9.3.1 Exercise:

(i) If we choose a hermitian metric h on our line bundle L, if we pick a point P and a neighborhood U_{P} and a local section s ∈ L(U_{p}) which is a generator for all points in U_{P} then we can form the expression

d′′d′ log h(s, s) = ω_{h}.

This is a (1,1)-form on S which is closed and it does not depend on the choice of the generator s. I claim that the class of this form in H^{2}(S, ℂ) is the Chern class c_{1}(L) = deg(L).

(ii) I refer to the proof of our Lemma: We consider the holomorphic 1-form function z_{P}^{−1}dz_{P} on the annulus U_{1} ∩ U_{2}. It is clear that we can extend this form to a C_{∞}-form η on the disc U_{2} = D_{P} , (we simply multiply it by a C_{∞}-function which is one on the annulus and zero in a neighborhood of P). If we consider d′′η we get a (1, 1) form on the disc D_{P} which has compact support because it vanishes on the annulus. Hence it defines a class in H_{c}^{2}(D_{P} , ℂ), this maps to H^{2}(S, ℂ) Show that this is the class δ([O_{S}(P)]). This way we can construct a form of type (1, 1) which represents the degree. This form can be written as a boundary on any open set in S which misses a small disc around P.

(iii) Let us assume that we have an arbitrary compact complex manifold X and a divisor D ⊂ X which is locally given on the open set of a covering 𝔘 = {U_{α}} by one equation f_{α} = 0. We choose a hermitian metric on X. We choose our covering in such a way that we place small balls around the points on D and choose a finite subcovering 𝔘 = {U_{α}} of D. Then we supplement it by an open set U_{0} which is the set of points having distance > ϵ from D. Construct a (1, 1) form ω_{D} which has support in the complement of U_{0} and which represents the class c_{1}(O_{X}(D)). Show that this form is a boundary outside of the support of D.

(iv) If we have divisors D_{1}, D_{2}, …, D_{d} (d = dim X) which intersect nicely then we can consider the intersection number D_{1} · D_{2} · . . . · D_{d}. Show that this intersection number can also be computed by the integral

∫_{X} ω_{D_{1}} ∧ ω_{D_{2}} ∧ … ∧ ω_{D_{d}}

(v) Of course we can attach to any line bundle L its Chern class c_{1}(L) ∈ H^{2}(X, ℤ). If we have d such bundles L_{1}, …, L_{d} we can compute their intersection number and we can take the cup product of their Chern classes which gives an element in H^{2d}(X, ℤ) = ℤ. Our exercise iv) gives us the equality of these numbers

L_{1} · . . . · L_{d} = c_{1}(L_{1}) ∪ … ∪ c_{1}(L_{d})

Show the equality of these numbers without using the de-Rham isomorphism ???.

1.3.2 The above Lemma implies: If we have a line bundle L and a non-zero section s ∈ H^{0}(S, L) then on a suitably small open set U we can write s = ft where t is a local generator and f is holomorphic. This function f defines a divisor on U, it is the divisor of its zeroes (See IV.1.1). Since we can do this everywhere we get a divisor D = div(s) and it is clear that L ∼ O_{S}(D). Then it follows from Lemma 1.3.1 that

deg(L) = deg(D)  (1.3.3.a)

. Hence we can conclude that the degree of a line bundle which has non zero sections must be positive.

If D is the divisor of a meromorphic function f then this function defines a section in O(D) and f^{−1} defines a section in O(−D) and consequently we must have

deg(Div(f)) = 0  (1.3.3.b)

1.3.4. We also can conclude that a line bundle of degree zero has a non zero section if and only if it is trivial. We can formulate the

Theorem of Riemann-Roch: If L is a line bundle on a compact Riemann surface S then

dim_{ℂ}H^{0}(S, L) − dim_{ℂ}H^{1}(S, L) = deg(L) + 1 − g.

We have

dim_{ℂ}H^{1}(S, L) = dim_{ℂ}H^{0}(S, L^{−1} ⊗ Ω_{S}^{1}).

Furthermore we have deg(Ω_{S}^{1}) = 2g − 2 and consequently dim_{ℂ}H^{1}(S, L) = 0 if deg(L) ≥ 2g − 1. This is now more or less obvious. We proved the finite dimensionality and the equality of the dimensions of the H^{0} and the H^{1} in the previous section. We write χ(L) for the left hand side. The assertion is true for L = O_{S} by definition. If we want to prove it for our given sheaf L we pick a point P and consider the exact sequence

0 → L → L(rP) → L(rP)/L → 0

for a large value of r. Then the dimension of the space of sections of the skyscraper sheaf becomes large and this space of sections is mapped to the finite dimensional H^{1}(S, L). This implies that eventually H^{0}(S, L(rP)) will be non zero. But then a non zero section gives us an inclusion O_{S} ⟶L(rP) with a scyscraper quotient S.(See proof of lemma 1.2.3). We have the exact sequence

0 → O_{S} → L(rP) → S → 0

and a glance at the resulting exact sequence yields that χ(L) − χ(O_{S}) = dim H^{0}(S, S). This is also the degree of L. (Lemma 1.3.1 iterated). Hence we have proved the first formula for L(rP). Then the same argument applied backwards proves it for L. It remains to prove the formula for the degree of Ω_{S}^{1}. To get this we apply the first formula in the theorem to the sheaf Ω_{S}^{1} We get

dim_{ℂ}H^{0}(S, Ω_{S}^{1}) − dim_{ℂ}H^{1}(S, Ω_{S}^{1}) = deg(Ω_{S}^{1}) + 1 − g.

The left hand side is equal g − 1 and the theorem is proved. I would like to stress again that the real difficulty is to show that H^{1}(S, O_{S}) is finite dimensional.

1.3.5. At this point we have proved a very strong finiteness result. We know that any line bundle L on a compact Riemann surface S has a very simple acyclic resolution: We simply take an effective divisor D = Σn_{p}P with sufficiently large degree and then

0 ⟶ L ⟶ L(D) ⟶ 𝕃_{D} ⟶ 0

is an acyclic resolution of L (see IV.1.1.). We get the exact sequence

0 ⟶ H^{0}(S, L) ⟶ H^{0}(S, L(D)) ⟶ H^{0}(S, 𝕃_{D}) ⟶ H^{1}(S, L) ⟶ 0.

We have seen that dim H^{1}(S, L) = dim H^{0}(S, L^{−1}⊗Ω_{S}^{1}) but we can prove a stronger result. We construct a non degenerate bilinear pairing

H^{1}(S, L) × H^{0}(S, L^{−1} ⊗ Ω_{S}^{1}) ⟶ ℂ.

To get this pairing we respresent an element ξ ∈ H^{1}(S, L) as the image unter the boundary map. We lift it to an element

ξ̃ = (… ξ̃_{p} …)_{P ∈|D|} ∈ H^{0}(S, 𝕃_{D})

where |D| is the support of D. We choose small discs D_{P} around these P such that we can trivialize the bundle L over these discs by non vanishing sections t_{P} ∈ H^{0}(D_{P} , L). Then the components ξ̃_{P} can be written as

ξ̃_{P} = (\frac{b_{−n}}{z_{P}^{n}} + ⋯ + \frac{b_{−1}}{z_{P}} ) t_{P}

where z_{p} is a local parameter at P. If now η ∈ H^{0}(S, L^{−1}⊗Ω_{S}^{1}), we can write the restriction of η to D_{p} in the form

η |_{D_{P}} = t_{P}^{−1} · f(z_{P})dz_{P} ,

and we can consider the product

ξ̃_{P} η = (\frac{b_{−n}}{z_{P}^{n}} + ⋯ + \frac{b_{−1}}{z_{P}} ) · f(z_{P})dz_{P} = (\frac{a_{−n}}{z_{P}^{n}} + ⋯ \frac{a_{−1}}{z_{P}} + ⋯) dz_{P} = ω_{P} ,

this is a holomorphic 1-form on D_{p} \ {P } which may have a pole (a meromorphic 1-form). To such a meromorphic 1-form we attach its residue at P, it is given by

Res_{P} (\frac{a_{−n}}{z_{P}^{n}} + ⋯ \frac{a_{−1}}{z_{P}} ⋯) dz_{P} = a_{−1}.

It is not clear a priori that this residue is well defined but everybody should know the formula

a_{−1} = \frac{1}{2πi} ∫_{Γ} ω_{p}

where Γ is a path in D_{P} \ {P } which winds counterclockwise aroung P just once. Then we define

⟨ξ, η⟩ = ∑_{P} Res_{P} (ξ̃_{P} η) = ∑_{P} Res_{P} (ω_{P}).

We have to show that the value of this pairing does not depend on the choice of the lifting. If we replace ξ̃ by ξ̃ + f where f ∈ H^{0}(S, L(D)), then fη = ω is a meromorphic 1-form on S, it is an element in H^{0}(S, Ω_{S}^{1}(D)). For such a form it is clear that the sum of the residues vanishes. We simply observe that we can take the D_{P} so small that they do not intersect and for the path Γ_{P} we take their boundaries with counterclockwise orientation. Then

∑_{P} Res_{P} (ω) = \frac{1}{2πi} ∑ ∫_{Γ_{P}} ω = \frac{1}{2πi} ∫_{S\∪D_{P}} dω = 0.

This proves that we get a well defined pairing

H^{1}(S, L) × H^{0}(S, L^{−1} ⊗ Ω_{S}^{1}) ⟶ ℂ.

But it is also clear that any non zero element η ∈ H^{0}(S, L^{−1} ⊗ Ω_{S}^{1}) induces a non zero linear form on H^{1}(S, L). To see that this is so we simply compute this linear form on 𝕃_{D}, and then it is obviously non zero. This non degenerate pairing is called Serre duality pairing but in this special case it was certainly known to Riemann. It expresses the fact that the existence of holomorphic differentials on a Riemann surface of higher genus provides an obstruction for a collection of Laurent expansions

ξ̃ ∈ H^{0}(S, O_{S}(D)/O_{S})

to come from a meromorphic function (see IV, Exercise 1.1.1(2)). It is stated in Riemann’s works (see [ ], ....) that ξ̃ comes from a meromorphic function if and only if for all holomorphic differentials ω we have

∑_{P} Res_{P} (ξ̃ω) = 0.

1.3.6. If the genus of the Riemann surface S is equal to zero and if P is any point, then it follows from Riemann Roch and Serre duality that

dim H^{0}(S, O_{S}(P)) = 2,

and we conclude that we can find a meromorphic function f which is holomorphic everywhere except at the point P and at P it has a simple pole.

We saw already that this function gives us a map

f : S ⟶ ℙ^{1}(ℂ),

I claim that this map is an isomorphism between Riemann surfaces. To see this we observe that there is exactly one point – namely the point P – which goes to infinity. If U = S\{P }, then we get for the restriction

f : U ⟶ ℂ = {(z, 1) | z ∈ ℂ} ⊂ ℙ^{1}(ℂ)

(see ......). For any c ∈ ℂ we know that the polar divisor of f − c is −P. Hence the zero divisor is of degree one and is equal to Q where f(Q) = c. Since S is compact it follows that this map is a homeomorphism.

For any point Q we can find a neighborhood D_{Q} ⊂ S such that

(D_{Q}, O_{D_{Q}}) ≃ (D, O_{D})

where D is the unit disc in ℂ. Let z be the resulting uniformizing element. Under the map f this neighborhood is mapped to an open set f(D_{Q}) which contains f(Q) ∈ ℙ^{1}(ℂ). We choose a uniformizing element z_{Q}, this is a holomorphic function defined in a neighborhood of f(Q) and which has a first order zero at z_{Q}. Then z_{Q} ∘ f is a holomorphic function on a smaller disc D′_{Q} ⊂ D_{Q} and hence a power series in z. Since the function z_{Q} ∘ f is injective we can conclude that

z_{Q} ∘ f = az + ⋯

with a ≠ 0.

Now O_{ℙ^{1},f(Q)} is the ring of convergent power series in z_{Q} and O_{S,Q} is the ring of convergent power series in z. We see that the map

O_{ℙ^{1},f(Q)} ⟶ O_{S,Q}

(see ........) is an isomorphism and this proves our assertion. We can also give examples of Riemann surfaces of genus one. If Ω ⊂ ℂ is a lattice, then the quotient S = ℂ/Ω is a compact Riemann surface (see III. 1.1.1). It is homeomorphic to ℝ^{2}/ℤ^{2}, and hence we have H^{1}(S, ℤ) = ℤ^{2} (see IV, ....) and hence we see that S has genus 1. We know that the space of holomorphic differentials is of dimension one and clearly the form ω = dz is a generator. If in turn S is a compact Riemann surface of genus one, then we may do the following: We pick a point s_{0} ∈ S and we consider the following space

S̃ = {(s, γ) | s ∈ S, γ homotopy class of a point starting in s_{0} and ending in s}.

We have the projection

π : S̃ ⟶ S

and locally this projection is a homeomorphism. (This construction can be done for any connected Riemann surface, then S̃ is the so called universal cover of S). It is also clear that we have a structure of a Riemann surface on S̃.

Now we can construct a holomorphic map h from S̃ to ℂ. We simply send

h : s̃ = (s, γ) ⟼ ∫_{γ} ω

where we choose a differentiable path in the homotopy class. I leave it as an exercise to the reader to show that this map is an isomorphism between S̃ and ℂ. It is also not difficult to show that h^{−1}(s_{0}) = Ω is a lattice in ℂ and the map factorizes over an isomorphism

S̃ ⟶^{h} ℂ
π ↓  ↓
S ⟶^{h̃} ℂ/Ω

This makes it clear that all compact Riemann surfaces are of the form ℂ/Ω.

1.4. The algebraicity of Riemann surfaces: 1.4.0: We are now able to show that compact Riemann surfaces may be considered as purely algebraic objects. More precisely we can say that compact Riemann surfaces are the same objects as smooth, connected, projective curves over ℂ. It will be discussed in the second volume of this book what this exactly means. The starting point is that the meromorphic functions on S form a field K = ℂ(S). We will show that this field is finitely generated over ℂ and it is of trancendence degree 1. We will reconstruct the Riemann surface S from this field. If for instance we consider the Riemann sphere S = S^{2} = ℙ^{1}(ℂ) (III.2.1. 1a) or 1.3.6)) then ℂ(S^{2}) = ℂ(z) is the rational function field in one variable. It is the quotient field of the polynomial ring ℂ[z] which is the ring of meromorphic functions which are holomorphic on U_{0} = ℙ^{1}(ℂ) \ {∞}.

1.4.1 If we pick a point P ∈ S and if we consider the line bundle O_{S}(nP) for n >> 0 then we see that we can find a non-constant function f ∈ H^{0}(S, O_{S}(nP)). As I explained in the introduction to this chapter a non-constant meromorphic function f on S provides a surjective map f : S → ℙ^{1}(ℂ) . We put U_{0} = S \ {P } and U_{1} will be the complement of the set of zeroes of f.

1.4.1.1 : If we have a point s ∈ S where f is holomophic, then the differential df is holomorphic at this point. If it is non-zero at s then we know from the theorem of implicit functions that f yields a biholomorphic map from a neighborhood of s to a neighborhood of f(s). We say that f is unramified or not ramified if df is nt zero at this point s. If f has a pole at s then we take g = \frac{1}{f}. If dg ≠ 0 then we have the analogous assertion for the functon g. In terms of f this can be reformulated: If f has a pole at s and if the pole of df is of second order, then f yields a biholomorphic map from a neighborhood of s to a neighborhood of ∞ ∈ ℙ^{1}(ℂ). Again we say that f is unramified at s.

The map f is called unramified at a point x ∈ ℙ^{1}(ℂ) if it is unramified in all points of the fibre f^{−1}(x). It is clear that the set of points where f is ramified is finite. Changing the coordinates allows us to assume that our map is unramified at 0 and ∞. We give it a new name and write

π : S → ℙ^{1}(ℂ).

Let V_{0} (resp. V_{1}) be the complement of 0 (resp. ∞), let U_{0} = π^{−1}(V_{0}), U_{1} = π^{−1}(V_{1}). For any set V ⊂ S (or in ℙ^{1}(ℂ)) which is the complement of a finite number of points we define O_{S}^{mer}(V) to be the ring of those holomorphic functions on V, which have at most poles in the points S \ V. For V_{0}, V_{1} ⊂ ℙ^{1}(ℂ) this yields

\begin{aligned}O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) & = ℂ[f] \\ O_{ℙ^{1}(ℂ)}^{mer}(V_{1}) & = ℂ[f^{−1}]. \end{aligned}

We may also consider the rings O_{S}^{mer}(U_{0}), O_{S}^{mer}(U_{1}) and these two rings are modules for O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) and O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) respecively.

1.4.2 Proposition: a) The modules O_{S}^{mer}(U_{0}) (resp. O_{S}^{mer}(U_{1})) over O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) (resp. O_{ℙ^{1}(ℂ)}^{mer}(V_{1})) are finitely generated.

b) If {a_{1}, …, a_{t}} ⊂ V_{0} is a finite subset and V′_{0} = V_{0} \ {a_{1}, …, a_{t}} and U′_{0} = π^{−1}(V′_{0}) then

O_{S}^{mer}(U′_{0}) = O_{S}^{mer}(U_{0}) · O_{ℙ^{1}(ℂ)}^{mer}(V′_{0}).

Proof: We show that O_{S}^{mer}(U_{0}) is a finitely generated O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) module. We consider the divisor D_{∞} = ∑_{P ∈π^{−1}(∞)} P, it is the divisor of poles of the function f pulled back to S.( Here we use that π is unramified at ∞, actually this is only technical). For n > 0 we consider the vector spaces H^{0}(S, O_{S}(nD_{∞})). They form an increasing sequence of vector spaces exhausting O_{S}^{mer}(U_{0}) if n → ∞. The dimension of these spaces is given by the theorem of Riemann-Roch: If n >> 0 then

dim_{ℂ}H^{0}(S, O_{S}(nD_{∞}) = ndeg(D_{∞}) + 1 − g.

We observe that the multiplication by z yields a linear map (multiplication by f??????)

×z : H^{0}(S, O_{S}(nD_{∞}) → H^{0}(S, O_{S}((n + 1)D_{∞})

and I claim that this map becomes surjective if we divide the space on the right hand side by the subspace H^{0}(S, O_{S}(nD_{∞}). We pick a funcion h ∈ H^{0}(S, O_{S}((n + 1)D_{∞}) its polar divisor is of the form D = ∑_{P ∈π^{−1}(∞)} m_{P} P with m_{P} ≤ n + 1. If even m_{P} ≤ n for all n then this function is in the subspace which we divide out. Now we observe that it follows from our assumption n >> 0 that

H^{0}(S, O_{S}(nD_{∞}) → H^{0}(S, O_{S}(nD_{∞})/O_{S}((n − 1)D_{∞}))

is surjective. Therefore we can find a function f ∈ H^{0}(S, O_{S}(nD_{∞}) which has an n-th order pole at a given point P where m_{P} = n + 1 and has at most a n − 1-th order pole at all the other points in π^{−1}(∞) For a suitable combination h − azf the number of m_{P} which are equal to n + 1 drops by one and our assertion follws by induction. Our claim implies that the O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) module O_{S}^{mer}(U_{0}) is generated by H^{0}(S, O_{S}(n_{0}D_{∞}) for some sufficiently large n_{0} and a) follows. Now b) is is not difficult anymore. Let f be meromorphic function in O_{S}^{mer}(U′_{0}). We can find a function h ∈ O_{ℙ^{1}(ℂ)}(V′_{0}) which has a zero in the points a_{1}, …, a_{t} and nowhere else (take the inverse of a function which has poles in exacly these points). If we pull it back to U_{0} it has zeroes in all points in the fibres π^{−1}(a_{i}) i.e. in all points in U_{0} \ U′_{0} and nowhere else. Hence f · h^{N} will be holomorphic in all points of U_{0} \ U′_{0} and this means fh^{N} ∈ O_{S}^{mer}(U′_{0}). Now we consider the function field K = ℂ(S) of meromorphic functions. It is clear that the function field of the Riemann sphere ℂ(ℙ^{1}((ℂ)) = ℂ(z) = ℂ(f) is the rational function field in one variable. The assertion b) in our proposition above implies that any meromorphic function h on S can be written as a quotient h = g/f where g ∈ O_{S}^{mer}(U_{0}) and f is a meromorphic function in ℂ(ℙ^{1}(ℂ)). Therefore we can conclude: If y_{1}, … y_{d} is a set of generators of the O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) module O_{S}^{mer}(U_{0}) then ℂ(S) is generated by these elements as a ℂ(ℙ^{1}(ℂ))-vector space. We conclude

1.4.3 Theorem The field of meromorphic functions on a compact Riemann surface is a finite extension of a rational funtion field ℂ(h) where h is any non constant meromorphic function on S. The only thing we have to observe that a field K which is a finite extension of a rational function field ℂ(x) is also finite over ℂ(h) for any h ∉ ℂ. We explain how we can reconstruct S from K. To do this we will use in an ad hoc manner some arguments from commutative algebra which will be explained in a more systematic way in chapter VII.

1.4.3.1: Integral elements and integral closures: The finiteness of O^{mer}(U_{0}) as a O_{ℙ^{1}}^{mer}(V_{0}) module implies by a standard argument of commutative algebra that any element h ∈ O^{mer}(U_{0}) is integral over O_{ℙ^{1}}^{mer}(V_{0}) and this means that it satisfies an equation

h^{n} + a_{1}h^{n−1} … + a_{n} = 0

where the a_{i} ∈ O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) and n > 0 (See Chap. VII ???? or [Ei], ???). But if in turn h ∈ ℂ(S) is integral over O_{ℙ^{1}(ℂ)}^{mer}(V_{0}) then it must be holomorphic on U_{0}, we simply look at the possible order of a pole. We conclude 1.4.3.2: The ring O_{S}^{mer}(U_{0}) is the integral closure of O_{ℙ^{1}}^{mer}(V_{0}) in K and this means that it consists of all the elements in K which are integral over O_{ℙ^{1}}^{mer}(V_{0}). The principal observation is that a point P ∈ S defines a subring O_{P}^{mer} ⊂ K, it is the ring of meromorphic functions which are regular at P. This ring is a valuation ring with quotient field K and this means: a) for any f ∈ K we have f or f^{−1} is in O_{P}^{mer}.

b) The ring is not equal to K and it contains the constant functions ℂ. Such a ring O_{P}^{mer} has a unique maximal ideal which consists of the elements

𝔪_{P} = {f ∈ O_{P}^{mer}|f^{−1} ∉ O_{P}^{mer}} = {f ∈ K|f vanishes at P }

This maximal ideal is non zero and it is generated by any function which has a first order zero at P. This means that the ring is even a discrete valuation ring. The elements which are not in the maximal ideal are invertible. The property a) implies that the quotient field of such a ring is K. Our aim is to show that we can identify S with the set Val(K) of all valuation rings in K, i.e. those subrings which satisfy a) and b).

1.4.4 Starting from the function field: We forget our Riemann surface completely and start from a field K over ℂ which is a finite extension of a rational function field ℂ(x). We can write

K = ℂ(x)[y]

where y satisfies an irreducible polynomial equation

y^{n} + a_{1}(x)y^{n−1} ⋯ a_{n}(x) = 0

with a_{i}(x) ∈ ℂ(x). Of course we know what Val(K) is.

Propositon 1.4.4.1 All A ∈ Val(K) are discrete valuation rings, i.e. the maximal ideal 𝔪_{A} is always a principal ideal. The composition ℂ → A → A/𝔪_{A} is an isomorphism and this means that the residue field is canonically isomorphic to ℂ. This also means that we can evalute an f ∈ A at A. The value f(A) = f mod 𝔪_{A}. Furthermore we will see that for any f ∈ K the set of A such that f ∉ A is finite. In accordance with our previous definitions we say that f is regular at A if f ∈ A. Again we have to invest a little bit of commutative algebra. If K_{0} = ℂ(x) then an A ∈ Val(K_{0}) contains ℂ[x] or ℂ[x^{−1}]. Let us assume that A ⊃ ℂ[x]. The maximal ideal 𝔪_{A} intersected with ℂ[x] gives us a non zero prime ideal in ℂ[x]. It is an elementary fact that the non zero prime ideals in ℂ[x] are of the form (x − α). This implies that the elements of Val(K_{0}) are in one to one correspondence with the points in ℙ^{1}(ℂ) = ℂ∪{∞} : For any α ∈ ℂ we have the ring

A_{α} = \{f = \frac{P (x)}{Q(x)} | Q(α) ≠ 0\} ,

and for ∞ we have

A_{∞} = \{f = \frac{P (x)}{Q(x)} | deg(P) ≤ deg(Q)\}

where P, Q are polynomials and f(A_{a}) = f(a). Clearly the valuation rings are discrete valuation rings. This must be well known. We consider the integral closures B_{0} (resp. B_{∞}) of ℂ[x] (resp. ℂ[x^{−1}]) in the field K. Then the theory of Dedeking rings (see [Ei ], [Neu], or Chap. VII, 2.4.4) implies that these integral closures are finitely generated modules over ℂ[x] (resp. ℂ[x^{−1}]). Since the polynomial rings have unique factorization, it follows that these modules are even free of rank [K : ℂ(x)]. This fact has the following consequence: 1.4.4.2 If 𝔭 is a prime ideal in B_{0} then 𝔭 is maximal and B_{0}/𝔭 = ℂ. The ring

A = \{\frac{f}{g} | f, g ∈ B_{0}, g ∉ 𝔭\}

is a discrete valuation ring. To see that this is so we consider 𝔭_{0} = 𝔪_{A} ∩ ℂ[x]. It is clear that 𝔭_{0} is non zero. Then B_{0}/𝔭 is an integral domain and a finite dimensional vector space over ℂ = ℂ[x]/𝔭_{0}. This implies that B_{0}/𝔭 = ℂ[x]/𝔭_{0} = ℂ. For the last assertion we refer to [Ei],??? or [Neu], or Chap. VII ) Now we pick an A ∈ Val(K) and let us assume A ⊃ ℂ[x], otherwise it contains the other ring. I claim that this implies A ⊃ B_{0}. If f ∈ B_{0} we write down the polynomial equation

f^{n} + a_{1}(x)f^{n−1} ⋯ a_{n}(x) = 0

with a_{i}(x) ∈ ℂ[x]. If now f ∉ A then f^{−1} lies in the maximal ideal 𝔪_{A} of A and our polynomial equation yields

1 = −a_{1}(x)f^{−1} ⋯ − a_{n}(x)f^{−n}

which gives a contradiction. Now A ⊃ B_{0}, we consider the prime ideal 𝔭 = 𝔪_{A} ∩ B_{0} and and 𝔭 must be maximal. Then

A = \{\frac{f}{g} | f, g ∈ B_{0}, g ∉ 𝔭\}.

This together with our considerations above make it clear that the elements of U_{0} are in one-to-one correspondence with the maximal ideals in B_{0} and these maximal ideals are also just the homomorphisms B_{0} → ℂ which are ℂ linear, i.e. the identity in ℂ. This proves the second assertion in 1.4.4.1 above. It is also clear that the complements of U_{0} and U_{∞} in Val(K) are finite because it is rather obvious that there are only finitely many prime ideals in B_{0} (resp. B_{∞} ) which lie above (x) (resp.(x^{−1}). This implies the finiteness assertion if we apply our consideration to x = f. Hence the proposition is proved. We define a topology on Val(K). The open sets U ⊂ Val(K) are defined as the complement of finite sets, and of course we have to add the empty set. This topology is called the Zariski topology.

We can define the sheaf of meromorphic functions. For any open set U ⊂ Val(K) we put

O(U) = ⋂_{A∈U} A,

this is the ring of functions which are regular on U and meromorphic on S. This gives (Val(K), Zar, O) the structure of a locally ringed space. If we take any f ∈ K which is not constant, i.e. f ∉ ℂ then D_{f} is the set of points where f is regular. Then

O(D_{f}) = the integral closure of ℂ[f] in K

and this follows from the fact that a Dedekind ring is the intersection of the discrete valuation in the quotient field which contain it (???) This object (Val(K), Zar, O) is almost what is called a smooth, projective, connected curve over ℂ. The only thing missing is the so called generic point. This generic point is simply the field K. We can just drop the assumption A ≠ K for our valuation rings and put Ṽal(K) = Val(K) ∪ {K}. We define the Zariski topology on Ṽal(K), the open sets are the complements of finite subsets in Val(K) and the empty set. We define the sheaf as before and now (Ṽal(K), Zar, O) is a locally ringed space and this is now really a smooth, connected, projective curve. The stalks of the structure sheaf are discrete valuation rings in the closed points and the stalk in {K} is K.

1.4.5 Back to the Riemann surface: Now we assume again that K is the field of meromorphic functions on our compact Riemann surface S. We observed earlier that we have a map

S → V al(K)

and we want to show that this is a bijection. Here it is clear that we have to use the compactness of the Riemann surface. This compactness will enter in the form that any holomorphic function on S must be constant. We pick a valuation ring A ⊂ K, let

𝔪_{A} = {f ∈ A | f^{−1} ∉ A}

be its maximal ideal. Our goal is to show that there is a unique point P ∈ S such that A = O_{P}^{mer}. We will show that this point P is the common zero of the f ∈ 𝔪_{A} and it also be characterized as the unique point where all the elements of A are regular. We pick a non zero f ∈ 𝔪_{A}. We consider the intersection A ∩ ℂ[f] then 𝔪_{A} ∩ ℂ[f] = (f) because the principal ideal (f) is maximal. The intersection

B = A ∩ ℂ(f) = {g/h | g, h ∈ ℂ[f], h ∉ (f)}

We consider the diagram

π : S  → ℙ^{1}(ℂ)
↓  ↓
π̃ : Val(K) → Val(ℂ(f))

induced by f. As before V_{0} = ℙ^{1}(ℂ) \ {∞} and U_{0} = π^{−1}(V_{0}). Then our ring B consists of those meromorphic functions on ℙ^{1}(ℂ) which are regular in 0. We have to show that the map π^{−1}(0) → π̃(0) is surjective. The integral closure A_{1} of B in K is a free module of rank [K : ℂ(f)] and

A_{1}/A_{1}f = ∑_{𝔭} A_{1}/𝔭^{e_{𝔭}}

where the 𝔭 are the maximal ideals in A_{1}, they are in one to one correspondence to the elements in π̃(0), we have ∑_{𝔭} e_{𝔭} = [K : ℂ(x)]. The divisor of f is of the form ∑_{𝔭∈π^{−1}(0)} e_{𝔭} and it has degree [K : ℂ(x)] and from this the equality of the two fibres follows.

1.4.6 The recovery of the analytic topology: The set S has some further structure, it has a topology and a sheaf of complex valued functions on it. We want to reconstruct this structure starting from K. Our Riemann surface is also a locally ringed space, and it is clear that the map

(S, O_{S}) ⟶ (Val(K), Zar, O^{mer})

is a morphism between locally ringed spaces. This is of course not an isomorphism because on the left hand side we have many more functions, the ring O_{S,P} is much larger than O_{S,P}^{mer} = A if P maps to A. We still go one step further. Again we forget the compact Riemann surface S, and we start from a function field

K = ℂ(x)[y]

where

y^{n} + a_{1}(x)y^{n−1} = + ⋯ + a_{n}(x) = 0.

We put S = Val(K) on this set, we have the Zariski topology and our sheaf O_{S} with respect to the Zariski topology. We want to construct a finer topology on S and S together with this finer topology will be called S_{an}. Of course the identity S_{an} → S will now be continuous. Furthermore we want to construct a sheaf O_{S}^{an} of ℂ valued functions on S_{an} such that we get a locally ringed space and such that

(S_{an}, O_{S}^{an})

will be a compact Riemann surface. Finally we can restrict meromorphic functions f ∈ O_{S}(U) to the open sets in S_{an}, and this will induce a morphism of locally ringed spaces

(S_{an}, O_{S}^{an}) → (S, Zar, O_{S})

(see III,......). We come to the construction of the analytic topology. For any open subset U ⊂ S we have the ring O_{S}(U), and we can interpret O_{S}(U) as ring of ℂ-valued functions on U. We introduce the coarsest topology on U for which all these functions are continuous. If we have two different points A, B ∈ S, then it is clear that we cannot have A ⊃ B or B ⊂ A. Hence we can find an f ∈ A for which f ∉ B. Since we can add a constant, we can assume f ∉ 𝔪_{A}. Then g = 1/f ∈ A but g ∉ 𝔪_{A} and g ∈ 𝔪_{B}. In other words, the element g is regular at A and at B and g(A) ≠ 0 and g(B) = 0. Hence we have A, B ∈ D_{f} and from the definition of the analytic topology follows that we can find neighborhoods of A and B whose intersection is empty and we have proved that our analytic topology is Hausdorff. We want to describe a neighborhood of a point A ∈ S, and we want to show that A has neighborhoods isomorphic to a disc in ℂ. This is of course clear if K = ℂ(x), in this case we could identify

Val(K) = ℙ^{1}(ℂ)

and the analytic topology is of course the usual topology on ℙ^{1}(ℂ). We reduce the general case to this one. We have our point A ∈ S. We choose an element f ∈ 𝔪_{A} which generates the ideal. Again we consider the integral closure O(D_{f}) of ℂ[f] in K. We have

ℂ[f] · f ⊂ ℂ[f]
∩  ∩
(f) ⊂ O(D_{f}).

In our special situation the ℂ[f]-module O(D_{f}) is free of rank n where n = [K : ℂ(f)] (see ......).

O(D_{f}) = ⊕ℂ[f]y_{i} = ℂ[f, y_{1},_{2} … y_{n}]

where the elements f, y_{1}, …, y_{n} satisfy some polynomial identities

P (f, y_{1}, …, y_{n}) = 0

with some polynomials P (F, Y_{1}, …, Y_{n}) from the polynomial ring ℂ[F, Y_{1}, … Y_{n}]. If is I the ideal generated by all these polynomals then we get an isomorphism

ℂ[F, Y_{1}, … Y_{n}]/I→^{∼}O(D_{f})

We introduce the evaluation map

\begin{matrix}E : D_{f} → ℂ^{a+1} \\ E : u ↦ (f(u), y_{1}(u), …, y_{n}(u)) \end{matrix}

The the elements of O(D_{f}) separate the points in D_{f} because the ponts corresond to the mximal ideals of O(D_{f}). Therefore the evaluation map is injective. The image consists of those points in (z_{0}, z_{1}, …, z_{n}) ∈ ℂ^{n+1} which satisfy P (z_{0}, z_{1}, …, z_{n}) = 0 for all elements P ∈ I. Our point A is mapped to an element (0, a_{1}, …, a_{n}) = (f(A), y_{1}(A), …, y_{n}(A)). We have a finite set of distinct points A = A_{0}, A_{1}, …, A_{m} in S for which f(A_{1}) = … = f(A_{m}) = 0. We can find an r > 0 such that for all i, α, β we have |y_{i}(A_{β}) − y_{i}(A_{α})| > 2r whenever these two numbers are not equal. We consider the open set U ⊂ S which is defined by the requirement

U = {B | |y_{i}(B) − y_{i}(A)| < r for all 1 = 1, 2, … n}.

We consider the projection to the first coordinate

\begin{matrix}p : U → ℂ \\ B ↦ f(B) \end{matrix}

and this projection is by construction a homeomorphism to the image. Now we observe that we can write any of our y_{i} in the form

y_{i} = y_{i}(A) + γ_{i}f + R_{i}

where γ_{i} ∈ ℂ and R_{i} = f^{2}g_{i}/h_{i} where g_{i}, h_{i} ∈ O(D_{f}) and h_{i}(A) ≠ 0. We represent these elements bv polynomials G_{i}, H_{i} ∈ ℂ[F, Y_{1}, … Y_{n}] and then we know that the ideal I above contains elements of the form

L_{i} = H_{i}(F, Y_{1}, …, Y_{n})(Y_{i}−y_{i}(A))−H_{i}(F, Y_{1}, ˙,Y_{n})γ_{i}F −F^{2}G(F, Y_{1}, …, Y_{n}) for i = 1, …, n.

The independent variables are F and the Y_{i} for i = 1 …, n and the partial Jacobi matrix

(\frac{∂L_{i}}{∂Y_{i}})(A)_{i,j}

is a diagonal matrix with non zero entries on the diagonal and therefore it has maximal rank. Hence we can conclude from the theorem of implicit functions that for a suitably small ε > 0 we can construct an inverse to the projection p above

\begin{matrix}q : D(ε) → U \\ z ↦ (z, y_{1}(z), …, y_{n}(z)) \end{matrix}

where now y_{1}(z), …, y_{n}(z) are convergent power series and q identifies D(ε) to an open neigborhood U(ε) of A in S. On this open neighborhood we can define the sheaf O_{S}^{an}(U(ε)) of holomorphic functions, this is simply the sheaf of holomorphic functions on our small disc. Hence we constructed the structure of a compact Riemann surface (S_{an}, O_{S}^{an}) and clearly the identity map

(S_{an}, O_{S}^{an}) → (S, Zar, O_{S})

is a morphism of locally ringed spaces. One word concerning the notation. Here we think that the algebraic object (S, Zar, O_{S}) is given first and to denote the analytic object we put the sub and superscripts and write S_{an}, O_{S}^{an}. In the beginning of this section we did the oppostite. There the Riemann surface was given and we had to introduce the sub- and superscripts Zar, mer .

Finally I want to say a few words about the connection to algebraic geometry. I come back to the description of

O(D_{f}) = ℂ[f, y_{1}, ⋯, y_{n}] = ℂ[F, Y_{1}, ⋯, Y_{n}]/I.

We described the image of D_{f} under the evaluation map as a set of solutions of polynomial equations

Y = E(D_{f}) = {(a_{0}, a_{1}, ⋯, a_{n}) | P (a_{0}, a_{1}, ⋯, a_{n}) = 0 for all P ∈ I},

and this means ( by defintion ) that this image is an affine algebraic variety over ℂ. I claim that for any point B = (a_{0}, ⋯, a_{n}) ∈ Y we can pick an index i such that y_{i} − a_{i} = ỹ_{i} is a local parameter: In a small neighborhood the other coordinates of a point b ∈ Y can be expressed as holomorphic functions in ỹ_{i}. We simply apply our arguments above to B. Therefore our variety is in fact one dimensional and smooth (See III.2 Example 5). Actually we can say even more. Since O(D_{f}) is the integral closure of ℂ[f] in the function field we know that the elements ỹ_{i} satisfy an equation

ỹ_{i}^{n_{i}} + a_{1}ỹ_{i}^{n_{i}−1} … + a_{n_{i}} = 0

where the coefficients a_{i} ∈ ℂ[f|. We may assume that this polynomial is irreducible. We must have a_{n_{i}}(A) = 0. It is not entirely obvious but true that the previous coefficient a_{n_{i}−1} does not vanish at A. We can conclude that for the points B in our small neighborhood of A the polynomial

Y^{n_{i}} + a_{1}(B)Y^{n_{i}−1} … + a_{n_{i}}(B)

has exactly one root which is close to one. This means in classical terms that ỹ_{i}^{n_{i}} is an algebraic function in the variable z = f(B), it is a root of the polynomial which is distinguished and depends analytically on z. Of course a few points are missing, namely, the points in S \ D_{f}. But we can find an element g ∈ K which is regular at these missing points. We have a second evaluation map which identifies

D_{g} ˜ ⟶Y_{1} ⊂ ℂ^{m+1}

and O(D_{g}) = ℂ[g, u_{1}, ⋯, u_{m}]. In Y we have the open subset Y_{g} where g is regular and in Y_{1} the open subset Y_{1,f} where f is regular and these two open sets are identifed to D_{f} ∩D_{g} under the evaluation maps. We have to say in terms of the two data what the regular functions on D_{f} ∩ D_{g} are. I claim that

O_{S}(D_{g} ∩ D_{f}) = ℂ[g, u_{1}, ⋯, u_{m}, f, y_{1}, ⋯, y_{n}],

and this means that the regular functions on D_{g} ∩ D_{f} can be written as sums of products of elements in O(D_{f}) and O(D_{g}). If h ∈ O_{S}(D_{g} ∩ D_{f}), then this function may have poles in T_{f} ∪ T_{g} where T_{f} = S \ D_{f}, T_{g} = S \ D_{g}. We want to modify h by sums of products of functions u_{1}u_{2} where u_{1} has poles only in T_{f} and u_{2} has poles only in T_{g}. Let us pick a point t ∈ T_{f} with t ∉ T_{g} and s ∈ T_{g}, s ∉ T_{f} such that h has a pole at t. If such a pair does not exist there is nothing to prove. We produce a function u_{1} which has a pole at t and nowhere else. This is possible by Riemann-Roch. We produce a function u_{2} which has a pole at s and nowhere else but which in addition has a simple zero at t. Then u_{1}u_{2}^{m} has a simple pole at t for a suitable choice of m. Now we can modify h by subtracting a suitable power of u_{1}u^{m}

h − γ · (u_{1}u^{m})^{k}

such that the pole order of h at t drops. This means that the total pole order at points in T_{f} \ (T_{f} ∩ T_{g}) drops. We repeat this process until h does not have any pole in the set T_{f} \ (T_{f} ∩ T_{g}), and then the modified function has only poles in T_{g}. Then we achieved our goal. I summarize: Our space S together with the sheaf O_{S} is covered by two affine varieties (or affine schemes) and the ring of regular functions is generated by the regualar functions on the two pieces. With a corn of salt this means that we constructed a separated scheme. (see Chap. VI ??) . Actually it is even projective this will be discussed later. We want to have a brief look at the case of Riemann surfaces of genus one. We have seen that they are of the form S = ℂ/Ω where Ω is a lattice in ℂ. In this case it is not so difficult to produce meromorphic functions on S, in a first semester course on function theory it is taught that we have the two meromorphic functions

℘(z) = \frac{1}{z^{2}} + ∑_{ω∈Ω,ω≠0}(\frac{1}{(z − ω)^{2}} − \frac{1}{ω^{2}} )

℘′(z) = −2 ∑_{ω∈Ω} \frac{1}{(z − ω)^{3}}

on S and they are related by an equation

℘′(z)^{2} = 4℘(z)^{3} + g_{2}(Ω)℘(z) + g_{3}(Ω)

where the coefficients g_{2}(Ω), g_{3}(Ω) can be expressed in terms of the lattice. ( They are modular forms see for instance [ ]). Furthermore we know that these two functions generate the field of meromorphic functions on S. We also get an embedding into the projective space, we map

z ↦ (℘′(z), ℘(z), 1) = (1, \frac{℘(z)}{℘′(z)}, \frac{1}{℘′(z)}) ∈ ℙ^{2}(ℂ)

this map provides an analytic isomorphism

S→^{∼}{(x, y, z) ∈ (ℂ^{3} \ 0)/ℂ^{∗}|y^{2}z − x^{3} − g_{2}(Ω)xz^{2} − g_{3}(Ω)z^{3} = 0} ⊂ ℙ^{2}(ℂ)

1.4.7 Geometrie Analytique et Geometrie Algebrique An analytic sheaf E^{an} on S_{an} is called a coherent sheaf if it is a sheaf of O_{S}^{an}–modules, and if for any point P ∈ S the O_{S,P}^{an} –module E_{P}^{an} is finitely generated. We have the same notion for Zariski sheaves on S and clearly any coherent Zariski sheaf E provides an analytic sheaf E^{an} = E ⊗_{O_{S}^{mer}} O_{S}^{an}.

Here we encounter the simplest case of the so called GAGA–principle. In our situation this principle says that this construction provides an equivalence of categories.

1.4.7.1 Proposition For any coherent sheaf E^{an} on S we can find a unique coherent Zariski sheaf E such that

E^{an} = E ⊗_{O_{S}^{mer}} O_{S}^{an}.

For any pair F, G of coherent Zariski sheaves the map

Hom_{S}(F, G) ⟶ Hom_{S^{an}}(F^{an}, G^{an})

is a bijection.

Proof: In either situation a coherent torsion sheaf is a sheaf where each stalk consists of torsion elements. Let us assume that our sheaf is a torsion sheaf. We pick a point P ∈ S and the generators t_{1}, t_{2}, …, t_{s} as above. They are annihilated by a non zero element f ∈ O_{P} regardless in which case we are. But then this element f can be extendet into a small neighborhood D′_{P} such that it is non zero at any point Q ≠ P, Q ∈ D′_{P}. Hence this restriction has finite support in P and we conclude that torsion sheaves are the scysraper sheaves. Now we observe that for any point P and any positive integer a > 0 we have the equality

O_{S,P}^{an} /(𝔪_{P}^{an})^{a} = O_{S,P} /(𝔪_{P})^{a},

and therefore analytic and Zariski torsion sheaves are the same objects. Since O_{S,P}^{an} is a discrete valuation ring we can find generators u_{1}, …, u_{m} such tht the stalk E_{S,P}^{an} is the direct sum

E_{S,P}^{an} = ⊕O_{S,P}^{an} u_{i}.

If we restrict to a still smaller neighborhood then we can express the u_{i} in terms of the t_{i} and vice versa. We may assume that the coefficients in these epressions are defined over this neighborhood and this means that can view the u_{i} as restrictions of some ũ_{i} which are defined in this neighborhood. We can replace the t_{i} by te ũ_{i}. Now some of the u_{i} are torsion elements and these elements define a torsion subsheaf if we restrict to this neighborhood. If we still shrink this neighborhood further then this torsion subsheaf has support in P and the quotient is free. This happens in a small neighborhood of an arbitrary point P shows us that we can define a finite skyscraper sheaf E_{tors}^{an} ⊂ E^{an} and the quotient E^{an}/E_{tors} = E^{′an} is locally free.

But if we have a locally free sheaf E^{an} and a section s which is defined in a punctured disc Ḋ_{P} = D_{P} \ {P }, then we know what it means that s has at most a pole at P. Hence for any Zariski open subset U ⊂ S we can define the O_{S}(U)–module of meromorphic sections E^{mer}(U) = E(U). We have defined E and the assertion E^{an} = E ⊗_{O_{S}} O_{S}^{an} follows from our strong finiteness results.

Now it is clear what we do in the general case. We have

E^{′an} = E^{an}/E_{tors},

and we define E(U) as the space of sections which go to meromorphic sections on E′(U). The rest is clear. Here we encountered a special case of a general principle which is called the GAGAprinciple (see the headline of this section). In a very rough form it says that compact complex manifolds are in fact algebraic, provided they have enough meromorphic functions. In such a case the coherent algebraic and the coherent analytic sheaves form equivalent categories (See [Se]). We will come back to this principle in the second half of this chapter.

1.4.8 Comparison of two pairings We have by the Hodge-Dolbeault theorem that

H^{1}(S, O_{S}) ≃ \overline{H^{0}(S, Ω_{S}^{1})}.

We have the Hodge decomposition of H^{1}(S, ℂ)→^{∼}H^{0}(S, Ω_{S}^{1}) ⊕ \overline{H^{0}(S, Ω_{S}^{1})} (1.1.3). If we compute the cohomology H^{1}(S, ℂ) using the de-Rham complex then the cup product

H^{1}(S, ℂ) × H^{1}(S, ℂ) → ℂ

on the cohomology is giving by integrating cup products of representing forms.(See IV.9.1.3. ) If we consider the above decompostion the two summands are isotropic and we get the ℂ-linear pairing

H^{0}(S, Ω_{S}^{1}) × \overline{H^{0}(S, Ω_{S}^{1})} → ℂ

which is given by

(ω_{1}, ω̄_{2}) ↦< ω_{1}, ω̄_{2} >= ∫ ω_{1} ∧ ω̄_{2}.

The combination of the isomorphism above and the pairing yields a ℂ-bilinear pairing

H^{0}(S, Ω_{S}^{1}) × H^{1}(S, O_{S}) → ℂ.

We will call this pairing the analytic pairing. We constructed the Serre duality pairing in section 1.3.5.

1.4.9 Theorem: The analytic pairing is −2πi times the Serre duality pairing.

To see this we need some simple considerations which in principle concern the comparison between Čzech-cohomology of sheaves and the cohomology groups obtained by injective (or acyclic) resolutions for instance the de-Rham resolution. We pick a point P ∈ S and an n >> 0 such that the map

H^{0}(S, O_{S}(nP)/O_{S}) → H^{1}(S, O_{S})

becomes surjective. We choose an element η ∈ H^{1}(S, O_{S}) and we lift it to an element ξ ∈ H^{0}(S, O_{S}(nP)/O_{S}). We choose a disc D_{P} around P and a local coordinate z_{P} which is zero at P. Now we represent an element ξ by a Laurent series

f(z) = \frac{a_{n}}{z^{n}} + … + a_{0} + a_{1}z + ….

We cover S by two open sets, one of them U_{1} = D_{p} and U_{2} is the complement of a smaller closed disc \overline{D_{p}(ϵ)} around P, hence U_{1} ∩ U_{2} is an annulus. We have that

f ∈ O_{S}(U_{1} ∩ U_{2})

and it defines a 1- cocycle for he covering S = U_{1} ∩ U_{2}. This cocycle maps to η under the edge homomorphism (See IV, 6.6.1) Now we proceed and use the de Rham resolution, we get a diagram

O_{S}(U_{1}) ⊕ O_{S}(U_{2}) ⟶ O_{S}(U_{1} ∩ U_{2})
↓  ↓
Ω_{∞}^{0}(U_{1}) ⊕ Ω_{∞}^{0}(U_{2}) ⟶ Ω_{∞}^{0}(U_{1} ∩ U_{2})
↓ d′′  ↓ d′′
Ω_{∞}^{01}(U_{1}) ⊕ Ω_{∞}^{01}(U_{2}) ⟶ Ω_{∞}^{01}(U_{1} ∩ U_{2}).

We send f to Ω_{∞}^{0}(U_{1} ∩ U_{2}) and I claim that we may write f|U_{1} ∩ U_{2} as the restriction of a C_{∞}-function h_{1} on U_{1} = D_{p}. To see this we simply multiply the function f which is actually defined on he punctured disk by a C_{∞}-function which is identically equal to 1 on the annulus and which isidentically zero in a neighborhood of zero. This C_{∞}-function on the disc is holomorphic on the annulus but if we go into the interior it certainly loses this property. This means that d′′h_{1} = ψ is an element in Ω_{∞}^{01}(U_{1}) which has compact support and therefore can be extended it by zero to S. Then ψ ∈ Ω_{∞}^{01}(S), it is closed and it represents our given class in H^{1}(S, O_{S}) via the Dolbeault isomorphism.

The integral

∫_{S} ω ∧ ψ

for a holomorphic 1-form ω on S gives the value of the analytic pairing between ξ and ω. We compute this integral. We observe that

∫_{S} ω ∧ ψ = ∫_{D_{p}} ω ∧ ψ

and the integrand has compact support in D_{p}. We choose a circle ∂D_{p}(r) which lies in the annulus, we still have

∫_{D_{p}} ω ∧ ψ = ∫_{D_{p}(r)} ω ∧ ψ.

But now we write again ψ = d′′h_{1} and we have ω ∧ dh_{1} = ω ∧ (d′h_{1} + d′′h_{1}) = ω ∧ d′′h_{1}. Therefore

∫_{D_{p}(r)} ω ∧ d′′h_{1} = ∫_{D_{p}(r)} ω ∧ dh_{1}
− ∫_{D_{p}(r)} d(ω ∧ h_{1}) = − ∫_{∂D_{p}(r)} h_{1}ω
= − ∫_{∂D_{p}(r)} fω = −2πiRes_{P} (fω).

and now the right hand side is by definition the value of the Serre duality pairing multiplied by 2πi. Since our pairings are non degenerate we conclude that we have two different ways of producing an identification H^{1}(S, O_{S})→^{∼}H^{0}(S, Ω_{S}^{1})^{∨} which differ by a factor −2πi. We could call the one produced by the cupproduct the analytic identification and the other one the algebraic identification. We will mostly use the analytic identification.

1.5. The Jacobian of a compact Riemann surface Let S be a compact Riemann surface. We defined Pic(S) = H^{1}(S, O_{S}^{∗}) to be the group of isomorphism classes of holomorphic line bundles on S. Our exact sequence in 1.3. provides the homomorphism

H^{1}(S, O_{S}^{∗}) = Pic(S) → H^{2}(S, ℤ) = ℤ

The kernel is denoted by Pic^{0}(S) and it is called the Jacobian of the curve and sometimes we write J = Pic^{0}(S). The exact sequence yields

Pic^{0}(S) = H^{1}(S, O_{S})/H^{1}(S, ℤ).

Here we divide a g dimensional ℂ-vector space by a free ℤ-module of rank 2g, I claim that we are in fact dividing by a lattice, i.e. the submodule is in fact discretely embedded. To see this we recall the Hodge decomposition (1.1.3) and get inclusions

H^{1}(S, ℤ) ↪ H^{1}(S, ℝ) ↪ H^{1}(S, ℂ) = H^{0}(S, Ω_{S}^{1}) ⊕ \overline{H^{0}(S, Ω_{S}^{1})}.

Since H^{1}(S, ℝ) = H^{1}(S, ℤ) ⊗ ℝ, we see that H^{1}(S, ℤ) is a lattice in H^{1}(S, ℝ). On the other hand it is clear that the projection of H^{1}(S, ℝ) to any of the two summands in the decomposition of H^{1}(S, ℂ) is an isomorphism since the summands are complex conjugate. This implies that the inclusions followed by the projection

H^{1}(S, ℤ) ↪ \overline{H^{0}(S, Ω_{S}^{1})} = H^{1}(S, O_{S})

maps H^{1}(S, ℤ) isomorphically to a lattice Γ in H^{1}(S, O_{S}). We want to denote this isomorphism by

j : H^{1}(S, ℤ)→^{∼}Γ.

Of course it is clear the the multiplication of line bundles in Pic^{0}(S) induces the addition on H^{1}(S, O_{S})/Γ and hence we see that the quotient

J = Pic^{0}(S) = H^{1}(S, O_{S})/Γ

has a natural structure of a connected, compact complex-analytic group of dimenson g. Such a group is called a complex torus.

1.6. The classical version of Abel’s Theorem In the previous section we described the group of line bundles Pic^{0} in terms of the cohomology group H^{1}(S, O_{S}^{∗}). Our main tool was the exact sequence

0 → H^{1}(S, ℤ) → H^{1}(S, O_{S}) → H^{1}(S, O_{S}^{∗}) → H^{2}(S, ℤ)

which allowed us to define the degree of the line bundle and gave us the description

Pic^{0}(S) = H^{1}(S, O_{S})/H^{1}(S, ℤ).

Now we recall ( see 1.3.)that the group of line bundles may also be described as the group of divisor classes

Div(S)/ principal divisors →^{∼}Pic(S).

We have seen that the degree of the line bundle δ(O_{S}(D)) = deg(O_{S}(D)) = deg(D) = ∑ n_{P} and by composition we get the isomorphism

Div^{0}(S)/ principal divisors →^{∼}H^{1}(S, O_{S})/H^{1}(S, ℤ).

We want to compute this isomorphism, i.e. if O_{S}(D) ∈ Pic^{0}(S), how can we compute the corresponding element in H^{1}(S, O_{S})/H^{1}(S, ℤ)? We reformulate our problem slightly. The analytic pairing gives us an identification H^{1}(S, O_{S})→^{∼}H^{0}(S, Ω_{S}^{1})^{∨}, the Poincare duality gives an identification H_{1}(S, ℤ)→^{∼}H^{1}(S, ℤ). The resulting embedding i_{1} : H_{1}(S, ℤ)⟶H^{0}(S, Ω_{S}^{1})^{∨} is obtained by the following rule: We represent a homology class [c] by a cycle c and to this class we attach the linear form

φ_{c} : ω ↦ ∫_{c} ω.

Then the homomorphism [c] ↦ φ_{c} is our embedding i_{1}. Hence our problem is to compute the isomorphism

Div^{0}(S)/ principal divisors →^{∼}H^{0}(S, Ω_{S}^{1})^{∨}/H_{1}(S, ℤ).

Let D be a divisor of degree zero. Then we can find a 1-chain 𝔷_{D} whose boundary ∂𝔷_{D} = D. This 1-chain provides a map

\begin{matrix}φ_{D,𝔷_{D}} : H^{0}(S, Ω_{S}^{1}) & → ℂ \\ φ_{D,𝔷_{D}} : ω & ↦ ∫_{𝔷_{D}} ω. \end{matrix}

If we have a second 1-cycle 𝔷′_{D} which also satisfies ∂𝔷′_{D} = D then 𝔷′_{D} = 𝔷_{D} + c_{D} where c_{D} is a closed one- cycle, i.e. ∂c_{D} = 0. Hence we see that

φ_{D,𝔷_{D}} − φ_{D,𝔷′_{D}} ∈ H_{1}(S, ℤ) ⊂ H^{0}(S, Ω_{S}^{1})^{∨}

Hence we see that D defines a well-defined element

φ̃_{D} ∈ H^{0}(S, Ω_{S}^{1})^{∨}/H_{1}(S, ℤ).

Theorem of Abel: The isomorphism

Div^{0}(S)/ principal divisors →^{∼}H^{0}(S, Ω_{S}^{1})^{∨}/H_{1}(S, ℤ).

is given by [D] ↦ φ̃_{D}.

To prove this it suffices to consider the case of two points P, Q on our Riemann surface S which lie in a small disc D_{P}. This is clear because our map D → φ_{D} is a homomorphism from the group of divisors of degree zero to H^{0}(S, Ω_{S}^{1})^{∨}/H_{1}(S, ℤ) and these divisors generate the group of divisors of degree zero. We assume that our local coordinate z is zero at P and 1 at Q. We want to compute the class of the line bundle O_{S}(Q − P) in H^{1}(S, O_{S})/H^{1}(S, ℤ). To be more precise we want to find a representative of this class in Ω_{∞}^{01}(S) and identify it as a linear form on the space of holomorphic differentials. We draw the straight path γ from P to Q in our disc and we cover S by U_{2} = S\γ and U_{1} = D_{P} . The meromorphic function \frac{z}{z−1} trivializes our bundle on U_{1} and the costant function 1 trivializes it on U_{2}. Hence the holomorphic function \frac{z}{z−1} on U_{1} ∩ U_{2} = D\γ defines a Czech cocycle with values in O_{S}^{∗} and its image in H^{1}(S, O_{S}^{∗}) is the class of O_{S}(Q − P). I claim that we can define the function log \frac{z}{z−1} on D\\{γ\}. This is so because we can write down integrals

∫_{a}^{z} \frac{dz}{z} and ∫_{a}^{z} \frac{dz}{z − 1}

along a path from a point a to z which avoids γ. The values of these integrals depend not only on z but also on the homotopy class of the path. But the multivaluedness drops out if we take the difference of the integrals, which then gives us the function log \frac{z}{z−1}. The element \frac{1}{2πi} log \frac{z}{z−1} ∈ O_{S}(D\\{γ\}) is a 1-cocycle with values in O_{S}. It defines a class in ξ_{P,Q} ∈ H^{1}(S, O_{S}) which maps to the class of O_{S}(Q − P) in H^{1}(S, O_{S}^{∗}). This class ξ_{P,Q} can be represented by a closed form of type (0, 1). To find such a form we shrink the set U_{2} a little bit to a set U′_{2} so that it is the complement of a little neighborhood N of γ. ( In the picture below this neigborhood is the ”cigar” containing the path from P to Q and U′_{2} is the complement of the ”cigar”. This ”cigar” is obtained by drawing have circles of radius ϵ > 0 around P, Q and then joining the endpoints by straight lines parallel to γ. The boundary is a C_{1} - manifold.)

By the same argument as in 1.4.8 we extend the restriction of \frac{1}{2πi} log \frac{z}{z−1} to U′_{2} ∩ D_{P} to a C_{∞}-function h on D_{P} and put μ = d′′h. This form μ has compact support in U, hence it can be extended to a (0, 1)–form on S which then represents ξ_{P,Q} ∈ H^{1}(S, O_{S}) (A special case of the argument in VI.6.1.1). Again we have that the pairing of this class with a holomorphic 1-form ω is given by

∫_{S} ω ∧ μ.

To compute this integral we observe that ω ∧ μ has support in the neighborhood N of γ hence it suffices to integrate over this neighborhood. But now we can write ω ∧ μ = ω ∧ dh and our integral becomes

− ∫_{∂N} ω ∧ h

where h = \frac{1}{2πi} log \frac{z}{z−1} on the boundary of N. Letting this neighborhood shrink to γ the values of log \frac{z}{z−1} differ by 2πi on the two sides of our path γ. Hence we get that

< ξ_{P,Q}, ω >= ∫_{γ} ω

and this is Abel’s theorem in the case that our divisor is Q − P, and P, Q close to each other. This theorem of Abel is the source for the so called self- duality of the Jacobian, which will be discussed in detail in a later section (V.2.3). We pick a point P_{0} and consider the morphism

i_{P_{0}} : S → J

which is given by

i_{P_{0}} : P ↦ (P) − (P_{0}).

This is clearly a holomorphic map and we want to explain how Abels theorem gives us its differential. The tangent space of J at any point is H^{1}(S, O_{S}) and hence we see that the space of holomorphic one forms on J is H^{0}(S, Ω_{S}). Hence i_{P_{0}} yields a ℂ-linear map between the spaces of holomorphic 1-forms

i_{P_{0}}^{∗} : H^{0}(J, Ω_{J}^{1}) = H^{0}(S, Ω_{S}^{1}) → H^{0}(S, Ω_{S}^{1})

and I claim that this map must be the identity. If ω ∈ H^{0}(J, Ω_{J}) is a holomorphic 1-form and if X ∈ T_{P} is a tangent vector at P_{0} we compute i_{P_{0}}^{∗}(ω)_{P} (X). We choose a local coordinate z at P, then we may assume that X = \frac{∂}{∂z}.

The i_{P_{0}}^{∗}(P + h · \frac{∂}{∂z}) is the linear form

ω → ∫_{P_{0}}^{P +h \frac{∂}{∂z}} ω = ∫_{P_{0}}^{P} ω + ∫_{P}^{P +h \frac{∂}{∂z}} ω

and this yields

i_{P_{0}}^{∗}(ω)_{P} (X) = i_{P_{0}}^{∗}(ω)_{P} (\frac{∂}{∂z}) = ω_{P} (\frac{∂}{∂z})

and hence i_{P_{0}}^{∗}(ω) = ω. Hence we see that i_{P_{0}} gives us an holomorphic embedding of the curve into its Jacobian. This map also induces a homomorphism between the Picard groups

i_{P_{0}}^{∗} : Pic(J) → P ic(S).

We will define the subgroup Pic^{0}(J) (V.2.1.1) and we will prove that the restriction

ti_{P_{0}} : Pic^{0}(J) → Pic^{0}(S)

is an isomorphism. This is the so called self-duality of J.

1.7. Riemann period relations The cup product <, >_{∪} defines a non degenerate alternating pairing on our lattice Γ . On the other hand we have Γ ⊗ ℝ→^{∼}H^{1}(S, O_{S}) and this identification provides a complex structure I on Γ ⊗ ℝ, namely the one which is induced by the multiplication by i on H^{1}(S, O_{S}). We will show that the complex structure I is an isometry for the extension of <, >_{∪} to Γ_{ℝ}. Therefore this pairing is the imaginary part of a hermitian form h on (Γ ⊗ ℝ, I) (See IV.10.2.1). We compute this form h and in addition it will turn out to be positive definite.

We can define a hermitian scalar product on \overline{H^{0}(S, Ω_{S}^{1})}. If we have two antiholomorphic forms ω̄_{1}, ω̄_{2} ∈ \overline{H^{0}(S, Ω_{S}^{1})} we put

h < ω̄_{1}, ω̄_{2} >= −2i ∫_{S} ω̄_{1} ∧ ω_{2}.

If we write locally

ω̄_{1} = f_{1}dz̄, ω̄_{2} = f_{2}dz̄

then the integrand becomes

f_{1}(dx − idy) ∧ f̄_{2}(dx + idy) = 2if_{1}f̄_{2}dx ∧ dy

hence we see that h is a positive definite hermitian form. Now we take to cohomology classes ξ, η ∈ H^{1}(S, ℤ). Using the de-Rham isomorphism, we can represent them by differential forms which we can decompose

\begin{aligned}ω_{ξ} & = \overline{ω_{ξ}^{01}} + ω_{ξ}^{01} \\ ω_{η} & = \overline{ω_{η}^{01}} + ω_{η}^{01}. \end{aligned}

The cup product pairing is given by integrating the representing differential forms

< ξ, η >_{cup}= ∫_{S} ω_{ξ} ∧ ω_{η} = − ∫_{S} ω_{η}^{01} ∧ \overline{ω_{ξ}^{01}} + ∫_{S} ω_{ξ}^{01} ∧ \overline{ω_{η}^{01}} = − Im h(ω_{ξ}^{01}, ω_{η}^{01}).

We have the isomorphism

j : H^{1}(S, ℤ)→^{∼}Γ ⊂ H^{1}(S, O_{S})→^{∼}\overline{H^{0}(S, Ω_{S}^{1})}

and it is clear the the classes j(ξ) (resp.j(η)) are represented by ω_{ξ}^{01} (resp. ω_{η}^{01}). We can transport the cup product pairing via j to Γ then we get the famous

Riemann period relations: The restriction of the imaginary part of the hermitian form h to Γ is the cup product times -1. Especially we can conclude that the values of Im h on Γ × Γ are integers and this form is a perfect pairing.

V.2. Line bundles on complex tori

2.1 Construction of line bundles: These period relations are of great importance, because they allow the construction of line bundles on J. The positivity of the form h will ensure that these bundles will be ample and this means roughly that high positive powers of this bundle have many sections (See below ???). To explain this construction we consider a more general situation. Let V be a complex vector space of dimension g and let Γ ⊂ V be a lattice in V, this means that Γ is a free ℤ-module of rank 2g which sits in V as a discrete submodule. The quotient A = V/Γ is a compact complex analytic variety which also carries the structure of a complex analytic abelian group, it is a complex torus (V.1.2). We have Γ_{ℝ}→^{∼}V as real vector space and as usual we denote by I the complex structure on Γ_{ℝ} induced by this isomorphism. We change our point of view slightly. Our starting point is a free abelian group Γ of rank 2g on which we have an alternating 2-form

⟨ , ⟩ : Γ × Γ ⟶ ℤ.

A second datum is a complex structure I : Γ_{ℝ} → Γ_{ℝ} which is an isometry for the pairing, i.e. ⟨Ix, Iy⟩ = ⟨x, y⟩ for all x, y ∈ Γ_{ℝ}. Then we can put V = (Γ_{ℝ}, I) and consider it as a complex vector space and V/Γ is our complex torus. Let H on V = (Γ_{ℝ}, I) be the hermitian form obtained from (⟨ , ⟩, I) (See IV.10.2.1). The pairing ⟨ , ⟩ allows us to construct certain vector bundles

L(⟨ , ⟩, η, φ)

which depend on additional data φ and η where

φ ∈ Hom(Γ, ℂ)

and where η is a map

η : Γ/2Γ → \frac{1}{2}ℤ/ℤ

which satisfies the compatibility relation

\frac{1}{2}⟨γ_{1}, γ_{2}⟩ mod ℤ + η(γ_{1} + γ_{2}) − η(γ_{1}) − η(γ_{2}) = 0  (∗ ∗ ∗)

for all γ_{1}, γ_{2} ∈ Γ. We say that η is adapted to the alternating form < , > . To define such a bundle we consider an open connected neighborhood U of a point x ∈ A which is so small that the connected components U_{α} in the inverse image of U map isomorphically to U under the projection

p : V → A.

For any two such components U_{α}, U_{β} ⊂ p^{−1}(U) there is exactly one element γ ∈ Γ such that γ + U_{α} = U_{β}. We define a sheaf L_{H}(⟨ , ⟩, η, φ) = L(⟨ , ⟩, η, φ) whose sections over U are the holomorphic functions

f : p^{−1}(U) → ℂ

which satisfy the transformation rule

f(z + γ) = f(z)e^{π(H(z,γ)+ \frac{1}{2} H(γ,γ))+2πi(φ(γ)+η(γ))}.

The reader should notice that e^{2πiη(γ)} is well defined and is equal to ±1. I claim that giving such a function is the same as giving a holomorphic function on any of the connected components U_{α} in p^{−1}(U) and then extending it to the other components by the transformation rule. To see this we have to check consistency which means we have to verify that

f(z + γ_{1} + γ_{2}) = f((z + γ_{1}) + γ_{2}).

To do this we compute both sides:

\begin{matrix}f(z + γ_{1} + γ_{2}) = f(z)e^{π(H(z,γ_{1}+γ_{2})+ \frac{1}{2} H(γ_{1}+γ_{2},γ_{1}+γ_{2}))+2πi(φ(γ_{1}+γ_{2})+η(γ_{1}+γ_{2}))} \\ = f(z)e^{π(H(z,γ_{1})+H(z,γ_{2})+ \frac{1}{2} (H(γ_{1},γ_{1})+H(γ_{2},γ_{2})+H(γ_{1},γ_{2})+H(γ_{2},γ_{1})))+2πi(φ(γ_{1})+φ(γ_{2})+η(γ_{1}+γ_{2}))}. \end{matrix}

For the other side we get

\begin{matrix}f((z + γ_{1}) + γ_{2}) = f(z + γ_{1})e^{π(H(z+γ_{1},γ_{2})+ \frac{1}{2} H(γ_{2},γ_{2}))+2πi(φ(γ_{2})+η(γ_{2}))} \\ = f(z) · e^{π(H(z,γ_{1})+ \frac{1}{2} H(γ_{1},γ_{1})+H(z,γ_{2})+H(γ_{1},γ_{2})+ \frac{1}{2} H(γ_{2},γ_{2}))+2πi(φ(γ_{1})+φ(γ_{2})+η(γ_{1})+η(γ_{2}))}. \end{matrix}

The exponential factors are equal because their quotient is

\begin{matrix}e^{π( \frac{1}{2} (H(γ_{2},γ_{1})−H(γ_{1},γ_{2})))+2πi(η(γ_{1}+γ_{2})−η(γ_{1})−η(γ_{2}))} \\ = e^{2πi( \frac{1}{2} Im H(γ_{2},γ_{1})+η(γ_{1}+γ_{2})−η(γ_{1})−η(γ_{2}))}. \end{matrix}

and this is equal to 1 since we required (∗ ∗ ∗) for η. If we consider the map

\begin{matrix}C_{H}(φ, η) : γ → O_{V} (V) \\ γ ↦ π(H(z, γ) + \frac{1}{2}H(γ, γ)) + 2πi(φ(γ) + η(γ)) \end{matrix}

as a map from the group Γ to the holomorphic functions on V then our computation says that this is a 1-cocycle modulo the constant functions with values in 2πiℤ. Hence we see that

L_{H}(⟨ , ⟩, η, φ)(U) ∼ O(U_{α})

for any component in U_{α} ⊂ p^{−1}(U). and this means that L_{H}(⟨ , ⟩, η, φ) is a line bundle. Now we see why the integrality of Im H on Γ × Γ is so important.

The data H and (< , >, I) determine eachother, therefore we may suppress the subscript H in the notation. On the other hand it follows from IV.10.2.1 that the pair of ℝ-bilinear forms < , >, H determines the complex structure I which is not directly visible in the definition of the line bundle. Hence it may be sometimes useful to keep the H. In IV.6.6.2 we have shown that the second cohomology group (See ???)

H^{2}(A, ℤ) = H^{2}(V/Γ) = Hom(Λ^{2}Γ, ℤ).

We have the exact sequence

0 → H^{1}(A, ℤ) → H^{1}(A, O_{A}) → H^{1}(A, O_{A}^{∗}) → H^{2}(A, ℤ)

and we leave it as an exercise to the reader to verify that

c_{1}(L_{H}(⟨ , ⟩, η, φ)) = Im H|Γ × Γ.

It is not too difficult to show that we can find an η for a given ⟨ , ⟩ . It is not unique, but it is not hard to see that for two choices η_{H}, η′_{H} we can find a homomorphism δ : Γ → ℂ so that δ(Γ) ⊂ \frac{1}{2}ℤ and δ(γ) = η_{H}(γ) − η′_{H}(γ) ∈ ℤ. Then it is clear that

L_{H}(⟨ , ⟩, η, φ)→^{∼}L_{H}(⟨ , ⟩, η′, φ + δ)

We want to show that this construction gives us all line bundles on A. In any case it is clear that the group of Chern classes of line bundles is the kernel of the homomorphism

H^{2}(A, ℤ) ⟶ H^{2}(A, O_{A}).

We have seen that H^{2}(A, ℤ) = Hom(Λ^{2}, ℤ) and it is an easy exercise in linear algebra that an element

c ∈ H^{2}(A, ℤ) = Hom(Λ^{2}Γ, ℤ)

goes to zero in H^{2}(A, O_{A}) if and only if the extension

c_{ℝ} : (Γ ⊗ ℝ) ∧ (Γ ⊗ ℝ) → ℝ

satisfies c_{ℝ}(Ix, Iy) = c_{ℝ}(x, y), i.e. the complex structure is an isometry. But for those classes c we gave an explicit construction of line bundles with Chern classe c. We can take any L(c, η, φ).

2.1.1 This means that our construction at least provides enough line bundles to fill up the module of possible Chern classes of line bundles. It is of course clear that these alternating forms c which satisfy c_{ℝ}(Ix, Iy) = c_{ℝ}(x, y) form a finitely generated subgroup NS(A) of Hom(Λ^{2}Γ, ℤ). This group is called the Neron-Severi group.

We should be aware that this group NS(A) can be trivial, actually this is the case for a generic choice of the complex structure on Γ_{ℝ}.

To get the group of all line bundles we return to its description as H^{1}(A, O_{A}^{∗}) and we define

Pic^{0}(A) = ker(δ : Pic(A) → H^{2}(A, ℤ))

From our familiar exact sequence we get

Pic^{0}(A) = H^{1}(A, O_{A})/H^{1}(A, ℤ).

Again we get from IV.6.6.2 that H^{1}(A, ℤ) = H^{1}(ℂ^{g}/Γ; ℤ) = Hom(Γ, ℤ). To compute H^{1}(A, O_{A})we consider the Dolbeault complex

0 → Ω_{∞}^{0}(O_{A})(A) → Ω_{∞}^{01}(O_{A})(A) → Ω_{∞}^{02}(O_{A})(A) →

The tangent bundle of A is trivial. Using the translations we can identify the tangent space at any point to T_{A,0} ≃ V the tangent space at zero. Hence the bundle of differentials is also trivial and at any point

Ω_{A,x}^{1} = Hom_{ℂ}(V, ℂ).

It is clear that the bundle Ω_{A,}^{01} is also trivial and if we give a basis to V and write z = (…, z_{α} …) ∈ ℂ^{g} = V then the global sections Ω_{∞}^{01}(A) are given by

ω = ∑ f_{α}dz̄_{α}

where f_{α} is a C_{α}-function on A. We apply the principles of Hodge theory: we choose a positive definite hermitian form on the tangent bundle, which we get from a hermitian form on V = T_{A,0}. We choose it in such a way that the basis vectors above form an orthonormal basis. Then it is an easy computation to show that

Δ′′ω = (d′′δ′′ + δ′′d′′)ω = (∑_{β} \frac{∂^{2}f_{α}}{∂z_{β}∂z̄_{β}} ) dz̄_{α}

and in IV.10.3 we proved that

ℍ^{01}(Ω_{∞}^{01}(A)) = {∑_{α} c_{α}dz̄_{α}|c_{α} ∈ ℂ}.

and that we get an isomorphism

{∑ c_{α}dz̄_{α}|c_{α} ∈ ℂ} ⟶^{∼} H^{1}(A, O_{A}).

It does not depend on the metric, it is induced from the embedding of the translation invariant differential form into the space of all differential forms.

The space on the left is canonically isomorphic to Hom_{ℂ−antilin}(V, ℂ) (See IV9.2.2) and we obtain for our complex torus A = V/Γ

Hom_{ℂ−antilin}(V, ℂ) = Hom_{ℂ}(V̄ , ℂ) = V̄^{∨} = H^{1}(A, O_{A})

To get the group Pic^{0}(A) we have to divide by the subgroup H^{1}(A, ℤ) and we have to remind ourselves how this group sits inside V̄^{∨} = H^{1}(A, O_{A}). This is clear: We know that H^{1}(A, ℤ) = Hom(Γ, ℤ) and

Hom(Γ, ℤ) ⊂ Hom(Γ, ℂ) = Hom(Γ ⊗_{ℤ} ℂ, ℂ)

and by construction

Γ ⊗_{ℤ} ℂ = V ⊕ V̄ = V^{10} ⊕ V^{01}

Now the embedding H^{1}(A, ℤ) ↪ H^{1}(A, O_{A}) is given by extending ϕ ∈ Hom(Γ, ℤ) to an element in Hom(Γ ⊗_{ℤ} ℂ, ℂ) and then restricting it to the summand V̄ = V^{01}. Hence we get an isomorphism

c : Pic^{0}(A)→^{∼} Hom(V^{01}, ℂ)/ Hom(Γ, ℤ)→^{∼} Hom(Γ, ℂ)/(Hom_{ℂ}(V, ℂ) + Hom(Γ, ℤ)).

We want to invert this isomorphism. We constructed the line bundles L_{H}(⟨ , ⟩, η, φ) where φ ∈ Hom(Γ, ℂ). We denote the restrictions of φ to V and V̄ respectively by φ_{10}, φ_{01} and hence φ = (φ_{10}, φ_{01}). It is clear from the construction that

2.1.2 Lemma : a) The two line bundles L_{H}(⟨ , ⟩, η, φ) and L_{H}(⟨ , ⟩, η, φ) are isomorphic if φ − φ′ = (ψ, 0).

b) The bundles L_{H}(⟨ , ⟩, η, φ) and L_{H}(⟨ , ⟩, η, φ′) are actually the same line bundles if

φ − φ′ ∈ Hom(Γ, ℤ)

Proof: To see a) we observe that e^{2πiψ(z)} is holomorphic on V and multiplication by this function provides an isomorphism between L_{H}(⟨ , ⟩, η, φ) and L_{H}(⟨ , ⟩, η, φ). The assertion b) is obvious because e^{2πiφ(γ)} = e^{2πiφ′(γ)} for all γ ∈ Γ.

If now the alternating form ⟨ , ⟩ = O is the trivial nullform then we choose η_{O} = 0. We find

L(O, 0, φ) ⊗ L(O, 0, φ′) = L(O, 0, φ + φ′),

our construction of line bundles yields a homomorphism

Hom(Γ, ℂ) → Pic^{0}(A).

which by the previous Lemma factors through Hom_{ℂ}(V, ℂ) + Hom(Γ, ℤ). Hence our construcion yields a homomorphism

d : Hom(Γ, ℂ)/(Hom_{ℂ}(V, ℂ) + Hom(Γ, ℤ)) → Pic^{0}(A).

I leave it as an exercise to show

Proposition 2.1.3 : These two homomorphisms c, d are inverse to each other.

Corollary 2.1.4: If A = V/Γ is a complex torus then the group Pic^{0}(A) has again the structure of a complex torus and is canonically isomorphic to V̄^{∨}/Γ^{∨}.

This torus is called the dual torus and denoted by A^{∨}. Our considerations also imply that the bundles with a given Chern class form a principal homogeneous space under

Hom_{ℂ}(Γ, ℂ)/(Hom_{ℂ}(V, ℂ) ⊕ Hom(Γ, ℤ)).

But this description requires a choice of an η adapted to < , >. We have seen that changing η can be corrected by the modification of the linear form φ. Now it is clear that all line bundles L on A are of the form L_{H}(⟨ , ⟩, η, φ).

2.1.5 We want to construct families of line bundles. We have the isomorphism

Pic^{0}(A) ⟶^{∼} Hom(Γ_{ℂ}^{01}, ℂ)/Γ^{∨}.

We consider the line bundles

L(⟨ , ⟩, η, (0, ψ))

where ψ ∈ Hom(Γ_{ℂ}^{01}, ℂ). It is clear that these line bundles can be seen as members of a family of line bundles on A × Hom(Γ_{ℂ}^{01}, ℂ): We consider the line bundle L̃(⟨ , ⟩, η) on A × Hom(Γ_{ℂ}^{01}, ℂ) whose sections over an open set

U × Hom(Γ_{ℂ}^{01}, ℂ)

are given by holomorphic functions

h : π^{−1}(U) × Hom(Γ_{ℂ}^{01}, ℂ) ⟶ ℂ

which satisfy

h(u + γ, ψ) = e^{π(H(u,γ)+ \frac{1}{2} H(γ,γ))+2πi(ψ(γ)+η(γ))}h(u, ψ).

Let

p_{2} : A × Hom(Γ_{ℂ}^{01}, ℂ)/Γ^{∨} = A × A^{∨}

be the projection on the second factor, we want to “push” this bundle down to a bundle P(⟨ , ⟩, η) on the quotient, i.e. we want that

p_{2}^{∗}(P(⟨ , ⟩, η))→^{∼}L̃(⟨ , ⟩, η)

To see how this can be done we have to find out what happens if we modify ψ by an element λ_{01} in Γ^{∨}. Such an element is the image of an element λ ∈ Hom(Γ, ℤ) under the restriction map Hom(Γ_{ℂ}, ℂ) → Hom(Γ_{ℂ}^{01}, ℂ). We can write

λ = λ_{10} + λ_{01},

and we saw in Lemma 2.1.2 that

L(⟨ , ⟩, η, (0, ψ)) = L(⟨ , ⟩, η, (0, ψ) + λ)

In the same lemma above we showed that we can write an explicit isomorphism

ψ_{λ} : L(⟨ , ⟩, η, (λ_{10}, ψ + λ_{01})) ⟶^{∼} L(⟨ , ⟩, η, (0, ψ + λ_{01}))

which on the sections was given by multiplication by e^{−2πiλ_{10}(z)}. Now it is clear that these isomorphisms are consistent. If we consider a sum λ′ + λ′′ = μ, then ψ_{μ} = ψ_{λ′} ∘ ψ_{λ′′}. This allows to “push” down the bundle. We can define the bundles

P(⟨ , ⟩, η, y)

where now y ∈ A^{∨} = Hom(Γ_{ℂ}^{01}, ℂ)/Γ^{∨} and they form a family P(< , >, η) on A × A^{∨}.

2.2 Homomorphisms between complex tori. If we have two such tori

V_{1}/Γ_{1} = A_{1}, V_{2}/Γ_{2} = A_{2}

then an analytic homomorphism ϕ : A_{1} → A_{2} is of course the same thing as a ℂ-linear map ϕ : V_{1} → V_{2} which maps the lattice Γ_{1} into Γ_{2}. We may also view ϕ as an element ϕ : Γ_{1} → Γ_{2} which after extension to a linear map Γ_{1} ⊗ ℝ → Γ_{2} ⊗ ℝ respects the complex structures on Γ_{1} ⊗ ℝ = V_{1}, Γ_{2} ⊗ ℝ = V_{2}. We summarize: 2.2.1 The module Hom(A_{1}, A_{2}) is a submodule of Hom(Γ_{1}, Γ_{2}). It consists of those elements which after extension to ℝ commute with the complex structures. A homomorphism φ : A_{1} → A_{2} also induces a homomorphism between the Picard groups

φ^{∗} : Pic(A_{2}) ⟶ Pic(A_{1})

which is induced by the pull back of line bundles.

We can restrict this homomorphism to the groups Pic^{0}(A_{2}) = A_{2}^{∨} and Pic^{0}(A_{1}) = A_{1}^{∨} and denote this restriction by

φ^{∨} : A_{2}^{∨} ⟶ A_{1}^{∨}.

A priori this is an homomorphism between abstract groups but from the explicit description of the isomorphism Pic^{0}(A_{i}) ˜ ⟶A_{i}^{∨} it becomes clear:

2.2.1.1. Proposition: The element φ^{∨} is a homomorphism of complex tori. This homomorphism– viewed as an element in Hom(Γ_{2}^{∨}, Γ_{1}^{∨})- is simply the adjoint of the element φ ∈ Hom(Γ_{1}, Γ_{2}). Especially we see that the function ϕ ↦ ϕ^{∨} is additve. To see that this is true, we consider an element x ∈ Pic^{0}(A_{2}). We gave an explicit construction of a line bundle L_{x} corresponding to x starting from a linear map

λ_{x} : Γ_{2} ⟶ ℂ,

which after extension to Γ_{2} ⊗ ℂ and restriction to V̄_{2} maps to x: For an open set V ⊂ A_{2} the space of sections

L_{x}(V) = {f : π^{−1}(V) → ℂ | f is holomorphic and f(z + γ) = f(z)e^{2πiλ_{x}(γ)}}

and the fibre of L_{x} in a point y ∈ H^{1}(S, O_{S})/Γ is given by

(L_{x})_{y} = {f : π^{−1}(y) → ℂ | f(y + γ) = f(y)e^{2πiλ_{x}(γ)}} .

If now φ : A_{1} → A_{2} and if y_{1} ∈ A_{1}, then

φ^{∗}(L_{x})_{y_{1}} = (L_{x})_{φ(y_{1})}.

If we consider the diagram

Γ_{1} ⟶^{φ} Γ_{2} ⟶^{λ_{x}} ℂ
↓  ↓
V_{1} ⟶^{φ} V_{2}

then we see that φ^{∗}(L) is the line bundle defined by the composition

λ_{x} ∘ φ : Γ_{1} ⟶ ℂ.

and this proves the desired formula.

We may also consider the induced map

\overline{φ}^{∗} : NS(A_{2}) ⟶ NS(A_{1}).

An element e ∈ NS(A_{2}) is an alternating form e : Γ_{2} × Γ_{2} → ℤ and φ^{∗}(e) is simply the form on Γ_{1} × Γ_{1} induced by φ, i.e.

φ^{∗}(e)⟨γ_{1}, γ′_{1}⟩ = e⟨φ(γ_{1}), φ(γ_{2})⟩.

This implies of course that the function ϕ ↦ ϕ̄^{∗} is not linear, it is in fact quadratic which means that we have

\overline{ϕ + ψ}^{∗} = ϕ̄^{∗} + ψ̄^{∗}+ < ϕ, ψ >  2.2.1.2

where (ϕ, ψ) →< ϕ, ψ > is bilinear.

2.2.2 We come to another interpretation of the Neron-Severi group. If we pick an element e ∈ NS(A), then this element defines a homomorphism

\begin{aligned}ϕ_{e} & : & Γ & ⟶ & Γ^{∨} \\ ϕ_{e} & : & γ & ⟶ & \{γ′ → e⟨γ, γ′⟩\}. \end{aligned}

It is clear that the condition that I is an isometry for the extension e_{ℝ} implies that it extends to a ℂ linear homomorphism

ψ_{e} : V → V̄^{∨}

We have the inclusions Γ ⊂ V and Γ^{∨} ⊂ V̄^{∨} and it is clear that ψ_{e} maps Γ into Γ^{∨} and induces ϕ_{e} on the lattices. Therefore we see that we have a canonical homomorphism

Φ : NS(A) ⟶ Hom(A, A^{∨}).

Any element ϕ : Γ → Γ^{∨} has a transpose

ϕ^{∨} : Γ^{∨∨} = Γ → Γ^{∨}.

We can define the alternating elements Hom_{alt}(A, A^{∨}) to be the elements which satisfy ϕ^{∨} = −ϕ and it is an easy exercise in linear algebra to show that our above map Φ provides an isomorphism

Φ : NS(A) ⟶ Hom_{alt}(A, A^{∨}).  (2.2.2.1)

The inverse of this homomorphism is given by the map that sends an alternating element ϕ to the form

e_{ϕ} < γ, γ′ >= ϕ(γ)(γ′).

( I hope that I got the sign right).

2.2.3 We also have a homomorphism Ψ : Hom(A, A^{∨}) → NS(A). To construct this homomorphism we start from the alternating form

τ : (Γ ⊕ Γ^{∨}) × (Γ ⊕ Γ^{∨}) → ℤ,

which is defined by

τ < (γ_{1}, ψ_{1}), (γ_{2}, ψ_{2}) >= ψ_{2}(γ_{1}) − ψ_{1}(γ_{2}).

If now ϕ : Γ → Γ^{∨} then this yields the homomorphism

Id ×ϕ : Γ → Γ ⊕ Γ^{∨}

and we can restrict τ via this inclusion to Γ. The resulting form

< γ_{1}, γ_{2} >_{ϕ}= ϕ(γ_{1})(γ_{2}) − ϕ(γ_{2})(γ_{1}) = e_{ϕ} < γ_{1}, γ_{2} > −e_{ϕ^{∨}} < γ_{1}, γ_{2} >

on Γ is alternating. It depends only on the alternating component of ϕ and for alternating ϕ the map ϕ ↦< , >_{ϕ} is twice the inverse of Φ.

We give a different construction of the homomorphism Φ which works with the bundles themselves rather than with their Chern classes.

To our element e ∈ NS(A) we choose a line bundle L with c_{1}(L) = e, in other words we choose an adapted η and a φ : Γ → ℂ and consider the line bundle

L = L(e, η, φ).

Any element x ∈ A induces a translation T_{x} : y ↦ x + y on A and we can consider the line bundle T_{x}^{∗}(L) ⊗ L^{−1}. To compute this line bundle we choose an element x̃ in the fibre p^{−1}(x). Let H_{e} the attached hermitian form then the fibre of T_{x}^{∗}(L) at a point z is equal to the fibre of L at x + z and therefore it is given by the functions which satisfy

f(z̃ + x̃ + γ) = f(z̃ + x̃)e^{π(H_{e}(z̃+x̃,γ)+ \frac{1}{2} H_{e}(γ,γ))+2πi(φ(γ)+η_{He} (γ))}

for all z̃ ∈ p^{−1}(z). Comparing this to the fibre of L at z yields that the fibre of T_{x}^{∗}(L) ⊗ L^{−1} is given by the functions

f(z̃ + γ) = f(z̃)e^{πH_{e}(x̃,γ)}

This line bundle is obtained from the linear form φx̃ : γ ↦ H_{e}(x̃, γ), in other words it is isomorphic to L_{O}(φx̃). An easy calculation shows that this linear form is of the type (0, φ_{10}) in other words it is trivial on the first component in the decomposition

Hom_{ℂ}(Γ ⊗ ℂ, ℂ) = Hom_{ℂ}(V ⊕ V̄ , ℂ)

The same calculation shows that the linear form ψx̃ : γ ↦ H(γ, x̃) is of type (ψ_{01}, 0). Therefore we do not change the isomorphism class of the line bundle if we replace φx̃ by ϕx̃(γ) = H(x̃, γ) − H(γ, x̃) = 2i Im H_{e}(x̃, γ) = 2ie < x̃, γ > . Hence we see that T_{x}^{∗}(L) ⊗ L^{−1}→^{∼}L(O, 0, e < x̃, >) where e < x̃, > is a linear map from Γ to ℂ.

Therefore it is clear that

2.2.4 The map

x ↦ T_{x}^{∗}(L) ⊗ L^{−1}

from A to A^{∨} is a homomorphism and this homomorphism is equal to Φ(e). This new description of Φ has the advantage that it is constructed in terms of the bundles rather than in terms of the Chern classes. It is of course important that this homomorphism depends only on the Chern class of the line bundle L. We have also a description of 2× the inverse of this homomorphism in terms of line bundles. We observe that the alternating form τ on Γ⊕Γ^{∨} defines an element in NS(A×A^{∨}) in other words the complex structure on (Γ ⊕ Γ^{∨})_{ℝ} is an isometry. Hence we can construct a line bundle N on A × A^{∨} whose Chern class is τ. If now ϕ : A → A^{∨} we get a homomorphism Id ×ϕ : A → A×A^{∨}. The pullback (Id ×ϕ)^{∗}(N) gives us a line bundle N_{ϕ} and Φ(N_{ϕ}) = 2ϕ.

2.2.5 If an element c = c_{1}(L) of the Neron-Severi group is called rationally non degenerate if the alternating pairing c_{ℚ} : Γ_{ℚ} × Γ_{ℚ} → ℚ is non degenerate then it is clear that the induced homomorphism ϕ_{c} : Γ → Γ^{∨} is injective and that the image ϕ_{c}(Γ) ⊂ Γ^{∨} has finite index. From our description of the complex tori it is immedeately clear that the kernel of

x ↦ T_{x}^{∗}(L) ⊗ L^{−1}

is canonically isomorphic to Γ^{∨}/ϕ_{c}(Γ) we have an isomorphism

ker(ϕ_{c})→^{∼}Γ^{∨}/ϕ_{c}(Γ).

2.3. The self duality of the Jacobian. We specialize these considerations to the Jacobian J of our Riemann surface S. I recall

J = H^{1}(S, O_{S})/Γ = \overline{H^{0}(S, Ω_{S}^{1})}/Γ

where we identify H^{1}(S, O_{S}) = \overline{H^{0}(S, Ω_{S}^{1})} by means of the Dolbeault isomorphism and Hodge theory. The submodule Γ is the image of H^{1}(S, ℤ) under the homomorphism j : H^{1}(S, ℤ) → Γ (see 1.5).

On this module Γ we have the priviledged alternating form given by the cup product e_{0} : Γ × Γ → ℤ. It provides an isomorphism

φ_{e_{0}} : Γ ⟶ Γ^{∨}.

We need to know that this isomorphism is compatible with the complex structures. This follows from the Riemann period relations (see 1.7) they imply that e_{0} is the imaginary part of a hermitian form

h_{e_{0}} : \overline{H^{0}(S, Ω_{S}^{1})} × \overline{H^{0}(S, Ω_{S}^{1})} ⟶ ℂ.

which then gives an isomorphism of the complex vector spaces - also denoted by φ_{e_{0}} - namely

φ_{e_{0}} : \overline{H^{0}(S.Ω_{S}^{1})} ⟶ H^{0}(S, Ω_{S}^{1})^{∨}

and this alltogether gives us an isomorphism

Φ(e_{0}) :  J  ⟶  J^{∨}
∥  ∥
H^{1}(S, O_{S})/Γ ⟶ H^{0}(S, Ω_{S}^{1})^{∨}/Γ^{∨}

Our isomorphism φ_{e_{)}} is of course the composition of the Dolbeault isomorphism and the analytic duality (See 1.4.8):

H^{1}(S, O_{S}) ⟶^{∼} \overline{H^{0}(S, Ω_{S}^{1})} ⟶^{∼} H^{0}(S, Ω_{S}^{1})^{∨}

The isomorphism - or what is the same - the class e_{0} ∈ NS(J) is called the canonical polarization of J. It is an additional datum attached to the complex torus. At the end of the discussion of Abel’s theorem we discussed the embedding

i_{P_{0}} : S ⟶ J

which provided a homomorphism

ti_{P_{0}} : Pic^{0}(J) ⟶ Pic^{0}(S)
∥  ∥
J^{∨}  ⟶  J

and now it is clear from these computations that ^{t}i_{P_{0}} is the inverse of the canonical polarization.

2.4 Ample line bundles and the algebraicity of the Jacobian. Let us assume that we have an alternating form e =< , > for which I is an isometry. So far it did not play any role that the hermitian form H attached to this form e was positive definite. We want to discuss the implication of te positivity and we will see that it implies that sufficiently high powers of this bundle will have many sections. I refer to the section on Kähler manifolds (See IV.10.2.2) There we attached a 2-form ω_{h} to any (positive definite) hermitian form h on the tangent bundle. In our case here the tangent bundle of A = V/Γ is trivial and isomorphic to Γ ⊗ ℝ at the origin. Then our 2form ω_{h} on A is invariant by translation and is our form e at the origin. It is clear that ω_{h} is closed, it defines a class [ω_{h}] ∈ H^{2}(M, ℤ) and of course

[ω_{h}] = e.

If now in addition the hermitian form H_{<,>} is positive definite, then <, > gives us a Kähler metric on M = V/Γ whose class is integral.

Now I want to formulate the famous embedding theorem of Kodaira.

2.4.1 Theorem: Let X be a compact complex manifold. Let us assume that we have a hermitian metric h on T_{X} whose corresponding class ω_{h} is closed and defines an integral class in H^{2}(X, ℤ). Then we can find a line bundle L on X whose Chern class c_{1}(L) = [ω_{h}]. For n >> 0 we have that H^{q}(X, L^{⊗n}) = 0 for all q > 0 and we get a projective embedding

\begin{matrix}Θ_{n} : X & ⟶ ℙ(H^{0}(X, L^{⊗n})) \\ x & ⟶ H_{x} = \{s ∈ H^{0}(X, L^{⊗n})|s(x) = 0\}. \end{matrix}

We have a tautological example for this theorem. If our manifold X is the projective space ℙ^{n}(ℂ) and the bundle is L = O_{ℙ^{n}(ℂ)}(1) then we can take n = 1 and the bundle provides an embedding and a closer look shows that this embeddeding is simply the identity. In the general case it is clear from the construction that the restriction by Θ_{n} of the bundle O_{ℙ(H^{0}(X,L^{⊗n}))}(1) to X is our bundle L^{⊗n} This theorem also applies to our complex tori, we need that our class e =< , > H = H_{<,>} is positive definite. Then any bundle L = L_{H}(⟨ , ⟩, η, φ) is of the type as in the theorem. If we can find such an alternating form e for which the attached hermitian form is positive definite, then we say that our complex torus is an abelian variety. We will almost prove the above theorem of Kodaira in the special case of abelian varieties. This will be done by showing that the bundles have a lot of sections. After that we will make it more precise what a projective embedding is.

2.4.2 The spaces of sections and Theta functions: We want to compute the space of global sections in our line bundles L(⟨ , ⟩, η, φ). To do this we give a different description of these bundles: we modify the cocycle C_{H}(z, ω) by a boundary. To get this modifiction we choose a sublattice G ⊂ Ω of rank g such that Ω/G is also free and the alternating form <, > is trivial on G. This is possible because our form is alternating. Then our hermitian form H_{<,>} restricted to G takes real values and is symmetric. Since we have G ⊕ IG = V we can extend this restriction to a symmetric ℂ-bilinear form h on V, so we have H(z, ω) = h(z, ω) for all ω ∈ G. For simplicity I want to assume that the restrictions of η and φ to G are trivial. Actually we can assume this without loss of generality. The function η|G satisfies η(e_{1} + e_{2}) = η(e_{1}) + η(e_{2}) for e_{1}, e_{2} ∈ G. We can construct a linear form φ′ : Ω → \frac{1}{2}ℤ such that η|G = φ′|G mod 2. Now we modify η by φ′ such that η(ω) = 0 for all ω ∈ G. Once we have done this we also modified φ. Now we can restrict the form φ : Ω → ℂ to G and extend this φ to a linear ℂ-form ψ on V. We have seen that L(< , >, η, φ) ≃ L(< , >, 0, φ − ψ) and hence we may also assume that φ restricted to G is trivial. We look at our 1- cocycle mod 2πiℤ

ω ↦ π(H(z, ω) + \frac{1}{2}H(ω, ω)) + 2πi(φ(ω) + η(ω)) = C_{H}(z, ω).

This cocycle is uniquely determined by Ω, < , > and the complex structure I and η. We change our notation slightly and we denote the resulting bundle by L(C_{H}, φ). Now we consider global sections in this bundle and this means that we consider holomorphic functions which which satisfy

f(z + ω) = f(z)e^{C_{H}(z,ω)+2πiφ(ω)}

We modify these functions and consider

f̃(z) = f(z) · e^{− \frac{π}{2} h(z,z)}.

These functions can be considered as sections in a new bundle which is isomorphic to the given one but which is described by a different 1-cocycle. If we put

C_{hol}(z, ω) = π(H(z, ω) − h(z, ω)) + \frac{π}{2} (H(ω, ω) − h(ω, ω))

then the section of the bundle L(C_{hol}, φ) are functions which satisfy

f̃(z + ω) = f̃(z) · e^{C_{hol}(z,ω)+2πi(φ(ω)+η(ω)}.

This new 1-cocycle has the disadvantage that it depends on the choice of G but it has several advantages.

1) We have H(z, ω) = h(z, ω) for all z ∈ V, ω ∈ G and η(ω) = 0, φ(ω) = 0 for all ω ∈ G. Hence we see that f̃(z + ω) = f̃(z) for all ω ∈ G, the function f̃ is periodic with respect to the sublattice G.

2) We will show that the cocycle depends “holomorphically” on I and this means that we can view the abelian varieties together with the bundles as a holomorphic family.

Now I give an indication how we can describe the space of sections in these bundles. We choose a basis e_{1} … e_{g} of G, this is also a ℂ-basis of V. We write the elements of V as linear combinations z = ∑ e_{ν}z_{ν} = (z_{1}, …, z_{g}).

The periodicity of f̃ with respect to G means that f̃(z_{1} + n_{1}, …, z_{g} + n_{g}) = f̃(z_{1}, …, z_{g}) for all n_{1}, …, n_{g} ∈ ℤ^{g}. Hence we may introduce the variables u_{ν} = e^{2πiz_{ν}} and write

f̃(z_{1}, …, z_{g}) = h(u_{1}, …, u_{g}).

We expand

h(u_{1}, …, u_{g}) = ∑ a_{m_{1}...m_{g}}u_{1}^{m_{1}} … u_{g}^{m_{g}}.

We apply the transformation formula defining the line bundle. The elements in Ω/G will provide recursion formulae for the coefficients. We will see that some of the coefficients can be chosen arbitrarily and then the other ones can be computed from these via recursion and depend holomorphically on I.

Before I carry out this computation in detail I want to make a remark on the moduli space of principally polarized abelian varieties.

2.4.3 The symmetric domain To explain this I start from a principally polarized abelian variety. I recall that this is a triplet

A = (Ω, ⟨ , ⟩, I)

where

(i) Ω is a free ℤ-module of rank 2g and ⟨ , ⟩ is a skew symmetric form

⟨ , ⟩ : Ω × Ω ⟶ ℤ

which is non degenerate over ℤ. This means that we can write our lattice

Ω = ⨁_{ν=1}^{g} ℤe_{i} ⊕ ℤf_{ν}

where ⟨e_{ν}, f_{ν}⟩ = −1 = −⟨f_{ν}, e_{ν}⟩ and where all other ⟨ , ⟩ between basis elements are zero.

(ii) The element I is a complex structure on Ω_{ℝ}, we have I^{2} = − Id and it respects the alternating form ⟨ , ⟩_{ℝ} : Ω_{ℝ} × Ω_{ℝ} → ℝ.

(iii) On the complex vector space V = (Ω_{ℝ}, I) we can define a hermitian form H_{I} on V by

Im H_{I}(x, y) = ⟨x, y⟩

for all x, y ∈ Ω_{ℝ}. It is part of our assumption that this form is positive definite.

We want to explain that these data can be viewed as points in a complex manifold.

If we consider such an element I, and if we extend the scalars to ℂ, then the ℂ-vector space Ω_{ℂ} = Ω ⊗ ℂ decomposes

Ω_{ℂ} = Ω_{ℂ}^{10} ⊕ Ω_{ℂ}^{01},

where Ω_{ℂ}^{10} is the eigenspace for I with eigenvalue i and Ω_{ℂ}^{01} is the eigenspace with eigenvalue −i. There exists an object which is called the Grassmann variety of maximal isotropic subspaces. I do not want to discuss this in detail. Its complex points X (ℂ) are the maximal isotropic subspaces of Ω ⊗ ℂ and in fact we have the structure of a complex manifold on this set. (We will give some local coordinates later.) To any element I we can attach a point in X (ℂ). Actually we have two choices – namely we can attach Ω_{ℂ}^{10} or Ω_{ℂ}^{01} to I – but in our situation we choose

I ⟶ Ω_{ℂ}^{01} = {u ∈ Ω_{ℂ} | Iu = −i ⊗ u}.

On X (ℂ) we have complex conjugation, it interchanges the two space in the decomposition and sends the element I into −I. This means that the two parabolic subgroups (the stabilizer of Ω_{ℂ}^{10} and Ω_{ℂ}^{01}) are in opposition. If in turn we have a point z ∈ X (ℂ), and the corresponding parabolic subgroup P_{z}, and if P_{z} and P\overline{z} are in opposition, then we get a decomposition

Ω_{ℂ} = W ⊕ \overline{W}

where W = z. Then we can consider the automorphism J which acts by multiplication by i on W and −i on \overline{W}. Clearly this defines a complex structur on Ω_{ℝ}: the elements of Ω_{ℝ} are the elements of the form

ω = w + \overline{w}

and Iω = w ⊗ i + \overline{w} ⊗ (−i) = w ⊗ i + \overline{w ⊗ i}. We conclude that we have a bijection

{I | I^{2} = − Id, ⟨Ix, Iy⟩ = ⟨x, y⟩} ⟶^{∼} X_{0}(ℂ)

where X_{0}(ℂ) is the set of points z for which z and \overline{z} are in opposition. This induces a complex structure on the set of all I. On X_{0}(ℂ) we have an action of G(ℝ) by conjugation, we want to determine the orbits. Recall that an element I defines a hermitian form H_{I} on the complex vextor space (Ω_{ℝ}, I) and the stabilizer of the element I is the unitary group U_{I} ⊂ G(ℝ) of the hermitian form This hermitian form H_{I} has a signature (p, q) with p + q = g and H_{I} ≃ U(p, q). Now it is an easy — or perhaps better — a well known theorem that: The orbits under G(ℝ) on X (ℂ) are given by the signatures (p, q) of the hermitian forms H_{I}. Especially we have the open orbit X ⊂ X_{0}(ℂ) where the form H_{I} is positive definite. This is the orbit which is hit by the principally polarized abelian varieties. We see that two such principally polarized abelian varieties (Ω, < , > I), (Ω, < , > I′) are isomorphic if we can find an automorphism of (Ω, < , >) which sends I into I′. The group of these automorphisms is the symplectic group G(ℤ) = Sp_{g}(ℤ) and this gives us a hint that the abelian varieties with a given type of polarization are parametrized by

G(ℤ)\X = universal family of principally polarized abelian varieties

We had chosen a basis e_{1} ⋯ e_{g}, f_{g} ⋯ f_{1} for Ω. Now we write

f_{ν} = ∑ (x_{νμ} + y_{νμ}I)e_{μ},  (†)

we put τ_{νμ} = x_{νμ} + y_{νμ} · i. The element I gives the decomposition Ω_{ℂ} = Ω_{ℂ}^{10}⊕Ω_{ℂ}^{01}. We attached the space Ω_{ℂ}^{01} ∈ X (ℂ) to I, and we want to write ”coordinates” this point (or space). To compute this space we have to observe that we have

Ω_{ℂ} = G_{ℂ} ⊕ Ω_{ℂ}^{01}.

The map V → Ω_{ℂ}/Ω_{ℂ}^{01} is by construction an isomorphism and hence

f_{ν} − ∑ τ_{νμ} ⊗ e_{μ} ∈ Ω_{ℂ}^{01}.  (††)

The assumption that Ω_{ℂ}^{01} is maximal isotropic is equivalent to the symmetry of the matrix

(τ_{νμ})

and our hermitian form is positive definite if and only if the real part of this matrix is positive definite. Therefore the τ_{νμ} are the holomorphic coordinates for the possible choices of I.. We see that the space of possible complex structures I which provide a positive definte form H_{I} can be identified to the points in the Siegel half space

H = {Z|Z = X + iY }

where Z is symmeric and Y is positive definite.

2.4.4 We consider the cocycle ω → C_{hol}(z, ω) and I want to explain that this cocycle depends holomorphically on I. To be more precise we can fix an element ω ∈ Ω and consider this cocycle as a function in the variables z and I. Then we want to show that this cocycle is holomorphic in both variables. We have the two forms

\begin{aligned}H & : & Ω_{ℝ} × Ω_{ℝ} & ⟶ & ℂ \\ h & : & Ω_{ℝ} × Ω_{ℝ} & ⟶ & ℂ, \end{aligned}

which H is hermitian with respect to the element I and where h is linear in both variables with respect to I. Now we extend these forms to Ω_{ℂ} = Ω_{ℝ} ⊗ ℂ bilinearly, i.e. we have

\begin{aligned}H_{ℂ}(ω ⊗ z, η ⊗ w) & = & zw H(ω, η) \\ h_{ℂ}(ω ⊗ z, η ⊗ w) & = & zw h(ω, η). \end{aligned}

for ω, η ∈ Ω and z, w ∈ ℂ We observe that the inclusion Ω_{ℝ} → Ω_{ℂ} induces an isomorphism

Ω_{ℝ} ⟶ Ω_{ℂ}/Ω_{ℂ}^{01},

and this map is ℂ-linear if we give Ω_{ℝ} the complex structure where multiplication by i is given by I. We can decompose

Ω_{ℂ} = G_{ℂ} ⊕ Ω_{ℂ}^{01},

and hence we can write any element ω ∈ Ω_{ℂ} as a sum

ω = ω_{G} + ω_{01}.

Now we consider the expression

H_{ℂ}(z, ω) − h_{ℂ}(z, ω),

and we observe that this depends only on ω mod G_{ℂ} in the second variable. On the other hand we see: If z ∈ Ω_{ℂ} is in Ω_{ℂ}^{01}, then

H_{ℂ}(z, ω) = h_{ℂ}(z, ω) = 0.

this is clear because z = η + Iη ⊗ i with some η ∈ Ω_{ℝ} and

H_{ℂ}(η + Iη ⊗ i, ω) = H(η, ω) + i H(Iη, ω) = H(η, ω) − H(η, ω) = 0,

and the same holds for h_{ℂ}. Hence we conclude that

H_{ℂ}(z, ω) − h_{ℂ}(z, ω)

defines a bilinear form

Ω_{ℂ}/Ω_{ℂ}^{01} × Ω_{ℂ}/G_{ℂ} ⟶ ℂ.

We can express this form in terms of the original alternating form: We write as above

ω = ω_{G} + ω_{01},

then I claim that

H_{ℂ}(z, ω) − h_{ℂ}(z, ω) = 2i⟨z, ω_{01}⟩_{ℂ}

where ⟨ , ⟩_{ℂ} is of course the bilinear extension of ⟨ , ⟩ to Ω_{ℂ}. To see that this is the case it suffices to show that

H_{ℂ}(z, ω_{01}) − h_{ℂ}(z, ω_{01}) = 2i⟨z, ω_{01}⟩_{ℂ}.

In this case

ω_{01} = η + Iη ⊗ i

and because h_{ℂ} is bilinear with respect to I, we get

h_{ℂ}(z, η + Iη ⊗ i) = 0

as above. Hence we have to show that

H_{ℂ}(z, ω_{01}) = 2i⟨z, ω_{01}⟩_{ℂ}.

We may assume that z ∈ Ω_{ℝ} and again we write ω_{01} = η + Iη ⊗ i. Then

H_{ℂ}(z, η + Iη ⊗ i) = H(z, η) + i · H(z, Iη) = i H(z, Iη) + H(z, η).

We invoke our formulae for H and get

i(⟨z, η⟩+i⟨z, Iη⟩) − ⟨z, Iη⟩ + i⟨z, η⟩ =
2i⟨z, η⟩ − 2⟨z, Iη⟩ = 2 (⟨z, η ⊗ i⟩_{ℂ} − ⟨z, Iη⟩) =
2i (⟨z, η⟩ + ⟨z, Iη ⊗ i⟩_{ℂ}) = 2i⟨z, ω_{01}⟩_{ℂ}.

This function (z, I) ↦ 2i⟨z, ω_{01}⟩_{ℂ} is now clearly holomorphic in the variables z, I. Now we pick an ω and we get for our 1-cocycle

C_{hol}(z, ω) + 2πiφ(ω) = 2πi⟨z, ω_{01}⟩_{ℂ} + πi⟨ω, ω_{01}⟩_{ℂ} + 2πi(η(ω) + φ(ω))

We write z = ∑ z_{ν}e_{ν} ∈ Ω_{ℝ} and we want exploit the formula

f(z + ω) = f(z)e^{C_{hol}(z,ω)+2πi(φ(ω)+η(ω))}

for the section of the bundle. If ω ∈ G then the factor on the right hand side is equal to 1 and this means that f is periodic in G which in our coordinates means that f(z_{1} + n_{1}, …, z_{g} + n_{g}) = f(z_{1}, ….z_{g}) where the n_{i} are integers. Hence we have to consider elements ω = ∑_{i} n_{i}f_{i}. Then the relation (††) says that

p_{01}(ω) = ∑ n_{μ}(f_{μ} − ∑ τ_{μν}e_{ν}).

If z = ∑ z_{μ}e_{μ}, then it follows that

H(z, ω) − h(z, ω) = 2i ∑ z_{μ}n_{μ},

and

\begin{matrix}H(ω, ω)− & h(ω, ω) = 2i⟨ω, p_{01}(ω)⟩ = \\ 2i⟨∑_{μ} n_{μ}f_{μ}, ∑_{μ} n_{μ}(f_{μ} − ∑_{ν} τ_{μν}e_{ν}) = −2i ∑_{ν,μ} n_{ν}n_{μ}τ_{νμ}. \end{matrix}

and hence our recursion formula will be

f(z + ω) = f(z)e^{2πi(∑_{μ} z_{μ}n_{μ})−πi(∑_{ν,μ} n_{ν}n_{μ}τ_{νμ})+2πi(φ(ω)+η(ω))}

We introduce the new variables

u_{ν} = e^{2πiz_{ν}} and q_{νμ} = e^{πiτ_{νμ}}.

The q- variables are not independent, we have the relation

q_{νμ} = q_{μν}.

We return to the problem of the computation of the space of sections. We start from our data A = (Ω, ⟨ , ⟩, I), and we consider the d-th power of our line bundle L(C_{hol}, φ)^{⊗d} = L(dC_{hol}, dφ). The periodicity with respect to G allows us to write our sections f as functions in the variables u_{ν}

f(z_{1}, …, z_{g}) = h(u_{1}, …, u_{g})

and the transformation rules becomes for an element ω = n_{1}f_{1} + … + n_{g}f_{g}

\begin{matrix}h(u_{1} ∏_{ν} q_{ν1}^{2n_{ν}} , …, u_{g} ∏_{ν} q_{νg}^{2n_{ν}}) \\ = h(u_{1}, …, u_{g}) ∏_{ν=1}^{g} u_{ν}^{dn_{ν}} ∏_{λκ} q_{λκ}^{−n_{λ}dn_{κ}}e^{2πd(φ(ω)+η(ω))} \end{matrix}

If we now expand the function h into a Laurent series

h(u_{1}, …, u_{g}) = ∑ a_{m_{1},...,m_{g}}u_{1}^{m_{1}} … u_{g}^{m_{g}}

then transformation rule for our element ω = ∑_{ν} n_{ν}f_{ν} yields the following recursion

a_{m_{1},...,m_{g}} ∏_{νμ} q_{νμ}^{2n_{ν}m_{μ}} = a_{m_{1}−dn_{1},...,m_{g}−dn_{g}} ∏_{λκ} q_{λκ}^{−n_{λ}dn_{κ}}e^{2πi(φ(ω)+η(ω))}

From this we conclude that the coefficients a_{ν_{1},...,ν_{g}} for 0 ≤ ν_{i} ≤ d − 1 determine the rest of the coefficients. On the other hand we can choose values for the coefficients a_{α_{1}}, …, a_{α_{g}} for the indices 0 ≤ α_{i} ≤ d − 1 and then we vary the n_{ν} we get

a_{α_{1}−dn_{1},...,α_{g}−2dn_{g}} = a_{α_{1},...,α_{g}}q_{νμ}^{n_{ν}α_{μ}+dn_{ν}n_{μ}}e^{2πid(φ(ω)+η(ω))}

Now we make the fundamental observation that the positive definiteness of our matrix Y above implies that we can prove an estimate

|a_{α_{1}−dn_{1},...,α_{g}−dn_{g}}| < e^{−c(n_{1}^{2}+...n_{g}^{2})}

with some constant c > 0 depending on Y. This implies that the Laurent series will be convergent for all u_{1}, …, u_{g} ∈ ℂ^{∗} and we conclude 2.4.5 We can write down explicitly all sections in a line bundle L(C_{hol}, φ)^{⊗d} as infinite Laurent series, if the corresponding hermitian form is positive definite. The dimension of the space of section is d^{g}. With a little bit more efforts in linear algebra linear algebra it is not difficult to show

Theorem: If e ∈ NS(J) is an alternating form on Γ which for which the corresponding hermitian form H_{e} is positive definite, then

dimH^{0}(V/Γ, L_{H_{e}}(φ, η_{H})) = det <, > .

If we return to our Jacobian J then we have the cup product pairing on H^{1}(S, ℤ) ≃ Γ. If H_{0} is the corresponding hermitian form then we can form the line bundle P = L_{H_{0}}(φ, η_{H}) with an arbitrary φ and suitable η_{H}. Our theorem yields

dim_{ℂ}H^{0}(J, P) = 1.

If we take powers of this line bundle then det(r <, >) = r^{2g} and it follows that

dim_{ℂ}H^{0}(J, P^{⊗r}) = r^{2g}.

This can be used to construct an embedding into the projective space

Theorem (Lefschetz): If we take r = 3 then the morphism

\begin{matrix}J & ⟶ ℙ(H^{0}(J, P^{⊗3})) \\ x & ⟶ H_{x} = \{s ∈ H^{0}(J, P^{⊗3})|s(x) = 0\} \end{matrix}

is everywhere defined and yields an embedding of J into the projective space. I want to comment on this theorem without proving it, its proof will be discussed in the second volume in the section on Jacobians. At first we need to know is that for any x ∈ J we can find a section s ∈ H^{0}(J, P^{⊗3}) which does not vanish at this point. Secondly we have prove that for any pair of points x ≠ y we can find a section which vanishes at x but not at y. And finally we need to know the following: If we pick a point x and a section s_{0} which does not vanish at x then the ratios s/s_{0} are function on J which are defined in a suitable neighborhood of x. Then we have to show that we can find sections s_{1}, …, s_{d} which vanish at x such that the differentials d(s_{1}/s_{0}), …, d(s_{d}/s_{0}) generate the dual tangent space. This implies that the local ring O_{(}J, x) is the ring of convergent power series in s_{1}/s_{0}, …, s_{d}/s_{0}. If all this is shown then it is clear that the image of J under Θ is a complex analytic submanifold in Y ⊂ ℙ(H^{0}(J, P^{⊗3})) and

Θ : J→^{∼}Y

is in fact an analytic isomorphism.

Now we use the theorem of Chow which says that a smooth and closed submanifold of ℙ^{n}(ℂ) is in fact a smooth projective algebraic variety. (See III.1.2.1,5d). Hence we can define the image Θ(J) = Y as the set of common zeroes of a finite number of homogeneous polynomials. Then the pair (J, P) becomes an object in algebraic geometry. To make this precise we have to say a few words about the comparison between algebraic and analytic geometry. As in the case of Riemann surfaces we define a new topology on Y, namely the Zariski toplogy. If we have a homogeneous polynomials f(z_{0}, …, z_{n}) then we can look at the set V (f) ⊂ ℙ^{n}(ℂ) where it vanishes and the set D(f) ⊂ ℙ^{n}(ℂ) where it does not vanish. These sets D(f) form a basis for the Zariski topology on ℙ^{n}(ℂ) , i.e. the Zariski open subsets in ℙ^{n}(ℂ) are unions of sets of the form D(f). The Zariski open subsets in Y are the intersections of Zariski open subsets in ℙ^{n}(ℂ) with Y. As in the case of Riemann surfaces we know that the identity map Y_{an} → Y_{Zar} is continous. If now U ⊂ ℙ^{n}(ℂ) is a Zariski open subset we say that a holomorphic function f : U → ℂ is meromorphic if for any point y ∈ U we can find homogenous polynomals g, h of the same degree, such that h(y) ≠ 0 and such that f = g/h on the open set U ∩ D(h). We put as before

O_{ℙ^{n}(ℂ)}^{mer}(U) = {f : U → ℂ|f is meromorphic}

We can do the same thing with Y and define the sheaf O_{Y}^{mer}. As in the case of Riemann surfaces the identity map

(Y_{an}, O_{Y}) → (Y_{Zar}, O_{Y}^{mer})

is a morphism between locally ringed spaces. A meromorphic function on Y is an element in some O_{Y}^{mer}(U) where U ≠ ∅ is Zariski open in Y. Now Y was special namely it was the image of J under Θ. Hence it is connected as a toological space and from this it follows easily that the intersection of two non empty Zariski open sets is again non empty. This allows us to define the field ℂ(Y) = ℂ(J) of meromorphic functions on J. We state without proof: The field of meromorphic functions on J is a finitely generated extension of trancendance degree d As in the case of Riemann surfaces we can define coherent sheaves of O_{Y} -modules (resp. O_{Y}^{mer}) modules on Y_{an} (resp.Y_{Zar}). In both cases this are sheaves of modules under the structure sheaf which locally are finitely generated. It is the content of Serre’s paper “Geometry algebrique ét geometry analytique” (in short GAGA) that these two categories are equivalent. In short words: To any coherent O_{ℙ^{n},an}- sheaf F_{an} on ℙ^{n}(ℂ) we can find a unique subsheaf F of O_{ℙ^{n}}-modules (i.e. F(U) is an O_{ℙ^{n}}-modules for any U ⊂ ℙ^{n}, Zariski open) such that F_{an} = F ⊗_{O_{ℙ^{n}}} O_{ℙ^{n},an}.

A first consequence of the GAGA-principle get that the sheaf I_{an} which defines the analytic subspace Y is the extension of a sheaf of ideals I ⊂ O_{ℙ^{n}}, and this is of course the statement of Chows theorem.

If we now consider line bundles on J we have the freedom to look at them as complex analytic bundles or as line bundles on the scheme (J, O_{J}). Hence we will not make any distinction between these two kinds of line bundles, we identify

Pic_{Zar}(J) = Pic(J) = Pic(J_{an})
H_{Zar}^{1}(J, O_{J}^{∗}) = H^{1}(J, O_{J,an}^{∗})

where actually H^{1}(J, O_{J,an}^{∗}) was exactly what we called H^{1}(J, O_{J}^{∗}) before.

2.4.6 Degenerations of abelian varieties At this point we achieved also something else. We can consider the τ_{νμ} as complex analytic variables and then then we can consider our abelian varieties as varying in a holomorphic family of bundles. And we have in addition a holomorphic family of lines bundles and a holomorphic family of sections. If the matrix Y tends to infinity then the q_{νμ} tend to zero and our abelian variety degenerates into a product of ℂ^{∗} in a certain sense we have written down a universal degenerating family of abelian varieties. The point is that this degeneration can be given an arithmetic meaning. This will be explained in a special case.

2.4.7 The case of genus 1 I want to discuss these constructions in the special case of curves of genus one. We can assume that the Jacobian is of the form

J = ℂ/{1, τ}

where τ ∈ ℂ, Im(τ) > 0 and where {1, τ} = Ω is the ℤ-lattice generated by the elements 1, τ.

We have an alternating pairing ⟨ , ⟩ on Ω, it is determined by its value on the basis elements, and we put

⟨1, τ⟩ = −1.

All other alternating pairings are of the form d⟨ , ⟩ with some integer d. In this case it is clear that ⟨ , ⟩ is the imaginary part of a hermitian form H on ℂ. If y = Im(τ), then this form is given by

H(z_{1}, z_{2}) = \frac{1}{y} z_{1}\overline{z}_{2}.

Now we consider line bundles

L(d⟨ , ⟩, φ, η)

where we have changed the notation in the sense of 6.1.1 (???).

Let us look at the case d = 1 first. In this case we must have a non trivial η. One possibility is to take

η(1) = η(τ) = η(1 + τ) = \frac{1}{2},

and there are 3 other choices, namely, taking the value \frac{1}{2} on exactly one of the elements {1, τ, 1 + τ} and zero on the two others.

I want to stick to the first choice, it is in a sense the most canonical. We investigate the line bundle

L(C_{H}, 0),

i.e. we take a linear form to be trivial. We have to look at functions which satisfy

f(z + ω) = e^{\frac{π}{y} z\overline{ω}+ \frac{π}{2y} ω\overline{ω}+2πiη(ω)}f(z).

Now we apply the strategy explained in 6.4. We choose the submodule G to be the module generated by 1. Now η is not trivial on the vector 1 this forces us to make some minor modifications. Again we put

f̃(z) = e^{− \frac{π}{2y} z^{2}}f(z)

and then we find

f̃(z + 1) = −f̃(z).

(Here we have to take into account that η(1) = \frac{1}{2}.)

A simple computation shows

\begin{aligned}f̃(z + nτ) & = e^{− \frac{π}{2y} (z+nτ)^{2}}f(z + nτ) = e^{− \frac{π}{2y} (z+nτ)^{2}}e^{\frac{π}{y} zn\overline{τ}+ \frac{π}{2y} n^{2}τ\overline{τ}+πin}f(z) \\ & = e^{−2πinz}e^{−πin^{2}τ+πin}f(z) \end{aligned}

In this situation we modify the definitions of q and u and we put u = e^{πiz} and q = e^{πiτ}, then our first relation above says that our function f has a Laurent expansion

∑_{m=1 mod 2} a_{m}u^{m},

and as in 6.4. the second relation gives a recursion for the coefficients a_{m}.

It says

f̃(uq^{n}) = (−1)^{n}u^{−2n} · q^{−n^{2}} f̃(u)

and looking at the expansion we get

a_{m}u^{n}q^{mn} = (−1)^{n}u^{−2n}q^{−n^{2}}(… + a_{m+2n}u^{m+2n} …),

and hence for any choice n, m

a_{m+2n} = (−1)^{n}q^{mn+n^{2}}a_{m}.

Since the coefficients with an even index are vanishing, we see that the coefficient a_{1} determines all the others. We put it equal to one and then we get

f̃(u) = ∑_{m∈ℤ}(−1)^{m} · q^{m+m^{2}}u^{1+2m}.

Since we have Im(τ) > 0 we have |q| < 1 and hence our function converges for all u ∈ ℂ^{∗}.

This function is one of the Jacobi Theta functions. We change the notation and write

ϑ(u, q) = ∑(−1)^{m}q^{m+m^{2}}u^{1+2m}.

We have seen that in modern language this ϑ-function is a section in a line bundle on the Jacobian ℂ/{1, τ}.

We can ask ourselves whether we have a different description of this line bundle. Clearly it is of degree one so on our Riemann surface J it should be of the form O_{J}(P) with some point P ∈ ℂ/{1, τ}. The bundle O_{J}(P) has a non trivial section which vanishes at P. Hence we see that our ϑ(u, q) must vanish for some value of u. A simple computation yields

ϑ(1, q) = 0,

and hence we conclude

L(⟨ , ⟩, 0, η) ≃ O_{J}(O)

where O ∈ ℂ/{1, τ} is the zero element.

Now it becomes clear what the other choices of η will give. In section 6.1. I explained that different choices of η can be compensated by changing φ. In this case we can consider φ : Ω → ℂ such that φ(Ω) ⊂ \frac{1}{2}ℤ, of course what matters is the resulting homomorphism

\overline{φ} : Ω/2Ω ⟶ \frac{1}{2}ℤ/ℤ.

We have three non zero such homomorphisms and

L(⟨ , ⟩, 0, η′) = L(⟨ , ⟩, φ, η)

if η = \overline{φ} + η. We could carry out the same calculations and get three more ϑ functions

\begin{aligned}ϑ_{01} & = & ∑(−1)^{m}q^{m^{2}}u^{2m} \\ ϑ_{10} & = & ∑ q^{m^{2}+m}u^{2m+1} \\ ϑ_{11} & = & ∑ q^{m^{2}}u^{2m} \end{aligned}

and they correspond to the linear forms with \overline{φ}_{ij}(1) = \frac{i}{2} mod ℤ, \overline{φ}_{ij}(τ) = \frac{j}{2} mod ℤ. These give the four Jacobi θ-functions. The kernel of φ_{ij} defines a two torsion point P_{ij} ∈ J, and we must have that ϑ_{ij} is a non zero section in

H^{0}(J, L(⟨ , ⟩, φ_{ij}, η)

and

L(⟨ , ⟩, φ_{ij}, η) = O_{J}(P_{ij}).

We make another change of notation and put

Θ_{1} = ϑ.

we know how we can write down sections in

H^{0}(J, L(dC_{hol}, 0,).

We know from the Riemann-Roch theorem that this space of sections has dimension d which we also get from our considerations above.

We always have for sections in H^{0}(J, L(d⟨ , ⟩, 0, dη))

\begin{aligned}f̃(−u) & = & (−1)^{d}f̃(u) \\ f̃(uq^{m}) & = & (−1)^{dm}u^{−2md}q^{−dm^{2}} f̃(u) \end{aligned}

and if we expand into a Laurent series in u the non zero coefficients with even (resp. odd) indices vanish if d is odd (resp. even). Later we will consider the case where d is even more closely. Then it is usefull to replace d by 2d and

u by u = e^{2πiz} and qbyq = e^{2πiτ}

then the recursion formula becomes

f(uq^{m}) = u^{−2md}q^{−dm^{2}} f̃(u)

If d = 2 then

Θ_{1}^{2} = … − ( ∑_{m∈ℤ} q^{2m^{2}+2m})u^{0} + ( ∑_{m∈ℤ} q^{2m^{2}})u^{2} + …,

and we find a second section where the coefficient a_{0} = 1 and a_{2} = 0.

Θ_{2} = ∑_{m∈ℤ} q^{2m^{2}}u^{4m}.

We consider d = 3, we have already two sections, namely, Θ_{1}^{3} and Θ_{1}Θ_{2}, and we can write a third section

Θ_{3} = ∑_{m∈ℤ}(−1)^{m}q^{3m^{2}+m}u^{1+6m}.

We are now in exactly the same situation as in ???. We have the sections

\begin{aligned}θ_{1} & ∈ & H^{0}(J, O_{J}(O)) & ⊂ & H^{0}(J, O_{J}(3O)) \\ θ_{2} & ∈ & H^{0}(J, O_{J}(2O)) & ⊂ & H^{0}(J, O_{J}(3O) \end{aligned}

and

θ_{3} ∈ H^{0}(J, O_{J}(3O)).

We must have linear relations among the monomials θ_{3}^{2}, θ_{3}θ_{2}θ_{1}, θ_{3}θ_{1}^{3}, θ_{2}^{3}, θ_{2}^{2}θ_{1}^{2}, θ_{2}θ_{1}^{4}, θ_{1}^{6}. Now we take into account that our curve depends on a parameter τ and hence on e^{πiτ} = q, and we want to find out what these relations are. I refer to the discussion of the Weierstraß equation in section ??? .

We have the two division points \frac{1}{2}, \frac{τ}{2} and \frac{1+τ}{2} in ℂ/\{1, τ\}, we call them P_{10}, P_{01} and P_{11} respectively. Then P_{00} = O. The ratios

ξ_{νμ} = \frac{ϑ_{νμ}^{2}}{ϑ^{2}}

are meromorphic functions on ℂ/{1, τ} and div(ξ_{νμ}) = 2P_{νμ} − 2O.

We choose one of these points, say P_{10}, and we put x = ξ_{10} and we consider the function

x(x − x(P_{01})) · (x − x(P_{11})).

The function y = \frac{ϑ_{10}ϑ_{01}ϑ_{11}}{ϑ^{3}} has the divisor P_{01} + P_{10} + P_{11} − 3O and hence

div(y^{2}) = 2P_{01} + 2P_{10} + 2P_{11} − 6O.

We have div(x − x(P_{01})) = P_{01} + Q − 2O, but then we conclude Q = P_{01} because if not the divisor could not be a principal divisor.(??) The same argument holds for the third factor. We get div(y^{2}) = div(x(x − x(P_{01}))(x − x(P_{11})).

We conclude that

y^{2} = a · x(x − x(P_{01}))(x − x(P_{11})).

Our division points are P_{10} = \frac{1}{2}, P_{11} = \frac{1+τ}{2} and P_{01} + \frac{τ}{2} . A simple calculation shows that

ϑ_{10}(P_{10}) = 0. Hence x = \frac{ϑ_{10}^{2}}{ϑ^{2}} (this shows that the choice of indices is correct) and we have

ϑ_{10} (\frac{τ}{2} ) = ∑_{m∈ℤ} q^{m^{2}+m}e^{\frac{πiτ}{2} (2m+1)} = ∑_{m∈ℤ} q^{m^{2}+m}e^{πiτm}e^{\frac{πiτ}{2}} =
(∑_{m∈ℤ} q^{m^{2}+2m}) e^{\frac{πiτ}{2}} = (∑ q^{m^{2}})e^{− \frac{πiτ}{2}}

ϑ_{10}(\frac{1+τ}{2} ) = ∑ q^{m^{2}+m}e^{πi \frac{τ+1}{2} ·(2m+1)} = (∑ q^{m^{2}+m}e^{πiτm} · e^{πim})e^{\frac{πi}{2}} e^{\frac{πi}{2}} =
(∑(−1)^{m}q^{m^{2}+2m}) · e^{\frac{πiτ}{2}} · e^{\frac{πi}{2}} ) = (∑(−1)^{m+1}q^{m^{2}})e^{− \frac{πiτ}{2}} e^{\frac{πi}{2}} .

The same calculation for ϑ yields

\begin{aligned}ϑ(P_{10}) & = & (∑ q^{m^{2}+m}) e^{\frac{πi}{2}} \\ ϑ(P_{01}) & = & (∑(−1)^{m}q^{m^{2}+2m}) e^{\frac{πiτ}{2}} \\ ϑ(P_{11}) & = & (∑ q^{m^{2}+2m}) e^{\frac{πiτ}{2}} e^{\frac{πi}{2}} . \end{aligned}

We get

y^{2} = a · x (x − \frac{(∑ q^{m^{2}})^{2}}{(∑(−1)^{m+1}q^{m^{2}})^{2}} ) (x − \frac{(∑(−1)^{m+1}q^{m^{2}})^{2}}{(∑ q^{m^{2}})^{2}} ) .

We can compute the factor a. As in (???) we look at the leading term in the expansion for x + (\frac{ϑ_{10}}{ϑ} )^{2} and y + (\frac{ϑ_{10}ϑ_{11}ϑ_{01}}{ϑ} )^{3} at z = 0. If

ϑ(z) = βz + higher order terms

then

x = \frac{ϑ_{10}(0)}{β^{2}z^{2}} + … higher order terms
y = \frac{ϑ_{10}ϑ_{11}ϑ_{01}(0)^{3}}{β^{3}z^{3}} + higher order terms

and we get

a = \frac{ϑ_{11}(0)^{2} · ϑ_{01}(0)^{2}}{ϑ_{10}(0)^{4}} .

If we now replace

\begin{matrix}y & by & y & · & ϑ_{10}(0)^{2} · ϑ_{01}(0)ϑ_{11}(0) · \frac{ϑ(P_{11})^{3}}{ϑ_{10}(P_{11})^{3}} \\ x & by & x & · & \frac{ϑ(P_{11})^{2}}{ϑ_{10}(P_{11})^{2}} \end{matrix}

then our equation becomes

y^{2} = x(x − 1) (x − \frac{(Σq^{m^{2}}(−1)^{n})^{4}}{(Σq^{m^{2}})^{4}} ) .

This is now an equation in Legendre normal form. We put

λ(q) = \frac{(Σq^{m^{2}}(−1)^{m})^{4}}{(Σq^{m^{2}})^{4}} .

We can change our point of view. So far we consider q as a complex number or perhaps better as a function depending on a complex variable τ. But the power series λ(q) is in fact a power series in ℤ[[q]] and we have

ℤ[λ(q)] ⊂ ℤ[[q]].

We can comfortably interprete our θ-series as elements in the ring of Laurent series

ℤ[[q]][[u, u^{−1}]].

We have constructed a curve – namely

E : y^{2} = x(x − 1) (x − \frac{(∑_{m} q^{m^{2}}(−1)^{m})^{4}}{(∑_{m} q^{m^{2}})^{4}} )

over ℤ[[q]] and its discriminant is Platz fuer Formel

We conclude that this curve is smooth over the ring ℤ [\frac{1}{2}] [[q]] [\frac{1}{q}]. In principle we can say that we are in the same situation as in (???) where we wrote the Legendre normal form for an elliptic curve with a 2-torsion point and a 1-form ω. But here we can write down a formula for the points. So we may for instance consider the morphism

E
↓ ↖^{θ}
Spec (ℤ [\frac{1}{2}] [[q]]) ⟵ Spec (ℤ [\frac{1}{2}] [[q]][[u, u^{−1}]])

and the θ-series; can be seen as a tautological section θ. We can write down points with values in extensions of ℤ [\frac{1}{2}] [[q]], so for instance we get n-torsion points on E if we evaluate at roots of unity ζ or elements of the form ζq^{1/2n}. If we consider a homomorphism

ℤ [\frac{1}{2}] ⟶ k

where k is any field, then we can consider the base change of our curve to k[[q]] and to the quotient field K = k[[q]] [\frac{1}{q}], then it can be shown that

E(K) ≃ K^{∗}/⟨q^{2}⟩

(Tate’s theorem), and we have an explicit description of the group of rational points.

V.3. Towards the algebraic theory

3.1 Introduction During our discussion of the Jacobian J of a Riemann surface S and the description of the Picard group of J we made heavily use of transcendental methods. This means we worked in the category of (compact) complex manifolds. We used the cohohomology groups H^{i}(J, ℤ) and the construction of line bundles was done in the analytic context. On the other hand we have seen that the Riemann surface can be viewed as the set of ℂ valued points of a non singular projective curve C/ Spec(ℂ) and we also have stated the result that J is projective algebraic variety which is equipped with a line bundle P which has very specific properties. In this section we will aim at an algebraic formulation of our central results, we still use the trancendental arguments in the proofs. It is the content of chapter X in the second volume that the main results of the present chapter here can be formulated and proved in algebraic terms and hence they belong to algebraic geometry. Especially they make sense in characteristic p > 0. The key to a deeper understanding of J and also of the Riemann surface itself is the investigation of the Picard group of varieties of the form

S × S, S × J, J × J and J × J^{∨}.

Let X be any smooth, projective, connected variety over ℂ (see III,???), we denote its set of complex valued points also by X. We use the above mentioned principles from GAGA. Then we have

0 → H^{1}(X, ℤ) → H^{1}(X, O_{X}) → H^{1}(X, O_{X}^{∗}) → H^{2}(X, ℤ) →

and from here

0 → Pic^{0}(X) → Pic(X) →^{δ} H^{2}(X, ℤ).

The class δ(L) is the Chern class of L and the subgroup generated by the Chern classes is called the Neron-Severi group NS(X). 3.1.1 The algebraic definition of the Neron-Severi group: If X is any smooth prjective algebraic variety then he group H^{2}(X, ℤ) is of course a trancendental object, it needs the concept of continuity in its definition. But if we believe in GAGA then the group Pic(X) is defined in the context of algebraic geometry. We also can give an algebraic definition of subgroups which are close to Pic^{0}(X). For instance we can define the subgroup Pic^{00}(X) of those line bundles which are algebraically equivalent to zero: We say that a line bundle L on X is algebraically equivalent to zero if we can find a connected projective algebraic variety T over ℂ and a line bundle L̃ on X ×T such that there are two poits t_{1}, t_{0} on T for which L_{t_{1}} = L̃|X × t_{1}→^{∼}L and L_{t_{0}} = L̃|X × t_{0}→^{∼}O_{X}. Naively speaking this means that we can deform our bundle into the trivial bundle. It is of course clear that during such a deformation process the Chern classes do not change. This means that the group Pic^{00}(X) of lines bunles algebraically equivalent to zero is always contained in Pic^{0}(X).

If we divide Pic(X) by this subgroup we get a modified Neron-Severi group which is defined in the context of algebraic geometry. Our results (for instance 2.1.5 ) imply that for abelian varieties A over ℂ we have in fact Pic^{0}(A) = Pic^{00}(A) and thus we have an algebraic definition of NS(A) for abelian varieties. 3.1.2 : The algebraic definition of the intersection numbers. At this point I want to outline how we can define in purely algebraic terms the intersection numbers of line bundles on a smooth connected projective variety X ⊂ ℙ^{d}(ℂ). If we have d line bundles L_{1} ⋯ L_{d} on X, then we have their Chern classes c_{1}(L_{1}), ⋯, c_{1}(L_{d}). We can form the cup product

c_{1}(L_{1}) ∪ ⋯ ∪ c_{d}(L_{d}) ∈ H^{2d}(X, ℤ) = ℤ,

and the result is a number. We have already seen that under certain favorable circumstances we can interprete this number as the number of points in the intersection of d smooth divisors (see IV, 9.3.2)

c_{1}(L_{1}) ∪ ⋯ ∪ c_{1}(L_{d}) = Y_{1} ∩ ⋯ ∩ Y_{d}.

I want to explain that it is always possible to interprete this cup product of Chern classes as intersection numbers. I have to appeal to some theorems in projective algebraic geometry which will be discussed in more detail later. Our projective space ℙ^{d}(ℂ) has the line bundle O_{ℙ^{d}(ℂ)}(1) = H on it. We will show that for any bundle L on X we can find an integer n > 0 and a non zero section s ∈ H^{0}(X, L⊗H^{⊗n}) such that [s = 0] is a smooth divisor (see IV ...). We take our bundle L_{1} and construct such a section

s_{1} ∈ H^{0}(X, L_{1} ⊗ H^{⊗n})

and another section

t_{1} ∈ H^{0}(X, H^{⊗n})

which also provides a smooth divisor on X. If we look at the cup product of the Chern classes, we find

c_{1}(L_{1}) ∪ ⋯ ∪ c_{d}(L_{d}) = c_{1}(L_{1} ⊗ H^{⊗n}) ∪ ⋯ ∪ c_{d}(L_{d}) − c_{1}(H^{⊗n}) ∪ ⋯ ∪ c_{d}(L_{d}).

Now the two divisors [s_{1} = 0] and [t_{1} = 0] are again smooth projective varieties. They are perhaps not connected but their connected components Z_{1} ⋯ Z_{k} ⋯ are also smooth projective varieties by the theorem of Chow. We can restrict the remaining line bundles L_{1} ⋯ L_{d} to these components. Now we assume by induction that we have an algebraic definition of the intersection number L′_{2} ⋯ L_{d} of d − 1 line bundles on smooth projective varieties of dimension d − 1. Then the above argument gives us a definition for d line bundles on X. Here we have to observe that in view of our result in IV. 9.3.2 we know that this definition does not depend on the choices of n and of the sections s_{i} and t_{1} because the intersection numbers are also given by the cup product. But later in the context of algebraic geometry, when now cohomology groups are available, then we have to work a little bit more.

If X is equal to one of our four varieties above and if we write X = Y × Z then we get for the cohomology in degree one We get from 2.3 and the Künnet formula

\begin{aligned}H^{1}(J, O_{J}) & = & H^{1}(S, O_{S}) \\ H^{1}(Y × Z, O_{Y ×Z}) & = & H^{1}(Y, O_{Y}) ⊕ H^{1}(Z, O_{Z}) \end{aligned}

The second isomorphism yields

Pic^{0}(X) = Pic^{0}(Y × Z)→^{∼}Pic^{0}(Y) ⊕ Pic^{0}(Z)

We say that Pic^{0} is linear. We recall the notation Γ ≃ H^{1}(S, ℤ) and then

H^{2}(J, ℤ) = Hom(Λ^{2}Γ, ℤ)
H^{2}(S × S, ℤ) =H^{2}(S, ℤ) ⊕ H^{1}(S, ℤ) ⊗ H^{1}(S, ℤ) ⊕ H^{2}(S, ℤ) =
ℤ ⊕ Γ ⊗ Γ ⊕ ℤ.
H^{2}(S × J, ℤ) =H^{2}(S, ℤ) ⊕ H^{1}(S, ℤ) ⊗ H^{1}(J, ℤ) ⊕ H^{2}(J, ℤ) =
ℤ ⊕ Γ ⊗ Γ^{∨} ⊕ Hom(Λ^{2}Γ, ℤ).

We have to find out what the Neron-Severi group will be in our four cases. It decomposes into

NS(Y) ⊕ A(H^{1}(Y, ℤ) ⊗ H^{1}(Z, ℤ)) ⊕ NS(Z).

To see this we observe that we have the Chern classses of the pull backs p_{1}^{∗}(L_{1}), p_{2}^{∗}(L_{2}) of line bundles on the two factors, which have Chern classes (c_{1}, 0, 0), (0, 0, c_{2}) with respect to the above decomposition. On the other hand we can choose point y_{0} ∈ Y, z_{0} ∈ Z and restrict a bundle L on Y × Z to y_{0} × Z, Y × z_{0}. The Chern classes of these restrictions do not depend on the selected points ( because Y, Z are connected ) and if we modify L by the product of the inverses of the pull backs we get a bundle whose Chern class is (0, c_{2}, 0). We put

NS′(Y × Z) = \{\underline{c} ∈ NS(Y × Z)|\underline{c} = (0, c_{2}, 0)\}

The first and the third summand are considered as less interesting at this point since they are filled up by the Chern classes of line bundles which are pull backs from the two factors. We are interested in the summand in the middle.

We have the morphisms

S × S ⟶^{Id ×i_{P_{0}}} S × J ⟶^{i_{P_{0}} ×Id} J × J ⟶^{Id ×P D} J × J^{∨}.

which induces a sequence of isomorphisms

Γ ⊗ Γ ⟵ Γ ⊗ Γ^{∨} ⟵ Γ^{∨} ⊗ Γ^{∨} ⟵ Γ^{∨} ⊗ Γ^{∨}

where the isomorphism is always the tensor product of the identity and the Poincaré duality pd. It is clear that this sequence of isomorphisms also induces homomorphisms between the corresponding subgroups NS′(Y × Z) and

3.1.3 Proposition:With the obvious notation we get a sequence of isomorphisms

A(Γ ⊗ Γ) ⟵ A(Γ ⊗ Γ^{∨}) ⟵ A(Γ^{∨} ⊗ Γ^{∨}) ⟵ A(Γ^{∨} ⊗ Γ^{∨})

To see that this is indeed the case we recall that the Neron-Severi group is always the kernel of

H^{2}(X, ℤ) → H^{2}(X, O_{X}).

In our situation we have to apply the Künneth-formula and look at the kernel of

H^{1}(Y, ℤ) ⊗ H^{1}(Z, ℤ) → H^{1}(Y, O_{Y}) ⊗ H^{1}(Z, O_{Z})

and then the claim follows because the maps

H^{1}(J^{∨}, O_{J^{∨}})→^{∼}H^{1}(J, O_{J}) → H^{!}(S, O_{S})

are isomorphisms. There is a slightly different way of looking at this. We have seen that have to study the alternating 2-forms on Γ ⊕ Γ, Γ ⊕ Γ^{∨}, Γ^{∨} ⊕ Γ^{∨}. If our two summands are Γ_{1}, Γ_{2} then we denote the space of those alternating 2-forms which are trivial on the two summands by Alt′_{2}(Γ_{1} ⊕ Γ_{2}, ℤ). Then we have an ispmorphism

Γ_{1}^{∨} ⊗ Γ_{2}^{∨}→^{∼}Alt′_{2}(Γ_{1} ⊕ Γ_{2}, ℤ)

An element ψ_{1} ⊗ ψ_{2} = Ψ is sent to the form

e_{Ψ} : ((γ_{1}, γ_{2}), (γ′_{1}, γ′_{2})) ↦ ψ_{1}(γ_{1})ψ_{2}(γ′_{2}) − ψ_{1}(γ′_{1})ψ_{2}(γ_{2}).

To get the Neron-Severi group we have to look at those alternating forms which are after tensorzation by ℝ compatible with the complex structure and then we have to translate what this means for the corresponding elements in Γ_{1}^{∨} ⊗ Γ_{2}^{∨}

If for instance Γ_{1} = Γ^{∨} and Γ_{2} = Γ then the we get Γ ⊗ Γ^{∨} = End(Γ). Now it is obvious that the subspace of alternating forms which satisfy in addition condition i) are exactly those which under the above map Ψ correspond to elements in End(Γ) which are compatible with the complex structure. Hence we have shown

NS′(J × J^{∨})→^{∼} End(J)

(To get this identification we did not use the polarization). Now we consider the case J × J, in this case we have to look at Γ ⊗ Γ and this is the module of bilinear forms on Γ^{∨} and via Poincaré duality this is the same as the module of bilinear forms on Γ. Following the identifications we see that an element γ_{1} ⊗ γ_{2} ∈ Γ ⊗ Γ gives us the bilinear form (η_{1}, η_{2}) ↦ e_{0} < γ_{1}, η_{1} > e_{0} < γ_{2}, η_{2} >. Now it is an easy exercise that the identification

Γ ⊗ Γ→^{∼}Γ ⊗ Γ^{∨}→^{∼} End(Γ)

the element Id ∈ End(Γ) corresponds to the polarization form e_{0} ∈ Γ ⊗ Γ→^{∼}Γ^{∨} ⊗ Γ^{∨}

This element e_{0} therefore defines an element E_{0} in Alt′_{2}(Γ ⊕ Γ, ℤ) which is given by

E_{0}⟨⟨(γ_{1}, γ_{2}), (γ′_{1}, γ′_{2})⟩⟩ = e_{0}⟨γ_{1}, γ′_{2}⟩ − e_{0}⟨γ_{2}, γ′_{1}⟩.

The alternating form provides the line bundle

L(E_{0}, 0, 0) = N  on  J × J

whose Chern class is E_{0}. More generally it is now obvious that under the identification Γ ⊗ Γ→^{∼}Γ ⊗ Γ^{∨}→^{∼} End(Γ) an element α ∈ End(Γ) corresponds to the bilinear form E_{α} < γ_{1}, γ_{2} >= E_{0} < γ_{1}, α(γ_{2}) > We can summarize this discussion and say

3.1.4 Theorem: We have a canonical identification

NS′(J × J)→^{∼} End(J)

which is given by the map which sends an elemet ψ ∈ End(J) to the line bundle (Id ×ψ)^{∗}(N).

This should be seen in conjunction with our earlier result

NS(J)→^{∼}End_{sym}(J)

I want to point out that the construction of our line bundle N on J × J an algebraic terms only using the bundle P.: We consider the product and three maps

J × J  ⟶^{p_{1}} ; ⟶^{m} ; p_{2}^{⟶}  J

and we put

N = m^{∗}(P) ⊗ p_{1}^{∗}(P)^{−1} ⊗ p_{2}^{∗}(P)^{−1}.

It is quite clear that this bundle has a Chern class, the class E_{0}. But it is also clear that this bundle does not depend on the choice of P: If we modify P by a line bundle L which has Chern class zero then this amounts to changing the linear form in the construction . But this change cancels in the construction of N, this means that N is a canonical bundle on J × J.

The bundles P provides the isomorphism

φ_{P} : J ⟶ J^{∨} = Pic^{0}(J)

which was given by

x ⟶ T_{x}(P) ⊗ P^{−1}.

This isomorphism can also be interpreted in terms of N: If we restrict N to J × {x}, then this restriction is a line bundle N_{x} on J and clearly by definition we have

N_{x} ≃ T_{x}(P) ⊗ P^{−1}.

The bundle N gives us a still better understanding of the self duality of the Jacobian. The map

\begin{matrix}y & ⟼ & N′ | S × \{y\} \\ J & ⟶ & Pic^{0}(S) = J \end{matrix}

is the identity.

But the bundle N has a still stronger property which is called universality. Let us assume that we have a line bundle L on J × T where T is a complex analytic variety. We assume that T is connected, and we also assume that L | J × t_{o} is in Pic^{0}(J) for some point t_{0}. Now we can define a map

ψ : T ⟶ J^{∨} = Pic^{0}(J)

which is defined by

L | J × {t} ≃ N_{ψ(t)}.

I claim that ψ is indeed an analytic map, and that in addition for any point t_{1} ∈ T we can find a neighborhood V of t_{1} such that

L | J × V ≃ (Id ×ψ^{∗})(N) | J × V.

We introduce the following notation: If we have two line bundles L_{1}, L_{2} on X × T then we write

L_{1} ∼_{T} L_{2} is these bundles are isomorphic locally in T which means that for any point t ∈ T we can find a neighborhood V such that

L_{1} | X × V ≃ L_{2} | X × V.

This amounts to that we can find a lim bundle M on T such that

L_{1} ≃ L_{2} ⊗ p_{2}^{∗}(M).

This looks very plausible but it is not so easy to prove, and I will not give the proof here. The proof requires some deeper results in complex analysis, for instance some finiteness results.

In the following chapter (???) we will prove an analogous statement in the context of algebraic geometry. In that case the truth of the assertion will be a consequence of the construction, and we will need the full strength of the finiteness results in algebraic geometry.

3.2 The structure of End(J):

Since we have an inclusion End(J) ⊂ Hom(Γ, Γ), we know that End(J) is a finitely generated algebra over ℤ.

For any φ ∈ End(J) defined an endomorphism

φ^{∗} : Pic(J) ⟶ Pic(J)

of the Picard group which is given by the pull back of line bundles. We denote by ^{t}φ the restriction of φ^{∗} to the subgroup Pic^{0}(J), and use the selfduality of the Jacobian we get the transposed

tφ : J ⟶ J.

I want to point out that this result makes sense in the context of algebraic geometry. We have seen that the group NS′(J × J) has an algebraic definition and this is also the case for End(J).

We have seen that ^{t}φ corresponds to the transpose of φ : Γ → Γ with respect to e_{0}.

The map φ →^{t} φ is called the Rosati involution (with respect to the standard polarization). It has the property

\begin{aligned}^{t}(φ + ψ) & = & ^{t}φ +^{t} ψ \\ ^{t}(φψ) & = & ^{t}ψ^{t}φ \end{aligned}

It is of course clear that φ^{∗} also induces an endomorphism

\overline{φ}^{∗} : NS(J) ⟶ NS(J).

We had the identification

NS(J) ≃ End_{sym}(J)

(see ???) and we saw that

\overline{φ}^{∗} : ψ ⟶ φψ^{t}φ.

Especially it follows that φ → \overline{φ}^{∗} is quadratic. This means that we can consider φ_{1} + φ_{2} and then

(\overline{φ_{1} + φ_{2}})^{∗} = \overline{φ}_{1}^{∗} + \overline{φ}_{2}^{∗} + ⟨φ_{1}, φ_{2}⟩

where ⟨φ_{1}, φ_{2}⟩ : NS(J) → NS(J) depends bilinearily on the two variables.

To any φ ∈ End(J) we can define tr(φ) and deg(φ) simply as the trace and the determinant of φ considered as endomorphism of Γ, i. e.

tr(φ) = tr(φ : Γ → Γ)
deg(φ) = det(φ : Γ → Γ).

These functions have the obvious properties

tr(^{t}φ) = tr(φ)
det(^{t}φ) = det(φ)
det(φ_{1}φ_{2}) = det(φ_{1}) det(φ_{2})

We have the following

3.2.1 Theorem (Positivity of the Rosati-Involution): For any φ ∈ End(J), φ ≠ 0 we have

tr(φ^{t}φ) > 0.

Proof: We give a proof which uses the lattice Γ. We have seen in IV.8.7 that we can find a basis

{e_{1}, f_{1}, e_{2}, f_{2}, ⋯ e_{g}, f_{g}}

such that Iev̇ = fv̇ and ⟨ev̇, fv̇ = −1 where the other scalar products vanish. Then it is clear that for any endomorphism ψ we have

tr(ψ) = − ∑⟨ψ(e_{i}), f_{i}⟩ + ∑⟨ψ(f_{i}), e_{i}⟩.

If ψ =^{t} φφ then we get

\begin{aligned}tr(^{t}φφ) & = & − ∑_{i=1}^{g}⟨φ(e_{i}), φ(f_{i})⟩ + ∑_{i=1}^{g}⟨φ, f_{i}⟩, φ(e_{i})⟩ \\ & = & −2 ∑_{i=1}^{g}⟨φ, e_{i}), φ(Ie_{i})⟩. \end{aligned}

Since φ commutes with I, the last sum is equal to

−2 ∑⟨φ(e_{i}), Iφ(e_{i})⟩ = + ∑ 2h_{⟨ , ⟩}(φ(e_{i}), φ(e_{i})).

The terms are ≥ 0 and since at least one of the φ(e_{i}) ≠ 0 (φ(f_{i}) = φ(Ie_{i}) = φ(e_{i})!) we conclude that the sum must be strictly positive. The existence of the trace and the positivity of the Rosati involution have a dramatic influence on the structure of the algebra End(J) this will be discussed in the second volume Chap. ???

We defined the degree of an endomorphism as deg(φ) = det(φ|Γ), this definition uses the trancendental object Γ. But we can easily define this degree in purely algebraic terms. We consider

J ⟶^{φ} J
↑  ↑
H^{1}(S, O_{S})/Γ ⟶^{φ} H^{1}(S, O_{S})/Γ.

and then we see easily:

3.2.2 Proposition: The degree of φ is non zero if and only if the morphism φ is finite. If the degree of φ is non zero, then we have

deg(φ) = [ℂ(J) :_{φ} ℂ(J)] = number of points in φ^{−1}(0).

Proof: This is rather clear: The determinant of φ is equal to the index of φ(Γ) in Γ. If Γ′ ⊂ H^{1}(O_{S}, ℤ) is the inverse image of Γ, then Γ′/Γ ≃ Γ/φ(Γ) and this proves that the order of the kernel φ^{−1}(0) is also equal to det(φ).

It is also clear that φ induces an isomorphism of the tangent spaces at zero and hence we can conclude that the fibre φ^{−1}(0) is reduced. The morphism φ induces an inclusion of the field of franctions ℂ(J) ↪ ℂ(J) and the subfield is the field of invariants under Γ′/Γ, the last assertion follows from Galois theory.

We defined the degree deg(φ) and the trace tr(φ) on J using the lattice Γ but now we found an algebraic definition of the degree. From the definition of the degree as a determinant it follows that

deg(φ + n Id) = a_{0}(φ) + ⋯ a_{2n−1}(φ)n^{2g−1} + n^{2g},

then

tr(φ) = a_{2n−1}(φ).

But it not clear how we can derive this formula from the algebraic definition as the degree of a field extension.

The point in the following considerations will be that for an element ψ ∈ End(A) the element

ψ^{∗}(c) ∈ NS(A)

contains decisive information on the endomorphism ψ. Especially we will see that we can express the degree of the endomorphism in terms of this class. The fundamental point is that its g-th power with respect to the cup product is an element in H^{2g}(J, ℤ) = ℤ by Poincare duality. But if we think of ψ^{∗}(c) ∈ NS(A) then we can define the g fold selfintersection of this line bundle in the context of algebraic geometry (See ???) this is also a number. We explained in (IV.8.6 ) show us that these numbers are the same.

The element c is an alternating form on Γ^{∨}. If dim A = g, then we can raise this element into the g’th power in the cohomology ring and we have seen (See IV.6.6.2.1) that this means that we have to take its g’th exterior power

c^{g} = c ∧ c … ∧ c ∈ Hom_{alt}^{2g}(Γ^{∨}, ℤ) ≃ ℤ,

(this is the selfintersection number of the class c.) Now the class

ψ^{∗}(c)(γ_{1}, γ_{2}) = c(ψ(γ_{1}), ψ(γ_{2})),

and it is an elementary excercise in linear algebra that

ψ^{∗}(c)^{g} = deg(ψ) · c^{g},

where det(ψ) is of course the determinant of the endomorphism ψ on the free module Γ which is of rank 2g.

Now the formula above gives us the degree of ψ in terms of the elements c and ψ^{∗}(c) which are in NS(A) and this makes sense- as explained above- in the context of algebraic geometry.

Of course this formula will give us an expression for the trace of ψ ∈ End(A). We assume that A has a polarization c ∈ NS(A) such that a corresponding bundle is ample. Now we consider the map

(ψ + n Id)^{∗} : NS(A) ⟶ NS(A).

We have seen that this map is

ψ^{∗} + n⟨ψ, Id⟩ + n^{2} · Id .

We have the formula for the degree

deg(ψ + n Id) · c^{g} = ((ψ + n Id)(c))^{g} =
= (ψ^{∗}(c) + n⟨ψ, Id⟩(c) + n^{2} · c)^{g} =
⋯ + gn^{2g−1}c^{g−1} ∧ ⟨ψ, Id⟩(c) + n^{2g} · c^{g}

and hence we get the formula

tr(ψ) · c^{g} = gc^{g−1} ∧ ⟨ψ, Id⟩(c),

and this gives us the trace as a cup product of classes in the cohomology evaluated on the fundamental cocycle. Let us assume that A = J and c is the canonical polarization. Let us assume that the endomorphism ψ is a product of the form ψ = ^{t}φφ. Then

\begin{aligned}c⟨(ψ + Id)γ_{1}, (ψ + Id)γ_{2}⟩ & = (ψ + Id)^{∗}c⟨γ_{1}, γ_{2}⟩ = \\ & = ψ^{∗}c⟨γ_{1}, γ_{2}⟩ + c⟨γ_{1}, γ_{2}⟩ + c⟨γ_{1}, ψγ_{2}⟩ + c⟨ψγ_{1}, γ_{2}⟩ \end{aligned}

and the sum of the last two terms is ⟨ψ, Id⟩(c)⟨γ_{1}, γ_{2}⟩. Hence we get

⟨ψ, Id⟩(c)⟨γ_{1}, γ_{2}⟩ = 2c⟨φγ_{1}, φγ_{2}⟩,

and this means that

⟨ψ, Id⟩(c) = φ^{∗}(c).

This gives us the formula

tr(^{t}φφ) · c^{g} = 2gc^{g−1} ∧ φ^{∗}(c).

We return to the interpretation in terms of algebraic geometry. We assumed that c should be the class of an ample line bundle and we have seen that the highest intersection numbers of line bundles are equal to the highest cup product of their Chern classes. Hence we can say that

tr(^{t}φφ) · P^{g} = 2gP^{g−1} ∧ φ^{∗}(P).

Since the right hand side must be positive for φ ≠ 0 we get a formula in the context of algebraic geometry which implies the positivity of the Rosati involution.

Finally we want to give a formula for the trace of an endomorphism in terms of intersection number of two divisors on these surface S × S. We return to our bundle

N = m^{∗}(P) ⊗ p_{1}^{∗}(P)^{−1} ⊗ p_{1}^{∗}(P)^{−1}

on J × J has Chern class zero when restricted to e × J and J × e and its Chern class is E_{0}.

If we pick an element ψ ∈ End(J), then we can consider the bundle

(Id ×ψ)^{∗}(N)

on J × J. We have the inclusion

i_{P_{0}} × i_{P_{0}} : S × S ⟶ J × J

and get the line bundle

(i_{P_{0}} × i_{P_{0}})^{∗} ∘ (Id ×ψ)^{∗}(N) = L_{ψ}

on S × S. The Chern class of this line bundle sits in NS′(S × S) ⊂ H^{1}(S, ℤ) ⊗ H^{1}(S, ℤ) = Γ ⊗ Γ.

Now we have the famous trace formula

Δ · L_{ψ} = − tr(ψ).

This is a rather formal consequence of the definitions. We have seen that the intersection product of two divisors is equal to the cup product of the Chern classes and this evaluated on the fundamental class. The cup product of the classes

ξ_{1} ⊗ η_{1}, ξ_{2} ⊗ η_{2} ∈ Γ ⊗ Γ ⊂ H^{2}(S_{1} × S_{1}, ℤ)

is given by

−ξ_{1}, ξ_{2} ∪ η_{1}η_{2}

where now ξ_{1}ξ_{2} ∈ H^{2}(S, ℤ) ⊗ H^{0}(S, ℤ) and η_{1}η_{2} ∈ H^{0}(S, ℤ) ⊗ H^{2}(S, ℤ).

Since we have the alternating 2-form, we can choose as a standard basis on Γ a ℤ-basis

u_{1} ⋯ u_{g} v_{1} ⋯ v_{g}

such that e_{0}⟨u_{i}, v_{i}⟩ = −e_{0}⟨v_{i}, u_{i}⟩ = 1 and all other products are zero.

Under this identification the element

E = ∑ u_{i} ⊗ v_{i} − ∑ v_{i} ⊗ u_{i} ∈ Γ ⊗ Γ

becomes the identity element in Γ^{∨}⊗Γ = Hom(Γ, Γ): The element E applied to an element γ ∈ Γ yields

E(γ) := ∑⟨u_{i}, γ⟩v_{i} − ∑⟨v_{i}, γ⟩u_{i}.

Then it is clear that E(u_{i}) = u_{i} and E(v_{i}) = v_{i}.

Then the Chern class of L_{ψ} is given by

c_{1}(L_{ψ}) = ∑ u_{i} ⊗ ψ(v_{i}) − ∑ v_{i} ⊗ ψ(u_{i})

and

E ∪ c_{1}(L_{ψ}) = (∑ u_{i} ⊗ v_{i} − ∑ v_{i} ⊗ u_{i}) ∪ (∑ u_{i} ⊗ ψ(v_{i}) − ∑ v_{i} ⊗ ψ(u_{i}))
= + ∑⟨u_{i}, v_{i}⟩ · ⟨v_{i}, ψ(u_{i})⟩ = ∑⟨u_{i}, v_{i}⟩ · ⟨u_{i}, ψ(v_{i})⟩
= ∑⟨v_{i}, ψ(u_{i})⟩ − ∑⟨u_{i}, ψ(v_{i})⟩ = − tr(ψ).

Let us consider the Abel map

i_{p_{0}} : S ⟶ J

which induces a map on the first cohomology

i_{P_{0}}^{∗} : H^{1}(J, ℤ) ⟶ H^{1}(S, ℤ)
∥  ∥
Γ^{∨}  ⟶  Γ

which we identified as the inverse of the polarization map. It induces a map

H^{2}(J, ℤ) ⟶ H^{2}(S, ℤ)
∥  ∥
Λ^{2}Γ^{∨} ⟶  ℤ

and this map is just the evaluation by the dual form

φ ∧ ψ ⟼ e^{∨}(φ, ψ).

This linear form on H^{2}(J, ℤ) can be viewed as an element [S] in H_{2}(J, ℤ) or as an element in H^{2g−2}(J, ℤ). This class [C] is of course the Chern class of our Riemann surface S, it is the image of the fundamental class in H_{2}(S, ℤ) in H_{2}(J, ℤ) under the map induced by the Abel map.

If we choose a basis e_{1}, ⋯, e_{g}, f_{1} ⋯ f_{g} on Γ such that ⟨e_{i}, f_{i}⟩ = 1 = −⟨f_{i}, e_{i}⟩ and all other pairings give zero, then the −f_{1}, ⋯, −f_{g}, e_{1}, ⋯, e_{g} are the elements of the dual basis if we identify Γ and Γ^{∨} by the polarization map. Then the form e_{v} is given by

∑ e_{i} ∧ f_{i}

again and our form in H^{2g−2}(J, ℤ) which is the fundamental class of S is given by

∑_{i} e_{1} ∧ f_{i} ⋯ ê_{i} ∧ f_{i} ∧ ⋯ e_{g} ∧ f_{g},

i.e. the factor e_{i} ∧ f_{i} is left out.

The polarization class e^{∨} ∈ Λ^{2}Γ^{∨} itself maps to g in H^{2}(S, ℤ), and it is clear that

(e^{∨})^{g−1} = (∑ e_{i} ∧ f_{i}) ∧ ⋯ ∧ (∑ e_{i} ∧ f_{i}) = (g − 1)![S]

and

(e^{∨})^{g} = g!.

Now recall the formula for the trace

tr(ψ) = g \frac{P^{g−1} · ⟨Id, ψ⟩(P)}{P^{g}}.

We have seen that P^{g−1} = (g − 1)![S] and P^{g} = g!, hence we get

tr(ψ) = [S] · ⟨Id, ψ⟩(P) =

I think that at this point the role of the polarization becomes clear. First of all we have seen that it allows the constructionof a line bundle P on J which is ample and hence it provides a projective embedding of J and hence (J, P) becomes an object in algebraic geometry. The reader should be aware that the bundle P is not unique since it depends on the choice of a linear form. Now we consider the diagram

S  ⟶^{Δ} S × S
↓ i_{P_{0}}  ↓ i_{P_{0}} × i_{P_{0}}
J  ⟶^{Δ} J × J

The ring of correspondences We have the isomorphism Pic(S × S)/p_{1}^{∗}(Pic(S)) + p_{2}^{∗}(Pic(S)) ≃ End(J). We want to explain how we can define a ring structure on the left hand side directly.

If we have an irreducible divisor D ⊂ S ×S we can look at it as a so called correspondence: To any point z ∈ S we can consider the points(z, z_{i}) ∈ D and call the points z_{i} counted with multiplicity as the points corresponding to z. We can form the free group of these divisors and mod out by the divisors of the form S × D′ or D′ × S where D′ (resp. D′′) is a divisor in the first or second factor, let us call this R. After we have done this, we can introduce a product on this group: We choose suitable representatives D_{1}, D_{2} of two elements and consider the divisors in

D_{1} × S, S × D_{2} on S × S × S.

Now we take the intersection – this makes sense if we made a careful choice – and project this intersection to the two outer factors.

This induces a ring structure on R with identity which is given by the class of the diagonal.

It is clear that this ring has an involution which is obtained by interchanging the two factors.

We can also define a trace: For any [D] ∈ R we choose a representative D for which D | z_{0} × S and D | z_{0} × S are both in Pic^{0}(S). Then we put

− tr([D]) = Δ · D.

Now it is at least plausible that

− tr(^{t}[D] ∗ [D]) = (^{t}[D] × [D]) · Δ = D · D,

and this last number is strictly < 0 if D ≠ 0 by the Hodge index theorem. (See ???)

We know of course that R ≃ End(J). These considerations show that we can define this ring without reference to the Jacobian.

(Vielleicht später an eine andere Stelle!) 3.3 An algebraic sustitute for the cohomology I think that I convinced the reader that the cohomology groups H^{1}(S, ℤ) = Γ, H^{1}(J, ℤ) = Γ^{∨} play a fundamental role in understanding the structure of S and J. Therefore we should have a substitute for these cohomology groups in the algebraic context. This will be explained later (see ???) but we give an indication how we can get cohomology in the algebraic context.

We have

J = H^{1}(S, O_{S})/H^{1}(S, ℤ) = H^{1}(S, O_{S})/Γ.

The module Γ does not make sense in the context of algebraic geometry. Now we consider the endomorphism

n Id : J ⟶ J,

and we consider the kernel

J[n] = ker(n Id : J ⟶ J).

This kernel is obviously isomorphic to

\frac{1}{n} Γ/Γ ≃ (ℤ/nℤ)^{2g}.

But this kernel can be defined in the context of algebraic geometry. We consider J as a projective variety over ℂ which has the structure of an abelian algebraic group and then the kernel of n Id is a finite group scheme over ℂ.

Once we have done this, we observe that we have for n | n_{1}

J[n] ⟶ J[n_{1}],

and we can define

Tors(J) = lim_{\overrightarrow{n}} J[n],

where the ordering on ℕ is given by divisibility. Of course it is clear that

Tors(J) = Γ ⊗ ℚ/ℤ

and we conclude:

The module Γ cannot be defined in terms of algebraic geometry, but the module

Γ ⊗ ℚ/ℤ

is an algebraic geometric object. We can pass to the dual module, we consider

Hom(Tors(J), ℚ/ℤ).

It is an elementary fact that

Hom(ℚ/ℤ, ℚ/ℤ) ≃ lim_{←} ℤ/n ℤ = ℤ̂,

and therefore we get

Hom(Tors(J), ℚ/ℤ) = Γ ⊗ ℤ̂,

and this is the so called Tate-module

T (J) = lim_{\overleftarrow{n}} J[n]

where now for n/n_{1} the map

J[n_{1}] ⟶ J[n]

is given by multiplication by n_{1}/n. (Verweise auf Kap. I ???) The Chinese remainder theorem yields

ℚ/ℤ = ⨁_{ℓ:ℓpine} ℚ_{ℓ}/ℤ_{ℓ}

where ℓ runs over the primes and ℚ_{ℓ} is the ℓ-adic completion and ℤ_{ℓ} is the ring of ℓ-adic integers. This yields a decomposition

Tors(J) = ⨁_{ℓ} Tors(J)_{ℓ}

where Tors(J)_{ℓ} = lim J[ℓ^{α}] and dually

ℤ̂ = ∏_{ℓ} ℤ_{ℓ}

and

T (J) = ∏_{ℓ} T_{ℓ}(J).

Chapter VI Basic notions in algebraic geometry

VI.1.Affine schemes

We consider commutative rings A, B … with identity (1A, 1_{B}, …), morphisms ϕ : A → B are homomorphisms sending the identity of A into the identity of B. We always assume that the identity in a ring is different from zero. For any such ring A we have the group of invertible elements (units): A^{∗} = {a ∈ A | ∃ a′ ∈ A such that aa′ = 1_{A}}. A proper ideal 𝔞 ⊂ A is an ideal with 1_{A} ∉ 𝔞, prime ideals are always proper. For any ring and any f ∈ A we use the standard notation (f) for the principal ideal Af.

VI.1.1 Localization

If we have a subset S ⊂ A which is closed under multiplication and contains the identity 1_{A} ∈ S we can define a quotient ring A_{S} and a map ϕ_{S} : A → A_{S} such that the elements of S become invertible.

To do this we consider pairs (a, s) ∈ A × S and introduce an equivalence relation

(a, s) ∼ (a′, s′) ⟺ ∃ s′′ ∈ S such that (as′ − a′s) · s′′ = 0.

We consider the quotient A_{S} of A × S by this relation, let π_{S} : A × S → A_{S} be the projection to this quotient. We define a ring structure on A_{S} by

π_{S}((a, s)) + π_{S}((a′.s′)) = π_{S}((as′ + a′s, ss′))
π_{S}((a, s)) · π_{S}((a′.s′)) = π_{S}((aa′, ss′)).

We have a homomorphism of rings

ϕ_{S} : \begin{matrix}A & ⟶ & A_{S} \\ a & ⟼ & π_{S}((a, 1)) \end{matrix} .

We will write the elements of A_{S} simply as π_{S}((a, s)) = \frac{a}{s} = as^{−1}. Of course

\frac{a}{s} = \frac{as′}{ss′} = \frac{as′s′′}{ss′s′′} . . ..

The quotient ring has a universal property: For any ring B we can consider

HomRings, S to units(A, B) = {ϕ : A → B|ϕ(S) ∈ B^{∗} for all s ∈ S}

and this set of homomorphisms is equal to

Hom_{Rings}(A_{S}, B)

where the identification is given by the diagram

\begin{aligned}A & ⟶^{ϕ_{S}} & A_{S} \\ ϕ ↘ & & ↙ ϕ′ \\ & B \end{aligned} .

If 0 ∈ S then A_{S} = {0}. If f ∈ A then we write

A_{f} = A_{{f^{n}}_{n=0,1...}}.

VI.1.2 The spectrum of a ring A. If A is a commutative ring with identity then we define the spectrum of A as

Spec(A) = {𝔭| 𝔭 prime ideal in A}.

The spectrum of A is ordered the ordering is given by the inclusion among prime ideals. It is functorial in A, if we have a homomorphism

ϕ : A → B

then it induces

tϕ : Spec(B) → Spec(A)
tϕ(𝔭) = ϕ^{−1}(𝔭) = {f | ϕ(f) ∈ 𝔭}

and ^{t}ϕ respects the order relation.

The spectrum Spec(A) always contains as a subset the set of maximal ideals. A maximal ideal 𝔪 ⊂ A is an ideal with 1_{A} ∉ 𝔪 and for any ideal 𝔪′ with

𝔪 ⊆ 𝔪′ ⊆ A

we have 𝔪 = 𝔪′ or 𝔪′ = A. We have a different characterization

Lemma VI.1.2.1 : An ideal 𝔪 ⊂ A with 1_{A} ∉ 𝔪 is maximal if and only if A/𝔪 is a field. Maximal ideals are prime ideals.

I think this should be clear.

The set of maximal ideals is denoted by Specmax(A) ⊂ Spec(A).

Zorn’s lemma implies:

1.2.1.1.If A is a commutative ring with 1_{A} ≠ 0, then Specmax(A) ≠ ∅.

I want to give brief indications of the proof: A chain of proper ideals is a totally ordered subset 𝔎 of the set of proper ideals, this means that for any pair 𝔞, 𝔟 ∈ 𝔎 we have 𝔞 ⊆ 𝔟 or 𝔟 ⊆ 𝔞. For any chain we can form

⋃_{𝔞∈K} 𝔞 = 𝔞^{∗},

this is an ideal with 1_{A} ∉ 𝔞^{∗} and 𝔞^{∗} ⊃ 𝔞 for all 𝔞 ∈ 𝔎. Hence we see that for any chain of proper ideals we can find a proper ideal which contains all elements of the chain. Now it is simply the assertion of Zorn’s lemma that this implies the existence of a (proper) maximal ideal. This has as a consequence:

Lemma VI.1.2.2 : If f ∈ A is not nilpotent, then Spec(A_{f}) ≠ ∅. Hence we get

Rad (A) = Ideal of nilpotent elements = ⋂_{𝔭∈Spec(A)} 𝔭.

A commutative ring is called a local ring if it has a unique maximal prime ideal.

If 𝔭 ∈ Spec(A) then the complement S = A \ 𝔭 is closed under multiplication. Then we write (abuse of notation)

A_{(A\𝔭)} =: A_{𝔭}.

The ring A_{𝔭} is called the local ring at 𝔭. The ideal

𝔪_{𝔭} = \{\frac{f}{g} |f ∈ 𝔭, g ∉ 𝔭\}.

is the unique maximal. The field k(𝔭) = A_{𝔭}/𝔪_{𝔭} is called the residue field at 𝔭.

Lemma VI.1.2.3 : If we consider any multiplicatively closed S ⊂ A and the localization ϕ : A → A_{S} then ^{t}ϕ is an inclusion. We get

tϕ : Spec(A_{S}) ⟶^{∼} {𝔭 ∈ Spec(A)|𝔭 ∩ S = ∅}.

If especially S = {f^{n}}_{n=0,1...} then

Spec(A_{f}) = {𝔭 ∈ Spec(A)|f ∉ 𝔭}.

If 𝔭 ∈ Spec(A) then

Spec(A_{𝔭}) = {𝔮 ∈ Spec(A) | 𝔮 ⊂ 𝔭}.

The proof is left to the reader.

VI.1.2.4 Heuristical remarks:

(i) The spectrum of a ring is a geometric object attached to the ring. At this point it is simply an ordered set, but soon we will put a topology onto this space (The Zariski Topology). We already spoke of maximal ideals. If our ring A is integral then the zero ideal (0) is also a prime ideal which then is the unique minimal element. This ideal (0) is called the generic point of Spec(A).

(ii) Intuitively we want to consider A as a ring of functions on Spec(A). This is not quite the case because these functions do not have a common domain of values. But it makes sense to say that f ∈ A ”vanishes” at 𝔭 ∈ Spec(A): By this we mean that f ∈ 𝔭. Sometimes we will write f(𝔭) = 0, (resp. f(𝔭) ≠ 0) for f ∈ 𝔭 (resp. f ∉ 𝔭).

Examples.

(1) The ring ℤ and the polynomial ring k[X] are principal ideal domains. This implies immediately that the maximal ideals are of the form

𝔭 = (p) resp. 𝔭 = (p(x))

where p is a prime number (resp. p(X) ∈ k[X] is a non-constant irreducible polynomial). Both rings contain one more prime ideal namely 𝔭 = (0) because they are integral. Hence

Spec(ℤ) = (0) ∪ {(2), (3), (5), …} Spec(k[X]) = (0) ∪ {(X), (X − 1), …} .

(Of course not all irreducible polynomials are linear, but I cannot write down any other polynomial which is irreducible regardless what the field k is).

(2) Let us assume that k is a field. We consider the polynomial ring A = k[X_{1}, X_{2}, …, X_{n}] in n variables. For any point P = (a_{1}, a_{2}, …, a_{n}) ∈ k^{n} we get an evaluation homomorphism

\begin{matrix}ϕ_{P} : A → k \\ ϕ_{P} : f ↦ f(P), \end{matrix}

whose kernel is the maximal ideal 𝔪_{P} = (X_{1} − a_{1}, X_{2} − a_{2}, …, X_{n} − a_{n}).

If our field k is algebraically closed then the Nullstellensatz of Hilbert (It will be discussed in the next section on commutative algebra) says that we get an identification

Specmax(k[X_{1}, X_{2}, …, X_{n}]) = k^{n}.

In other words the maximal ideals are exactly the ideals of the form 𝔪 = 𝔪_{P}.

Exercise (1):

(i) Prove the Nullstellensatz in the case of a polynomial ring in one variable.

(ii) Try to prove it in the case of a polynomial ring A = k[X, Y] in two variables.

Hint: Let 𝔪 be a maximal ideal. It cannot be the zero ideal.

Step 1:

Assume it contains an element of the form F (X, Y) = Y^{m} + g_{1}(X)Y^{m−1} + … g_{m}(X) where the g_{i} are polynomials in X. Now we get a diagram

A  →  A/(F) = B
↘  ↙
A/𝔪

The ring B contains the polynomial ring k[X] = B_{0} and over this ring it is generated by an element y which satisfies the relation y^{m} + g_{1}(X)y^{m−1} + … g_{m}(X) = 0. The maximal ideal 𝔪 ⊂ A has as its image a maximal ideal 𝔪̄ in B.

Prove that 𝔪̄∩k[X] is a maximal prime ideal! (In this case it suffices to show it is not zero.) Hence B_{0}/𝔪̄ = k and since B/𝔪 is a finite extension of k it follows that B/𝔪 = k.

Step 2: We know that 𝔪 contains a non zero polynomial

F (X, Y) = ∑_{νμ} a_{ν,μ}X^{ν}Y^{μ}.

Write this as a polynomial in Y with coefficients polynomials in X. Now it will not be the case in general that the highest power of Y occuring in the polynomial has a constant coefficient not involving X. But if we make a substitution

\begin{matrix}X → X + Y^{m} = X′ \\ Y → Y \end{matrix}

then k[X′, Y = k[X, Y] for m >> 0 the new polynomial will satisfy the assumption in step 1 .

It is known that a polynomial ring k[X_{1}, …, X_{n}] has unique factorization ( See [Ei],???). This implies that any non constant irreducible polynomial f ∈ k[X_{1}, …, X_{n}] defines a prime ideal 𝔭 = (f). This means of course that for instance Spec(k[X, Y]) contains many more elements than just (0) and the maximal ones: Any irreducible polynomial

p(X, Y) = X + Y or X^{2} + Y^{3} or …

defines a prime ideal 𝔭 = (p(X, Y)).

If k is algebraically closed then the Nullstellensatz allows us to identify 𝔭 = (p(X, Y)) with

V (𝔭) = {(a, b) ∈ k^{2}|f(a, b) = 0 for all f ∈ 𝔭}

this is the set of common zeroes of the elements in 𝔭 or the set of zeroes of p(X, Y) and also the set of maximal ideals containing 𝔭. Hence we get an injection into the power set

Spec(k[X, Y]) ⟶ 𝔓(Specmax(k[X, Y]))𝔭 ⟼ V (𝔭).

Then the maximal ideals correspond to the sets consisting of one element, the prime ideals 𝔭 = (p(X, Y)) give hypersurfaces and 𝔭 = (0) gives us the entire plane.

(3) Let k be arbitrary and

A = k[X, Y](XY) = k[x, y].

The elements x, y satisfy xy = 0. Hence this ring has zero divisors.

A prime ideal 𝔭 in A has to contain either x or y. On the other hand the principal ideals 𝔭 = (x) and 𝔮 = (y) are prime because after dividing by them we get polynomial rings in the other variable.

We see that

Spec(k[x, y]) = Spec(k[x]) ∪_{(0,0)} Spec(k[y])

where the two spectra are identified at (x, y) = (0, 0).

This is an example of a reducible spectrum.

VI.1.3. The Zariski Topology on Spec(A) = X. We define a topology on the space X. To do so we have to define what open sets are.

At first we declare the sets of the form

X_{f} = Spec(A_{f}) ⊂ X

open. We saw that X_{f} was the set of prime ideals 𝔭 which do not contain f. In our remark (VI.1.2.4..) we said that this means ”f does not vanish at 𝔭.” Hence our topology has the property that the sets where a given f ∈ A is not zero are open set.

This system of sets is closed under finite intersection because

X_{f_{1}} ∩ … ∩ X_{f_{s}} = X_{f_{1}...f_{s}}.

These open sets are called affine open sets the reason is that they are again equal to a spectrum of a ring. This system of affine open sets forms a basis for the Zariski topology and this means that a set U ⊂ X is open if and only if it is the union of the affine open sets which are contained in U.

A subset Y ⊂ X is closed if the complement X\Y is open. Of course this means that Y is the set of common zeroes of a collection of elements in A.

The Zariski topology is not Hausdorff in general. It has other strange properties one has to get used to:

Exercise (2):

If 𝔭 ∈ Spec(A) then the closure of the set {𝔭} is given by

\overline{\{𝔭\}} = \{𝔮 ∈ Spec(A)|𝔮 ⊃ 𝔭\}.

A point 𝔭 ∈ Spec(A) is closed if and only if 𝔭 is maximal.

If 𝔮 is in the closure of {𝔭} then we say that 𝔮 is a specialization of 𝔭.

Examples: In our rings ℤ and k[X] the closed points are the principal ideals (p) resp. (p(x)) where p is a prime (resp. p(x) a non-constant irreducible polynomial). The generic point (0) is dense in Spec(A) in both cases. The open sets are the complements of finite sets of closed points and the empty set.

Here it becomes quite clear that Spec(ℤ) and Spec(k[X]) are not Hausdorff.

For an integral ring A the generic point (0) always dense in the space Spec(A). Every prime ideal 𝔭 ∈ Spec(A) is a specialization of the generic point.

VI.1.3.1.General remark:

We have put some further structure onto the set Spec(A) : Now it is a topological space. But still this space does not yet contain a lot of information on the original ring A. If for instance A is a field then Spec(A) is a single point which will never be able to recover the field A.

This is different for finitely generated algebras A = k[x_{1}, x_{2}, …, x_{n}] over an algebraically closed field k. In the next section we will see that the Nullstellensatz implies (See CA.1.3.)

⋂_{𝔪∈Specmax(A)} 𝔪 = Rad(A).

If our k-algebra is reduced, i.e. if Rad(A) = 0 then this implies that we can view A as an an algebra of k-valued functions on Specmax(A) This discussion is resumed in VI. 2.5. and 2.6.

Exercise (3):

Let k be an algebraically closed field and let

A = k[x_{1}, x_{2}, …, x_{n}], B = k[y_{1}, y_{2}, …, y_{m}]

be two finitely generated reduced k-algebras. Let ϕ : A → B be a homomorphism which is the identity on k ( a k- algebra homomorphism). Show that this induces a map ϕ^{∗} : Specmax(B) → Specmax(A). We assume in addition that A and B do not contain nilpotent elements.

Prove: (a) ϕ^{∗} is injective ⟺ ϕ is surjective.

(b) If ϕ^{∗} is surjective then ϕ is injective.

(c) The map ϕ^{∗} determines ϕ.

The following property of the topological space Spec(A) is very important and perhaps a little bit surprising at the first glance

Proposition VI.1.3.2: The space X = Spec(A) is quasicompact, this means that for any covering ⋃_{i∈I} U_{i} by open sets U_{i} we can find a finite subcovering, i. e. we can find a finite subset E ⊂ I such that X = ⋃_{i∈E} U_{i}

Proof: The U_{i} are open, hence we can cover each of them by open sets of the form X_{f_{i,ν}}. Therefore it is clear that we may assume that the U_{i} themselves are of this form U_{i} = X_{f_{i}}. Now we consider the ideal generated by the f_{i} it consists of the finite linear combinations

𝔞 = {∑_{i∈I} g_{i}f_{i} | almost all g_{i} = 0}

This cannot be a proper ideal because otherwise we could find a maximal ideal 𝔪 containing 𝔞 (See 1.2.1.1.). Then we have f_{i}(𝔪) = 0 for all i ∈ I and hence 𝔪 ∉ ⋃_{i∈I} X_{f_{i}}. This implies that 𝔞 = A and hence the identity 1_{A} is in our ideal. We can find a finite linear combination 1_{A} = ∑_{i∈E} g_{i}f_{i} with E ⊂ I finite. But then it is clear that X = ⋃_{i∈E} X_{f_{i}} because if there would be a 𝔭 not contained in this union then we would have f_{i}(𝔭) = 0 for all i ∈ E and hence 1_{a}(𝔭) = 0 which cannot be the case if our ring is not the zero ring. But for this last ring the spectrum is empty so the claim is also clear.

Our next goal will be to put more structure on X = Spec(A). Since it is already a topological space we have the notion of a sheaf on this space. We will construct the sheaf of regular functions on O_{X} = Spec(A) and then (X, O_{X}) will be a locally ringed space.

VI.1.4 The structure sheaf on Spec(A): We want to introduce the structure of a locally ringed space on X = Spec(A). This

means that we want to construct a sheaf of rings Ã on X which plays the role of the sheaf of regular functions on X. It will turn out- but this will be a theorem- that the ring of regular functions on the total space is again A.

We make the following Ansatz: If we have an open set X_{f} ⊂ X then the element 1/f ∈ A_{f} should be a regular function on the affine open set X_{f}. Hence we define Ã′(X_{f}) = A_{f}. If we have h = gf then A_{h} = (A_{f})_{g} then the map ϕ_{{g^{n}}}_{{n=0,1,...}} : A_{f} → A_{h} gives us a restriction map Ã′(X_{f}) → Ã′(X_{gf}) = X_{h}. This obviously satisfies the transitivity relation for presheaves this means that the restriction from Ã′(X_{f}) to X_{fg} and then composed with the restriction to X_{fgl} is equal to the restriction from X_{f} to X_{fgl}. We will denote the restriction map ϕ_{{g^{n}}}_{{n=0,1,...}} : A_{f} → A_{h} also by | X_{h}. Hence our Ã′ is something like a presheaf except that it is not yet defined it on all open set but only on the affine open sets of the form X_{f}.

We now have a proposition which says that Ã′ satisfies the axioms (Sh1),(Sh2) when we restrict them to these special open sets.

Proposition VI.1.4.1..: If we have an arbitrary covering X = ⋃_{i∈I} X_{f_{i}} then the sequence

Ã′(X) ⟶^{p_{0}} ∏_{i∈I} Ã′(X_{f_{i}}) ⟶^{p_{1}} ; ⟶_{p_{2}} ∏_{(i,j)∈I×I} Ã′(X_{f_{i}f_{j}}),

is exact, this means that the first arrow is injective and its image is exactly the set of elements which become equal under p_{1} and p_{2}.

This proposition is really central.

Proof: Since our space is quasicompact, we can find a finite subset E ⊂ I such that already X = ⋃_{i∈E} = X. I assume that I proved exactness for this finite covering. I want to show that then we have exactness for the original covering. We get a map from the diagram above to the corresponding diagram for our finite covering.

Ã′(X) ⟶^{p_{0}} ∏_{i∈I} Ã′(X_{f_{i}}) ⟶^{p_{1}} ; ⟶_{p_{2}} ∏_{(i,j)∈I×I} Ã′(X_{f_{i}f_{j}})
↓  ↓  ↓
Ã′(X) ⟶^{p_{0}} ∏_{i∈E} Ã′(X_{f_{i}}) ⟶^{p_{1}} ; ⟶_{p_{2}} ∏_{(i,j)∈E×E} Ã′(X_{f_{i}f_{j}}),

The injectivity of the first arrow this is quite obvious, because the arrow p_{0} for the second diagram is the composite of the p_{0} for the first diagram and the projection from the product over I to the product over E. Now let us take an element (…, \frac{g_{i}}{f_{i}^{n_{i}}} , …)_{i∈I} in the first diagram with p_{1}((…, \frac{g_{i}}{f_{i}^{n_{i}}} , …)) = p_{2}((…, \frac{g_{i}}{f_{i}^{n_{i}}} , …)). If we project it to the second diagram then the image is also equalized by the corresponding two arrows. Hence by our assumption it comes from an element g ∈ A, this means that the image of g in A_{f_{i}} is equal to \frac{g_{i}}{f_{i}^{n_{i}}} for all i ∈ E. We have to show that this implies that g actually maps to (…, \frac{g_{i}}{f_{i}^{n_{i}}} , …)_{i∈I}. This means that we have show that g maps to \frac{g_{i}}{f_{i}^{n_{i}}} for all i ∈ I. Let us pick an i ∈ I. We know that X_{f_{i}} = ⋃_{e∈E}(X_{f_{i}} ∩ X_{f_{e}}) = ⋃_{e∈E} X_{f_{i}f_{e}}. But then we have g | X_{f_{i}f_{e}} = (g | X_{f_{e}}) | X_{f_{e}f_{i}} = \frac{g_{e}}{f_{e}^{ne}} | X_{f_{e}f_{i}} = \frac{g_{i}}{f_{i}^{n_{i}}} | X_{f_{e}f_{i}} for all e ∈ E. Now we need a little remark. The open set X_{f_{i}} is again the spectrum of a ring. Hence everything we proved for X is also valid for X_{f_{i}}. Especially we can assume that A_{f_{i}} → ∏_{e∈E} A_{f_{i}f_{e}} is injective. We have seen that g | X_{f_{i}f_{e}} = \frac{g_{i}}{f_{i}^{n_{i}}} | X_{f_{e}f_{i}} for all e ∈ E hence we conclude g | X_{f_{i}} = \frac{g_{i}}{f_{i}^{n_{i}}} | X_{f_{i}} for all i ∈ I. Hence the reduction to the case of a finite covering is complete and therefore we assume now that our covering is finite.

If the homomorphism p_{0} is not injective then we have have an element f ∈ A and f|X_{f_{i}} = 0 for all i ∈ I. This means that we can find exponents n_{i} so that ff_{i}^{n_{i}} = 0 in the ring A. Since I is finite we can assume that all these exponents are equal to a fixed integer n.

We have seen in the proof of the compactness of X that we can find g_{i} ∈ A such that

∑ g_{i}f_{i} = 1.

Raising this to a suitable high power N we get a relation

∑ G_{i}f_{i}^{n} = 1

and hence f = f1 = ∑_{i∈I} G_{i}f_{i}^{n}f = 0 and proves injectivity.

Now let us assume we have an element

(…, \frac{g_{i}}{f_{i}^{n_{i}}} , …)_{i∈I}

for which

\frac{g_{i}}{f_{i}^{n_{i}}} |X_{f_{i},f_{j}} = \frac{g_{i}}{f_{j}^{n_{j}}} | X_{f_{i},f_{j}},

for all pairs (i, j) ∈ I × I. Again we may assume that all n_{i} are equal.

Then the equality means that we can find an integer N so that

(g_{i}f_{j}^{n} − g_{j}f_{i}^{n})(f_{i}f_{j})^{N} = 0.

We are searching an element g ∈ A which satisfies

g|X_{f_{i}} = \frac{g_{i}}{f_{i}^{n}}

for all i ∈ I. This is certainly true if

f_{i}^{n}g = g_{i} for all i ∈ I.

But again we can find H_{ν} ∈ A for ν ∈ I such that

∑_{ν∈I} H_{ν}f_{ν}^{n} = 1

and we see that g = ∑_{ν∈I} H_{ν}g_{ν} solves our problem.

Still we have not yet defined our sheaf Ã. For an arbitrary open set U ⊂ X we choose a covering U = ⋃_{i∈I} X_{f_{i}} and define (See SH xxx)

Ã(U) = (∏_{i∈I} A_{f_{i}} \overrightarrow{⟶} ∏_{(i,j)∈I×I} A_{f_{i}f_{j}})[p_{1} = p_{2}].

We have to verify that this does not depend on the covering and really defines a sheaf. I will not do this in detail, the proof is a little bit tedious. To prove the independence of the covering we first pass to a refinement of the covering: We have τ : J → I and

X_{f_{i}} = ⋃_{ν∈τ^{−1}(i)} X_{f_{i}h_{ν}} .

We put h̃_{ν} = f_{i}h_{ν} (the index ν determines the index i) and X = ⋃_{ν∈J} Xh̃_{ν}.

We get a diagram

∏_{i∈I} A_{f_{i}}  \overrightarrow{⟶} ∏_{(i,j)∈I×I} A_{f_{i}f_{j}}

↓  ↓
∏_{ν∈J} Ah̃_{ν}  \overrightarrow{⟶} ∏_{(ν,μ)∈J×J} A_{h̃_{ν},h̃_{μ}}

and is easy to check that if we apply the proposition VI.1.4.1. to the vertical arrows that we get an isomorphism between the [p_{1} = p_{2}] subrings.

Then we may compare two coverings by passing to a common refinement.

The fact that U → Ã(U) is actually a sheaf can be derived by similar arguments as those used in the sheafification process. The intuitive meaning of Ã(U) is clear: These are the ”regular functions” on U and these are ”functions” which can locally be written in the form \frac{g_{i}}{f_{i}^{n_{i}}} in such a way that \frac{g_{i}}{f_{i}^{n_{i}}} , \frac{g_{j}}{f_{j}^{n_{j}}} match on the intersection of their ”domains of

definition” which of course is X_{f_{i}f_{j}} .

Our proposition says that Ã(X_{f_{i}}) = A_{f_{i}} and this means that a ”regular function” on affine open sets X_{f_{i}} has always a kind of ”global” description which uses only denominators of the form f_{i}^{n_{i}}. Especially we have Ã(X) = A.

VI.1.5. Quasicoherent sheaves

Our considerations can be generalized. If we have an A-module M a set S ⊂ A which is closed under multiplication and contains 1_{A}, then we define

M_{S} = {(m, s)|m ∈ M, s ∈ S}/ ∼

where the equivalence relation is

(m, s) ∼ (m′, s′) ⟺ ∃s” ∈ S such that (ms′ − m′s)s” = 0.

It is quite clear that this defines an A_{S}-module M_{S}.

Now we can construct a sheaf M̃ of Ã-modules just by defining

M̃(X_{f}) = M_{f}

then verifying the proposition (just replace A by M everywhere) and then we put

M̃(U) = (∏_{i∈I} M_{f_{i}} \overrightarrow{⟶} ∏_{(i,j)∈I×I} M_{f_{i}f_{j}})[p_{1} = p_{2}].

The stalk of the sheaf Ã at a point 𝔭 is the local ring A_{𝔭}, the stalk of M̃ in 𝔭 is the A_{𝔭}-module M_{𝔭} = M_{(A\𝔭)}.

It can happen that the stalk M_{𝔭} vanishes in some points. This is so if for any m ∈ M we can find an s ∈ A\𝔭 such that sm = 0. The module M defines an ideal

Ann(M) = {f ∈ A|fM = 0}.

It is clear that M_{𝔭} ≠ 0 is equivalent to 𝔭 ⊃ Ann(M). The set of these 𝔭 is called the support of M̃ and is a closed subset in Spec(A).

It is not so that any sheaf M of Ã-modules is automatically of the form M̃ with some A-module M. On Spec(ℤ_{(p)}) we have the sheaf

\begin{aligned}M(\{(0)\}) & = ℚ \\ M(Spec(ℤ_{(p)})) & = 0 \end{aligned}

which is not of this form.

The sheaves M̃ which are obtained from an A-module M are called the quasicoherent sheaves on Spec(A). We can recover the A-module from the sheaf since M = M̃(X).

We will say that M̃ is the sheaf obtained from M

We get quasicoherent sheaves of ideals on X = Spec(A) by starting from an ideal I ⊂ A, this is an A-module and the sheaf

Ĩ ⊂ Ã

is a quasicoherent sheaf of ideals.

VI.1. 6 Schemes as locally ringed spaces:

An affine scheme is a locally ringed space of the form (X, O_{X}) = (Spec(A), Ã). (See SH x.x.x)

I recall the definition of a morphism between two locally ringed spaces. A morphism is a pair

(f, h) : (X, O_{X}) → (Y, O_{Y})

where f is a continuous map from X to Y and h is a map of sheaves of rings

h : f^{∗}(O_{Y}) → O_{X}

which induces local homomorphisms

h_{x} : O_{Y,f(x)} = f^{∗}(O_{Y})_{x} → O_{X,x}

on the stalks. This means that the maximal ideal of O_{Y,f(x)} is mapped into the maximal ideal of O_{X,x}. The locally ringed spaces form a category we have an obvious way of composing two morphisms.

VI.1.6.1.Heuristical remark:

The difficulty is as always that the sections of the sheaves are not actual functions, they are elements in very abstract rings. In our previous examples (C_{∞}-manifolds, complex manifolds (See SH xxx)) a continous map f : X → Y between the spaces gave us a map h_{0} from the sheaves of continuous functions on Y to the continuous functions on X. Then we made requirements that this map should respect certain distinguished subsheaves of functions which define a ”so and so” structure on X and Y. If this was the case we called f a ”so and so” map. The map h_{0} was determined by f in such a case. Especially it is clear in these examples that a germ f at a point y ∈ Y with f(y) = 0 is mapped by h_{0} to a germ at x ∈ f^{−1}(y) which vanishes at x. This means the h_{0} is automatically local.

Here the situation is different, the map is h : f^{∗}(O_{Y}) → O_{X} is an extra datum.

What is left from the notion of functions is the notion of vanishing of a section f ∈ O_{X}(U) in a point x ∈ U (VI.1.2.4.). Now the requirement that h induces local homomorphism in the stalks becomes clear: A germ in O_{Y,f(x)} which vanishes in f(x) must be sent by h to a germ in O_{X,x} which vanishes at x. The reader should observe that a germ in O_{Y,f(x)} which does not vanish at f(x) is a unit and hence it goes automatically to a germ in O_{X,x} which does not vanish in x.

End of the remark

The following theorem is fundamental.

Theorem. VI.1.6.2 Let (X, O_{X}) = (Spec(A), Ã) and (Y, O_{Y}) = (Spec(B), B̃) be affine schemes. A morphism

(f, h) : (X, O_{X}) → (Y, O_{Y})

defines a map h_{X} : O_{Y} (Y) → O_{X}(X) i.e. a homomorphism

h_{X} : B → A.

The map

Homaffine schemes((X, O_{X}), (Y, O_{Y})) → Hom_{Rings}(B, A)

given by

(f, h) → h_{X}

is a bijection.

Proof: We start by constructing a map in the other direction and then we show that the maps are inverse to each other.

Given ϕ : B → A we have defined a map

tϕ : Spec(A) → Spec(B) ^{t}ϕ(𝔭) = ϕ^{−1}(𝔭) .

If we have an element b ∈ B we get an open set Y_{b} = {𝔮|b ∉ 𝔮} in Y and it is clear that

tϕ^{−1}(Y_{b}) = X_{ϕ(b)}.

Hence our map is continuous and we get maps

ϕ_{b} : O_{Y} (Y_{b}) = B_{b} → f_{∗}(O_{X})(Y_{b}) = O_{X}(X_{ϕ(b)}) = A_{ϕ(b)}

which by the adjointness formula is nothing else than a map

ϕ̃: f^{∗}(O_{Y}) → O_{X}

and hence we constructed a morphism (^{t}ϕ, ϕ̃) between locally ringed spaces.

We have to show that these maps are inverse to each other. At first we start from ϕ : B → A we get (^{t}ϕ, ϕ̃). From this we construct again a homomorphism from B → A. According to our rules we have to evaluate ϕ̃ on the pair X, Y and get

ϕ̃_{X} : B → A

which is our original map.

Now we start from (f, h) The map h can be evaluated on X, Y this gives us

h_{X} : B → A.

We have to prove at first that the map

th_{X} : X → Y

is equal to f. We have ^{t}h_{X}(𝔭) = h_{X}^{−1}(𝔭) = 𝔮.

Since we know that h induces a morphism between the sheaves we get a diagram

\begin{matrix}h_{X} : B & ⟶ & A \\ ↓ & & ↓ \\ h_{𝔭} : B_{f(𝔭)} & ⟶ & A_{𝔭} \end{matrix} .

This implies that h_{X} has to map elements b ∈ B\f(𝔭) to elements h_{X}(b) ∈ A\𝔭 because b becomes invertible in B_{f(𝔭)} and hence h_{𝔭} has to map it to a unit in A_{𝔭}. This implies

B\f(𝔭) ⊂ B\h_{X}^{−1}(𝔭)

and hence h_{X}^{−1}(𝔭) ⊂ f(𝔭). We also know that h_{𝔭} maps the maximal ideal 𝔪_{f(𝔭)} into the maximal ideal 𝔪_{𝔭}. Hence it maps the elements of f(𝔭) into 𝔭 and this implies f(𝔭) ⊂ h_{X}^{−1}(𝔭) and we have the desired equality for f(𝔭) = h_{X}^{−1}(𝔭) =^{t} h_{X}(𝔭) .

The rest is clear, the map h̃_{X} which we construct from h_{X} is obviously equal to h since these two coincide on the global sections.

VI 1.6.1 Closed subschemes:

We start from an ideal I ⊂ A. We have the projection map π : A → A/I and we have Spec(A/I) = {𝔭 | 𝔭 ⊃ I} = V (I). If i : V (I) → Spec(A) is the inclusion then we consider the map

(i, π̃) : (V (I), Ã/I) → (Spec(A), Ã)

as a closed subscheme of (Spec(A), Ã).

At this point the reader might wonder: We made a lot of effort to show the following: something seemingly simple namely the category of commutative rings with identity is antiequivalent category of locally ringed spaces and this category consists of rather complicated objects and the morphisms are also not so easy to define.

The reason why we do this will become clear: These concepts allow us to glue affine schemes together so that we can build larger objects namely schemes. Locally these schemes look like affine schemes but globally they look different.

VI. 2. Schemes:

VI 2.1 The definition of a scheme:

An arbitrary scheme is a locally ringed space (X, O_{X}) which is locally isomorphic to an affine scheme. In other words we can find a covering X = ⋃_{ν} U_{ν} by open sets such that (U_{ν}, O_{X} | U_{ν}) is affine.

This implies of course that (U_{ν}, O_{X} | U_{ν}) = (Spec(A_{ν}), Ã_{ν}) where A_{ν} = O_{X}(U_{ν}) is the ring of regular functions on U_{ν}. But in contrast to the case of affine scheme the ring of regular functions on X may be to small to contain enough information to recover the scheme (X, O_{X}). This will be demonstrated in the next section on projective schemes.

Example VI.2.1.1:

Let us consider the polynomial ring A = k[f, g], let X = Spec(A) . We remove the point (0, 0) from X, the resulting space U inherits a topology and the structure of a locally ringed space. It is clearly a scheme since we may cover it by

U = X_{f} ∪ X_{g}

.

But U is not an affine scheme since we have O_{U}(U) = A and U ≠ Spec(A). We are not able to produce a regular function on U which is not regular on X.

This example shows that any open subset of an affine scheme is a scheme but not an affine scheme in general.

2.1.3 :Again we have the notion of a quasicoherent O_{X}-module M. This means that for all open subsets U the sections M(U) form an O_{X}(U)−module and for affine open subsets U the restriction M | U is obtained from the O_{X}(U) module M(U) (See 1.5.1.).

2 .1.4. At his point it is rather clear what a closed subscheme of a general scheme is. We know how to define a quasicoherent sheaf of ideals I ⊂ O_{X}: It is a sheaf of ideals, i.e. for any open subset U ⊂ X the sections I(U) ⊂ O_{X}(U) form an ideal in O_{X}(U) and for an affine U the restriction I | U is the sheaf associated to I(U). On this open affine subset U we have the closed subscheme ((V (I(U)),˜(A(U)/I(U)))),˜(A(U)/I(U))) ↪ (U, Ã(U)). We define V (I) to be the union of all these subsets (V (I(U)),˜(A(U)/I(U))) and a quotient sheaf O_{X}/I by its restriction to the affine pieces. This yields the closed subscheme (V (I), O_{X}/I) ↪ (X, O_{X}).

2.1.5. If we have a morphism f : X → Y between schemes ????

VI.2.2:Construction of quasicoherent sheaves

We have an important way of constructing quasicoherent O_{X}−modules on X. Let us assume we have a covering 𝔘 = {U_{ν}}_{ν∈N} of a scheme (X, O_{X}) by affine subschemes. Let us also assume that we have given an O_{X}(U_{ν})−module M_{ν} for all ν ∈ N. Each of them defines a quasicoherent sheaf M̃_{ν} on the corresponding subscheme U_{ν}. Now let us assume that for any pair (ν, μ) of indices we have an isomorphism

g_{ν,μ} : M̃_{ν} | U_{ν} ∩ U_{μ}→^{∼}M̃_{μ} | U_{ν} ∩ U_{μ}

such that this system of isomorphism satisfies

a) g_{ν,ν} = Id for all = ν

b)g_{ν,μ} ∘ g_{ν,μ} = Id for all pairs ν, μ

c) and for any three indices ν, μ, λ we have the relation g_{ν,μ} ∘g_{μ,λ} = g_{ν,λ} on U_{ν} ∩U_{μ} ∩U_{λ}.

Then we can construct a sheaf M̃ = (M_{ν}, 𝔘, g_{ν,μ}) on X by the glueing process: For an open set V ⊂ X which is contained in at least one of the U_{ν} we define M̃(V) to be the set of vectors m = (…, m_{ν}, …, m_{μ}, …) where the indices run over the subset of indices λ for which U_{λ} ⊃ V, where the m_{ν} ∈ M̃_{ν}(V) and where

g_{ν,μ}(m_{ν}) = m_{μ} for all pairs ν, μ

Of course any of the components determines all the others). Then for an arbitrary V we may cover it by the V ∩ U_{ν} and define M̃(V) by the conditions (SH1), (SH2) for sheaves (See 1.1.2.).

I will not discuss an example for this kind of construction, for this I refer to the chapter on projective spaces.

VI 2.3 Fibered products

The schemes form a category. It is very important to study relative situations. We consider a fixed scheme S. Then a scheme over S is a scheme X together with a morphism

\begin{matrix}X \\ ↓π \\ S \end{matrix} .

Sometimes we write X/S, the morphism π is called the structure morphism and S is called the base scheme. (For more motivation why we need this general concept I refer to the end of this section)

The schemes over S form a category again. If we have two schemes X/S, Y/S then the S-morphisms

Hom_{S}(X, Y)

are those morphisms φ from X to Y which render the following diagram

\begin{aligned}& φ \\ X & ⟶ & Y \\ ↘ & & ↙ \\ & S \end{aligned}

commutative.

Given two schemes X/S, Y/S we have the notion of the fibered product of these schemes over S. This fibered product is nothing else than the product in the category of schemes over S.

Hence the fibered product is an object Z/S together with (See Cat xx)

\begin{aligned}& p_{1} \\ Z & ⟶ & X \\ ↘ & & ↙ \\ & S \end{aligned}  \begin{aligned}& p_{2} \\ Z & ⟶ & Y \\ ↘ & & ↙ \\ & S \end{aligned}

such that for any scheme T/S we have

Hom_{S}(T, Z) ⟶^{∼} Hom_{S}(T, X) × Hom_{S}(T, Y)

where the identification is given by

Ψ ⟼ (p_{1} ∘ Ψ, p_{2} ∘ Ψ).

We can do this for any category. The reader is advised to consider the construction of fibered product in the category of sets as an example.

Theorem VI.2.3 : In the category of schemes fibered products exist. This theorem will not be proved here, I will prove it for affine schemes in this section

and at the end of the proof I will give some indication how to do it in general. ( See also Harthshornes book Chap. II, 3.3 or to EGA I, 3.2.6.) In the next section on projective schemes I will prove that the product of projective schemes is again projective and the existence of products in that case will be a byproduct.

We consider the category of affine schemes. If X = Spec(A) and S = Spec(R) then π : X → S is the same thing as a homomorphism of rings

φ : R → A.

At this point two remarks are in order

(i) The datum φ : R → A is the same as giving the additive group A the structure of an R-module, i.e. giving a composition

· : R × A → A,

which satisfies the usual rules, especially we want 1_{R} · a = a and we have to require in addition r · (a_{1}a_{2}) = (r · a_{1})a_{2}. This is clear because starting from φ we put

r · a = φ(r) · a.

On the other hand if we have given the R-module structure on A then

Ψ(r) = r · 1_{A}

gives us the ring homomorphism.

(ii) To simplify the notation we will drop the name of the morphism, this means we will only write R → A instead of φ : R → A. In view of the first remark this means: If we see R → A then this allows us to write r · a for r ∈ R on a ∈ A and this satisfies the obvious rules.

If we have R → A we say that A is a ring over R or a R-ring sometimes we also say that A is an R-algebra.

If we have given R-rings A, B then

Hom_{R}(A, B)

are exactly those homomorphisms which are linear with respect to R, i.e.

φ ∈ Hom_{R}(A, B)

means that φ satisfies in addition φ(r·a) = r·φ(b). (Now the two · have different meaning).

Of course we don’t make any assumption that R → A should be injective. For instance ℤ → ℤ/p make ℤ/pℤ to a ring over ℤ. Actually it is clear that any ring A is in a unique way a ring over ℤ, we simply send 1 → 1_{A}.

I come back to the construction of fibered products in the category of affine schemes. We describe the problem in the category of rings and therefore we turn the arrows around. We have

\begin{aligned}A & & B \\ ↖ & & ↗ \\ & R \end{aligned} .

We are looking for a ring C over R together with two R-homomorphism α : A → C, β : B → C such that for any other ring T over R we get: In the following diagram

C
α ↗  ↖ β
A  ↑  B
↖  ↗
R ⟶ T

a R homomorphism from C to T is the same thing as a pair of R-homomorphisms f : A → T, g : B → T.

How do we get such a C? Starting from f, g we get a map

\begin{matrix}A × B & ⟶ T \\ (a, b) & ⟼ f(a) · g(b) \end{matrix} .

This is an R-bilinear map from A × B to T. We have to verify

(r · a, b)
↘
r · (f(a)g(b))
↗
(a, r · b)

but

(r · a, b) → f(r · a)g(b) = f((r · 1_{A})g(b)
= f(r · 1_{A})f(a) · g(b) = (r · f(1_{A}))f(a)g(b)
= (r · 1_{C})f(a)g(b) = r · (f(a)g(b))

and the distributivity is clear.

But this tells us that the pair f, g provides us an R-linear map

f ⊗ g : A ⊗_{R} B ⟶ T

where A ⊗_{R} B is of course the tensor product of the two R-modules A, B.

We define a ring structure on A ⊗_{R} B: The elements of the tensor product are finite sums

a_{1} ⊗ b_{1} + a_{2} ⊗ b_{2} … + a_{s} ⊗ b_{s}

where we have the following rules

\begin{matrix}(r · a) ⊗ b & − a ⊗ r · b = 0 \\ (a_{1} + a_{2}) ⊗ b & − a_{1} ⊗ b − a_{2} ⊗ b = 0 \\ a ⊗ (b_{1} + b_{2}) & − a ⊗ b_{1} − a ⊗ b_{2} = 0 \end{matrix} .

We introduce as multiplication

(a ⊗ b)(a′ ⊗ b′) = aa′ ⊗ bb′,

we extend this by distributivity. Then we have to check that this is compatible with the rules above.

We put C = A ⊗_{R} B with this ring structure, we have the homomorphism

r ⟶ r · 1_{A} ⊗ 1_{b} = 1_{A} ⊗ r · 1_{b},

we have

\begin{aligned}α : A & → A ⊗_{R} B \\ a & → a ⊗ 1_{B} \\ β : B & → A ⊗_{R} B \\ b & → 1_{A} ⊗ b. \end{aligned}

Starting from f, g we already had the R-linear map from the R-module A ⊗_{R} B to T. But the ring structure on A⊗_{R} B is made in such a way that f ⊗g is a ring homomorphism. On the other hand, if h : A⊗_{R}B → T is a R-homomorphism then we may put f = h∘α, g = h∘β and it is clear that

h(a ⊗ b) = h((a ⊗ 1_{B}) · (1_{A} ⊗ b))
= h((a ⊗ 1_{B})) · h((1_{A} ⊗ b)) = f(a) · g(b)
= (f ⊗ g)(a ⊗ b).

After all this it should be clear that the diagram

Spec(A ⊗_{R} B)
p_{1} ↙  ↘ p_{2}
Spec(A)  Spec(B)
↘  ↙
Spec(R)

is a fibered product of Spec(A) and Spec(B) over the base Spec(R).

2.3.1. Examples: 1)If A and B are polynomial rings over R, i.e.

\begin{aligned}A & = R[X_{1} … X_{n}] \\ B & = R[Y_{1} … Y_{m}] \end{aligned}

then A, B are free R modules with a basis consisting of monomials

X_{1}^{ν_{1}} … X_{n}^{ν_{n}}, Y_{1}^{μ_{1}} … Y_{m}^{μ_{m}}.

Then A ⊗_{R} B is free again and has as basis

X_{1}^{ν_{1}} … X_{n}^{ν_{n}} ⊗ Y_{1}^{μ_{1}} … Y_{m}^{μ_{m}}.

But then it is obvious that A ⊗_{R} B is actually isomorphic to the polynomial ring in X_{1} … X_{n}, Y_{1} … Y_{m}, i.e.

R[X_{1} … X_{n}] ⊗ R[Y_{1} … Y_{m}] = R[X_{1} … X_{n}, Y_{1}, … Y_{m}].

The scheme Spec(R[X_{1} … X_{n}]) is called the n-dimensional affine space over Spec(R) and if S = Spec(R) we write 𝔸_{S}^{n} for this scheme. Hence we get the truly exciting formula

𝔸_{S}^{n} ×_{S} 𝔸_{S}^{m} = 𝔸_{S}^{n+m}.

2) If we have two R-algebras R → A, R → B and we have given two ideals I ⊂ A, J ⊂ B then these ideal define closed subschemes

Spec(A/I)  ↪  Spec(A) Spec(B/J)  ↪  Spec(B) ↘  ↙  ↘  ↙ Spec(R)  Spec(R) .

Hence we get a morphism from the fibered products

Spec(A/I) ×_{Spec(R)} Spec(B/J)  →  Spec(A) ×_{Spec(R)} Spec(B)
↘  ↙
Spec(R)

and I claim that this is again a closed embedding. I leave it to the reader as an exercise to show, that the arrow gives us an isomorphism of the fibered product of the subschemes to the subscheme defined by the ideal (A ⊗_{R} B)(1_{A} ⊗_{R} J) + (A⊗_{R})(I ⊗_{R} B) ⊂ A ⊗_{R} B.

3) If k is a field and K/k is a finite extension, then we have a map

Spec(K) ; ↓ ; Spec(k) which on the underlying sets is just a map from a point to a point. But we have different rings of regular functions on these points hence this morphism is not an isomorphism.

If we take the fibered product we get

Spec(K) ×_{Spec(k)} Spec(K) = Spec(K ⊗_{k} K)

and K ⊗_{k} K will not be a field in general. If for instance K/k is a separable normal extension then

K ⊗_{k} K = ⨁_{σ∈Hom_{k}(K,K)} K.

(Main theorem of Galois theory) and hence

Spec( ⨁_{σ∈Hom_{k}(K,K)} K) = Hom_{k}(K, K)

as a set. Hence we have an example where the underlying set of X ×_{S} Y may differs from the set theoretic fibered product which in our case is still a point.

As I explained earlier, we always have a canonical morphism Spec(A) → Spec(ℤ) and we may define the absolute product of two affine schemes as

Spec(A) × Spec(B) = Spec(A ⊗_{ℤ} B).

We can consider the situation that we have an R-algebra R_{1} and two R_{1} algebras, i.e. we have a digram

\begin{aligned}& & A \\ & ↗ \\ R → R_{1} \\ & ↘ \\ & & B. \end{aligned}

Then we have a morphism

ρ : A ⊗_{R} B → A ⊗_{R_{1}} B

(in the ring on the right hand side we have the rule r_{1}a ⊗ b = a ⊗ r_{1}b for r_{1} ∈ R which is not valid in the ring on the right hand side).

The map ρ is clearly surjective, hence we have

Spec(A) ×_{Spec(R_{1})} Spec(B) ↪ Spec(A) ×_{Spec(R)} Spec(B).

2.3.2. I want to say a few words concerning the construction of fibered products for general schemes. First of all we have to cover the base scheme S by affines and to construct the fibered product over these affine subschemes and to glue the fibered products over the intersections. Over an affine base S we cover the schemes X/S, Y/S by affines and constructs the fibered products for the pairs of affine covering sets. These will be glued together. For the details I refer to the references given above.

In the following section on projective schemes I will discuss this construction in a special case.

VI.2.4 Points, T -valued points and geometric points

In the theory of schemes we have to be careful with the notion of a point. If we have a scheme (X, O_{X}) then X is a set and the points of the scheme are the elements of this set. But we have seen that these points do not behave well under fibered products.(2.3.1. Example 3)

If we have a scheme X/S and another scheme T/S, i.e. we have a diagram

X ; ↓ ; S ← T

then the T -valued points of X are simply the S-arrows from T to X i.e.

X_{S}(T) = Hom_{S}(T, X).

If S = Spec(R) and T = Spec(B) then we denote the set of T = Spec(B)-valued points also by X(B) and speak of B-valued points.

Therefore we see that a scheme X/S defines a contravariant functor

F_{X} : Schemes/S → Ens.,

where F_{X}(T) = X_{S}(T).

It is the definition of the fibered product that the T valued points behave well under fibered products. We have

(X ×_{S} Y)_{S}(T) = X_{S}(T) × Y_{S}(T).

I want to discuss this concept in a couple of examples. Let k be a field and we consider

A = k[X_{1} … X_{n}]/I = k[x_{1}, … x_{n}]

where I is an ideal in the polynomial ring which is generated by polynomials

F_{1}(X_{1} … X_{n}), … F_{r}(X_{1} … X_{n}].

We have

Spec(A) ↓ Spec(k) \underleftarrow{id} Spec(k)(= T) .

In this case the T valued points are the k-homomorphisms

φ : k[x_{1} … x_{n}] → k.

Such a φ is determined by its values (φ(x_{1}), …, φ(x_{n})) = (a_{1}, …, a_{n}) ∈ k^{n}. But we have constraints on the n-tuples of values because

F_{1}(x_{1}, …, x_{n}) = … = F_{r}(x_{1}, …, x_{n}) = 0

and hence also (a_{1}, … a_{n}) has to satisfy

F_{1}(a_{1}, … a_{n}) = … = F_{r}(a_{1}, … a_{n}) = 0.

Hence the point a = (a_{1}, … a_{n}) has to be a solution of the polynomial equations F_{1} = … = F_{r} = 0. If in turn a = (a_{1}, … a_{n}) solves the system of equations then we can look at the diagram

k[X_{1} … X_{n}] ⟶ k[X_{1} … X_{n}]/I = A
Ψa ↘↖  ↗
k

where Ψ_{a}(X_{i}) = a_{i}. This homomorphism vanishes on the generators (F_{1}, …, F_{r}) of the ideal I and hence its factors over A. All this tells us that

Spec(A)(k) = {a ∈ k^{n}|F_{1}(a) = … = F_{r}(a) = 0},

the k-valued points are the solutions of the system of polynomial equations given by the F_{1}, …, F_{r}.

2.4.1. Geometric points: We can embed our field k into an algebraic closure k̄. Then this defines a morphism Spec(k̄) → Spec(k) hence we have the diagram

X ↓ Spec(k) ← Spec(k̄) .

Then X(k̄) is the set of solutions of the system of equations over k̄. These are the so-called geometric points of the scheme X → Spec(k).

Of course it is much easier to find geometric points than k-valued points. If we have a polynomial y^{2} = x^{3} − x − 1 in ℚ[x, y] then we may start from a value a ∈ ℚ for x but then it requires a good portion of luck if we want to find a point (a, b) ∈ ℚ satisfying the equation because we have to find a square root of a^{3} − a − 1 in ℚ. But if we look for geometric points we do not have this problem.

VI.2.4.2. To conclude this section I want to say a few words why we want to consider such relative situations

\begin{matrix}X \\ ↓π \\ S \end{matrix} .

For any point s ∈ S we can consider the stalk O_{S,s} this is a local ring and let 𝔪_{s} be its maximal ideal. Then k(s) := O_{S,s}/𝔪_{s} is a field and we have a morphism S ← Spec(k(s)) and hence we get a base change X ×_{s} Spec(k(s)) and this is a scheme over Spec(k(s)). It is called the fiber of X/S over s. Hence we get a family of schemes which is parametrized by the points of S. We may even go one step further and embed k(s) into an algebraically closed field k̄. This gives us a morphism Spec(k(s)) ← Spec(k̄) and the composition S ← Spec(k) is a geometric point of S. The base change S ×_{S} Spec(k̄) is the geometric fiber over the geometric point.

VI2.5.Closed points and geometric points on schemes of finite type:

A scheme X = Spec(k[X_{1}, X_{2}, …, X_{n}]/(F_{1}, …, F_{r})) = Spec(A) is called an affine scheme of finite type. If we have a geometric point P : A → k̄ then the kernel of P is a maximal ideal 𝔪_{P} of A hence it is also a closed point. Hence we get a map X(k̄) → Specmax(A). If we have in turn a maximal ideal 𝔪 in A then it follows from The Nullstellensatz that k(𝔪) = A/𝔪 is a finite extension of k. We get a diagram

A  →  A/𝔪 = k(𝔪)
↖  ↗
k → k̄

and it is clear that the k-homomorphisms from k(𝔪) → k̄ correspond one to one to the geometric points of X which lie above 𝔪. Hence we get from the results of Galois theory that

2.5.1. (1) The map X(k̄) → Specmax(A) is surjective with finite fibers. The points in the fibre are in one to one correspondence with the prime ideals in the local k̄ -algebra k(𝔪) ⊗ k̄.

(2) The cardinality of the fibre of 𝔪 of this map divides the degree [k(𝔪) : k] and it is equal to this degree if and only if k(𝔪)/k is a separable extension.

(3) The Galois group of k̄/k acts transitively on the geometric points in the fibre over 𝔪. We may also speak of integral solutions. If we have a scheme

A = ℤ[X_{1}, …, X_{n}]/I

then we may consider

X = Spec(A) ↓ Spec(ℤ) ← Spec(ℤ) .

and X(ℤ) = X(Spec(ℤ)) are the integral solutions of the system of equations. For instance we can try to find ℤ-valued points on Spec(ℤ[x, y]/(y^{2} − x^{3} + x + 1)) which is even harder than finding ℚ-valued points.

VI.2.6 Morphisms, T -valued points and geometric points We start from the following diagram of schemes

\begin{aligned}X & & Y \\ ↘ & & ↙ \\ & S & ← T. \end{aligned}

It is clear that morphisms ϕ ∈ Hom_{S}(X, Y) induce maps between the sets of T -valued points : ϕ(T) : X(T) → Y (T). On the other hand if we have a consistent system of maps ϕ(T) : X_{S}(T) → Y_{S}(T) (consistent means that the obvious diagrams commute) then we get from this a morphism ϕ ∈ Hom_{S}(X.Y) . We simply plough in X itself for T and get the morphism as the image of the identity in Hom_{S}(X, X). (This is a Yoneda type argument).

At this point one can always ask the question whether a given functor

F : Schemes/S → Ens

is representable. Developing tools to answer this question in concrete situations turns out to be a fundamental problem in algebraic geometry.

Hence if we have two schemes X/S, Y/S and a system of maps between their T valued points for all possible T, then this is induced by a unique morphism between the schemes. In some cases it is not necessary to have a map between the T valued points for all T.

If for instance S = Spec(k) where k is an algebraically closed field. Let us assume in addition that X = Spec(k[x_{1}, x_{2}, …, x_{n}]), Y = Spec(k[y_{1}, y_{2}, …, y_{m}]) are affine schemes of finite type over Spec(k) and we assume moreover that the two k-algebras A = k[x_{1}, x_{2}, …, x_{n}], B = k[y_{1}, y_{2}, …, y_{m}] do not have nilpotent elements (they are reduced).

Then the Nullstellensatz guarantees that the sets of geometric points X(k̄) = Specmax(A), Y (k̄) = Specmax(B). In this case we can interprete the elements of the rings A, resp. B as k̄ valued functions on X(k̄) = Specmax(A), resp. Y (k̄) = Specmax(B). (To see this we have to notice that an element f in one of these algebras which vanishes in all geometric points lies in the intersection of the maximal ideals. It follows from the Nullstellensatz that the intersection of the maximal ideals is the radical and hence zero. (How ?).)

Now it is clear that a map ψ between the sets of geometric points comes from morphism X → Y if and only if the induced map between the k-valued functions maps the elements of B into elements of A. Hence we get a homomorphism of rings which then in turn induces the map between the geometric points from which we started.

Exercises : (1) Let us assume that k is a field of characteristic p > 0. We take A = B = k[X]. Then the set of geometric points is k̄ and we have the bijective map x → x^{p} on the set of geometric points. Show that this map comes from a morphism but its inverse does not.

This teaches us that a morphism between affine schemes of finite type over k which induces a bijection between the sets of geometric points is not necessarily an isomorphism.

(2) We go back to the general situation that we have two reduced affine schemes of finite type over Spec(k). We assume now in addition that k is perfect ( this means that the map x → x^{p} is surjective and hence bijective if our field has positive characteristic p > 0.) Let us assume that we have a morphism ϕk̄ : X ×_{k} k̄ → Y ×_{k} k̄ which induces a map ^{t}ϕ : X(k̄) → Y (k̄). Now we can define an action of the Galois group Gal(k̄/k) on the two set of geometric points. Show that ϕk̄ is defined over k, i.e. comes from a morphism ϕ : B → A if and only if it commutes with the action of the Galois group.

To summarize this section we can say that for reduced finitely generated k-algebras over an algebraically closed field the set of geometric points together with the algebra A of regular k - valued functions contains all the information. This is of course tautological, because the algebra A determines everything but its realization as algebra of k-valued functions gives us some geometric picture which is less abstract than the concept of a locally ringed space.(Exercise (3), VI.1.2.)

If our ground field is not algebraically closed but perfect we get this information if we consider the geometric points together with the action of the Galois group. But if our ground field is not perfect or if the schemes are not reduced this breaks down.

VI.2.7.Flat morphisms

Let us consider a commutative ring A with identity and its category of A-modules. In this category we have the tensor product of two modules, which is again an A-module.

Small remark: Here we need the commutativity of A since we want

m ⊗ (fg)n = (fg)m ⊗ n = f(gm) ⊗ n = gm ⊗ fn = m ⊗ g(fn) = m ⊗ (gf)n.

If we fix a module M we can consider the functor

N → N ⊗_{A} M

of the category into itself. This is a so-called right exact functor: If we have an exact sequence

0 → N′ → N → N′′ → 0

then the sequence

N′ ⊗_{A} M → N ⊗_{A} M → N′′ ⊗_{A} M → 0

is still exact. But the functor is not exact in general, it can happen that the first arrow is not injective anymore.

Exercise.

1) Prove the right exactness (I think one has to use the definition by generators and relations).

(2) To construct an example which shows that the functor is not exact, let us consider an element m ∈ M, m ≠ 0 which has a non-trivial annulator in A, i.e. there is an f ∈ A, f ≠ 0, and fm = 0. Consider the sequence of A modules

0 → Af → A → A/Af → 0.

Show: The element f ⊗_{A} m ∈ Af ⊗ M goes to zero in A ⊗_{A} M = M. Construct an example where you know that f ⊗_{A} m ≠ 0.

2.7.1. The concept of flatness

An A-module M is called flat if the functor N → N ⊗_{A} M is exact. A simple example of a flat module is the free A-module over an arbitrary index set I

A^{I} = {(…, a_{i}, …)_{i∈I}| almost all a_{i} = 0}.

This is the direct sum

A^{I} = ⊕_{i∈I}A

but the direct product ∏_{i∈I} A is also flat.

An A-module M is called faithfully flat if it is flat and if in addition N ⊗_{A} M = 0 implies N = 0.

The above examples of the direct sum and direct product are indeed faithfully flat. We will see flat modules which are not faithfully flat in a minute.

Since an A-algebra A → B is also an A-module, we can speak of flat A-algebras. Since we may view A → B also as a morphims of affine schemes we can speak of flat morphisms (Spec(B), B̃) → (Spec(A), Ã) of affine schemes. We also observe that for an A-module N the module B ⊗_{A} N has an obvious natural structure as a B-module, we simply put b(b_{1} ⊗_{A} n) = bb_{1} ⊗_{A} n

A morphism of affine schemes is called faithfully flat if the A-module B is faithfully flat. 2.7.1.1. Examples and exercises :

(1) Of course the polynomial ring A[X_{1}, …, X_{n}] is faithfully flat over A because it is free as an A-module.

(2) An important case of a flat algebra is given by the localization. It is not so difficult to check that for any subset S ⊂ A, which is closed under multiplication the A algebra

A → A_{S}

is flat. To see this one should also observe that N ⊗_{A} A_{S} ≃ N_{S} where the isomorphism is simply given by

n ⊗ \frac{f}{s} ↦ \frac{fn}{s}.

Now it is clear that if N′ ⊂ N then

N′_{S} ⊂ N_{S}.

(3) Let us assume that S = {f^{n}} where f ∈ A and not nilpotent. We have just seen that

A → A_{f}

is flat. But it will not be faithfully flat in general. To see this one should prove

A/fA ⊗_{A} A_{f} = 0.

But on the other hand:

(4) If we have a covering X = Spec(A) = ⋃_{i∈I} X_{f_{i}} where we may assume that I is finite (prop,VI.1.3.2.) then

A → ∏_{i∈I} A_{i}

is faithfully flat. This corresponds to a map

X ← ⨆ X_{f_{i}}.

and this is a faithfully flat morphism of schemes.

(5) If B is the quotient of A by an ideal

A → B = A/I,

then Spec(B) = Spec(A/I) → Spec(A) is a closed subscheme. In this case one can show that this is almost never flat. (I think that in the case that A is noetherian this morphism is flat if and only if Spec(B) → Spec(A) is also open.)

(6) If A is a Dedekind ring ( See VII. 2.3.1 ), then an A-module N is flat if and only if it is torsion free, i.e. if fn = 0 with f ∈ A, n ∈ N then f or n is zero.

If we have A → B → C and an A-module N then we have a canonical isomorphism

C ⊗_{B} (B ⊗_{A} N)→^{∼}C ⊗_{A} N

We leave it to the reader to verify

7). If we have A → B → C and B is a flat A-algebra and C a flat B-algebra then C is also a flat A-algebra.

8). If we have A → B → C and if C is a flat A-algebra and if C is a faithfully flat B-algebra then B is a flat A-algebra.

The second assertion is important because it allows us to check flatness locally.

Proposition VI.2.7.2.

A → B → ∏_{i∈I} B_{f_{i}}

with some f_{i} in B such that the X_{fi} ⊂ X = Spec(B) form a covering of X and if A → B_{fi} is flat for all i then B is a flat A-algebra.

This is obvious from the exercise.

The same principle can be applied if we want to check whether an A-module is flat: If A → B is faithfully flat, then an A-module M is flat if and only if the B-module M ⊗_{A} B is flat.

Now we can define when a morphism between to schemes

\begin{matrix}X \\ ↓ π \\ S \end{matrix}

is (faithfully) flat: It is flat if for any open affine subscheme V ⊂ S and any open affine subscheme U ⊂ π^{−1}(V) the O_{S}(V)-algebra O_{X}(U) is flat. It is faithfully flat if it is also surjective.

To justify the last definition we need to solve the following

2.7.2.1 Exercise: Show that for a flat A-algebra B the two conditions are equivalent

(1) B is faithfully flat

(2) Spec(B) → Spec(A) is surjective.

In section VI.2.5. we explained that a scheme

\begin{matrix}X \\ ↓ π \\ S \end{matrix}

provides a contravariant functor from the category of schemes over S to the category of sets

T → Hom_{S}(T, X) = X(T).

This functor determines X/S up to a canonical isomorphism. We also mentioned the problem to decide whether such a functor is representable.

Now we have the following fundamental fact ( See III.1.2. )

Theorem VI.2.7.3:

Let X/S be a scheme. Then for any faithfully flat morphism

S ← S′

we get an exact sequence of sets

X(S) ⟶^{p_{0}} X(S′) ⟶^{p_{1}} ; ⟶_{p_{2}} X(S′ ×_{S} S′)

Proof: It is clear that it suffices to prove this for affine schemes. Then we have S = Spec(A), X = Spec(C) and A → C. If now S′ = Spec(B) → S is faithfully flat this means that A → B is faithfully flat. We have to show that the sequence

Hom_{A}(C, A) ⟶^{p_{0}} Hom_{A}(C, B) ⟶^{p_{1}} ; ⟶_{p_{2}} Hom_{A}(C, B ⊗_{A} B)

is exact. Of course A → B is injective, so the first arrow is indeed an injection. Now we have to show that an element ϕ : C → B in Hom_{A}(C, B) which is sent to the same homomorphisms under the two arrows

\begin{aligned}& ↦ b ⊗ 1 \\ b \\ & ↦ 1 ⊗ b \end{aligned}

is actually an element in Hom_{A}(C, A). But this is clear once we know that

A = {b ∈ B | b ⊗ 1 − 1 ⊗ b = 0}

We put δ(b) : b ↦ b ⊗ 1 − 1 ⊗ b then last assertion says that the sequence

0 → A → B ⟶^{δ} B ⊗_{A} B

is exact. Since A → B is faithfully flat this is equivalent to the exactness of the same sequence tensorized by B:

0 → B → B ⊗_{A} B → B ⊗_{A} B ⊗_{A} B.

where the homomorphisms are b ↦ 1 ⊗ b and b_{1} ⊗ b_{2} ↦ (b_{1} ⊗ 1 − 1 ⊗ b_{1}) ⊗ b_{2}. This is now a sequence of B modules where B acts always on the last factor. We have an isomorphism

B ⊗_{A} B ⊗_{A} B→^{∼}B ⊗_{A} B ⊗_{B} B ⊗_{A} B

which is given by b_{1} ⊗ b_{2} ⊗ b_{3} ↦ b_{1} ⊗ 1 ⊗ b_{2} ⊗ b_{3}. ( In the fourfold tensor product we are allowed to move the entry at place 2 to place 4 and backwards since we take the tensor product over B). Hence we have to prove the exactness of

0 → B → B ⊗_{A} B → B ⊗_{A} B ⊗_{B} B ⊗_{A} B.

But now the inclusion B → B ⊗_{A} B admits a splitting given by b_{1} ⊗ b_{2} ↦ 1 ⊗ b_{1}b_{2} and hence we have B ⊗_{A} B = B ⊕ Y where Y is the kernel of that splitting. The elements of y ∈ Y are mapped to

δ(y) = 1 ⊗ y + y ⊗ 1 + … ∈ B ⊗ Y ⊕ Y ⊗ B ⊕ Y ⊗ Y

and now it is clear that δ is injective on Y.

This theorem plays an enormous role since it gives us a necessary condition for the representability of a functor (See I.3.3)

F : Schemes/S → Ens.

Recall that this means that can find a scheme X/S and an element ( the ”identity section” )

e_{X} ∈ F(X))

such that map

\begin{matrix}Hom_{S}(T, X) & ⟶ & F(T) \\ φ & ⟼ & φ^{∗}(e_{X}) \end{matrix}

is a bijection for all T → S. Here φ^{∗} is an abbreviation for F(φ) and e_{X} is called the ”identity section”, because it is equal to φ^{∗}(Id_{X}).

The reader should have noticed that the sequence above is rather similar to the condition which have to be satisfied by sheaves on topological spaces. To make this analogy clear we start from a covering of a topological space X = ⋃_{i∈I} U_{i} by open sets. We consider the disjoint union of the open set we get a continous map X ⟵^{p} X′ = ⨆ U_{i}. We can form the fibered product

X′ ⟵_{p_{2}}^{\underleftarrow{p_{1}}} X′ ×_{X} X′

Then the conditions SH1,SH2 for a presheaf F is exactly the exactness of the sequence

F(X) ⟶^{p_{0}^{∗}} F(X′) ⟶^{p_{1}^{∗}} ; ⟶_{p_{2}^{∗}} F(X′ ×_{X} X′),

where the p^{∗} are the restriction maps.

A. Grothendieck introduced a much more general concept of topologies. For instance we have the flat topology on a scheme S. This means roughly that we replace coverings by Zariski open sets just by faithfully flat morphisms S ← S′. In this more general context we can still define sheaves: These are functors

F : (Faithfullyflatschemes S′ → S) → Ens

which satisfy the extra condition that for all S′ ⟶^{p_{0}} S we get an exact sequence

F(S) ⟶^{p_{0}^{∗}} F(S′) ⟶^{p_{1}^{∗}} ; ⟶_{p_{2}^{∗}} F(S′ ×_{S} S′),

where again the p^{∗} are called the restriction maps, these are the maps induced by the functor.

We can summarize:

If we have a scheme S and a contravariant functor

F : Schemes/S ⟶ Ens
F :  T  ⟶ F(T)

then a necessary condition for this functor to be representable is that its restriction to faithfully flat topology is a sheaf.

We can say even more. For any such functor F and any scheme T ⟶^{f} S we can define the restriction

F_{T} : Schemes/T ⟶ Ens,

which is defined by the obvious definition

F_{T} (T′) = F(T′)

for any scheme T′ → T, which then by the composition with f becomes a scheme over S. We hve tautologically

If our functor is representable by a scheme X/S, and if T → S is a scheme over S, then the restriction F_{T} is represented by X ×_{S} T.

In the last chapter of this book we will discuss the representability of the Picard functor or more precisely a modified Picard functor. There we will encounter a problem of the following type.

Let S be a scheme and let F be a contravariant functor from schemes over S to Ens

F : T ⟶ F(T)

which satisfies our sheaf condition for the faithfully flat topology.

Let us assume that we can find a faithfully flat scheme S′ → S such that the restriction F′ = F_{S′} of our functor becomes representable over S′, i.e. we have a scheme X′/S′ and an element the ”identity section” e_{X′} ∈ F′(X′) which represent the functor in the above sense. Under what conditions can we conclude the already F itself is representable?

This question has been analyzed by Grothendieck and we will describe the techniques which allow to construct- under certain conditions - an X/S which represents F.

We introduce some notation, we put

S′ ×_{S} S′ = S′′, S′ ×_{S} S′ ×_{S} S′ = S′′′

We have the two projections

S′ ⟵^{p_{1}}_{\overleftarrow{p_{2}}} S′′,

and we can take the pullback of X′/S′ by these two arrows, i.e. we consider

X′′_{1} = X′ ×_{S′,p_{1}} S′′  and  X′′_{2} = X′ ×_{S′,p_{2}} S′′.

These two schemes together with the restrictions e_{X′′_{1}}, e_{X′′_{2}} of e_{X′} represent the two restrictions of F′ to S′′ . But these restrictions are also the restrictions of F to S′′ via the composition of p_{1} and p_{2} with p : S′ → S. Since these two compositions are equal we see that F′′_{1}(T) = F′′_{2}(T) for any object T → S′′. This means that the two restrictions of functors are the same, hence have uniquely determined isomorphisms of schemes (inverse to each other)

X′′_{1} ⟵^{φ_{12}}_{\overrightarrow{φ_{21}}} X′′_{2}

which send the restrictions e_{X′′_{1}}, e_{X′′_{2}} of the section e_{X′} into eachother.

Now we go one step further and consider

S′ ⟵^{p_{1}}_{\underleftarrow{p_{2}}} S′′ _{\underleftarrow{p_{12}} ; \underleftarrow{p_{13}} ; \underleftarrow{p_{23}}} S′′′

and can consider the pullback

p_{ij}^{∗}(φ_{12}) : (X′ ×_{S′,p_{1}} S′′) ×_{S′′,p_{ij}} S′′′ → (X′ ×_{S′,p_{2}} S′′) ×_{S′′,p_{ij}} S′′′

and the same for p_{ij}^{∗}(φ_{21}).

The composition p_{ν} ∘ p_{ij} is always a projection to a factor (the α-th factor)

π_{α} : S′′′ → S′.

and we have

\begin{aligned}p_{1} ∘ p_{12} & = & π_{1}  p_{1} ∘ p_{13} & = & π_{1}  p_{1} ∘ p_{23} & = & π_{2} \\ p_{2} ∘ p_{12} & = & π_{2}  p_{2} ∘ p_{13} & = & π_{3}  p_{2} ∘ p_{23} & = & π_{3}. \end{aligned}

This allows us to identify always two of the two step pullbacks to a one step pullback, we get

(X′ ×_{S′,p_{1}} S′′) ×_{S′′,p_{12}} S′′′ = X′ ×_{S′,π_{1}} S′′′ = (X′ ×_{S′,p_{1}} S′′) ×_{S′′,p_{13}} S′′′,
(X′ ×_{S′,p_{2}} S′′) ×_{S′′,p_{12}} S′′′ = X′ ×_{S′,π_{2}} S′′′ = (X′ ×_{S′,p_{1}} S′′) ×_{S′′,p_{23}} S′′′,
(X′ ×_{S′,p_{2}} S′′) ×_{S′′,p_{13}} S′′′ = X′ ×_{S′,π_{3}} S′′′ = (X′ ×_{S′,p_{2}} S′′) ×_{S′′,p_{23}} S′′′,

Our φ_{ij} induce isomorphisms amoung these S′′′ schemes, for example

X′ ×_{S′,π_{1}} S′′′  ⟶^{p_{12}^{∗}(φ_{12})}  X′ ×_{S′,π_{2}} S′′′
↘  ↙
S′′′

and the assumption that X′ represents the restriction F′ of our functor F implies that the resrictions of the ”identity sections” are mapped into eachother. Hence we get the following 1-cocycle relation:

p_{13}^{∗}(φ_{12})^{−1} ∘ p_{23}^{∗}(φ_{12}) ∘ p_{12}^{∗}(φ_{12}) = Id .

Let us forget about the functor F for a moment, let us assume that we have a faithfully flat scheme

S′ ⟶ S

and a scheme f′ : X′ → S′.

If we have an isomorphism

φ_{12} : X′ ×_{S′,p_{1}} S′′ ⟶ X′ ×_{S′,p_{2}} S′′

which satisfies the cocycle rule

p_{13}^{∗}(φ_{12})^{−1} ∘ p_{23}^{∗}(φ_{12}) ∘ p_{12}^{∗}(φ_{12}) = Id

where we made the 3 identifications between the corresponding two step pullbacks, then we call this after A. Grothendieck a descend datum.

Such a descend datum is called effective if we can find a scheme X ⟶^{f} S and an isomorphism

X ×_{S} S′ ⟶^{h} X′
↘  ↙_{f′}
S′

such that: Firstly

X ×_{S} S′′ = (X ×_{S} S′) ×_{S′,p_{1}} S′′ = (X ×_{S} S′) ×_{S′,p_{2}} S′′,

and secondly the diagram

X′ ×_{S′,p_{1}} S′′
↗
X ×_{S} S′′  ↓^{φ_{12}}
↘
X′ ×_{S′,p_{2}} S′′

is commutative.

Then we say that (X′/S′, φ_{12}) descends to S and (X, h) is the realization of the descend datum. To be able to say this, we should convince ourselves that such a realization is unique up to a canonical isomorphism.

Let us assume we have two such realizations (X/S, h) and (X_{1}/S, h_{1}). Then we get from the definition an isomorphism

h_{1}^{−1} ∘ h : X ×_{S} S′ ⟶ X_{1} ×_{S} S′

and from the compatibility with the descend datum we conclude that the two pullbacks

p_{1}^{∗}(h_{1}^{−1} ∘ h_{2}) : (X ×_{S} S′) ×_{S′,p_{1}} S′′ → (X_{1} ×_{S} S′) ×_{S′,p_{1}} S′′

and

p_{2}^{∗}(h_{1}^{−1} ∘ h_{2}) : (X ×_{S} S′) ×_{S′,p_{2}} S′′ → (X_{1} ×_{S} S′) ×_{S′,p_{2}} S′′

must be equal. Hence we get from 2.7.3 that h_{1}^{−1}∘h is the pullback of a uniquely determined isomorphism.

Now we come back to our functor F whose restriction to X ×_{S} S′ was supposed to be representable by a scheme X′/S′. Then we constructed a descent datum for X′/S′ from this information and we can summarize:

Proposition 2.7.3.1: If our functor

F : Schemes/S ⟶ Ens

is a sheaf for the flat topology and if it is representable by a scheme X′/S′ for some faithfully flat S′ → S, then it is representable by a scheme X/S if and only if the descent datum is effective.

Here we can also discuss the descend for sheaves. If we have a flat morphism S′ → S and if we have a sheaf for the flat topology F′/S′. We want to know under which conditions we can find a sheafF/S which restricts to F′. We may also define descend data: We take the two restriction F′_{1}, F′_{2} of F′ by the two morphisms p_{1}, p_{2} from S′ ×_{S} S′ to S′ and we assume that we have and isomorphism of sheaves

φ_{12} : F′_{1}→^{∼}F′_{2}

We say that this isomorphism is a descent datum for the sheaf F′ if its pulbacks via the different projections from S′ ×_{S} S′ ×_{S} S′ to S′ ×_{S} S′ satisfy the 1-cocycle relation

p_{13}^{∗}(φ_{12})^{−1} ∘ p_{23}^{∗}(φ_{12}) ∘ p_{12}^{∗}(φ_{12}) = Id .

It is now easy to see that now the sheaf F′ descends to a sheaf F, if we have such a descend datum on it. The sheaf F is unique up to a canonical isomorphism.

We can now look at our previous discussion of descend data on schems from a different point of view. If we have a scheme X′/S′ then it defines a scheme for the flat topology. A descend datum defines also a descend datum for the sheaf. Then the sheaf descends to a sheaf over S. Now the question whether the descend datum is effective is whether this resulting sheaf is representable.

Remarks:

1) Of course it is clear that for a scheme X/S we have a descend datum on X ×_{S} S′, we can simply take the pullbacks of the identity.

2) Just to avoid possible misunderstandings: If we have a scheme X′/S′, and if we can find a scheme X/S such that X ×_{S} S′ ⟶^{∼} X′, then this scheme X/S is not necessarily unique. Only if the isomorphism X ×_{S} S′⟶^{∼}X′ is compatible with the given descend datum, then we have uniqueness.

We have a simple case where we have effectiveness of descend data. If all our schemes S = Spec(A), S′ = Spec(A′) and X′ = Spec(B′) are affine, then we have the diagram

\begin{aligned}& & B′ \\ & & ↑ \\ A & ⟶ & A^{′.} \end{aligned}

We have the two homomorphisms and we assume that we have an isomorphism of A′⊗_{A}A′- algebras

B′ ⊗_{A′,i_{1}} A′ ⊗_{A} A′ ⟶^{φ} B′ ⊗_{A′,i_{2}} A′ ⊗_{A} A′.

We consider the pullbacks

φ ⊗ i_{vμ} : (B′ ⊗_{A′,i_{1}} A′ ⊗_{A} A′) ⊗_{i_{vμ}} A′ ⊗ A′ ⊗ A′ → B′ ⊗_{A′,i_{2}} A′ ⊗_{A} A′.

The i_{vμ} send an a′ ⊗ a′′ to a three fold tensor with a1_{A′} at the right place. Now we say that φ is a descend datum (we simply have to translate) if

(φ ∘ i_{12}) ∘ (φ ⊗ i_{23}) ∘ (φ ⊗ i_{13})^{−1} = Id .

Now we want to show that such a descend datum is always effective. We consider the algebra B ⊂ B′ consisting of elements (???)

B = {b ∈ B′ | φ(b ⊗ i_{1}) = (b ⊗ i_{2})}.

This is an A-algebra and checks easily that

B ⊗_{A} A′ ≃ B′

In this book we need only a very special case for applying this method of descend. This case is also the historic origin of the method.

Let us assume that we have a field k and a finite separable normal extension L/k. These are our two schemes S = Spec(k), S′ = Spec(L).

Let us assume that we have an affine scheme X′/ Spec(L) it is given as an affine Lalgebra A′, i.e. X′ = Spec(A′). Let us even go one step further: We consider X′ as a closed subscheme of A_{L}^{n}/L, this means that

X′ = Spec(L[X_{1} ⋯ X_{n}]/I′)

where I′ ⊂ L(X_{1} ⋯ X_{n}] is the defining ideal. Now we look at the action of the Galois group Gal(L/k). It is clear that it acts on L[X_{1} ⋯ X_{n}] via the action on the coefficients and we can define the conjugate of X′ as

(X′)^{σ} = Spec(L[X_{1} ⋯ X_{n}]/I^{σ})

where of course

I^{σ} = {Σa_{v_{1}···v_{n}}^{σ}X_{1}^{v_{1}} ⋯ X_{n}^{v_{n}} | Σa_{v_{1}···v_{n}}X_{1}^{v_{1}} ⋯ X_{n}^{v_{n}} ∈ I} .

If we consider the set of geometric point X′(\overline{k}), then we see

(X′)^{σ}(\overline{k}) = (X′(\overline{k}))σ̃

where σ̃ ∈ Gal(\overline{k}/k) maps to σ. If we want that X′/ Spec(L) is obtained by a base extension of an affine scheme over k, then the two affine L-schemes X′ and (X′)^{σ} must be isomorphic for any σ ∈ Gal(L/k).

What does it mean to have an isomorphism? This means that we should have an Lalgebra isomorphism

f_{σ} : L[X_{1} ⋯ X_{n}]/I ⟶ L[X_{1} ⋯ X_{n}]/I^{σ}
”  ”
f_{σ} :  A  ⟶  A^{σ}.

Now we see that the element in the Galois group defined an isomorphism

σ : h ⟼ h^{σ}

between the two rings A and A^{σ}, the point is that this isomorphism is not L-linear, but only L − σ-linear, this means that

σ(λh) = λ^{σ}h^{σ}

for all λ ∈ L. But this remark allows us to write the L-algebra in a different way.

We say A^{σ} is the L-algebra which as a ring is equal to A but where the scalar λ ∈ L acts by

λ ∗_{σ} h = λ^{σ}h

where on the right hand side λ^{σ} act by the original L-algebra structure. The point is that we gave a definition of the conjugate scheme X^{′σ} without reference to the embedding.

Now we reformulate the concept of descend datum for this special case. We say that a Galois-descend datum is a family of isomorphisms

f_{σ} : A ⟶ A^{σ}

which satisfies a compatibility condition.

We observe that for any τ and any σ we get an isomorphism

f_{σ}^{τ} : A^{τ} ⟶ (A^{τ})^{σ} = A^{στ}

which is given by

h ⟼ f_{σ}(h)

(we have f_{σ}^{τ} (λ^{τ} h) = f_{σ}(λ^{τ} h) = (λ^{τ})^{σ}f_{σ}(h)) = λ^{στ} f_{σ}(h)). Then we require (1) f_{1} = Id.

(2) For any pair σ, τ of elements in the Galois group the diagram

A ⟶^{f_{τ}} A^{τ}
↘^{f_{στ}} ↙^{f_{σ}^{τ}}
(A^{τ})^{σ}

commutes.

Now I want to explain two things:

(i) If we have such a Galois-descend datum, then the k-algebra

A_{0} = {a ∈ A | f_{σ}(a) = a}

defines an affine scheme over k (of course) and

\begin{matrix}A_{0} ⊗_{k} L & ⟶ & A \\ h ⊗ λ & ⟼ & λh \end{matrix}

is an L-algebra isomorphism.

(ii) Such a Galois-descend datum is nothing else than a descent datum in the previous sense.

The first assertion is a consequence of the main theorem in Galois theory. If h ∈ A and if λ ∈ L, then we can form

∑_{σ∈Gal(L/k)} λ^{σ}f_{σ}(h).

If τ ∈ Gal(L/k), then

\begin{aligned}f_{τ} (∑_{σ} λ^{σ}f_{σ}(h)) = & ∑_{σ}(λ^{σ}(λ^{σ})^{τ} f_{τ} f_{σ}(h) = \\ & ∑ λ^{τσ}f_{τ}^{σ}f_{σ}(h) = ∑ λ^{τσ}f_{τσ}(h) \end{aligned}

and hence we see that this element lies in A_{0}. But we know that we can find d = [L : k] elements λ_{1} ⋯ λ_{d} ∈ L such that the determinant of the matrix

(λ_{i}^{σ})_{i=1···d,σ∈Gal(L/k)} is non-zero. (Linear independence of the elements in the Galois group.) Hence we can write h as a unique linear combination of the elements

∑ λ_{i}^{σ}f_{σ}(h),

and hence the assertion is clear.

We can show: A Galois-descend datum for affine schemes is always effective.

We come to assertion (ii): Again we apply the main theorem of Galois theory which can be summarized to

L ⊗_{k} L ⟶^{∼} ⨁_{σ∈Gal(L/k)} L

where

a ⊗_{k} b ⟶ (⋯, aσ(b), ⋯)_{σ∈Gal(L/k)}.

If we look at the two arrows

i_{1} : L ⟶ L ⊗ L, i_{2} : L ⟶ L ⊗ L

then

\begin{aligned}i_{1}(λ) & = & (⋯ λ ⋯)_{σ∈Gal(L/k)} \\ i_{2}(λ) & = & (⋯ λ^{σ} ⋯)_{σ∈Gal(L/k)} \end{aligned}

and k = {λ | i_{1}(λ) = i_{2}(λ)} (see ???). Now I skip some details. I say it is quite clear for an L-algebra A the datum of a descend datum

φ : A ⊗_{L,i_{1}} L ⊗ L ⟶ A ⊗_{L,i_{2}} (L ⊗_{k} L)

is exactly the same as the datum of a Galois-descent datum, we simply have to use the above description of L ⊗_{k} L.

Giving a descent datum on a scheme X′/L of finite (???) can also be interpreted in a different way.

Let us assume that X′/L is reduced. It is of course clear that we have an action of the Galois group Gal(\overline{k}/L) on the set X′(\overline{k}) of geometric points. We can speak of L-rational points. Obviously a point x ∈ X′(\overline{k}) which has already in X′(k_{s}) is defined over L if and only if it is a fixed point under the action of Gal(\overline{k}/L).

If we want that X′/L descends to a scheme over k or in other words, if we want to construct a scheme X/k such that X ×_{k} L ≃ X′/L, then we have to be able to extend the Galois action of Gal(\overline{j}/L) to an action of Gal(\overline{k}/k).

First of all we can define a “conjugation” map

σ : X′(\overline{k}) ⟶ (X′)^{σ}(\overline{k})

which on affine pieces is defined as follows: If U′ ⊂ X′ is affine on U′ = Spec(A′), then a geometric point is an L-homomorphism

x : A′ ⟶ \overline{k}
↖  ↙
L

Then the composition σ ∘ x : A′ → \overline{k}, (σ ∘ x)f = x(φ(f)) is clearly a homomorphism of the L-algebra (A′)^{σ} to \overline{k}. This gives us our conjugation for affine pieces and hence also for X′/L itself. But of course our descend datum {f_{σ}}_{σ∈Gal(L/k)} yields a bijection

X′(\overline{k}) →^{f_{σ}} (X′)^{σ}(\overline{k}). For any σ̃ ∈ Gal(\overline{k}/k) with restriction σ to Gal(L/k) we define

σ̃(x) = f_{σ}^{−1} ∘ σ(x)

for all x ∈ X(\overline{k}). Then it is clear that the cocycle condition for the descend datum implies that this defines an action of Gal(\overline{k}/k) on X(\overline{k}).

It is also clear that in case of effectiveness of our descend datum this action of Gal(\overline{k}/k) on X′(\overline{k}) is exactly the action which is provided by h : X ×_{k} L → X′ and the action of Gal(\overline{k}/k) on X(\overline{k}).

Moreover it is also clear that the descend datum is uniquely determined by the extension of the action to Gal(\overline{k}/k) if our scheme is reduced. A morphism between reduced schemes is determined by what it does on geometric points. (Do we need absolutely reduced ???)

We want to say a word on Galois descend on absolutely reduced schemes X/k which are of finite type. It is quite clear that we can prove effectiveness for a descend datum

(X′/L, f_{σ})_{σ∈Gal(L/k)}

if we know in addition that we can find a finite covering X′ = {U′_{i}}_{i∈E} by affine open subschemes which is compatible with the descend datum, i.e.

f_{σ} : U′_{i} ⟶ (U′_{i})^{σ}

for all σ and all indices i ∈ E. This is clear because then we apply that the datum on the affine pieces is effective, and we glue the descended affine pieces together. The detailed proof is pure routine.

There is a simple criterion that tells us that we can verify this assumption. If our scheme X′/k is absolutely reduced and separated, if we know that any finite set {x_{1}, ⋯, x_{m}} ∈ X′(\overline{k}) is contained in the set of geometric points of an affine open subset U′/L ⊂ X′/L, then our assumption is true. We simply start with any point x ∈ X′(\overline{k}) and look at its finitely many conjugates \{σ̃(x)\}_{σ̃∈Gal(\overline{k}/k)}. We pick an open affine set U′′ ⊂ X′/L which contains all these points. It is clear that the conjugates σ̃(U′′(\overline{k})) are also the sets of geometric points of an affine open subset, namely f_{σ}^{−1}((U′′)^{σ}) and we look at the intersection

U′ = ⋂_{σ∈Gal(L/k)} σ̃(U′′).

It is affine and it contains our given point x ∈ X′(\overline{k}). Moreover it is clear that f_{σ} : U′ → (U′)^{σ}, and we are finished.

2.7.4. The Artin-Rees theorem: Before I formulate the theorem I discuss its consequences and I also put it into its general framework.

Now we assume that A is a noetherian local ring (See CA.1.4.) with maximal ideal 𝔪. We can provide it with the so called 𝔪-adic topology: The open neighborhoods of m ∈ M are the sets m + 𝔪^{N}.

We have the completion

i : A → Â = lim_{←} A/𝔪^{N}.

We have also the completion of an A-module M we define M̂ = lim_{←} M/𝔪^{N}M = M ⊗_{A} Â. The theorem of Artin-Rees implies

If M is a finitely generated module over the noetherian local ring A then M → M̂ is injective

This is of course equivalent to

⋂_{N≥1} 𝔪^{N}M = (0).

Then the assertion above says that the 𝔪-adic topology is Hausdorff. The reader should verify that the module M̂ is indeed complete with respect to the 𝔪-adic topology, i.e. any Cauchy sequence is convergent.

We consider finitely generated A-modules M, M′ . . ..

The main implication of the Artin-Rees theorem is:

On the category of finitely generated A-modules the functor

M → M̂ = M ⊗_{A} Â

is faithfully flat which means that it is exact and M̂ = 0 ⇒ M = 0.

The faithfulness is Nakayama’s lemma (See CA.1.2.) To prove flatness we have to show: If M′, M are finitely generated A-modules and if

M′ → M is injective then

M̂′ → M̂

is again injective.

If one tries to prove this one sees that the following has to be shown: If we consider M′ ⋂ 𝔪^{N}M for a very large N then this must be contained in 𝔪^{n}M′ where n is also large. We cannot prove our theorem if we are not able to show that n goes to infinity if N goes to infinity.

The answer to our problem is the actual Artin-Rees theorem

There exists an r_{0} > 0 such that for all r > r_{0} we have

𝔪^{r}M ⋂ M′ = 𝔪^{r−r_{0}}(𝔪^{r_{0}}M ⋂ M′) ⊂ 𝔪^{r−r_{0}}M′

For the proof I refer to the literature.(Ref ?????)

It is clear that it is the technical answer to our problem and it implies that A → Â is faithfully flat on finitely generated modules.

We leave it as an exercise to verify that it also gives ⋂_{N≥1} 𝔪^{N}M = (0).

VI 2.8.Formal schemes and infinitesimal schemes

I want to discuss a situation which is in some sense opposite to the one discussed above. Let k be a field, consider a polynomial ring A = k[X_{1}, …, X_{n}] and we localize it at the maximal ideal 𝔪 = (X_{1}, X_{2}, …, X_{n}). This local ring A_{𝔪} is the ring of germs of regular functions at zero and hence we say that Spec(A_{𝔪}) is the germ of our affine space at zero. This ring is not of finite type over k anymore but it is still a direct limit of finitely generated k-algebras. This local ring sits in the ring Â = k[[X_{1}, …, X_{n}]] of power series in the variables X_{1}, …, X_{n} as a subring: If Q(X_{1}, …, X_{n}) ∈ A and a_{0} = Q(0, …, 0) ≠ 0 then we can write Q = a_{0} + R where R is a polynomial without constant term and

\frac{1}{Q(X_{1}, …, X_{n})} = \frac{1}{a_{0} + R} = \frac{1}{a_{0}} (1 − R/a_{0} + (R/a_{0})^{2} …).

It is easy to see that this power series ring can also be obtained as the projective limit lim_{←}_{N}A_{𝔪_{P}} /𝔪_{P}^{N}. It is also the completion of A_{𝔪} with respect to the 𝔪-adic topology. (See

subsection on the Artin Rees theorem)

The verification of the following assertions are left to the reader.

(1)

Hom_{k}(A_{𝔪}, k) = Hom_{k}(Â, k) = (0) ∈ k^{n}

(we lost all our geometric points except the origin. The geometric points are not dense and this does not contradict the Nullstellensatz because the local ring is not of finite type.)

Let us assume that B is a k-algebra which is local with maximal ideal 𝔫 and B/𝔫 = k. Then

Hom_{k}(A_{𝔪}, B) = 𝔫^{n}

where the identification is provided by ϕ ↦ (ϕ(X_{1}), …, ϕ(X_{n})).

(3) We can construct rings B as above such that we still have

Hom_{k}(Â, B) = (0)

(4) If in addition the ideal 𝔫 in our k algebra B consists of nilpotent elements then we have

Hom_{k}(A_{𝔪}, B) = Hom_{k}(Â, B) = 𝔫^{n}

.

We have the inclusion of rings:

A ↪ A_{𝔪} ↪ k[[X_{1}, …, X_{n}]]

which in turn induces morphism between schemes

Spec(k[[X_{1}, …, X_{n}]]) ↪ Spec(A_{𝔪}) ↪ Spec(A)

We call an algebra B as above for which the ideal 𝔫 consists of nilpotent elements an infinitesimal algebra. Its spectrum is a single point. But if we consider Spec(B)- valued points of Spec(k[[X_{1}, …, X_{n}]]) then we get n-tuples (ϵ_{1}, ϵ_{2}, …, ϵ_{n}) ∈ 𝔫^{n}. These are considered to be points which are infinitesimally small in the sense that the coordinates are not zero but if we raise them into a suitably high power they become zero. This leads us to the idea that Spec(k[[X_{1}, …, X_{n}]]) should be considered as an infinitesimal neighborhood of the origin in Spec(k[X_{1}, …, X_{n}]) = 𝔸^{n}.

The scheme Spec(k[[X_{1}, …, X_{n}]]) is also called the formal completion of Spec(k[X_{1}, …, X_{n}]) at the origin.

Chapter VII Some commutative algebra

I want to collect some standard facts from commutative algebra. Here I will be rather scetchy because there are many good references available. On the other hand I think the reader might be able to fill the gaps itself.

VII. 1. Finite A-algebras

A morphism ϕ : A → B between two affine schemes is called finite if one of the following two conditions is satisfied

(i) The A−module B is finitely generated. This means that we can find b_{1}, b_{2}, …, b_{r} such that any b ∈ B can be written as linear

combination b = a_{1}b_{1} + … a_{r}b_{r} with a_{i} ∈ A.

(ii) The A-ring B is finitely generated by elements x_{1}, …, x_{s} where each of the generators is a zero of a unitary polynomial in A[X].

This means that the A-module B is generated by the monomials x_{1}^{ν_{1}} … x_{s}^{ν_{s}} and that for any i we have a polynomial P_{i}|X] = a_{0} + a_{1}X … + X^{n_{i}} ∈ A[X] such that P_{i}(x_{i}) = 0. (We say that the x_{i} are integral over A.)

I think that is immediately clear that (ii) implies (i) since we can use the polynomials to reduce the degree of the generating monomials.

The proof that (i) implies (ii) is amusing, we leave it as an exercise. Item a) in the following exercise gives a hint.

Exercise 1:

a) We have to show that any b ∈ B is a zero of a unitary polynomial in A[X], i. e. it is integral over A. To see this we multiply the generators b_{i} by b and express the result again as A-linear combination of the b_{i}. This gives us a r × r-matrix M with coefficients in A. If \underline{b} is the column vector formed by the b_{i} we get a relation b\underline{b} = M\underline{b} or (M − b Id)\underline{b} = 0. From this we have to conclude that det(M − b Id) = 0. This would be clear if A would be integral. But it is also true in the general case ( I refrain from giving a hint ). Hence we see that b is a zero of the characteristic polynomial of the matrix M but this polynomial equation has highest coefficient 1 and the other coefficients are in A.

b) This argument generalizes: Let us consider any A-algebra A → B. Show that an element b ∈ B is integral over A if we can find a finitely generated A-submodule Y ⊂ B which satisfies some extra condition (?) with bY ⊂ Y

c) Let us assume that A → B are both integral and that K → L is the corresponding extension of their quotient fields. Let us assume that L/K is a finite extension. Let us also assume that A is integrally closed in K ( any element in K that is integral over A is already in A, such rings are called normal rings). Then an element x ∈ L induces by multiplication a linear transformation L_{x} of the K-vector space L and it is well known that x is a zero of the characteristic polynomial of L_{x}. Show that x is integral if and only if this polynomial has coefficients in A.

d) Under the assumptions of c) we have trace_{L/K}(x) ∈ A for any element x ∈ L which is integral over A.

e) Again under the assumptions of c) we can say: For any x ∈ L we can find a non zero element a ∈ A such that ax becomes integral over A

We have the following

VII.1.2 Theorem:

Assume that the ring homomorphism ϕ : A → B is finite and injective. Then the induced map ^{t}ϕ : Spec(B) → Spec(A) is surjective, has finite fibers and the elements in the fibers are incomparable with respect to the order on Spec(B)

This means in other words: For any 𝔭 ∈ Spec(A) we can find a 𝔮 ∈ Spec(B) such that A ∩ 𝔮 = 𝔭. The number of such 𝔮 is finite, whenever we have two of them 𝔮_{1}, 𝔮_{2} we have 𝔮_{1} ⊄ 𝔮_{2}. This theorem summarizes the so called going up and going down theorems in the books on commutative algebra.

To prove the theorem we need another famous result from commutative algebra Lemma of Nakayama: Let A be a local ring with maximal ideal 𝔪 and let M be a

finitely generated A-module. If

M ⊗ (A/𝔪) = M/𝔪M = 0

then M = 0.

To see this one uses the same trick as above: Express a system of generators of M as a linear combination of these generators but now with coefficients in 𝔪. We find that 1_{A} is a zero of a characteristic polynomial of a matrix with coefficients in 𝔪 which is only possible for the 0 × 0 -matrix.

Now I sketch the proof of the theorem. We pick a prime 𝔭 ∈ Spec(A). The residue class ring A/𝔭 is integral, we have Spec(A/𝔭) ↪ Spec(A) and the zero ideal (0) is mapped to 𝔭. We localize at (0) and we get the quotient field (A/𝔭)_{(0)}. Taking fibered products we get a diagram of affine schemes

Spec(A)  ←  Spec(B)
↑  ↑
Spec(A/𝔭) ← Spec(B) ⊗_{A} (A/𝔭)
↑  ↑
Spec(A/𝔭)_{(0)} ← Spec(B) ⊗_{A} (A/𝔭)_{(0)}

The vertical arrows are inclusions and it is clear that the prime ideals 𝔮 ∈ Spec(A) for which 𝔮 ∩ A are exactly the elements in Spec(B) ⊗_{A} (A/𝔭)_{(0)} To prove the surjectivity we have to show that this scheme is not empty. This follows from the lemma of Nakayama because we can obtain Spec(A/𝔭)_{(0)} also as the residue field of the local ring A_{𝔭}/𝔪_{𝔭}. We have B ⊗_{A} (A_{𝔭}) ≠ 0 (only the zero divisors of S = A \ 𝔭 go to zero in this tensor product) and hence we get by Nakayama that B ⊗_{A} A_{𝔭}/𝔪_{𝔭} ≠ 0 and his implies B ⊗_{A} A_{𝔭}/𝔪_{p} ≠ ∅. Now we have that B ⊗_{A} (A/𝔭)_{(0)} is a finite dimensional vector space over the field (A/𝔭)_{(0)} it is a finite (A/𝔭)_{(0)}− algebra. This implies that any prime ideal 𝔮 ∈ Spec(B ⊗_{A} (A/𝔭)_{(0)} is maximal because the residue ring is automatically a field. Then it is also clear that Spec(B) ⊗_{A} (A/𝔭)_{(0)} must be finite. (The map B ⊗_{A} (A/𝔭)_{(0)} → ∏_{𝔮} B ⊗_{A} (A/𝔭)_{(0)}/𝔮 is easily seen to surjective.) Hence we have proved that the fibers are finite and non empty and we have seen that the prime ideals in the fibers are incomparable.

The following lemma is extremely important, we leave the proof to the reader a special case of it was discussed in exercise (1) V.1.2

Noethersches Normalisierungslemma:

Let A = k[x_{1}, x_{2}, …, x_{n}] be a finitely generated k-algebra (k a field. Then we can replace the system of generators by another system y_{1}, … y_{r}, y_{r+1}, …, y_{m} such that B = k[y_{1}, y_{2}, …, y_{r}] is a polynomial ring in the variables y_{1}, …, y_{r} and the other variables (or the algebra A) are integral over B .

For the proof I refer to the standard books. The basic idea is explained in the exercise mentioned above.

The theorem above has several important applications. One of them is the

Nullstellensatz von Hilbert:

Let k be a field and let A = k[x_{1}, x_{2}, …, x_{r}] be a finitely generated k-algebra. For any maximal ideal 𝔪 ∈ Spec(A) the residue field A/𝔪 is a finite field extension of k. Especially if k is algebraically closed then A/𝔪 = k.

We divide by 𝔪 in other words we may assume that A itself is already a field. We proceed by induction on r. If r = 0 then the theorem is obvious. Let r ≥ 1. We apply the lemma above and find that that A is finite over a polynomial ring B = k[y_{1}, ⋯, y_{s}] ⊂ A If s ≥ 1 then B has non trivial prime ideals which by our theorem VII.1.2. extend to non zero prime ideals in A which is not possible. Hence we see that B = k and therefore A is a finite field extension of k.

VII. 1.3 The Nullstellensatz implies :

For a finitely generated k-algebra A/k the intersection of the maximal prime ideals is equal to the radical

Proof: This has been mentioned earlier. Note that A_{f} is still a finitely generated kalgebra.

VII 1.2. Dimension theory for finitely generated k algebras

If such a k-algebra A is integral we define the dimension dim(A). The number dim(A)+1 is equal the maximal length a chain of prime ideals (0) ⊂ 𝔭_{1} ⊂ … ⊂ 𝔭_{r} where all the inclusions are proper. We say say that the dimension is ∞ if chains of arbitrary length exist.

A maximal chain is a chain that cannot be refined. We also define the height h(𝔭) of a prime ideal 𝔭. The number h(𝔭) + 1 is the maximal length of a chain (0) ⊂ 𝔭_{1} ⊂ … ⊂ 𝔭 ending with 𝔭. There exists a theory of dimensions for arbitrary noetherian rings (See for instance At-McD) which is more general than what we are doing here.

Before we prove the main theorem in this section I want to mention

Proposition 1.2.1. If we have an inclusion i : A → B of two commutative rings with identify and if B is integral over A then for any chain (0) ⊂ 𝔭_{1} ⊂ … ⊂ 𝔭_{r} of prime ideals in A we can find a chain (0) ⊂ 𝔭′_{1} ⊂ … ⊂ 𝔭′_{r} of prime ideals such that 𝔭_{i} = 𝔭′_{i} ∩ A. Especially we can say that the two rings have the same dimension.

Proof: Obvious from VII. 1.2.

VII.1.3 Theorem

The dimension of an integral finitely generated k-algebra A is finite and any maximal chain has the same length. The dimension of the polynomial ring k[X_{1}, X_{2}, …, X_{n}] is n.

To see this we proceed again by induction on the number of generators. If we have two integral k-algebras A ⊂ B and if B is finite over A then dim(A) = dim(B). Using the Noether Normalisierung we can write our algebra A as a finite extension of an algebra with less generators as long as we have non trivial relations among the generators. Hence we are reduced to the case of a polynomial ring A = k[X_{1}, …, X_{n}]. At this point the reader should be aware that it is not clear at this point that we can find minimal non zero prime ideals, it could be possible that we can always to smaller and smaller non zero prime ideals. This would of course imply that the dimension is ∞.

To see that this is not the case we pick a an arbitrary non zero element F (X_{1}, X_{2}, …, X_{n}) ∈ A. We consider prime ideals 𝔭 containing the principal ideal (F (X_{1}, X_{2}, …, X_{n})).

We will show:

We can find minimal prime ideals 𝔭 ⊃ (F (X_{1}, X_{2}, …, X_{n})) and each such ideal is also a minimal non zero prime ideal (Hauptidealsatz of Krull).

Proof: Again we apply the Noether Normalisierung which tells us that after some change of variables we can assume that this polynomial is of the form

F (X_{1}, X_{2}, …, X_{n}) = a_{0}(X_{1}, …, X_{n−1}) + a_{1}(X_{1}, …, X_{n−1})X_{n} + … + X_{n}^{m},

i.e. it is unitary in X_{n} over the ring k[X_{1}, …, X_{n−1}].

We rewrite our polynomial ring by increasing the number of generators

A = k[X_{1}, …, X_{n}, Y]/(F (X_{1}, X_{2}, …, X_{n}) − Y).

This gives us an embedding

k[X_{1}, …, X_{n−1}, Y] ⊂ k[X_{1}, …, X_{n}, Y]/(F (X_{1}, X_{2}, …, X_{n}) − Y)

where now k[X_{1}, …, X_{n−1}, Y] is again a polynomial ring. (This requires a little argument.) We exchanged the variable X_{n} against the polynomial F. Now it is very easy to see that the principal ideal (Y) = Y k[X_{1}, …, X_{n−1}, Y] is a non zero prime ideal and it is minimal with this property. Our original algebra A is integral over k[X_{1}, …, X_{n−1}, Y]. Our theorem tells us that we can find a prime ideal 𝔭 for which we have 𝔭∩k[X_{1}, …, X_{n−1}, Y] = (Y) and hence 𝔭 ⊃ F (X_{1}, …, X_{n}) and 𝔭 is minimal with this property. It is even clear that 𝔭 is a minimal non zero prime ideal because a smaller one would induce a non zero prime ideal contained in (Y). Now we see that chains of prime ideals in which the non zero members contain F (X_{1}, …, X_{n}) have the same length as the induced chains in k[X_{1}, …, X_{n−1}, Y] where the non zero members contain (Y). But these last chains are chains in k[X_{1}, …, X_{n−1}] and it follows from the induction hypothesis that the maximal chains in this ring have length n. Hence we see that maximal chains in A have length n+1 and the theorem is proved.

Finally I want to comment briefly on the Hauptidealsatz von Krull. We proved in the course of the proof of the previous theorem for a polynomial ring, but the general case follows easily again by Noether normalization. In our situation it says that

Hauptidealsatz von Krull

For a finitely generated k-algebra A which is integral and a non zero element f ∈ A and a minimal prime ideal 𝔭 ⊃ (f) we have dim A − 1 = dim(A/𝔭).

The general version of this theorem holds for noetherian rings.(See Atiyah- Macdonald p. 122)

1.3.1. We need the integrality of our algebra because otherwise Spec(A) can if several irreducible components (See VII. 2.) and these components may have different dimensions. This would have the effect that we can find non refinable chains of prime ideals which have different length.

On the other hand we know that Spec(A) = Spec(A/ Rad(A)) and this implies that we need is the integrality of A/ Rad(A) to define the dimension.

VII.1.4.Rings with finiteness conditions

In this section I collect some facts about rings. I will not give any proof because these facts are easily available in the literature. On the other hand it may be a good exercise if the reader tries to find the proofs itself.

A commutative ring A with identity is called noetherian if it satisfies one of the following four conditions

(1) Any ideal 𝔞 ⊂ A is finitely generated.

(2) Any submodule N of a finitely generated A-module M is finitely generated.

(3) Any ascending chain

𝔞_{ν} ⊆ 𝔞_{ν+1} ⊆ … 𝔞_{n} ⊆ …

becomes stationary, i.e. there exists an n_{0} such that

𝔞_{n_{0}} = 𝔞_{n_{0}+1} … = 𝔞_{n_{0}+m}

for all m ≥ 0.

(4) Any ascending chain of A-submodules

N_{ν} ⊆ N_{ν+1} ⊆ …

of a finitely generated A-module M becomes stationary.

The ring ℤ is noetherian and of course we know that fields are so too.

Hilbertscher Basissatz:

If A is a noetherian ring then the polynomial ring A[X] is also noetherian.

This implies that polynomial ring A[X_{1}, …, X_{n}] are noetherian.

VII.2. Minimal prime ideals and decomposition into irreducibles

For any noetherian ring A we can consider the set of minimal prime ideals. Of course for an integral ring A the set of minimal prime ideals consists of just one element (0) ∈ Spec(A), this is the generic point. In the general case it is not a priory clear that minimal prime ideals exist

A ring is called reduced if Rad(A) = 0, we know Spec(A) = Spec(A/ Rad(A)) hence if we want to say something about the ordered topological space Spec(A) we may very well restrict our attention to the case that A is reduced. I state a theorem which is a weak form of a theorem proved by E. Lasker.

Theorem VII.2.1. Let A be a reduced noetherian ring. Then the set of minimal prime ideals is finite. To any minimal prime ideal 𝔭 we can find an f ∈ A such that

(𝔭) = Ann_{A}(f) = {x ∈ A|xf = 0}.

I want indicate the steps of the proof and leave it to the reader to fill the gaps.

1) We prove that there exist minimal prime ideals. This is clear if A is integral. If not we find

f, g ∈ A \ {0} such that fg = 0.

1a) Consider Ann_{A}(f) = 𝔞 and prove: If 𝔞 is not prime then we can find an x ∈ A such that f_{1} = xf ≠ 0 and such that 𝔞_{1} = Ann_{A}(f_{1}) is strictly larger than 𝔞 = Ann_{A}(f).

1b) Show that this implies that we can find an y ∈ A such that Ann_{A}(fy) = 𝔭 is a prime ideal and that this prime ideal is minimal. Hence we see that minimal primes exist.

Let us write yf = f_{𝔭}. It is clear that f_{𝔭} ∉ 𝔭.

2) Prove that any prime ideal 𝔮 contains a minimal prime ideal of the form Ann_{A}(f_{𝔭}) and hence all minimal prime ideals are of this form.

Let us assume we picked an f_{𝔭} for any minimal prime ideal.

3) Prove that for two minimal prime ideals 𝔭 ≠ 𝔭_{1} the product f_{𝔭}f_{𝔭_{1}} = 0.

4) Consider the ideal generated by these f_{𝔭} and combine the fact that this ideal is finitely generated and 3) above to show that these f_{𝔭} form a finite set.

5) Let A be an arbitrary noetherian ring, let 𝔭_{1}, …, p_{r} be the set of minimal prime ideals. Let us also assume that the spaces Spec(A/𝔭_{i}) are disjoint. Then there is a unique collection of elements e_{1}, …, e_{r} such that

e_{i} ∉ 𝔭_{i}and e_{i} ∈ 𝔭_{j}for all j ≠ i
e_{i}^{2} = e_{i}for all i
e_{i}e_{j} = 0 for all i ≠ j

∑ e_{i} = 1_{A}

I want to give a hint: Our assumption that the spaces Spec(A/𝔭_{i}) are disjoint implies that we can find e′_{i} such that e′_{i} ≡ 1 mod 𝔭_{i} and e_{i} ∈ 𝔭_{j} for all j ≠ i. These e′_{i} satisfy all the relations if we compute modulo the radical Rad(A). Now we can modify e′_{i} → e′_{i} + r_{i} = e_{i} such that we have the idempotency e_{i}^{2} = e_{i}. (See the following exercise 6)). Then all the other requirements are also fulfilled.

6) If we have any noetherian ring R and if we consider the homomorphism R → R/ Rad(R) then the group R^{∗} of units of R is the inverse image of the units in R/ Rad(R))

This decomposition of 1_{A} = e_{1} + … + e_{r} is called the decomposition into orthogonal idempotents. It gives a decomposition of the ring

A = ⨁_{i} Ae_{i}

If our ring has no radical the the f_{𝔭_{i}} are equal to the e_{i}.

An affine scheme X = Spec(B) is called irreducible if cannot be written in a non trivial way as a union of two closed subschemes X = X_{1}∪X_{2}. (Non trivial means both subschemes X_{1} and X_{2} are not equal to X.)

If A = B/ Rad(B) is not integral then we get a non trivial decomposition Spec(B) = ∪ Spec(A/𝔭_{i}) and hence we see that Spec(B) is not irreducible in this case. If on the other hand we have a non trivial decomposition Spec(B) = X_{1} ∪ X_{2} then we can find elements f_{1}, f_{2} which are non zero such that f_{i} vanishes on X_{i}. Then f_{1}f_{2} vanishes on Spec(B), hence it must be in the radical. (Ref ???). This implies that A has zero divisors. We proved

2.1.2.The spectrum Spec(B) of a noetherian ring B is irreducible if and only if B/ Rad(B) is integral.

It follows from Laskers theorem that for a noetherian ring A we have a unique finite decomposition of Spec(A) into irreducible subschemes Spec(A/𝔭_{i}). These irreducible subschemes are called the irreducible components. If our ring A has no radical then the elements f_{𝔭} have the property that the vanish on all components except the one which they define. This makes it clear why we have 3) in the exercise above.

2.1.3. For our given noetherian ring A we consider the natural homomorphism

π : A → ∏ A/𝔭_{i}

where the 𝔭_{i} run over the minimal prime ideals. The kernel is the radical of A but it is not always surjective because the irreducible components may have non empty intersections.

A non zero element f ∈ A is called a non zero divisor if fg = 0 implies g = 0. The non zero divisors form a multiplicatively closed subset S ⊂ A. It is clear that f ∈ A is a non zero divisor if and only if all its components under the projection map πare non zero or what amounts to the same if f is not contained in any of the minimal prime ideals.

The total quotient ring of A is the localization A_{S} = Quot(A). The spectrum of A_{S} is simply the set of generic points of the irreducible components. If we pick a minimal prime ideal 𝔭_{i} then the localization A_{𝔭_{i}} is a local artinian ring this means that its unique maximal ideal is also minimal ( and the ring itself is noetherian). We get

The ring A_{S} is the direct product of the finitely many local artinian rings A_{𝔭_{i}} and our map π_{S} is defined componentwise as

∏ A_{𝔭_{i}} → Quot(A/𝔭_{i})

It is surjective the kernel is Rad(A)_{S}

In principle this is the situation of 5) in our exercise, since we localize at the generic points of the components, we make the irreducible components disjoint. Instead of taking this radical step we could have chosen non zero divisors f_{ij} ⊂ A which are zero on Spec(A/𝔭_{i})∩ Spec(A/𝔭_{j}). If we take the product of all these f_{ij} we get a non zero divisor F ∈ A and then Spec(A_{F}) has now the virtue that its irreducible components are disjoint. Hence we may apply exercise 5) and write

A_{F} →^{∼} ⨁ A_{F} e_{i}.

Here we replaced the generic points by actual open sets which looks a little bit more geometric.

An element f ∈ Quot(A) can be written as f = g/h where h is a no zero divisor. Then we have of course f ∈ A_{h} and f is a regular function on Spec(A_{h}). We we look at all different ways to write f as a quotient, the we see that f is defined on the union of all the Spec(A_{h}). his open set is called the domain of definition for f. It is a dense subset of Spec(A).

2.1.4. Decomposition into irreducibles for noetherian schemes: If we have an arbitrary scheme X we say that this scheme is a noetherian scheme if we have a finite open covering X = ⋃ U_{i} by affine schemes U_{I} = Spec(A_{i}) where the rings A_{i} are noetherian rings. Again we can speak of irreducible schemes. If one of the covering set U_{i} is not irreducible then we write it non trivially as U_{i} = V_{i} ∪ W_{i}. If we take the closures of these subschemes in X we clearly get two closed subschemes which are not equal to X. Then we get a non trivial decomposition of X = V̄_{i} ∪ W̄_{i} ∪ Z_{i} where Z_{i} is the complement of U_{i}.

Hence we see: Our noetherian scheme X = ∪U_{i} is irreducible if and only if all the U_{i} are irreducible.

From this we can deduces very easily that X itself has always a unique finite decomposition into irreducible subschemes.

This is usually proved directly without reference to affine schemes.

If we have an affine scheme X = Spec(k[X_{1}, …, X_{n}]/I) = Spec(A) then it is automatically noetherian. Hence our previous considerations apply.

VII.2.2.Affine schemes over k and change of scalars In this section I want to consider k-algebras A/k and I call such a k-algebra an affine k-algebra if it is finitely generated over k as an algebra.

For these schemes over a field we always have the option of base change to k̄. This means that we can consider the algebra A ⊗_{k} k̄. We have to discuss what happens to irreducibility, reducedness and dimensions under such a base change. We also want to discuss their behavior under the formation of products.

First of all it can happen that A = K is finite field extension. In this case Spec(K) is irreducible but K ⊗_{k} k̄ is not if the extension is non trivial and separable. Hence irreducibility is not preserved under base change to k̄.

We call A/k absolutely irreducible if A ⊗_{k} k̄ is irreducible.

We call a k-algebra A/k reduced if it does not contain non zero nilpotent elements. Again it can happen that a reduced k-algebra A/k becomes non reduced if we extend the ground field. To see this we start from a non separable finite extension K/k and then we know that K ⊗_{k} k̄ contains nilpotent elements. We call call A/k absolutely reduced if A⊗_{k} k̄ does not contain non trivial nilpotent elements.

2.2.1..A k algebra is called an affine k- variety if it is absolutely irreducible and absolutely reduced.

In the following exercise we will show that these examples are typical: If an integral k-algebra becomes reducible or reduced after extension of scalars, then we can find a finite extension of k which is contains in the field of fractions of A and which is responsible for that.

Exercise 2

Let A/k be an integral finitely generated algebra. If it is not absolutely irreducible or not absolutely reduced then we can find a non zero f ∈ A such that the localization A_{f} contains a non trivial finite extension of k.

Hint: Reduce the problem to a base change A ⊗_{k} L where L = k[X]/(p(X)) with an irreducible polynomial p(X) ∈ k[X].

a) Investigate under which conditions the algebra A ⊗_{k} L can become reducible or non reduced. This means that we can find two polynomials g_{1}(X), g_{2}(X) ∈ A[X] whose degrees are less than the degree of p(X) such that p(X) divides the product g_{1}(X)g_{2}(X). Look at all such pairs and pick one where the degree of say g_{1}(X) is minimal. Using the arguments you learned in your first algebra course to show that there must be a non zero element f ∈ A such that g_{1}(X) divides fp(X). This means that g_{1}(X)/f ∈ A_{f}[X] divides p(X). Now conclude that the coefficients of g_{1}(X)/f must be algebraic over k. But not all of them are in k and we found the non trivial extension.

b) We call an element y ∈ A_{f} a pseudoconstant if it generates a finite extension of k, i.e. if it is algebraic over k. We call it a constant if it is actually in A .( It looks a little bit bizzar that we can have pseudoconstants which are not constants, but this may be the case). I claim:

b) The pseudoconstants generate a finite algebraic extension of K/k.

To see this we apply the Normalisierungslemma and write A as a finite extension of a polynomial ring B. Hence A is a finite B module let m be the number of elements in a system of generators of this B-module. It is easy to see that for any non zero f ∈ A we can find an F ∈ B such that A_{f} ⊂ A_{F} and A_{F} is still integral over B_{F}. If our field k is infinite then we can find a k rational point a ∈ Spexmax(B_{F}) let 𝔪_{a} be the corresponding maximal ideal. We get an injection of the field of constants of A_{F} to A_{F} /𝔪_{a}A_{F} and the degree of this extension is bounded by m. Hence we see that the degree of the field of pseudoconstants is bounded. This proves the assertion if k is infinite. If k is finite then we can replace k by an extension k_{1}/k of prime degree where this prime is larger than m and such that we find a k_{1} valued point a of B_{F}. Then we see that the tensor product A_{F} ⊗_{k} k_{1} is still a field and injects into A_{F} ⊗ B_{F} /𝔪_{a} and we have the same argument as before.

To give an example we consider the R-algebra A = ℝ[x, y|/(x^{2} + y^{2}). Then our scheme Spec(A) is irreducible but A ⊗ ℂ is not. The elements x/y, y/x are pseudoconstants but they are not constants. The geometric points are given by two lines intersecting in the origin. If we remove the origin, the value of say x/y is ±i on the two lines. In the origin the function x/y is just a little bit confused and does not know which value to take.

If k is a non perfect field of characteristic 2 and if a ∈ k is not a square then A = k[x, y|/(x^{2} + ay^{2}) is integral. Again we see that x/y, y/x are pseudoconstants but not constants.

2.2.1.1 It is clear that a pseudoconstant has a dense domain of definition. Hence we see that we have an open dense subset U ⊂ Spec(A)which is the common domain of all pseudoconstants.

2.2.2.. If we consider an arbitrary finitely generated k-algebra we can consider its total quotient Quot(A)/k and define the subalgebra of pseudoconstants L/k ⊂ Quot(A)/k as being the ring of elements which are finite over k. It is obvious that this k-algebra L/k is exactly the inverse image of the pseudoconstants under the map

π : Quot(A) → ⨁_{𝔭_{i} minimal} Quot(A/𝔭_{i}),

where now on the right hand side we have a sum of fields.

We conclude using part b) of the exercise that L/k is finite dimensional, because the radical is of finite dimension over k. It is also clear that the map

Spec(Quot(A)) → Spec(L)

is bijective, the left hand side is just the set of generic points of the irreducible components of Spec(A).

We also see that all the pseudoconstants lie already in a suitable subalgebra A_{f} where f is a non zero divisor.

Now we have the following

Lemma 2.2.2.1. The k-algebra L/k of pseudoconstants is preserved under change of scalars, i.e. L ⊗ k̄ is the k̄-algebra of pseudoconstants of A ⊗ k̄ . We can find a finite separable extension K/k such that the irreducible components of A ⊗ K will be absolutely irreducible.

We can give this a slightly different formulation. We can consider the k-algebra homomorphism

L ⊗ k̄ → Quot(A) ⊗ k̄.

The lemma says that this induces a bijections

Spec(L ⊗ k̄) ← Spec(Quot(A) ⊗ k̄)
Rad(L ⊗ k̄) → Rad(Quot(A) ⊗ k̄)

The proof is based on an observation which will also play a role in the next section. 2.2.2.2. Let us consider an element F ∈ A ⊗ k̄. We can write this element in the form

F = ∑ f_{i} ⊗ α_{i}

where the α_{i} are taken from a finite normal extension K/k. This extension has a maximal separable subextension K_{sep} in it. We know that any element α ∈ K raised to a sufficiently high power of the characteristic p of our field k will fall into K_{sep}. Hence we get that

F^{p^{r}} = ∑ f_{i}^{p^{r}} ⊗ α_{i}^{p^{r}} ∈ A ⊗ K_{sep}

If we know form the norm of this power, this means we form the product over all conjugates by elements σ ∈ Gal(K_{sep}/k) then its norm

∏_{σ} σ(F^{p^{r}}) = ∏_{σ}(∑ f_{i}^{p^{r}} ⊗ σ(α_{i}^{p^{r}})) ∈ A.

This has consequences: If for instance F is a non zero divisor then we see that we can find a multiple F F_{1} = G ∈ A ⊂ A ⊗ k̄ which is still a non zero divisor. Hence we see that (A ⊗ ¯)_{F} ⊂ A_{G} ⊗ k̄ and this implies that

Quot(A) ⊗ k̄ = Quot(A ⊗ k̄).

Now let us assume that F ∈ Quot(A) ⊗ K is a pseudoconstant. This is so if and only if G = F^{p^{r}} is a pseudoconstant. We rewrite G in the form

G = ∑ g_{i} ⊗ β_{i}

where the β_{i} ∈ K_{sep} form a basis over k. We have the trace map tr : K_{sep} → k and consider the elements

tr(Gβ_{j}) = ∑_{i} g_{i} tr(β_{i}β_{j}).

These elements are pseudoconstants and since the determinant of the matrix tr(β_{i}β_{j}) is non zero we get that the coefficients g_{i} ∈ Quot(A) must be pseudoconstants. We have the decomposition of Quot(A) ⊗ k̄ into a direct sum of artinian algebras and we have the decomposition of the identity into orthogonal idempotents 1_{A} = ∑ e_{i}. The e_{i} are clearly pseudoconstants in Quot(A) ⊗ k̄. We apply our previous argument and find that suitably high powers e_{i}^{p^{r}} lie in L ⊗ k̄. But then we have e_{i} ∈ L ⊗ k̄ and more precisely we see that the e_{i} are already in L ⊗ K where K/k is separable.

In our discussion above we gave an argument which also can be used to show exercise

c) If K/k is separable then Rad(A ⊗ K) = Rad(A) ⊗ K

The only thing that has to be proved is the equality concerning the radicals in the reformulation of the Lemma. For this we may assume that our field k is already separably closed and that A/k is integral. Now we repeat the argument in the exercise above. We reduce the problem to the case of an extension K = k(α) where α^{p} = a ∈ k. Then this argument yields: If A ⊗ K has nilpotent elements then we can find a β ∈ Quot(A) such that β^{p} = α^{p} = a. This shows the desired equality of radicals for this small extension. Then we proceed by looking at (A ⊗ K)/ Rad(A ⊗ K) and apply the same argument.

2.2.3. Let us consider two affine k̄-algebras A_{1}/k̄, A_{2}/k̄ then we have the following Lemma2.2.3.1. We have

Rad(A_{1} ⊗ A_{2}) = Rad(A_{1}) ⊗ A_{2} + A_{1} ⊗ Rad(A_{2})

and especially the tensor product A_{1} ⊗ A_{2} is reduced if the factors are so.

If the two algebras are irreducible then then tensor product A_{1} ⊗ A_{2} is also irreducible. Proof: Choose a basis {g_{1}, g_{2}, …} of A_{2}/k where the first say t basis vectors form a basis

of Rad(A_{2}). Let h = ∑ f_{i} ⊗ g_{i} be an element in the radical. We evaluate this element at geometric points x ∈ Specmax(A_{1}) and then the element ∑ f_{i}(x)g_{i} ∈ A_{2} must be in the radical of A_{2}. This implies that for all such points x we have f_{i}(x) = 0 for all indices i > t. The Nullstellensatz gives us f_{i} ∈ Rad(A_{1}) for all these i > t and this proves the first assertion.

The proof of the second assertion is similar. We can assume that both algebras are actually integral. We take the same basis and consider a pair of zero divisors

(∑ f_{i} ⊗ g_{i})(∑ f′_{i} ⊗ g_{i}) = 0

If we evaluate at the geometric points x as above then for each such x we must have that one of the factors vanishes. (Irreducibility of Spec(A_{2})). Hence we may consider the two closed subschemes of Spec(A_{1}) which are defined by the vanishing of of the f_{i} or the f′_{i} respectively. There union contains all geometric points hence they cover Spec(A_{1}). Since Spec(A_{1}) is irreducible we get that one of them must be the whole Spec(A_{1}) and this proves that the corresponding factor in the product must be zero because the product is absolutely reduced.

Proposition 2.3.2. If A_{1}, A_{2} are two irreducible k̄-algebras then we have

dim(A_{1} ⊗ A_{2}) = dim(A_{1}) + dim(A_{2}).

Proof: This is rather clear. We apply Noether Normalisierungssatz two our algebras and write them as finite extensions of polynomial rings. Then the number of variables is equal to the dimension in either case (VII. prop. 1.2.1. and Thm. VII. 1.3.). The tensor product is finite over the polynomial ring in the disjoint union of the variables.

VII. 2.3. Local irreducibility

Now we assume that we have a geometric point P ∈ Hom_{k}(A, k̄). It induces a maximal ideal 𝔪_{P} in A ⊗_{k} k̄. We say that A ⊗ k̄ is locally integral at P if the local ring (A ⊗_{k} k̄)_{𝔪_{P}} is integral. The point P also induces a maximal ideal 𝔪_{P}^{0} in A. Under these assumptions we have the following technical Lemma

Lemma 2.3.1. If A ⊗ k̄ is locally integral at P then the local ring A_{𝔪_{P}^{0}} is also integral, we have an injection i : A_{𝔪_{P}^{0}} → (A ⊗_{k} k̄)_{𝔪_{P}} and equality of dimensions dim(A_{𝔪_{P}^{0}} ) = dim((A ⊗_{k} k̄)_{𝔪_{P}})). If 𝔭^{0} is the unique minimal prime ideal in A_{𝔪_{P}^{0}} which is contained in 𝔪_{P}^{0} then these dimensions are also equal to dim(A/𝔭^{0}).

Proof: We prove the integrality of A_{𝔪_{P}^{0}} , Let x ∈ A_{𝔪_{P}^{0}} be an element that maps to zero in (A ⊗_{k} k̄)_{𝔪_{P}}. We can assume that x ∈ A and via the inclusion A → A ⊗_{k} k̄ we can view it as element in the ring A ⊗_{k} k̄. We now that we can find an element y_{P} ∉ 𝔪_{P} such that xy_{P} = 0. We have a finite number of geometric points P = P_{1}, P_{2}, …, P_{s} which lie above 𝔪_{P}^{0} and the Galois group of k̄/k permutes these points transitively (See 2.5.). We can find an element z_{P} ∉ 𝔪_{P} such that z_{P} ∈ 𝔪_{P_{i}} for all i = 2, …, s. If we take an element σ_{i} in the Galois group which maps P to P_{i} then σ(y_{P} z_{P}) = y_{P_{i}}z_{P_{i}} ∉ 𝔪_{P_{i}} but it lies in all the other 𝔪_{P_{j}} for j ≠ i. We always have xy_{P_{i}}z_{P_{i}} = 0 and if we put s = ∑ y_{P_{i}}z_{P_{i}} then we have s ∉ 𝔪_{P_{i}} for all i and of course we still have xs = 0. We apply the argument from 2.2.2.1. to s and see that we can produce an element t^{p^{N}} = Norm(s^{p^{N}}) ∈ A.Then we still have xt^{p^{N}} = 0 and t^{p^{N}} ∉ 𝔪_{P_{i}} for all i and hence t^{p^{N}} ∉ 𝔪_{P}^{0} . But this tells us that x maps to zero in A_{𝔪_{P}^{0}} and since we assumed A integral we must have x = 0.

Now we prove the assertions concerning the equality of dimensions. It is clear that we can find an f ∉ 𝔪_{P}^{0} such that A_{f} is integral and then we have dim(A_{f}) = dim(A_{𝔪_{P}^{0}} ) this follows from Theorem VII.1.3. and the fact that Spec(A_{𝔪_{P}^{0}} ) = {𝔭 ∈ Spec(A_{f}) | 𝔭 ∈ 𝔪_{P}^{0} }. (ref/????). Now we have seen that we can find a finite extension K/k such that A_{f} ⊗ K = ∑_{i} A_{f} ⊗ Ke_{i} such that the summands are absolutely irreducible. For this we may have to change the f. But now we have that A_{f} → A ⊗ K is finite, then it is clear that (??? e_{i} conjugate under galois) A → Ae_{i} is injective and finite and hence we have dim(A) = dim(Ae_{i}).Now we have exactly one index i such that e_{i}(P) = 1 and then we have (A ⊗_{k} k̄)_{𝔪_{P}} = (Ae_{i})_{𝔪_{𝔓}} and the equality of dimensions follows.

But if we extend the ground field to k̄ then this is a union of finite extensions and the equality follows from our first theorem in this chapter.

The property that A is locally integral at the geometric point P implies that P lies on exactly one irreducible component of Spec(A ⊗_{k} k̄) and this implies that it also lies on exactly one irreducible component of Spec(A). The converse is not true since the local ring at P may still have nilpotent elements. But for the definition of the dimension at P we only need that P lies on exactly one component. In this case we will say that A is locally irreducible at P.

2.3.1.2. If we have an integral affine k-algebra A/k and a maximal ideal 𝔪 then we know that the dimension of A is equal to the dimension of the local ring A_{𝔪}(Thm VII.1.3.) If our k-algebra is not integral and if 𝔪 is a maximal ideal then we may still speak of dim(A_{𝔪}) provided this local ring is irreducible which means that 𝔪 lies on exactly one irreducible component, i.e. if there is exactly one minimal prime ideal 𝔭_{i} ⊂ 𝔪. If we pick an element f ∉ 𝔪 which lies in all the other minimal prime ideals then Spec(A_{f}) is open and irreducible in Spec(A). We again have the equality of dimensions dim(A_{f}) = dim(A_{𝔪}). If we have another maximal ideal 𝔪_{1} ⊃ 𝔭_{i} and which does not lie in any other irreducible component then we can choose f so that f ∉ 𝔪_{1}. This shows that the local dimension dim(A_{𝔪}) stays constant as long as moves within one component and avoids the set of points which lie in several components.

If A/k is irreducible then we can speak of dim(A). But what happens if A/k is not absolutely irreducible. If claim that nothing happens. We have seen that we can find a finite separable extension K/k such that the irreducible components of A ⊗ K will be absolutely irreducible (Lemma 2.2.2.). Then we see as in the proof Lemma 2.3.1. that dim(A) = dim(A ⊗ K)/𝔭 for any minimal prime ideal. We can summarize by saying that the dimension is stable under the extension of scalars.

VII.2.4. Low dimensional rings

A noetherian ring is of dimension zero if every prime ideal is maximal (and minimal). In this case it is clear from the assertion above that Spec(A) = {𝔭_{1}, …, 𝔭_{t}} is a finite set. Then the local rings A_{𝔭_{i}} are also of dimension zero and A_{𝔭_{i}} has only one prime ideal namely m_{𝔭_{i}} and hence m_{𝔭_{i}} is also the radical of this local ring. We have a map

A ⟶ ∏_{i=1}^{t} A_{𝔭_{i}}.

This map is an isomorphism. We have done this in even greater generality in 2.1.3. but here the irreducible components are automatically disjoint.

The rings A_{𝔭_{i}} and hence also A are artinian which means that every descending chain of ideals becomes stationary.

Now we consider integral rings A with dim(A) = 1. This means that every non-zero prime ideal 𝔭 is already maximal. If we have any ideal (0) ≠ 𝔞 ≠ A , then dim(A/𝔞) = 0 and

Spec(A/𝔞) ⊂ Spec(A)

is a finite subset by the previous results.

Hence we see that for a one dimensional ring A the open sets U ⊂ Spec(A) are the complements of a finite set of closed points (maximal prime ideals) and of course the empty set.

If A is integral and of dimension one and local, then Spec(A) consists of two points {𝔭, (0)}. Such a ring is A is called a discrete valuation ring if 𝔭 is a principal ideal, i.e. we can find an element π_{𝔭} such that

𝔭 = A · π_{𝔭} = (π_{𝔭}).

The element π_{𝔭} is called an uniformizing element. It is of course not unique in general, it can be multiplied by a unit and is still an uniformizing element. It is quite clear that any element a ∈ A can be written as

a = ϵ · π_{𝔭}^{ν_{𝔭}(a)}

where ϵ is a unit and where ν_{𝔭}(a) is an integer. This exponent is called the order of a and can be considered as the order of vanishing a at 𝔭.

The elements of the quotient field K are of the form

x = \frac{b}{c} = \frac{ϵ · π_{𝔭}^{ν_{𝔭}(b)}}{ϵ′ · π_{𝔭}^{ν_{𝔭}(c)}} = ϵ^{”} · π_{𝔭}^{ν_{𝔭}(a)−v_{𝔭}(b)} = ϵ′′ · π_{𝔭}^{ord_{𝔭}(x)}.

We clearly have ν_{𝔭}(x) ≥ 0 if and only if x ∈ A. We may say that x has a pole of order −ν_{𝔭}(x) if ν_{𝔭}(x) < 0.

A very important class of one dimensional rings is provided by the Dedekind rings. We have the following

2.3.1.Proposition/Definition:

A one-dimensional integral ring A is called a Dedekind ring if one of the following conditions is satisfied.

(1) The ring A is integrally closed in its quotient field K, i.e. if x = \frac{b}{c} ∈ K and we can find a polynomial

x^{n} + a_{1}x^{n−1} + … a_{n} = 0

with a_{i} ∈ A then x ∈ A.

(2) For every prime ideal 𝔭 ≠ (0) the local ring A_{𝔭} is a discrete valuation ring.

The inclusion (2) ⇒ (1) is quite clear. We consider our element x ∈ K. I claim that for any 𝔭 ≠ (0) we must have x ∈ A_{𝔭}. Otherwise we could write x = ϵπ_{𝔭}^{−n} with n > 0 and ϵ a unit in A_{𝔭}. But then it is not possible that x satisfies the polynomial equation. But if x ∈ A_{𝔭} for all 𝔭 then this means that is regular at all points in Spec(A) and the assertion follows from proposition V. 1.4.1.

The direction (1) ⇒ (2) is a little bit tricky. Of course we may assume that A is already local. If 𝔭 is the maximal ideal then we consider the A-module 𝔭^{−1} of all elements x ∈ K which satisfy x𝔭 ⊂ A. We clearly have 𝔭^{−1} ⊃ A. The decisive point is to show that we can find an element y ∈ 𝔭^{−1} which is not in A. To see this we pick a non zero element b ∈ 𝔭. The ring A/(b) has dimension zero and therefore the image of 𝔭 in this ring is equal to the radical. This implies that a suitable power 𝔭^{n} ⊂ (b) we choose n minimal with this property. Then we know that we can find elements p_{1}…. , p_{n−1} ∈ Liep such that the element y = p_{1}p_{2} … p_{n−1}/b ∉ A. But if we multiply by any further element element in 𝔭 then the result lies in A. Now we conclude y𝔭 = A or y𝔭 = 𝔭. But the second case is impossible, because by exercise 1) b) in VII. 1. this would imply that y is integral over A and since A is integrally closed we get y ∈ A in contrast to what we know. The rest is clear: We can find a π ∈ 𝔭 such that yπ = 1. Now it is clear that 𝔭 = (π) because if p ∈ 𝔭 then yp = a ∈ A and this gives p = πa.

This proposition is fundamental for the foundation of the theory of algebraic numbers. If we have a Dedekind ring A and a non-zero ideal (0) ≠ 𝔞 ⊂ A, then the quotient A/𝔞

has dimension zero and we just saw that

A/𝔞 = ∏_{𝔭⊃𝔞}(A/𝔞)_{𝔭}.

If 𝔞_{𝔭} is the image of 𝔞 in the localization A_{𝔭} then (A/𝔞)_{𝔭} = A_{𝔭}/𝔞_{𝔭}. Now we know that A_{𝔭} is a discrete valuation ring hence we have 𝔞_{p} = (π_{p}^{ν_{𝔭}(𝔞)}) and ν_{𝔭}(𝔞) is called the order of 𝔞 at 𝔭. It is not difficult to show that A/𝔭^{ν_{𝔭}(𝔞)} = A_{𝔭}/(π_{p}^{ν_{𝔭}(𝔞)}) and hence we get

A/𝔞 = ∏_{𝔭⊃𝔞} A/𝔭^{ν_{𝔭}(𝔞)}.

Exercise 3:

a) this assertion implies 𝔞 = ∏_{𝔭⊃𝔞} 𝔭^{ν_{𝔭}(𝔞)}. (Hint: What is in general the relation between the product 𝔞𝔟 and the intersection of two ideals 𝔞, 𝔟 in an arbitrary ring A? Show that there is always an inclusion in one direction. Then verify that this inclusion becomes an equality if the two ideals generate the ring, or in other words if Spec(A/𝔞) ∩ Spec(A/𝔟) = ∅.)

b) To any maximal prime ideal 𝔭 we can find an x ∈ K (the field of fractions) such that ord_{𝔭}(x) = −1 and ord_{𝔮}(x) ≥ 0 for all the other maximal ideals. Then xp ∈ A for all p ∈ 𝔭.

A fractional ideal 𝔟 of a Dedekind ring A is a finitely generated A non zero submodule in the field of fractions K. For any such fractional ideal 𝔟 we can find an x ∈ K^{∗} such that x𝔟 ⊂ A becomes an integral (ordinary) ideal. We can multiply such fractional ideals and our previous results imply that

2.3.1.1The fractional ideals in a Dedekindring form a group under multiplication.

The neutral element is obviously given by the ring A itself and (exercise 2 b) above gives the inverse 𝔭^{−1} = (1, x). This group is the free abelian group generated by the prime ideals. It is also called the group of divisors Div(A). This group of divisors contains the subgroup of principal divisors P (A), these are the ideals of the form (x) with x ≠ 0. The quotient group

Pic(A) = Div(A)/P (A)

is the so called ideal class group of A. It is an important invariant of the ring. By definition it is trivial if and only if A is a principal ideal domain.

If we have an Dedekindring A with quotient field K and if L/K is an extension of finite degree, then we may consider the integral closure of A in L. This is the ring B consisting of those elements b which satisfy an equation b^{n} + a_{1}b^{n−1} + … a_{0} = 0 with a_{i} ∈ A. (These elements can also be characterized as those elements which leave a finitely generated non zero A-submodule M ⊂ L invariant under multiplication, i.e. aM ⊂ M. Then it is easy to see that the form indeed a ring).

We have the

2.4.2.Theorem(Krull - Akizuki) The integral closure of a Dedekindring in a finite extension of its quotient field is again a Dedekindring

This is not an easy theorem for the proof I refer to the book of J. Neukirch (Algebraische Zahlentheorie).The main problem is to show that B is again noetherian.

The following fundamental theorem is easier

2.4.3.a) If A is a Dedekindring with quotient field K and if L/K is a finite separable extension, the the integral closure B of A in L is a finitely generated A- module. Hence B is clearly noetherian and therefore again a Dedekindring.

b) If A is a Dedekindring which is a finitely generated algebra over a field k, i. e. A = k[x_{1}, …, x_{n}] and if L is any finite extension of the quotient field K of A then the integral closure B of A in L is again a finitely generated algebra over k and hence noetherian and Dedekind.

Exercise 4

To see a) we start from a basis a_{1}, ⋯, a_{n} of the field L over K which consists of integral elements over A. Write an element b ∈ B as linear combination b = a_{1}x_{1} + a_{2}x_{2} … a_{n}x_{n} with x_{i} ∈ K. We use the seperapabilty to invert this system of equations for the x_{i}. The traces trace_{L/K}(ba_{ν}) are integral ( Use Exercise 1 c) ) and we find the relations

trace_{L/K}(ba_{ν}) = ∑ trace_{L/K}(a_{i}a_{ν})x_{i}.

Conclude that we can find an element a ∈ A which does not depend on b such that a_{i}a ∈ A, hence B ⊂ A\frac{a_{1}}{a} + … + A\frac{a_{n}}{a} and therefore is finitely generated.

To prove b) we check that we may assume that L/K is normal. Then we have a maximal inseparable subextension L_{i}/K which is purely inseparable. This is obtained by successive extraction of p-th roots. Hence we prove the assertion for extensions of the form L = K[r^{1/p}] (not so easy) and proceed by induction and apply a) at the end.

The theorem above has the following implication: Let us assume that we have a Dedekindring A with quotient field K and a finite extension L/K and we assume that the assumptions of a) or b) are valid. Then we know that the integral closure B of A in L is a finitely generated A module. Let us pick a maximal prime ideal 𝔭 ⊂ A. We consider the A/𝔭 algebra B/𝔭B. First of all I claim that the dimension of B/𝔭B as a A/𝔭-vector space is equal to the degree [L : K] = dim_{K} L. To see this we may assume that A is local and then B must be a free A-module of rank [L : K]. Now we have seen that

B/𝔭B = ∏_{𝔓⊃𝔭B} B/𝔓^{ν_{𝔓}(𝔭B)}.

For a 𝔓 ⊃ 𝔭 we get a finite extension of residue fields (B/𝔓)/(A/𝔭) and we denote its degree by f_{𝔓} = [B/𝔓 : A/𝔭]. Moreover we know that for any integer m that the quotient 𝔓^{m}/𝔓^{m+1} is a B/𝔓 vector space of dimension one and hence a A/𝔭 vector space of dimension f_{𝔓}. Hence we get that B/𝔓^{ν_{𝔓}}(𝔭B) is a A/𝔭 vector space of dimension f_{𝔓}ν_{𝔓}(𝔭B) We call the numbers ν_{𝔓}(𝔭B) = e_{𝔓} they are the so called ramification indices. Counting the dimensions yields the formula

[L : K] = ∑_{𝔓⊃𝔭B} f_{𝔓}e_{𝔓}

A finite extension L of ℚ is called an algebraic number field. The ring ℤ is a Dedekindring and hence its integral closure O_{L} in L is so too. This is the called the ring of integers in L. The study of these rings of integers is the subject of algebraic number theory.

Let me just state the theorem that Pic(O_{L}) is a finite abelian group.

VII.2.4.Finiteness properties of Tor The functor T or^{R}. has a certain finiteness property which will become important later. If we have an R-module M which is not flat, then this means that we can find another

R-module N such that T or_{1}^{R}(N, M) ≠ 0.

Proposition 1: If M, N arc R-modules such that T or_{1}^{R}(N, M) ≠ 0 then we can find a finitely generated submodule N′ ⊂ N such that T or_{1}^{R}(N′, M) ≠ 0.

Proof: We start from a projection resolution of N

0 → X → P_{0} → N → 0

and we get from our assumption that

X ⊗_{R} M → P_{0} ⊗_{R} M

is not injective. Hence we find an element y = ∑ ×_{ν} ⊗m_{ν} ∈ X ⊗_{R} M which is not zero but which goes to

zero in P_{0} ⊗_{R} M. Using the construction of the tensor product by generators and relations we see that we can find a finitely generated submodule Y ⊂ P_{0} which contains the X_{ν} and such that y goes already to zero in Y ⊗_{R} M. But then we get a sequence

0 → Y ∩ X → Y → N′ → 0

where N′ ⊂ N is finitely generated. Our element y ∈ (Y ∩ X) ⊗ M is still non zero and maps to zero in Y ⊗_{R} M. Hence we see that

(Y ∩ X) ⊗_{R} M → Y ⊗_{R} M

is not injective and this implies T or_{1}^{R}(N′, M) ≠ 0. This has the consequence that for a non flat A-module M we can find an ideal 𝔞 ⊂ A

such that T or_{1}^{R}(A/𝔞, M) ≠ 0.

If we have a flat A-module M which is not faithfully flat, then we get by similar arguments that we can find an ideal 𝔞 ⊂ A, ≠A such that (A/𝔞 ⊗ M) = 0.

Hence we can say that the modules of the form A/𝔞 recognize flat resp. faithfully flat A-modules.

Proposition 2.4.1.:Let A be a noetherian ring and assume that M is a finitely generated A-module. For 𝔭 ∈ Spec(A) the A_{𝔭} module M_{𝔭} = M ⊗ A_{𝔭} is free if and only if it is flat. Then the set of points 𝔭 ∈ Spec(A) where M_{𝔭} is a free A_{𝔭}-module is an open subset U ⊂ Spec(A).

Proof: We pick a point 𝔭 then M ⊗ k(𝔭) is a vector space of finite dimension over k(𝔭). If we lift the elements of a basis of this vector space to elements m_{1}, …, m_{s} then these elements generate the A_{𝔭} module M_{𝔭} (Lemma of Nakayama). Hence we get an exact sequence

0 → R → A_{𝔭}^{s} → M → 0.

It is clear that M_{𝔭} is free if and only if R = 0 and then this module is also flat.

Now we assume that M_{𝔭} is flat. Taking the tensor product with k(𝔭) gives us an exact sequence

0 → R ⊗ k(𝔭) → A_{𝔭}^{s} ⊗ k(𝔭) → M ⊗ k(𝔭) → 0

where now the second arrow in this sequence is an isomorphism by construction. Hence we that the first arrow is the zero map. Our assumption implies that R ⊗ k(𝔭) = 0. But since R is finitely generated (remember we assumed that A is noetherian) we get R = 0.

Now the rest follows because it is clear that if M_{𝔭} is free then we can find an open neighborhood U = Spec(A_{f}) of 𝔭 such that M_{f} = M ⊗_{A} A_{f} is free.

VII.2.5. Dominant morphisms: If we have an arbitrary morphism π : X → Y between two schemes X/k, Y/k of finite

type (???) then we would like to have some answers for the following questions: (a) What is the structure of the image?

(b) What can we say about the dimension of the fibers? 2.5.1 To attack these questions we have the following reduction process. (i)e cover X, Y by affine schemes U_{i}, V_{i} such that π : U_{i} → V_{i}, then we see that we can

reduce these questions to the case of affine morphisms.

(ii) If now ϕ : A → B, Spec(B) → Spec(A) is a morphism of affine schemes then we see that we may assume that ϕ is injective, because it factors over Spec(A/𝔞), where 𝔞 is the kernel of ϕ.

(iii) Now we decompose both afine schemes into irreducible components and then we see that we may assume that our the algebras are integral.

Let us assume that we have two integral affine k-algebras and a morphism

φ: A ⟶ B
↖  ↗
k

Let us assume in addition that the morphism φ is injective, under these assumptions the morphism φ is called dominant. We get a morphism X = Spec(B) → Y = Spec(A). We can pass to the quotient fields K of A and L of B and we get an injection

φ_{1}: K ⟶ L
↖  ↗
k

and now we get a surjective Spec(L) → Spec(K) map and the extension L/K is faithfully flat.

The following proposition asserts that these two facts remain true after the restriction to suitable nonempty subsets U ⊂ X, V ⊂ Y, i.e. we get a faithfully flat morphism U → V.

Proposition 2.5.1: Let φ: A → B be a dominant morphism between two affine kalgebras. Then we can find an f ≠ 0 in A and a g ≠ 0 in B such that

A_{f} ⟶ B_{fg}

is faithfully flat and then Spec(B_{fg}) → Spec(A_{f}) is surjective.

Proof: We consider the field of fractions K of A. Then B ⊗_{A} K becomes a finitely generated K-algebra, it is still integral. We have B ⊗_{k} K = K[x_{1}, …, x_{m}] let x_{1}, …, x_{r} a maximal subset of elements such that K[x_{1}, …, x_{r}] is the polynomial ring in r variables. The remaining generators x_{r+1}, …, x_{m} satisfy algebraic equations

P_{ν0}(x_{1}, …, x_{r})x_{ν}^{m_{ν}} + P_{ν1}(x_{1}, …, x_{r})x_{ν}^{m_{ν}−1} + … P_{νm_{ν}} (x_{1}, …, x_{r}) = 0.

where ν = r+1, …, m and the P_{ν0}(x_{1}, …, x_{r}) ∈ K[x_{1}, …, x_{r}] are not zero. If H(x_{1} … x_{r}) is the product of the P_{ν0} then we can find f ∈ A, f ≠ 0 such that fH(x_{1}, …, x_{r}) ∈ A[x_{1} … x_{r}] and hence we see that B_{fH} = B⊗A_{f}[x_{1}, …, x_{r}, \frac{1}{fH}] is integral over A_{f}[x_{1}, …, x_{r}, \frac{1}{fH}]. We look at the coefficients of fH, which are elements in A_{f}. At least one of these coefficients c is not zero. We replace f by fc then one of the coefficients of fH = g is a unit.

Now we have

A_{f} ⊂ A_{f}[x_{1}, …, x_{r}, \frac{1}{g}] ⊂ B_{fg}

and B_{fg} is finite over A_{f}[x_{1} … x_{r}, \frac{1}{g}].

I claim that under these conditions the A_{f}-algebra A_{f}[x_{1}, …, x_{r}, \frac{1}{g}] is faithfully flat.

First of all we observe that

A_{f}[x_{1}, …, x_{r}, \frac{1}{g}] = A_{f}[x_{1}, …, x_{r}, y]/(yg − 1)

The ideal (yg − 1) is a free A_{f}[x_{1}, …, x_{r}, y]− module and hence we see that the A_{f}- modules A_{f}[x_{1}, …, r, y] and (yg − 1) are free. Hence we have the sequence of A_{f}-modules

0 → (yg − 1) → A_{f}[x_{1}, …, x_{r}, y] → A_{f}[x_{1}, …, x_{r}, \frac{1}{g}] → 0

To check flatness it suffices to show that this sequence stays exact if we tensorize by quotients A/𝔞. We get

(yg − 1) ⊗ A/𝔞 → A/𝔞[x_{1}, …, x_{r}, y] → A/𝔞[x_{1} … x_{r}, \frac{1}{g}] → 0

but now it is not difficult to see that the first arrow is still injective because yg − 1 is not a zero divisor in A/𝔞[x_{1}, …, x_{r}, y]

To check that A_{f}[x_{1}, …, x_{r}, \frac{1}{g}] is faithfully flat, we need that A/𝔞[x_{1}, …, x_{r} \frac{1}{g}] ≠ 0 for all ideals 𝔞 ∈ A, 𝔞 ≠ A. This is obviously the case since we assumed that g ≢ 0 mod 𝔞 for all ideals of this kind.

Now we observe that the quotient field of B_{fg} is a finite extension of the quotient field of A_{f}[x_{1}, …, x_{r}, \frac{1}{g}]. Hence we see that B_{fg} is a free in the generic point of Spec(A_{f}[x_{1}, …, x_{r}, \frac{1}{g}]) and hence it is is flat over a non empty open subset of Spec(A_{f}[x_{1}, …, x_{r}, \frac{1}{g}]) and then it is automatically locally free hence faithfully flat. This open set contains some Spec(A_{f}[x_{1}, …, x_{r}, \frac{1}{g}])_{g_{1}} but then we see that this again contains a subset of the form Spec(A_{f}, [x_{1}, …, x_{r}, \frac{1}{g},]). Hence we proved the assertion concerning faithful flatness.

The surjectivity follows from the faithfulness but can also easily derive from the fact that B_{fg} is integral over A_{f}[x_{1}, …, x_{r}, \frac{1}{g}].

Of course our consideration imply:

Proposition 2.5.2: If φ: A → B be a dominant morphism between two affine k-algebras, if r is the difference of the trancendence degrees of their fraction fields. Then we can find a non empty open subset V = Spec(A_{f}) ⊂ Spec(A) such that for any point y ∈ V the dimensions of the irreducible components of the fiber f^{−1}(y) are greateror equal to r.

Proof: This follows from the proof of the previous proposition. We can even say that the components of f^{−1}(y) which itersect non trivial with Spec(B_{fg}) have dimension = r.

2.5.3 : Now we can derive a general statement concerning the image of a morphism between arbitrary k schemes of finite type. Actually what we would like to have

Under some reasonable conditions the image of f : X → Y is open in the Zariski closure of the image.

Let us start from a dominant morphism f : X → Y between affine schemes. Then we have seen that the image contains a non empty open subset U ⊂ Y. Let us consider the complement Y′ of U, this is a closed subscheme, we get a morphism X′ = f^{−1}(Y′) → Y′. We consider two irreducible componets Y′_{1}, X′_{1} of Y′, X′ respectively, such that X′_{1} → Y′_{1} and we take open affine subsets V′_{1}, U′_{1} such that U′_{1} → V′_{1} and we get a homomorphism between the corresponding integral affine algebras A′ → B′. But now there is no reason why this morphism is still dominant, we get a diagram

A′/𝔭  ↪  B′
↖  ↗
A′

and this tells us that our morphism may factor through a proper closed irreducible subscheme V′′_{1} of V′_{1}. This morphism U′_{1} → V′′_{1} is again dominant and therefore its image contains a non empty open subset Ṽ′′_{1} ⊂ V′′_{1}. Again we can take its complement and proceed.

This consideration tells us that the image under a morphism between schemes of finite type over a field k is always a so called constructible subset of Y. The family of constructible subsets is the smallest family of subsets which contains the open subsets and is closed under finite intersections and the formation of complements. So open and closed subsets are constructible but also open subsets of closed subsets and so on.

Exercise Construct an example of the form

k[U, V]⟶k[U, V, Z]/(P (U, V, Z))

where the image is the affine plane minus ( one of the axes minus the point (0, 0))

2.5.4 If we analyse our reasoning in 2.5.3. and the example in the exercise. Then we see that taking the restriction from Y = Spec(A) to the complement of the open set U we may loose the dominance. Keeping the notations from 2.5.3 this means that the tensor product with the quotient field of A′

A′_{(0)} ⊗_{A′} B′

may become zero. But now we look at the sequence

0 → A′ → A′_{(0)} → A′_{(0)}/A′ → 0

and take the tensor product with B′. We get

0 → B′ → A′_{(0)} ⊗_{A′} B′ → A′_{(0)}/A′ → 0.

Now if B′ is not zero (i.e. X′ ≠ ∅ ) then we see that B′ is not a flat B′ algebra if A′_{(0)} ⊗_{A′} B′ = 0. This gives us the decisive hint.

Proposition 2.5.5 If f : X → Y is a flat morphism between to k-Schemes of finite type, then the image f(X) is open in Y.

Proof: Again we easily reduce this to the case X = Spec(A), Y = Spec(B) and ϕ : A → B is flat. It is a formal consequence of the concept of flatness that for any ideal 𝔞 ⊂ A the algebra homomorphism A/𝔞 → B ⊗_{A} A/𝔞 is flat. It is also clear that we can restrict the morphism to the inverse image of a connected component of Y. A component is given by a minimal prime ideal 𝔭. We remarked that A/𝔭 → B ⊗_{A} A/𝔭 is flat. Then we may have that B ⊗_{A} A/𝔭 = 0, then the inverse image of Spec(A/𝔭) is empty and nothing has to be proved. Otherwise we observed that (A/𝔭)_{(0)} ⊗_{(A/𝔭)} B ≠ 0 and hence we see that B contains a minimal prime ideal 𝔮 such that ^{t}ϕ(𝔮) = 𝔭. I other words A/𝔭 → B/𝔮 is injective. We are in the situation of proposition 2.5.1. and can conclude that Spec(B) → Spec(A/𝔭) has an open image. Now we proceed as indicated, we pass to the complement of the open image and use the fact that the restriction of the morphism to this complement is still flat.

VII.3.Smooth points

Let k be a field and X/k be a scheme of finite type. This means that we can find a finite covering X = ⋃ U_{i} by affine schemes whose rings of regular functions O_{X}(U_{i}) are finitely generated k-algebras. We consider the set of geometric points X(k̄) = Hom_{Spec(k)}(X, Spec(k̄)) of X.

I want to introduce the notion of a smooth point P ∈ X(k̄). To give a first idea of what it should be a give examples of non smooth or singular points.

Examples:

Consider the two algebras A = k[x, y]/(x^{2} − y^{3}) and B = k[x, y]/(xy). The point P = (0, 0) is a geometric point on both of them and it is singular. If you now take k = ℝ and draw a picture of the ℝ-valued points then you get an idea what a singular point is.

The property of a point to be smooth will be local, this means that by definition a point will be smooth if and only if for all affine open neighborhoods U ⊂ X, P ∈ U(k̄) the point is smooth on U. Hence we may assume that X = Spec(A) we could also pass to the germ of X at P. The notion of a smooth point is also ”geometric”: If we embed k ↪ k̄ and form the base extension X̄ = X ×_{Spec(k)} k̄ = Spec(A ⊗_{k} k̄) then P is also a geometric point of X̄. The definition of a smooth point will be such that P is smooth as a geometric point on X if and only if it is smooth as a geometric point of X̄. It will even turn out that the smoothness of the point is a property of the formal completion of our scheme at P (See section on infinitesimal schemes in the section on affine schemes).

Our point P is now a homomorphism

\begin{aligned}P : & A ⊗_{k} k̄ & → & k̄ \\ & ↖ & & ↗ \\ & & k \end{aligned} .

It defines a maximal ideal 𝔪_{P} ⊂ A ⊗_{k} k̄. We have a decomposition of the k̄-vector space

A ⊗_{k} k̄ = k̄ ⊕ 𝔪_{P} ,

which is given by f = f(P) + (f − f(P)) for f ∈ A ⊗_{k} k̄. Now we give the

VII.3.1. Definition: The point P ∈ Hom_{k}(A, k̄) is smooth if and only if the local ring (A ⊗ k̄)_{𝔪_{P}} is integral and if the dimension of the k̄-vector space 𝔪_{P} /𝔪_{P}^{2} is equal to the dimension of the local ring (A ⊗_{k} k̄)_{𝔪_{P}}.

3.1.1. The integrality of the local ring implies that our point lies on exactly one irreducible component, i.e.the local ring is irreducible. Actually we will see that it would be sufficient to assume that the local ring is irreducible. Then we can speak of its dimension (See VII.???) and we will see that the required equality of dimensions will imply that the local ring is reduced. hence integral.

We also know that dim((A ⊗ k̄)_{𝔪_{P}}) = dim((A ⊗ k̄)/𝔭) (See 2.3.1.2.). The maximal ideal 𝔪_{P} induces also a maximal ideal 𝔪_{P}^{0} in A itself. I recall that we proved that the local ring A_{𝔪_{P}^{0}} is also integral, we have an injection i : A_{𝔪_{P}^{0}} → (A ⊗_{k} k̄)_{𝔪_{P}} and equality of dimensions dim(A_{𝔪_{P}^{0}} ) = dim((A ⊗_{k} k̄)_{𝔪_{P}})). If 𝔭^{0} is the unique prime ideal in A_{𝔪_{P}^{0}} which is contained in 𝔪_{P}^{0} then these dimensions are also equal to dim(A/𝔭^{0}). (See VII.2.3.1. and 2.3.1.2. ) It is quite clear that we can say that the point 𝔪_{P}^{0} ∈ Specmax(A) is smooth if one or all the geometric points lying above this point are smooth. (See V 2.5.1)

Now a want to discuss the implications of the smoothness of a point. For the following discussion we assume that k = k̄ and hence A ⊗_{k} k̄ = A.

I want to explain the meaning of the equality

dim_{k}𝔪_{P} /𝔪_{P}^{2} = dim(A_{𝔪_{P}}) = d.

We pick elements t_{1} … t_{d} ∈ 𝔪_{P} which provide a basis t̄_{1} … t̄_{d} in the quotient 𝔪_{P} /𝔪_{P}^{2} . These elements generate the A-module 𝔪_{P} by Nakayamas lemma. We consider the polynomial ring with the same number of generators B = k[X_{1}, …, X_{d}] and we consider its maximal ideal 𝔪_{0} = (X_{1}, …, X_{d}). We get a k-algebra homomorphism B_{𝔪_{0}} → A_{𝔪_{P}} sending X_{i} to t_{i}. We compute modulo powers of the maximal ideals, i.e. we look at the rings

B_{𝔪_{0}}/𝔪_{0}^{N}, A_{𝔪_{P}} /𝔪_{P}^{N}.

Clearly our homomorphism induces maps B_{𝔪_{0}}/𝔪_{0}^{N} → A_{𝔪_{P}} /𝔪_{P}^{N}. It is easy to see that these maps are surjective: The t_{i} generate the A_{𝔪_{P}} −module 𝔪_{P} hence any f ∈ A_{𝔪_{P}} can be written as f = a + ∑ g_{i}t_{i} with a ∈ k. Apply the same to the g_{i} we get f = a + ∑ b_{i}t_{i} + quadratic terms with b_{i} ∈ k. Since this goes on forever and the assertion becomes clear.

We have projective systems

B_{0}/𝔪_{0}^{N+1} ⟶ B_{0}/𝔪_{0}^{N} ⟶
↓ ↓ ↓
A_{𝔪_{P}} /𝔪_{P}^{N+1} ⟶ A_{𝔪_{P}} /𝔪_{P}^{N} ⟶

and if we consider the completions B̂ = lim_{←}B_{0}/𝔪_{0}^{N} and Â_{𝔪_{P}} = lim_{←}A_{𝔪_{P}} /𝔪_{P}^{N}. The ring B̂_{𝔪_{0}} is isomorphic to the ring of formal power series k[[X_{1}, …, X_{d}]]. (Ref. to Affine schemes)

We get a diagram

B_{𝔪_{0}}  → A_{𝔪_{P}}
↓  ↓
i_{P} : k[[X_{1}, …, X_{d}]] → Â_{𝔪_{P}}

Theorem: VII.3.2 Let A/k be an affine algebra over the algebraically closed field k, let P be a geometric point such that A_{𝔪_{P}} is irreducible, i.e. P lies on exactly one irreducible component. We always have

dim_{k}𝔪_{P} /𝔪_{P}^{2} ≥ dim(A_{𝔪_{P}}) = d.

The vertical maps in the diagram above are always injective. We have equality in our inequality if and only if the horizontal arrow in the bottom line i_{P} is an isomorphism. This is the case if and only if P is smooth.

This would be obvious if we had proved that the ring of formal power series is noetherian and if we had the general theory of dimensions of noetherian rings. But since we do not have this we give a different argument.

Proof: The injectivity of the vertical maps follows from the Artin-Rees theorem (without any assumption on P). It is also clear that the bottom line arrow is surjective since we have the surjective maps B_{0}/𝔪^{N} → A_{𝔪_{P}} /𝔪_{P}^{N}.

Now we prove the inequality. We proceed by induction on d′ = dim(A_{𝔪_{P}}). Here we use our assumption that P lies on exactly one irreducible component. Let 𝔞 be the kernel of i_{P}. For d′ = 0 the maximal ideal 𝔪_{P} is nilpotent and we have d ≥ d′ = 0. If d′ ≥ 1 then 𝔪_{P} ≠ Rad(A_{𝔪_{P}}) we pick one of the generators say t_{1} which is not in the radical. We choose a minimal prime ideal 𝔪_{P} ⊃ 𝔭 ⊃ (t_{1}). The Hauptidealsatz tells us that dim((A/𝔭)_{𝔪_{P}}) = d′ − 1. Now the number of generators of the image of 𝔪_{P} in the local ring (A/𝔭)_{𝔪_{P}} has dropped by a number r ≥ 1. Hence we get d − r ≥ d′ − 1 and this proves that always d ≥ d′.

If our point P is smooth then it is also a smooth point on the subscheme A/𝔭 because in this case we have d − r ≥ d′ − 1 and since d = d′ we must have r = 1. We apply our construction above to (A/𝔭)_{𝔪_{P}} and get a homomorphism

k[[X_{2}, …, X_{d}]] → ˆ (A/𝔭)_{𝔪_{P}} .

This homomorphism is a bijection by the induction hypothesis. The kernel of

k[[X_{1}, X_{2}, …, X_{d}]] → k[[X_{2}, …, X_{d}]]

is the principal ideal (X_{1}) and hence 𝔞 ⊂ (X_{1}). Assume 𝔞 ≠ (0). If d = 1 we get that 𝔞 = (X_{1}^{m}) for some m, but this is impossible because this implies t_{1}^{m} = 0 and the dimension of A_{𝔪_{P}} would be zero. If d > 1 then we could replace t_{1} by any linear combination of the generators t_{i} and conclude that 𝔞 is contained in any principal ideal (f) where f = a_{1}X_{1} + a_{2}X_{2} + … a_{d}X_{d} + higher order terms with non zero linear term. But this tells us that any element h ∈ 𝔞 must be divisible by any such power series f which is clearly impossible unless h = 0. Once we proved that i_{P} is an isomorphism we conclude that the local ring (A/𝔭)_{𝔪_{P}} must be integral and we proved that P is smooth.

If in turn i_{P} is an isomorphism then we observe that X_{1} is sent to t_{1} ∈ 𝔪_{P}. Hence we see that (t_{1}) is a prime ideal and

k[[X_{2}, …, X_{d}]] → ˆ (A/(t_{1})_{𝔪_{P}}

is an isomorphism. By induction this implies d′ − 1 = d − 1 hence d′ = d and our point must be smooth.

The theorem means that in an ”infinitesimal neighborhood” of a smooth point a scheme X/k over an algebraically closed field k looks like the ”infinitesimal neighborhood” of the origin of the the an affine space of dimension dim(X). (V.2.8.)

Now we assume that our k-algebra is given as a quotient of a polynomial ring, in other words we consider X = Spec(A) as embedded into an affine space. We write

0 → I → k[X_{1}, …, X_{n}] → A → 0

where I is the defining ideal. Again we look at our smooth point P : A → k and for simplicity we assume that P (X_{i}) = 0 for all i, i.e. our point P is the origin. We want to show that the smoothness of P has the consequence that the system of equations which defines X locally at P as a subscheme has certain nice properties ( Jacobi criterion).

Let 𝔪_{0} be the maximal ideal in A defined by P and 𝔐_{0} = (X_{1}, …, X_{n}) the maximal ideal in the polynomial ring. We get an exact sequence (localization is flat)

0 → I_{𝔐_{0}} → k[X_{1}, …, X_{n}]_{𝔐_{0}} → A_{𝔪_{0}} → 0.

Let d = dim(A_{𝔪_{0}}) then we may assume that X_{1}, … X_{d} map to a basis in 𝔪_{0}/𝔪_{0}^{2}. We write x_{i} for the image of X_{i} in A_{𝔪_{0}}. We get a diagram

0 ⟶ I_{𝔐_{0}} ⟶ k[X_{1}, …, X_{n}]_{𝔐_{0}} ⟶  A_{𝔪_{0}}  ⟶ 0
↓  ↓  ↓
0 ⟶ Î_{𝔐_{0}} ⟶ k[[X_{1}, …, X_{n}]] ⟶ k[[x_{1}, …, x_{d}]] ⟶ 0

where Î_{𝔐_{0}} is the completion of I_{𝔐_{0}} and the exactness on the left in the second sequence is again the Artin-Rees theorem. We also use our previous theorem which gives us that Â_{𝔪_{P}} is indeed the ring of power series in the given d variables.

Now we can find certain specific elements in I_{𝔐_{0}} which turn out to be generators of the ideal. We observe that for any i = d + 1, …, n the element x_{i} ∈ 𝔪_{0} must be up terms of higher order (i.e. elements in 𝔪_{0}^{2}) a linear combination of the x_{ν} with ν = 1, …, d. In other words for any such i we get a relation in A_{𝔪_{0}} :

x_{i} = L_{i}(x_{1}, … x_{d}) + G_{i}(x_{1}, …, x_{n})

where L_{i} is linear in the x_{1}, …, x_{d} and where G_{i} is an expression which contains only terms of degree ≥ 2. If we lift this to the variable X_{1}, …, X_{n} this provides elements in I_{𝔐_{0}}

F_{i} = X_{i} − L_{1}(X_{1}, …, X_{d}) − G_{i}(X_{1}, …, X_{n}) ∈ I_{𝔐_{0}} i = d + 1, …, n.

I claim:

The elements F_{d+1}, …, F_{n} generate the ideal I_{𝔐_{0}}.

This is rather clear: If we look at the completion then we see that for any i = d+1, …, n we can express the image of x_{i} as power series in the variables x_{1}, …, x_{d}. ( Just substitute the expression for x_{i} into the G_{i} and you get an expression for the x_{i} in terms of a quadratic term in the x_{1}, x_{2}, …, x_{d} and an expression in all x_{i} where the terms are of degree ≥ 3. and so on) Hence we can write the x_{i} for i = d + 1, …, n as power series in the x_{1}, …, x_{d}. Hence we get that the F_{i} viewed as elements in the ideal Î_{𝔐_{0}} can be written as

F_{i}(X_{1}, …, X_{n}) = X_{i} − P_{i}(X_{1}, X_{2}, …, X_{d}) i = d + 1, …, n,

where the P_{i}(X_{1}, X_{2}, …, X_{d}) are power series. If we divide the power series ring k[[X_{1}, …, X_{n}]] by these F_{i} then the resulting ring is the power series ring in the variables X_{1}, …, X_{d} and this is also the result if we divide by Î_{𝔐_{0}}. This makes it clear that these F_{i} generate the ideal Î_{𝔐_{0}}. Now I_{𝔐_{0}}/𝔐_{0}I_{𝔐_{0}} = Î_{𝔐_{0}}/𝔐_{0}Î_{𝔐_{0}} and the Nakayama lemma implies that the ideal I_{𝔐_{0}} is also generated by these F_{i}

This has several consequences. It is clear that we can find an affine open neighborhood U ⊂ 𝔸_{k}^{n} of our point P such that the restriction Ĩ(U) of I to U contains the F_{i} and is generated by them. Hence

(1) In a suitable Zariski neighborhood of a smooth point P can define a scheme X/k ↪ 𝔸_{k}^{n} by n − dim(X) equations, namely the above F_{i}.

(2) If we form the Jacobi matrix of the relations F_{i} at our smooth point P then the matrix

(\frac{∂F_{i}}{∂X_{j}} (P))_{i=d+1,...,n;j=1,...n}

has rank n − d.

This leads us to the famous Jacobi criterion which gives a characterization of the smooth points of an embedded affine scheme X ⊂ 𝔸^{n}.

VII.3.3. The Jacobi criterion :From now on we drop the assumption that k should be algebraically closed. We consider an affine scheme A = k[X_{1}, …, X_{n}]/I. A geometric point P ∈ Hom_{k}(A, k̄) defines a maximal ideal 𝔪_{P} in A but it can happen that different P give the same maximal ideal. We have also the maximal ideal 𝔐_{P} in the polynomial ring which lies above 𝔪_{P}. We want to formulate a criterion for the smoothness of P in terms of the data over k , i.e. we do not extend the base to k̄. This will imply for instance that all geometric point lying over a given maximal ideal 𝔪 will be smooth if one of them is so.

Jacobi criterion:

Let I be an ideal in k[X_{1}, X_{2}, …, X_{n}] and A = k[X_{1}, X_{2}, …, X_{n}]/I. Let P be a geometric point of Spec(A), let 𝔪_{P} (resp 𝔐_{P}) be the corresponding maximal ideal in A (resp. k[X_{1}, X_{2}, …, X_{n}]). Let F_{1}, …, F_{r} be a system of generators of I_{𝔐_{P}}. We assume that the Jacobi matrix

(\frac{∂F_{i}}{∂X_{j}} (P))_{i=1,...,r;j=1,...n}

has rank r. Then P is smooth and dim(A_{𝔪_{P}}) = n − r.

If conversely P is smooth and dim(A_{𝔪_{P}}) = n − r then we can find generators G_{1}, …, G_{r} of the ideal I_{𝔐_{P}} such that the Jacobi matrix built in these generators has rank r.

Before I give an outline of the proof, I want to make a few comments. First of all we remark that the evaluation of an element F ∈ k[X_{1}, …, X_{n}] at P has the result is in k̄.

Hence we did not keep our promise not to extend the ground field. But our statement concerning the rank is equivalent to the assertion that a determinant of a suitable r × rmatrix is non zero. This matrix has entries \frac{∂F_{i}}{∂X_{j}} ∈ k[X_{1}, …, X_{n}] and hence the determinant is an element F ∈ k[X_{1}, …, X_{n}]. Now F (P) ≠ 0 is equivalent to F ∉ 𝔐_{P} we restated our condition on the Jacobi matrix without extending the ground field. Then it is also clear that all the geometric points lying over a given maximal ideal are all smooth or none of them is.

Finally I want to say that this criterion tells us that in a Zariski neighborhood of a smooth maximal ideal we can describe a subscheme X/k ⊂ 𝔸_{k}^{n} always by n − dim(A_{𝔪_{P}}) equations which satisfy the Jacobi criterion and vice versa a set of equations satisfying the Jacobi criterion in an open set U ⊂ X tells us that all the geometric points in U are smooth.

Proof: Since we have a geometric definition of the smoothness of a point we have to extend the ground field to k̄. We consider the ideal I ⊗_{k} k̄ ⊂ k̄[X_{1}, X_{2}, …, X_{n}]. After making a substitution we may assume that P = (0, …, 0). Hence it is reasonable to denote the maximal ideals in A ⊗_{k} k̄ resp. the polynomial ring by 𝔪_{0} resp. 𝔐_{0}. We put d = n − r. If we renumber our variables we can assume that the partial Jacobi matrix

(\frac{∂F_{i}}{∂X_{j}} (P))_{i=1,...,r;j=d+1,...n}

has rank r. If we expand the polynomials at P = (0, …, 0) we get expressions

F_{i}(X_{1}, …, X_{n}) = ∑_{j=1}^{n} \frac{∂F_{i}}{∂X_{j}} (P)X_{j} + higher order terms

for i = 1, …, r. Let us denote the homomorphism of the polynomial ring to our algebra A by Φ. Then we get in (A ⊗_{k} k̄)_{𝔪_{P}} the relations

0 = ∑_{j=1}^{n} Φ(\frac{∂F_{i}}{∂X_{j}})x_{j} + higher order terms in the x_{i}

If we ignore higher terms and evaluate at P then we get a system of linear for the x_{i} ∈ 𝔪_{0}/𝔪_{0}^{2} and our assumption on the partial Jacobi matrix implies that we can express the images x_{i} for i = d + 1, …, n as linear combinations of the x_{i} with i = 1, …, d. We get that the x_{i} with i = 1, …, d generate 𝔪_{0}/𝔪_{0}^{2}.

To prove the smoothness of P it suffices to show that (A ⊗_{k} k̄)_{𝔪_{0}} is integral, it has dimension d and that x_{i} with i = 1, …, d form a basis 𝔪_{0}/𝔪_{0}^{2}.

To prove these facts we embed (A ⊗_{k} k̄)_{𝔪_{0}} into its 𝔪_{0}- adic completion ˆ (A ⊗_{k} k̄)_{𝔪_{0}}. In this completion our defining equations just say that the x_{i} with i = d + 1, …, n can be expressed as power series in the x_{i} with i = 1, …, d and hence it is clear that ˆ (A ⊗_{k} k̄)_{𝔪_{0}} is the power series ring in the variables x_{i} with i = 1, …, d. This implies integrality of (A ⊗_{k} k̄)_{𝔪_{0}} and it shows that dim(𝔪_{0}/𝔪_{0}^{2}) = d.

As in our previous considerations we can can construct an injective homomorphism from the localized polynomial ring

k̄[X_{1}, …, X_{d}]_{𝔐_{0}^{0}} → (A ⊗_{k} k̄)_{𝔪_{0}}

Then it is clear (see 2.5.3.) that dim((A ⊗_{k} k̄)_{𝔪_{0}}) ≥ d.Now it follows from our theorem above that dim(A_{𝔪_{0}}) = d. This proves the first half of the theorem.

To prove the converse we observe that we have seen that the ideal I_{𝔐_{P}} ⊗ k̄ can be generated by G_{1}, …, G_{r} ∈ k̄[X_{1}, …, X_{n}] such that these have non vanishing Jacobi matrix at P. Let F_{1}, …, F_{t} ∈ I be a system of generators. Then we can find an H ∈ k̄[X_{1}, …, X_{n}] with H(P) ≠ 0 such that we can write

G_{i}H = ∑_{j} L_{ij}F_{j}

where L_{ij} ∈ k̄[X_{1}, …, X_{n}]. Taking partial derivatives and evaluating at P yields a system of equations

H(P)\frac{∂G_{i}}{∂X_{ν}} (P) = ∑_{j} L_{ij}(P)\frac{∂F_{j}}{∂X_{ν}} (P)

This makes it clear that the Jacobi matrix built out of the F_{j} evaluated at P has rank r. But this means that the Jacobi matrix (\frac{∂F_{j}}{∂X_{ν}}) has rank r if we send it to the residue field A/𝔪_{P}. Taking a submatrix having the right number of columns gives us a subset of the set of generators say F_{1}, …, F_{r} which satisfies the Jacobi criterion. This subset will necessarily generate I_{𝔐_{P}} .

3.3.1. The Jacobi criterion implies that the set of smooth geometric points of a scheme X/k of finite type is always open. To see this we consider the case of an affine k-algebra A = k[X_{1}, …, X_{n}]/I. We may assume that it is irreducible because smooth point lie on exactly one irreducible component. We pick generators F_{1}, F_{2}, …, F_{t} of our ideal and consider the open set where the Jacobi matrix has rank equal to n − dim(A). This set can be empty. If for instance A = k[X]/(X^{2}) then the only geometric point is x = 0 the dimension is zero but the number d = 1.

VII.3.4. Theorem: Generic smoothness

Let A/k be a finitely generated k-algebra, assume that A/k is absolutely reduced. Then we can find a non empty open subset U ⊂ Spec(A) such that the morphism π : U → Spec(k) is smooth.

Proof: Since localization does not destroy reducedness we may assume that A is integral. We also may assume that k̄ = k. We write A = k[x_{1}, x_{2}, …, x_{n}] = k[X_{1}, X_{2}, … X_{n}]/I and let F_{1}, F_{2}, … F_{t} be a set of generators of the ideal I. Let \frac{∂F_{i}}{∂x_{j}} be the image of \frac{∂F_{i}}{∂X_{j}} ∈ k[X_{1}, X_{2}, …, X_{n}] modulo the ideal I. Let r be the rank of the Jacobi matrix (\frac{∂F_{i}}{∂x_{j}}) considered as a matrix in the field of fractions of A and assume that

(\frac{∂F_{i}}{∂x_{j}})_{i=1,...,r;j=n−r+1,...n}

has non zero determinant. Then this determinant is a unit in a suitable localization O_{X}(U) = Ã(U). Let d = n − r. I claim that the subalgebra k[x_{1}, …, x_{d}] is a polynomial ring in d variables. If not the the intersection k[X_{1}, …, X_{d}] ∩ I ≠ 0. For all F in k[X_{1}, …, X_{d}] ∩ I we have \frac{∂F}{∂x_{j}} = 0 for all j = 1, …, d because otherwise the rank of the Jacobian would be greater than r. This is equivalent to the assertion that for all F ∈ k[X_{1}, …, X_{d}] ∩ I we have \frac{∂F}{∂X_{j}} ∈ I for all j = 1, …, d. Let us consider a non zero element F ∈ k[X_{1}, …, X_{d}] ∩ I with lowest total degree, i.e. the sum of the exponents in the highest monomial occuring in F is minimal. Then \frac{∂F}{∂X_{j}} ∈ I for all j = 1, … d and these polynomials have a strictly smaller total degree. Hence they are zero and F must be constant or of the form

F (X_{1}, …, X_{d}) = ∑ a_{ν}X^{pν}

where p is the characteristic of k and ν = (ν_{1}, …, ν_{d}) is a multiindex. Of course F cannot be constant. But then we find nilpotent elements in k̄[x_{1}, …, x_{d}] because we can write

F (X_{1}, …, X_{d}) = ∑ a_{ν}X^{pν} = (∑ a_{ν}^{1/p}X^{ν})^{p}.

The element (∑ a_{ν}^{1/p}X^{ν})^{p} ∉ k[X_{1}, …, X_{d}] ∩ I because it has smaller total degree than F. Its image in A is a non zero nilpotent element. This gives a contradiction to our assumption that A is absolutely reduced.

But once we know that k[x_{1}, …, x_{d}] is a polynomial ring we see that the dimension of A is greater or equal to d (2.5.3.) and the Jacobi criterion yields that the open subscheme where the above determinant is not zero must be smooth.

I want to discuss a consequence of the smoothness of a geometric point for the original algebra A and the induced maximal ideal 𝔪_{P}. Let us assume we have an ideal I ∈ k[X_{1}, …, X_{n}] which is generated by F_{1}, …, F_{r}. Let X/k be the subscheme defined by the ideal, i.e. X = Spec(k[X_{1}, …, X_{n}]/I) = Spec(A). We assume that these generators satisfy the Jacobi criterion at a point P ∈ X(k̄), hence we know that this point is smooth. Let 𝔪_{P} be the maximal ideal induced by P in our algebra A, we know that A_{𝔪_{P}} is integral. Again we put d = n − r. Now we assume for the coordinates X_{d+1}, …, X_{n} that

det((\frac{∂F_{j}}{∂X_{μ}} (P))_{j=1...,r,μ=d+1,...,n}) ≠ 0

Let us denote by x_{ν} the image of the X_{ν} in A. The we consider the subalgebra B ⊂ A generated by the elements x_{1}, …, x_{d}. It will become clear in a minute that this algebra is indeed the polynomial algebra in these variables. We have the diagram

k[x_{1}, …, x_{d}] →  A
↓  ↓
k̄[x_{1}, …, x_{d}] → A ⊗_{k} k̄

The maximal ideal 𝔪_{P} induces a maximal ideal (geometric point) 𝔫_{P} in k[x_{1}, …, x_{d}]. Let 𝔫̄_{P} (resp. 𝔪̄_{P} be the maximal ideals induced by P in k̄[x_{1}, …, x_{d}] (resp. A ⊗_{k} k̄) We have seen in the beginning of this section that after passing to the completions we get an isomorphism

ˆk̄[x_{1}, …, x_{d}]𝔫̄_{P} →^{∼}ˆ(A ⊗_{k} k̄)𝔪̄_{P} .

and it follows from this that the subalgebra B ⊂ A is the polynomial algebra.

We have the inclusion of local rings

B_{𝔫_{P}} ↪ A_{𝔪_{P}}

We consider the residue field B/𝔫_{P} = k(𝔫_{P}).

I claim:

(i) The tensor product

A_{𝔪_{P}} ⊗_{B} k(𝔫_{P}) = A_{𝔪_{P}} /A_{𝔪_{P}} · 𝔫_{P}

is a field, it is a finite separable extension of k(𝔫_{P})

(ii) Moreover we have A_{𝔪_{P}} · 𝔫_{P} = 𝔪_{P} and the residue field k(𝔪_{P}) = A_{𝔪_{P}} ⊗_{B} k(𝔫_{P}) Proof: The second assertion follows from the first. To prove the first assertion we put

K = k(𝔫_{P}) and Ā = A_{𝔪_{P}} ⊗_{B} k(𝔫_{P}). We have that Ā = K[X̄_{d+1}, …, X̄_{n}]/(F̄_{1}, …, F̄_{r}) where the bar means taking the residue classes. Our point P induces a maximal ideal 𝔪̄_{P} ⊂ Ā. Of course we have that the generators F̄_{i}, i = 1, …, r satisfy the Jacobi criterion at P and this means

det((\frac{∂F̄_{j}}{∂X̄_{μ}} (P)_{j=1...,r,μ=d+1,...,n}) ≠ 0.

Then it follows that all points P_{i} which lie above P are smooth and the scheme Ā ⊗_{K} K̄ is of dimension zero and reduced. But this means that Ā is a separable extension of K.

In our situation above we will call the elements x_{1}, …, x_{d} a system of local parameters at the point P. We can interprete the inclusion B ⊂ A as the morphism from X/k to 𝔸^{r}/k which is induced by the projection of the ambient affine space 𝔸^{n}/k to the first r coordinates. If we extend the ground field to k̄ then this projection becomes an isomorphism between the infinitesimal neighborhoods of P in X × k̄ and its image in 𝔸^{r}/k̄. This is the algebraic version of the theorem of implicit functions. If we want to formulate what happens over k itself then this becomes a little bit less intuitive, we have to be aware that we may have non trivial residue fields which even may be non separable.

Our notion of a system of local parameters is given in terms of an embedding of our scheme X/k into an affine space and with the help of the Jacobi criterion. In the next section we will give a much more elegant formulation using differentials.

VII.3.4. Relative differentials: For any morphism

π : X → Y

we will define a quasicoherent sheaf Ω_{X/Y}^{1} on X which called the sheaf of relative differentials. Since it is a quasicoherent sheaf it is enough to define it in the case where X = Spec(B), Y = Spec(A). Let ϕ : A → B be the homomorphism of rings corresponding to π.

We consider the fibered product of X/Y by itself

\begin{matrix}B ⊗_{A} B & X ×_{Y} X \\ ↑ & ↓ \\ A & Y \end{matrix}

and we have the diagonal X → X ×_{Y} X defined by (Id, Id). The diagonal corresponds to the multiplication

B ⊗_{A} B ⟶^{m} B

and is defined by the ideal I which is the kernel von m, i.e. we have an exact sequence

0 → I → B ⊗_{A} B → B → 0.

Lemma 3.4.1. : The ideal I is generated as a B⊗_{A}B-module by the elements f⊗1−1⊗f. The ideal I/I^{2} is a B-module which is generated by the elements dg = f ⊗1−1⊗f mod I^{2}

Proof: If Σf_{ν} ⊗ g_{ν} ∈ I, i.e. Σf_{ν}g_{ν} = 0 then

Σf_{ν} ⊗ g_{ν} = Σ(f_{ν} ⊗ 1)(1 ⊗ g_{ν}) = Σ(f_{ν} ⊗ 1 − 1 ⊗ f_{ν})(1 ⊗ g_{ν}).

The B-module structure on I/I^{2} is induced by the B ⊗_{A} B module structure of I.

The B-module I/I^{2} is denoted by Ω_{B/A}^{1} and is called the module of relative differentials. We have the product rule

dfg = gdf + fdg

and

da = 0 for a ∈ A

which follows from

fg ⊗1−1⊗gf = fg ⊗1−f ⊗g +f ⊗g −1⊗fg = (f ⊗1)(g ⊗1−1⊗g)+(g ⊗g)(f ⊗1−1⊗f) and

a ⊗ 1 = 1 ⊗ a.

We collect some facts where proofs can be found in the book by Matsumura but since they are all not difficult to prove, the reader could try to find them himself.

a) The B-module Ω_{B/A}^{1} is a universal module of A-differentials: Whenever we have a B-module M and an A-linear map d_{1} : B → M such that d_{1}(fg) = fd_{1}g + gd_{1}f then we get a commutative diagram

Ω_{B/A}^{1} ⟶^{φ} M
d ↖  ↗ d_{1}
B

where φ is unique and B-linear. It is clear that ϕ(f ⊗ 1 − 1 ⊗ f) = d_{1}(f) but it is not so clear why it is well defined. This can be checked by using d) below and reducing to the case of a polynomial ring.

b) We start from our algebra homomorphism A → B and we pick an element f ∈ B. Then we have A → B → B_{f} and hence we know what Ω_{B_{f} /A}^{1} is. It is easy to see that we

have a canonical isomorphism Ω_{B_{f} /A}^{1}→^{∼}(Ω_{B/A}^{1})_{f} which in principle comes from the rule

d\frac{1}{f} = −\frac{df}{f^{2}}

Therefore we can view Ω_{B/A}^{1} as a quasicoherent sheaf on Spec(B) and then we can define Ω_{X/Y}^{1} for an arbitrary morphism of schemes π : X → Y.

c) The module of relative differentials is compatible with base change. If we have a diagram

B → B′ = B ⊗_{A} A
↑  ↑
A →  A′

then we have a canonical isomorphism

Ω_{B/A}^{1} ⊗_{B} B′ ⟶^{∼} Ω_{B′/A}^{1}.

d) Finally we consider the situation that our algebra B is a quotient of an A-algebra C by an ideal J, i.e. we have

\begin{aligned}0 → J & → C ⟶^{π} & B → 0 \\ & ↑ & ↗ \\ & A \end{aligned} .

It is clear that we have a surjective C-module homomorphism

Ω_{C/A}^{1} → Ω_{B/A}^{1}

which sends gdh ∈ Ω_{C/A}^{1} to π(g)dπ(h). On the other hand we can map J to Ω_{C/A}^{1} by sending an f ∈ Y to df. This yields a sequence of B-modules

J/J^{2} → Ω_{C/A}^{1} ⊗_{C} B → Ω_{B/A} → 0

and I claim that this sequence is exact. We have to show that the elements in Ω_{C/A}^{1} ⊗_{C} B which go to zero come from J/J^{2}. To see this we start with the observation that

Ω_{C/A}^{1} ⊗_{C} B = Ω_{C/A}^{1}/JΩ_{C/A}^{1}.

If we look at

C ⊗_{A} C → B ⊗_{A} B

then we have seen that the kernel is the ideal J ⊗_{A} C + C ⊗_{A} J (See V.2.3.1.??). On the other hand we have the kernels of the multiplication maps

\begin{aligned}I_{C} & = ker(C ⊗_{A} C → C) \\ I_{B} & = ker(B ⊗_{A} B → B) \end{aligned} .

We consider elements in I_{C} which go to zero in I_{B}/I_{B}^{2} = Ω^{1}f_{B/A}. But it is so clear that I_{C}^{2} → I_{B}^{2} is surjective hence we can represent an element in Ω_{C/A}^{1} which goes to zero in Ω_{B/A}^{1} by an element

h ∈ I_{C} ∩ (J ⊗_{A} C + C ⊗_{A} J).

Let us write this element

∑ j_{ν} ⊗ c_{ν} + ∑ c′_{μ} ⊗ j′_{μ}

where the j_{ν}, j′μ ∈ J. We have ∑ j_{ν}c_{ν} + ∑ j′_{μ}c′_{μ} = 0. The sum is equal to

\begin{matrix}∑ j_{ν} ⊗ c_{ν} + ∑ j′_{μ} ⊗ c′_{μ} & + ∑(c′_{μ} ⊗ j′_{μ} − j′_{μ} ⊗ c′_{μ}) = \\ ∑_{ν}(1 ⊗ c_{ν})(j_{ν} ⊗ 1 − 1 ⊗ j_{ν}) & + ∑_{μ}(1 ⊗ c′_{μ})(j′_{μ} ⊗ 1 − 1 ⊗ j′_{μ}) \\ + ∑_{μ}(1 ⊗ j′_{μ})(c′_{μ} ⊗ 1 − 1 ⊗ c′_{μ}) & − ∑_{μ}(1 ⊗ c′_{μ})(j′_{μ} ⊗ 1 − 1 ⊗ j′_{μ}) \end{matrix} .

This yields an element in Ω_{C/A}^{1} namely

∑_{ν} c_{ν}dj_{v} + ∑_{μ} c′_{μ}dj′_{μ} + ∑ j′_{μ}dc′_{μ} − ∑ c′_{μ}dj′_{μ}

which is clearly in the image of J mod JΩ_{C/A}^{1}.

Example:

1) If B = A[X_{1} … X_{n}] then

B ⊗_{A} B = A[X_{1} … X_{n}, Y_{1}, … Y_{n}]

where X_{i} = X_{i} ⊗ 1 and Y_{i} = 1 ⊗ X_{i}. The ideal I is generated by the elements X_{i} − Y_{i} = X_{i} ⊗ 1 − 1 ⊗ X_{i} as a B ⊗_{A} B-module and I/I^{2} is the free B-module generated by dX_{i} = X_{i} − Y_{i} mod I^{2}.

(2) Assume that

B = A[X_{1}, …, X_{n}]/ < F_{1}, …, F_{r} >

where F_{i} = ∑_{ν} a_{iν}X^{ν} ∈ A[X_{1}, …, X_{n}]. Assume we have an f ∈ B such that for all maximal ideals 𝔪 ∈ Spec(B_{f}) the rank of

(\frac{∂F_{i}}{∂X_{j}})_{i=1,...,r,j=n−r+1,...,n} mod 𝔪

is maximal, i.e. equal to r. Then Ω_{B_{f} /A}^{1} is locally free of rank d = n−r. If at our maximal ideal 𝔪 the subdeterminant

det(\frac{∂F_{i}}{∂X_{j}})_{i=1,...,r,j=n+1−r,...,r−n} ≢ 0 mod 𝔪

then the differentials dx_{1}, …, dx_{d} (where the x_{i} are the images of the X_{i} in B) are free generators of Ω_{B/A,𝔪}^{1}.

This is clear from our consideration above.

3) We look at our examples of non-smooth points and we will see how the fact that they are not smooth is reflected by the sheaf of relative differentials.

a) If

A = k[X, Y]/(XY) = k[x, y]

then the set of geometric points is the union of the X and the Y -axis in 𝔸^{2}. The origin is not smooth. The A-module of differentials is generated by dx and dy and we have the relation

xdy + ydx = 0.

In this example the origin is not smooth because already the condition that the local ring should be integral is violated. But we could easily modify the example by looking at

A_{1} = k[X, Y]/(XY + X^{5} + Y^{7}) = k[x, y].

Now the local ring at (O, O) is integral and we have

(y + 5x^{4})dx + (x + 7y^{6})dy = 0.

The dimension of A_{1} is one but the module of differentials is not free of rank one. We will see later that this is another criterion for that the origin is not smooth.

This example is interesting for a different reason. If we look at the completions of our two rings then they become isomorphic

Â ≃ Â_{1}.

We leave this as an exercise to the reader, one has to show that one can construct power series

\begin{aligned}X̃ & = X + P (X, Y) \\ Ỹ & = Y + Q(X, Y) \end{aligned}

such that the relation XY + X^{5} + Y^{7} = 0 becomes X̃Ỹ = 0.

The reader should also prove that for k = ℂ and a small number ϵ > 0 the complex space

{(x, y) ∈ ℂ^{2}|xy + x^{5} + y^{7} = 0} ⋂ B(0, ϵ)

where B(0, ϵ) = {(x, y) ∈ ℂ^{2}||x|^{2} + |y|^{2} < ϵ} is indeed the union of two discs intersecting transversally at the origin.

Hence we see that depending on the microscope which we use to look at our singular point, the local ring may be integral (Zariski topology) or non-integral (analytic topology). The two branches of our complex space above in B(0, ϵ) will come together very far out again.

Such a singular point is called an ordinary double point.

b)

A = k[X, Y]/(X^{2} − Y^{3}).

Again we see that Ω_{A/k}^{1} at the origin is generated by dx, dy with the relation 2xdx − 3y^{2}dy = 0 the module of relative differentials is not free of rank 1.

In our three examples the scheme is smooth outside the origin and this is also the open set where the sheaf of differentials is free of rank one. We will see that this indeed a very general fact which will allow us define smooth morphisms in a very general context.

If we are in the situation of example (2) and if we consider a point

Spec(k(y)) ↪ J, i.e. Ψ : A → k(y),

then we can consider the base change

B ⟶ B ⊗_{A} k(y)
↑  ↑
A ⟶  k(y)
.

This scheme is called the fibre over the point y this is now a scheme over a field. It is clear that this scheme

Spec(B ⊗_{A} k(y)) → Spec(k(y))

is smooth of dimension d. Moreover it also follows that if we have given a system of local parameters at a closed point as above - say x_{1}, …, x_{d} - then the differentials dx_{1}, …, dx_{d} form a basis of Ω_{B⊗_{A}k(y))/ Spec(k(y))}^{1} at this point.

Now we have enough insight to formulate our general definition.

Let us assume we have an arbitrary ring A and a finitely generated A- algebra

B = A[x_{1}, …, x_{n}] = A[X_{1}, …, X_{n}]/I

Definition and Theorem VII.3.4.1f :

The morphism

Spec(B) = X ; π ↓ ; Spec(A) = S

is called smooth if

a) it is flat

b) For all point s ∈ S the fibre X_{s} → Spec(k(s)) is smooth.

The condition b) is equivalent to

b1) The fibres X_{s} are locally integral at each point and at any point x ∈ X_{s} the sheaf Ω_{X/S}^{1} is locally free of rank dim(O_{X_{s},x}).

Before I come to the proof I want to make two comments:

1) We do not require that the fibres are integral since we never did that before. Hence it can happen that the fibres are disjoint unions of irreducible ones which then may even have different dimensions

2) Again it is clear that the notion of smoothness can be checked locally at the points. Hence it is clear that our definition extends to an arbitrary morphism π : X → S where X is of finite type over S.

To prove the theorem we start with the case that S = Spec(k) is a field. In this case the assumption a) is automatically fulfilled. We have seen in the discussion of example 2) that b) ⇒ b1).

If (b1) is satisfied we write our algebra B = k[x_{1}, …, x_{n}] as a quotient

k[x_{1}, …, x_{n}]/I ⟶^{∼} B.

Let us assume that we have a geometric point P which induces the maximal ideals m_{p} (resp. 𝔐_{p}) in B (resp. C = k[X_{1}, …, X_{n}]). The module Ω_{B/k}^{1} is locally free at P and it is of course generated by dx_{1}, …, dX_{n}. We can choose a basis locally at P let us assume it is of the form

dx_{1}, …, dx_{d} where d = dim B_{m_{p}}.

As usual we put r = n − d. It is clear (see (d) above) that we can find F_{1}, …, F_{r} ∈ I such that dF_{1}, …, dF_{r} together with dX_{1}, …, dX_{r} provide a basis of Ω_{C/k}^{1} at P. But this implies that

rank (\frac{∂F_{i}}{∂X_{μ}}) mod 𝔐_{p} = r.

These F_{1}, …, F_{r} generate an ideal J and we get

C_{𝔐_{p}} = k[X_{1}, …, X_{n}]_{𝔐_{p}} ⟶ B_{m_{p}}
↘  ↗ Ψ
B̃m̃_{p}

where B̃m̃_{p} is the quotient by the ideal J. We know that B̃m̃_{p} is smooth at P by the Jacobi criterion and dim B̃m̃_{p} = r. But then it is clear that Ψ must be an isomorphism and it follows that B is smooth at P.

Now we pass to the general case. From what we know it is clear that b1) ⇒ b) because b) can be checked fibre by fibre.

Now we assume b), we pick a point s ∈ S and a closed point P ∈ X_{s}. We want to show that Ω_{B/A}^{1} is locally free and its rank is dim(O_{X_{s},P}).

We may assume that S = Spec(A) is local and s is closed. Then we can write

A[X_{1}, …, X_{n}]/I = B = A[x_{1}, …, x_{n}].

If we pass to the special fibre we get an exact sequence

0 → I_{0} → k(s)[X_{1}, …, X_{n}] → k(s)[x_{1}, …, x_{n}] = B ⊗_{A} k(y).

Now B is a flat A-algebra (we localized in between but this preserves flatness) and hence we get

I ⊗ k(s) ⟶^{∼} I_{0}.

Now we know that we can find

F_{1}^{(0)}, …, F_{r}^{(0)} ∈ I_{0}

which generate I_{0} and which satisfy the Jacobi criterion at P. These elements can be lifted to elements F_{1}, …, F_{r} in I and by Nakayama’s lemma they generate I locally at P. Hence we see that locally at P we have

B ≃ A[X_{1}, …, X_{n}]/(F_{1}, …, F_{r})

and we have seen in example 2) that this implies that

Ω_{B/A}^{1} is locally free of rank d at P.

We are now able to give a very intrinsic definition of a system of local parameters.

If we have a smooth morphism

A → B

and a point P then we say that

x_{1}, …, x_{d} ∈ B_{f} with f(P) ≠ 0

is a system of local parameters at P, if the dx_{1}, …, dx_{d} form a basis of Ω_{B_{f} /A}^{1}. We get a diagram

A  ⟶  A[X_{1}, …, X_{d}]
↘  ↙
B_{f}

where we send the X_{i} t to the x_{i}. We have seen in our previous proof that we can write

B_{f} = A[X_{1}, …, X_{d}, X_{d+1}, …, X_{n}]/I

where I = (F_{1}, …, F_{r}) and where we have

(\frac{∂F_{i}}{∂X_{μ}})(P)_{i=1,r,μ=d+1,...,n} ≠ 0.

(Actually if we modify f we can assume that this is true at all points in Spec(B_{f})).

Now we can show that

A[X_{1}, …, X_{r}] ↪ B_{f}

is flat, smooth and Ω_{B_{f} /A[X_{1},...,X_{r}]}^{1} is zero. This implies that the fibres are reduced zero dimensional.

This is an example of an etale morphism. A morphism A → B is called etale if it is smooth, if finite type and if the sheaf of relative differentials is zero.

If we have a closed point 𝔫 ∈ Spec(B) then k(𝔫) → B ⊗_{B} k(𝔫) is still smooth and the connected components are zero dimensional. Hence it follows that k(𝔫) → B ⊗_{B} k(𝔫) is a finite separable algebra, i.e a finite sum of separable field extensions of k(𝔫).

Chapter VIII

Projective schemes

VIII.1 Geometric constructions

1.1 The projective space ℙ_{A}^{n}

Let A be a commutative ring with 1_{A}. We consider the following n + 1 affine schemes

U_{i} = Spec(A[T_{i0} … T_{in}]/(T_{ii} − 1)).

The ring of regular functions of U_{i} is the polynomial ring in n + 1 independent variables variables T_{i0}, T_{i1} … T_{i,i−1}, T_{i,i}, T_{i,i+1} … T_{in} and we divide by the relation T_{ii} = 1. This means that all the U_{i} are copies of 𝔸_{Spec(A)}^{n}.

We denote by t_{ij} the images of the T_{ij} in the quotient ring

A[t_{i0} … t_{in}],

this means that we consider for a given i the t_{i0}, t_{i1} … t_{i,i−1}, t_{i,i+1} … t_{in} as independent and t_{ii} = 1.

For any index j we define the open subscheme

U_{ij} = Spec(A[T_{i0} … T_{in}]/(T_{ii} − 1))_{T_{ij}} = Spec(A[T_{i0} … T_{in}, T_{ij}^{−1}]/(T_{ii} − 1)) .

We have an isomorphism

f_{ij} : U_{ij} → U_{ji}

given on the level of rings by

\begin{aligned}ϕ_{ij} : A[T_{j0} … T_{jn}, t_{ji}^{−1}]/(T_{jj} − 1) & → A[T_{i0} … T_{in}, t_{ij}^{−1}]/(T_{ii} − 1) \\ ϕ_{ij}(T_{jν}) & = T_{iν} · t_{ij}^{−1} \end{aligned} .

We see that ϕ_{ij}(t_{jj}) = ϕ_{ij}(1) = t_{ij} · t_{ij}^{−1} = 1 and ϕ_{ij}(t_{ji}^{−1}) = t_{ii} · t_{ij} = t_{ij} as it must be. We allow i = j and then ϕ_{ii} is the identity on U_{i}.

Given three indices i, j, k we get a commutative diagram

U_{ij} ⊃ U_{ij} ∩ U_{ik} ⟶^{ϕ_{ij}} U_{ji} ∩ U_{jk} ⊂ U_{ji}
↘ ϕ_{ik}  ↙ ϕ_{jk}

U_{ki} ∩ U_{kj} of isomorphisms. This allows us to define an equivalence relation on the space

⨆_{i=0,...,n} U_{i}

namely u_{i} ∼ u_{j} if and only if u_{i} ∈ U_{j}, u_{j} ∈ U_{ji} and ϕ_{ij}(u_{i}) = u_{j}. We define the space

⨆ U_{i}/ ∼= ℙ_{A}^{n}.

The projection map

π : ⨆ U_{i} → ℙ_{A}^{n}

provides a homeomorphism from U_{i} to an open subset in ℙ_{A}^{n}, we identify U_{i} with this open subset, i.e. we consider U_{i} as an open subset in ℙ_{A}^{n}.

Now we define a sheaf O_{ℙ_{A}^{n}} on the space ℙ_{A}^{n} simply by putting

O_{ℙ_{A}^{n}}|U_{i} = A[T̃_{i0}, …, T_{in}]/(T_{ii} = 1)

and we use the ϕ_{ij} to glue O(U_{i})|U_{i} ∩ U_{j} with O(U_{j})|U_{i} ∩ U_{j}.

After constructing ℙ_{A}^{n} we can think of it in the following way: The scheme ℙ_{A}^{n} admits an open covering

ℙ_{A}^{n} = ∪_{i=0...n}U_{i}

such the U_{i} are affine spaces. The ring of regular functions on U_{i} is

O_{ℙ_{A}^{n}}(U_{i}) = A[t_{i0} … t_{in}].

where t_{ii} = 1 and t_{i0}, …, t_{i,i−1}, t_{i,i+1} …, t_{in} are independent.

The intersection U_{i} ∩ U_{j} is affine and

O_{ℙ_{A}^{n}}(U_{i} ∩ U_{j}) = A[t_{i0} … t_{in}, t_{ij}^{−1}] = A[t_{j0} … t_{jn}, t_{ji}^{−1}]

and

t_{iν} = t_{jν} · t_{ji}^{−1}, t_{jμ} = t_{iν} · t_{ij}^{−1}.

Theorem 1.1.1: The regular functions on ℙ_{A}^{n} are the constants, this can be stated briefly as

O_{ℙ_{A}^{n}}(ℙ_{A}^{n}) = A.

Proof: Let f ∈ O_{ℙ_{A}^{n}}(ℙ_{A}^{n}). We restrict f to the open sets U_{i} and get a polynomial

f|U_{i} = P_{i}(t_{i0} … t_{in}) ∈ A[t_{i0} … t_{in}]

For any pair of indices we have P_{i}|U_{i} ∩ U_{j} = P_{j}|U_{i} ∩ U_{j} and this means that ϕ_{ij}(P_{j}) = P_{i} and hence

P_{i}(t_{i0} … t_{in}) = P_{j}(t_{i0} · t_{ij}^{−1}, …, t_{in} · t_{ij}^{−1}).

Now we write P_{j} as polynomial

∑ a_{ν_{0}...ν_{n}}t_{j_{0}}^{ν_{0}} … t_{j,j−1}^{ν_{j−1}}t_{j,j+1}^{ν_{j+1}} … t_{jn}^{ν_{n}}.

We get

P_{i}(t_{i0} … t_{in}) = ∑ a_{ν_{0}...ν_{n}} (\frac{t_{i0}}{t_{ij}} )^{ν_{0}} … (\frac{t_{in}}{t_{ij}} )^{ν_{n}}

this is a polynomial in t_{i0} … t_{i,i−1}, t_{i,i+1} … t_{in}.

But if i ≠ j then this is only possible if all exponents ν_{0} = … ν_{i−1}, ν_{i+1}, … ν_{n} = 0 in other words, P_{i} has to be constant.

This proves the theorem.

1.1.2. We have a different way of looking at this argument.

We consider n + 1 new variables X_{0}, X_{1}, …, X_{n+1} and we endow the polynomial ring A[X_{0}, X_{1}, …, X_{n}] with its standard graduation: To any monomial X_{0}^{r_{0}}X_{1}^{r_{1}} … X_{n}^{r_{n}} we attach the degree d = ∑ r_{i}, then we define A[X_{0}, X_{1}, …, X_{n}](d) to be the A-module generated by the monomials of degree d. This is the module of homogeneous polynomials ( or forms ) of degree d. We have the direct sum decomposition

⨁_{d=0}^{∞} A[X_{0}, X_{1}, …, X_{n}](d) = A[X_{0}, X_{1}, …, X_{n}].

We can localize this ring by inverting any of the variables and in any of these localizations we can consider the subring of elements of degree zero, i.e. we consider

A[X_{0}, X_{1}, …, X_{n}]_{X_{i}}^{(0)} = \{\frac{F}{X_{i}^{d}} | F ∈ A[X_{0}, X_{1}, …, X_{n}](d)\}.

It is clear that the homomorphism sending t_{ij} ↦ X_{j}/X_{i} induces an isomorphism

A[t_{i0} … t_{in}]→^{∼}A[X_{0}, X_{1}, …, X_{n}]_{X_{i}}^{(0)}

and the ϕ_{ij} correspond to the obvious inclusion maps. We can always think that we have the relation t_{ij} = X_{j}/X_{i}.

Now our theorem above says that

⋂_{i=0...n} A[X_{0}, X_{1}, …, X_{n}]_{X_{i}}^{(0)} = A

1.1.3. Points: It is of course clear what an A-valued point on ℙ_{A}^{n} is. Any such A-valued point x ∈ ℙ_{A}^{n}(A) is represented by an element

x_{i} = (a_{i0}, …, 1, …, a_{in})

where a_{iv} ∈ A. Such a point lies also in U_{j} if a_{ij} is a unit in A and then

x_{j} = (\frac{a_{i0}}{a_{ij}} , …, 1, …, \frac{1}{a_{ij}} , … \frac{a_{in}}{a_{ij}} )

represents the same point.

Of course we can also say that we can represent a point by an element in

𝔸_{∗}^{n+1} = {(a_{0}, …, a_{n}) | at least one entry is a unit in A},

and we divide by the equivalence relation

(a_{0}, …, a_{n}) ∼ (a_{0}b, …, a_{n}b) with b ∈ A^{∗}.

1.2. Closed subschemes: Now we know of course what a closed subscheme of ℙ_{A}^{n} is (See V 2.1.2.). We simply pick a quasicoherent sheaf of ideals I in O_{ℙ_{A}^{n}}, then ((V (ℑ), O_{ℙ_{A}^{n}}) is a closed subscheme of ℙ_{A}^{n}. We get a commutative diagram

(V (ℑ), O_{ℙ_{A}^{n}}/ℑ) ↪  ℙ_{A}^{n}
↘  ↓
Spec(A)

hence this subscheme is automatically a scheme over Spec(A).

Such a quasicoherent sheaf of ideals I is simply a collection of ideals I_{i} ⊂ O_{ℙ^{n}}(U_{i}) such that I_{i} and I_{j} generate the same ideal in O_{ℙ^{n}}(U_{i} ∩ U_{j}).

We call an ideal Ĩ ⊂ A[X_{0}, X_{1}, …, X_{n}] homogeneous if for any F ∈ Ĩ its homogeneous components F_{d} are also in the ideal.

We can get quasicoherent sheaves of ideals from homogeneous ideals Ĩin A[X_{0}, X_{1}, …, X_{n}]. This is not difficult to see. To any homogeneous polynomial F ∈ A[X_{0}, X_{1}, …, X_{n}](d) ∈ Ĩ of degree d we attach a collection of elements {f_{i} ∈ O_{ℙ^{n}}(U_{i})}_{i=0,1...n} simply by substituting t_{i,ν} for X_{ν} into the polynomial. These f_{i} generate an ideal I_{i} in O_{ℙ^{n}}(U_{i}). It is clear that the restrictions of f_{i}, f_{j} to U_{i} ∩ U_{j} satisfy t_{ij}^{−d}f_{i}|U_{i} ∩ U_{j} = f_{j}|U_{i} ∩ U_{j} and since t_{ij} is a unit in O_{ℙ^{n}}(U_{i} ∩ U_{j}) it follows that I_{i} and I_{j} generate the same ideal in O_{ℙ^{n}}(U_{i} ∩ U_{j}). If F runs over the homogeneous elements of Ĩ these F give us a quasicoherent sheaf of ideals I.

As in the affine case we can attach the closed subset V (Ĩ)( resp. V (I)) which is the set of common zeroes of Ĩ (resp. I).

If in turn Z ⊂ ℙ_{A}^{n} is a closed subset, we may consider the ideal Ĩ generated by homogeneous polynomials which vanish on Z. It is clear that V (Ĩ) = Z (???).

Lemma 1.2.1.For a homogeneous ideal Ĩ ⊂ A[X_{0}, X_{1}, …, X_{n}] we have V (Ĩ) = ∅ if and only if Ĩ(d) = A[X_{0}, X_{1}, …, X_{n}](d) for d sufficiently large.

Proof: Of course we know that V (Ĩ) = ∅ if and only if V (Ĩ) ∩ U_{I} = ∅ for i = 0 … n. This is equivalent to the assertion: For all i we can find g_{ν} ∈ O_{ℙ^{n}}(U_{i}) and f_{ν} ∈ I_{i} such that

∑ g_{ν}f_{ν} = 1

(See ???). Now we remember that we should think of t_{i,j} as being X_{j}/X_{i} then we see that we can multiply this relation by a power of X_{i} and it yields a relation

∑ G_{ν}F_{ν} = X_{i}^{m_{i}}

where the G_{ν}, F_{ν} are homogeneous and F_{ν} ∈ I_{i}. Hence we conclude that V (Ĩ) = ∅ implies that for all i a suitable power of X_{i} is in the ideal which proves one direction of the Lemma. The other direction is obvious.

For our considerations above it was not necessary to assume that the base scheme Spec(A) is affine. All the constructions work over an arbitrary base scheme S.

We could have started with the ring A = ℤ and construct the scheme ℙ_{ℤ}^{n}. If S is any scheme then we have the absolute morphism S → Spec(ℤ) (???) and we could define ℙ_{S}^{n} = ℙ_{ℤ}^{n} × S.

1.2.2. Points again: Of course it is again very easy to describe A-valued points of closed subscheme X ⊂ ℙ_{A}^{n} if this subscheme is given by a homogeneous ideal I ⊂ A[X_{0}, …, X_{n}]. Then the B-valued points of X are given by

X(B) = {(b_{0}, …, b_{n}) ∈ B_{∗}^{n+1} | f(b_{0}, …, b_{n}) = 0 for all homogeneous f ∈ I}/B^{∗}.

1.3. Projective morphisms and projective schemes: We call a morphism X → S projective if we can find a commutative diagram

i : X → ℙ_{S}^{n}
↘ ↓
S

where i is a closed embedding, i.e. an isomorphism to a closed subscheme in ℙ^{n}. We also say that X/S is a projective scheme over S.

Now I want to show that the fibered product of projective schemes exists and is again a projective scheme.

I want to change the notation slightly. We write S = Spec(A). We have the two schemes

\begin{aligned}ℙ_{S}^{n} & & ℙ_{S}^{m} \\ ↘ & & ↙ \\ & S \end{aligned}

It will turn out that this fibered product can be written as a closed subscheme of ℙ_{S}^{nm+n+m}.

To construct this closed subscheme I start from the usual covering by affine spaces. But here I will change the numeration: We had n + 1 open affine spaces to cover ℙ_{S}^{n}. Here we have nm + n + m = (n + 1)(m + 1) − 1 hence we need (n + 1)(m + 1) open affine spaces for the covering.

We write for 0 ≤ i ≤ n, 0 ≤ j ≤ m

U_{ij} = Spec(A[… x_{ijνμ} …])

where of course 0 ≤ ν ≤ n, 0 ≤ μ ≤ m, the x_{ijνμ} except that we have the relation x_{ijij} = 1. We cover ℙ_{S}^{n} by

U_{i} = Spec(A[t_{i0} … t_{in}]) V_{j} = Spec(A[t′_{j0} … t′_{jn}]) .

We have already constructed the fibered product

U_{i} ×_{S} V_{j} = Spec(A[t_{i0} … t_{in}, t′_{j0} … t′_{jn}]).

We construct an isomorphism from U_{i} × V_{j} with a closed subscheme of U_{ij}. To do this we construct an A-homomorphism of rings

A[… x_{ijνμ} …] → A[t_{i0} … t_{in}, t′_{j0} … t′_{jn}])

and this homomorphism is given by

x_{ijνμ} → t_{iν} · t′_{jμ}.

We observe that x_{ijij} maps to 1. We also observe that

\begin{aligned}x_{ijiμ} & → t′_{jμ} \\ x_{ijνj} & → t_{iν} \end{aligned}

and hence it is clear that the kernel of this homomorphism has as its kernel the ideal I_{ij} generated by

x_{ijiμ} · x_{ijνj} − x_{ijνμ}.

If we divide by the ideal generated by these polynomials, then we get a polynomial ring generated by x_{ijνμ} with (ν, μ) ≠ (i, j)).

Hence this ideal defines a closed subscheme V (I_{ij}) ⊂ U_{ij} which is isomorphic to U_{i} ×_{S} V_{j}. This ideal defines a quasicoherent sheaf Ĩ_{ij} on U_{ij} and we will show that

Ĩ_{ij}|U_{ij} ∩ U_{i′j′} = Ĩ_{i′j′}|U_{ij} ∩ U_{i′j′}.

To see this we consider the following diagram

\begin{aligned}U_{i} ×_{S} V_{j} & ⟶^{∼} & V (I_{ij}) ⊂ & U_{ij} \\ & & ⋃ & ⋃ \\ & & V (I_{ij}) ∩ V (I_{i′j′}) & ⊂ U_{ij} ∩ U_{i′j′} \\ & & ⋂ & ⋂ \\ U′_{i} ×_{S} V_{j} & ⟶^{∼} & V (I_{i′j′}) ⊂ & U_{i′j′} \end{aligned} .

We can construct the fibered product (U_{i} ∩ U_{i′}) ×_{S} (V_{j} ∩ V_{j′}) since both factors are affine and of course we can place it into the middle of the left column of the diagram above and we get open subschemes (U_{i} ∩ U_{i′}) ×_{S} (V_{j} ∩ V_{j′}) ⊂ U_{i′} ×_{S} V_{j} and (U_{i} ∩ U_{i′}) ×_{S} (V_{j} ∩ V_{j′}). Of course we want a horizontal arrow from (U_{i} ∩ U_{i′}) ×_{S} (V_{j} ∩ V_{j′}) to V (I_{ij}) ∩ V (I_{i′j′}) which should be an isomorphism. To see that this works we rewrite the diagram in terms of rings, we drop the column in the middle

\begin{aligned}A[… t_{iν}, …, t′_{jμ} …] & ⟵ & A[… x_{ijνμ} …] \\ ⋂ & & ⋂ \\ A[… t_{iν}, …, t′_{j′μ}, t_{ii′}^{−1}, t_{jj′}^{−1}] & ⟵ & A[… x_{ijνμ} …, x_{iji′j′}^{−1}] \\ ∥ & & ∥ \\ A[… t_{i′ν}, …, t′_{j′μ}, t_{i′i}^{−1}, t_{j′j}^{−1}] & ⟵ & A[… x_{i′j′νμ} …, x_{i′j′ij}^{−1}] \\ ⋃ & & ⋃ \\ A[… t_{i′ν}, …, t′_{j′μ} …] & ⟵ & A[… x_{i′j′νμ} …] \end{aligned} .

The two arrows in the middle are obtained from the arrow on the top and the bottom by extending. We send x_{iji′j′}^{−1} (resp. x_{i′j′ij}^{−1}) to t_{ii′}^{−1} · t_{jj′}^{−1} (resp. t_{i′i}^{−1} · t_{j′j}^{−1}). But using the rules for changing the coordinates yield that the two arrows in the middle are equal. The kernel of this arrow in the middle is clearly (I_{ij})_{x_{iji′j′}} = (I_{i′j′})_{x_{i′j′ij}} and this proves Ĩ_{ij}|U_{ij} ∩ U_{i′j′} = Ĩ_{i′j′}|U_{ij} ∩ U_{i′j′}. For later references we write our diagram again, but now we write the completed diagram

\begin{aligned}U_{i} ×_{S} V_{j} & ⟶^{∼} & V (I_{ij}) ⊂ & U_{ij} \\ ⋃ & & ⋃ & ⋃ \\ (U_{i} ∩ U_{i′} ×_{S} (V_{j} ∩ V_{j′}) & ⟶^{∼} & V (I_{ij}) ∩ V_{I_{i′j′}}) & ⊂ U_{ij} ∩ U_{i′j′} \\ ⋂ & & ⋂ & ⋂ \\ U′_{i} ×_{S} V_{j} & ⟶^{∼} & V (I_{i′j′}) ⊂ & U_{i′j′} \end{aligned} .

Hence the I_{ij} define a coherent sheaf of ideals on ℙ_{S}^{nm+n+m} and this defines a closed subscheme y of ℙ_{S}^{nm+n+m}.

We can define projection maps

\begin{aligned}p_{1} : y & ⟶ & ℙ_{S}^{n} \\ & ↘  ↙ \\ & S \end{aligned}  \begin{aligned}p_{2} : y & ↪ & ℙ_{S}^{m} \\ & ↘  ↙ \\ & S \end{aligned}

which on U_{ij} ∩ y = U_{i} ×_{S} V_{j} are the projections to the first and second factor. (It needs a little computation that these projections match on U_{ij} ∩ U_{i′j′}).

Now it is clear that

y
p_{1} ↙  ↘ p_{2}
ℙ_{S}^{n}  ℙ_{S}^{m}
↘  ↙
S

is indeed a fibered product. If we have a scheme T → S and a pair of arrows

\begin{aligned}T & ⟶^{f} & ℙ_{S}^{n} \\ & ↘  ↙ \\ & S \end{aligned}  \begin{aligned}T & ⟶^{g} & ℙ_{S}^{m} \\ & ↘  ↙ \\ & S \end{aligned}

then we find for any t ∈ T an open neighborhood V such that

f(V) ⊂ U_{i}, g(V) ⊂ V_{j}

for some i, j. Hence on V we get a map

V ⟶ U_{i} ×_{S} V_{j} ≃ U_{ij} ∩ y
↘ ↙
S

and if we cover T by such V′s then the maps must match on the intersections. This follows from the completed diagram in our proof.

We could also use the description of subschemes by homogeneous ideals. We introduce the ring A[Z_{ij}] where i = 0, 1, …, n, j = 0, 1, …, n and in this ring we have the homogeneous ideal P̃ generated by the polynomials

Z_{ij}Z_{ab} − Z_{aj}Z_{ib}

for all quadruples of indices. Then the process of passing from homogeneous ideals to quasicoherent sheaves of ideals gives us the ideal describing ℙ_{S}^{n} ×_{S} ℙ_{S}^{m} as a subscheme of ℙ_{S}^{(n+1)(m+1)−1}. This is the Segre embedding.

1.3.1. Locally free sheaves on ℙ^{n}

At this point I return to the construction of sheaves by the gluing process. We want to construct locally free sheaves and line bundles on ℙ_{A}^{n}. To do this we start from the collection of free modules of a fixed rank m on the open schemes U_{i}:

O_{ℙ^{n}}(U_{i})^{m} = A[t_{i0}, …, t_{in}]^{m} = M_{i}.

They define sheaves M̃_{i} on the affine schemes U_{i}. Now we choose O_{ℙ^{n}}(U_{i} ∩ U_{j}) - linear isomorphisms

g_{ij} : M̃_{i}(U_{i} ∩ U_{j}) ⟶^{∼} M̃_{j}(U_{i} ∩ U_{j}),

this is nothing else than a collection of matrices

g_{ij} ∈ Gl_{m}(O_{ℙ^{n}}(U_{i} ∩ U_{j}))

which should satisfy the cocycle relation

\begin{aligned}g_{ii} & = Id \\ g_{ij}g_{ji} & = Id \\ g_{ij} · g_{je} & = g_{ie} on U_{i} ∩ U_{j} ∩ U_{e} \end{aligned} .

Then we get a locally free sheaf (vector bundle, see below) on ℙ_{A}^{n} by gluing:

M̃ = (M_{i}, g_{ij})_{i=0,...,u;j=0,...,n}.

I recall that for an open set a section s ∈ M̃(V) is a collection s = (s_{0}, …, s_{i}, …, s_{n}) where s_{i} ∈ M̃_{i}(U_{i}∩V) and where g_{ij}s_{i} = s_{j} on V ∩U_{i}∩U_{j}. Especially we have M̃(U_{i}) = M_{i} because in this case the i-th component determines the others.

In the simple case m = 1 our matrix becomes a unit in O_{ℙ^{n}}(U_{i} ∩ U_{j}) and we may choose for instance an integer r and define

g_{ij} = t_{ij}^{−r}.

This yields a locally free sheaf of rank one (a line bundle or invertible sheaf) which is called

O_{ℙ^{n}}(r)

on ℙ_{A}^{n}. Again I point out that O_{ℙ^{n}}(r)(U_{i}) = O_{ℙ^{n}}(U_{i}) = A[t_{i0}, …, t_{in}].

1.3.2. Exercise

(1) Compute H^{0}(ℙ_{A}^{n}, O_{ℙ^{n}}(r)) and show that this is isomorphic to the A-module of homogeneous polynomials in n + 1-variables of degree r. To be more precise: A section in H^{0}(ℙ_{A}^{n}, O_{ℙ^{n}}(r)) is by definition a collection

s = (f_{0}, …, f_{n})

where f_{i} ∈ A[t_{i0}, …, t_{in}] which satisfies

t_{ij}^{−r} · f_{i} = f_{j}.

on U_{i} ∩ U_{j}. A homogeneous polynomial

F (X_{0}, …, X_{n}) = Σa_{ν_{0},...,ν_{n}}X_{0}^{ν_{0}}, …, X_{n}^{ν_{n}}

with Σν_{i} = r provides such a section if we define

f_{i}(t_{i0}, …, t_{in}) = F (t_{i0}, …, 1, …, t_{in}).

Show that this gives us the isomorphism.

(2) Any homogeneous polynomial F (X_{0}, …, X_{n}) defines a sheaf of ideals

O_{ℙ^{n}}(F) ↪ O_{ℙ^{n}}

which is defined by O_{ℙ^{n}}(F)(U_{i}) = O_{ℙ^{n}}(U_{i})f_{i}, in other words: the restriction of the ideal to the open sets U_{i} is the principal ideal (f_{i}).

Show: If for all 𝔭 ∈ Spec(A) this polynomial is non-zero in A/𝔭[X_{0}, …, X_{n}] then this is a locally free sheaf on ℙ_{A}^{n} which is isomorphic to O_{ℙ^{n}}(−d) where d = degree of F.

Hint: Write a section O_{ℙ^{n}}(F)(V) as acollection of elements (…, h_{i}f_{i}, …) and show that we must have t_{ij}^{d}h_{i} = h_{j}. Where do we use our assumption. Can it be replaced by a weaker assumption.

I refer to the exercise (2) The sheaf of ideals defines a closed subscheme V (F) which is called a hypersurface of degree d. If we take for instance simply F = X_{0} then this closed subscheme is the reduced scheme H_{0} = ℙ^{n} \ U_{0}, it is isomorphic to ℙ^{n−1}/ Spec(A). It is called the hyperplane at infinity (from the point of view of somebody who lives in U_{0}). We as well consider the other hyperplanes H_{i} = V (X_{i}).

The tensor product of locally free sheaves is again a locally free sheaf and especially we have O_{ℙ^{n}}(r) ⊗ O_{ℙ^{n}}(s)→^{∼}O_{ℙ^{n}}(r + s). The inclusion O_{ℙ^{n}}(X_{0}) ⊂ O_{ℙ^{n}} combined with the isomorphism O_{ℙ^{n}}(X_{0})→^{∼}O_{ℙ^{n}}(−1) gives us an inclusion O_{ℙ^{n}}(−1) ↪ O_{ℙ^{n}} and this induces after taking tensor products O_{ℙ^{n}}(r − 1) ↪ O_{ℙ^{n}}(r) and we get a chain of inclusions

O_{ℙ^{n}} ↪ O_{ℙ^{n}}(1) ↪ O_{ℙ^{n}}(2) ↪ … O_{ℙ^{n}}(d) …

where all the embeddings are obtained from O_{ℙ^{n}}(X_{0}) ↪ O_{ℙ^{n}}.

Of course it should be clear that we have many ways of mapping the sheaf O_{ℙ^{n}} into O_{ℙ^{n}}(d). To be more precise we can look at

Hom_{O_{ℙ^{n}}} (O_{ℙ^{n}}, O_{ℙ^{n}}(d))

and to give a homomorphism among sheaves we only need to know what happens to 1 ∈ H^{0}(ℙ^{n}, O_{ℙ^{n}}) because this section generates the stalk on each point. Hence we get

Hom_{O_{ℙ^{n}}} (O_{ℙ^{n}}, O_{ℙ^{n}}(d)) = H^{0}(ℙ^{n}, O_{ℙ^{n}}(d))

and this is the A-module of homogeneous forms of degree d in A[X_{0}, …, X_{n}]. Hence giving such an embedding O_{ℙ^{n}} ↪ O_{ℙ^{n}}(d) amounts to pick a form of degree d. We have chosen at this point the form X_{0}^{d}.

1.3.3. O_{ℙ^{n}}(d) as sheaf of ”meromorphic functions”

If we look at

O_{ℙ^{n}} ↪ O_{ℙ^{n}}(d)

and restrict this map to U_{i} then we get

O_{ℙ^{n}}(U_{i}) ⟶ O_{ℙ^{n}}(d)(U_{i}) = O_{ℙ^{n}}(U_{i})

and this map is given by

f_{i} ⟶ t_{0i}^{d}f_{i}.

We can embed O_{ℙ^{n}}(U_{i}) into the module \frac{1}{t_{0i}^{d}} O_{ℙ^{n}}(U_{i}) which is the O_{ℙ^{n}}(U_{i})-module of ”meromorphic functions on U_{i}” which are regular on U_{i} ∩ U_{0} and have at most ”a pole of order d along the hyperplane U_{i}\U_{i} ∩ U_{0} = U_{i} ∩ V (X_{0})”.

Then the map above gives an isomorphism

\frac{1}{t_{i_{0}}^{d}} O_{ℙ^{n}}(U_{i}) ⟶^{∼} O_{ℙ^{n}}(d)(U_{i}).

Especially for i = 0 we have t_{00} = 1 which means that V (X_{0}) does not meet U_{0}. We glue these modules \frac{1}{t_{0i}^{d}} O_{ℙ^{n}}(U_{i}) over the intersections of two affine sets and hence we get the sheaf O_{ℙ^{n}}(−dH_{0}) and call this the sheaf of ”meromorphic functions on ℙ^{n}” which are regular on U_{0} and have at most a pole of order d along the hyperplane at infinity. We have a diagram

O_{ℙ^{n}} ↪ O_{ℙ^{n}}(−dH_{0})→^{∼} O_{ℙ^{n}}(d)

This also gives a slightly different view of the above chain of inclusions: The chains arise simply because we allow higher and higher orders of poles.

Of course we have also an interpretation for r < 0: In this case the sections are the germs of regular functions which have a zero of order ≥ −r along the hyperplane.

1.3.4. Vector bundles attached to locally free modules:

If we have any scheme X and a locally free O_{X}−module M̃ of finite constant rank n then we may cover X by suitable affine open subschemes V_{j} such that M̃|V_{j} becomes trivial in other words we may choose a basis e_{1,j}, …, e_{n,j} of the O_{X}(V_{j})−module M̃(V_{j}). If we use this basis to identify ψ_{j} : O(V_{j})^{n}→^{∼}M(V_{j}) by sending the canonical basis to the e_{ν,j}, then g_{j,l} = ψ_{l} ∘ ψ_{j}^{−1} : O_{X}(V_{j} ∩ V_{l})^{n}→^{∼}O_{X}(V_{j} ∩ V_{l})^{n} gives gluing data which in turn give a locally free module which is isomorphic to M̃. Hence any locally free module is obtained by gluing.

We stick to the situation above. We consider the dual module M̃^{∨}. Locally on the V_{j} we may consider the symmetric tensor algebra Sym(M^{∨}(V_{j})) which in this situation is nothing else than the polynomial algebra O(V_{j})[X_{1j}, …, X_{nj}] where the X_{νj} form a dual basis of the basis e_{μj}. Hence we get a scheme

Spec(Sym(M^{∨}(V_{j}))) ⟶^{∼} Spec(O(V_{j})[X_{1j}, …, X_{nj}])
↘ ↙
Spec O(V_{j})

We have Spec(Sym(M^{∨}(V_{j} ∩ V_{l})) ⊂ Spec(Sym(M^{∨}(V_{j}), Spec(Sym(M^{∨}(V_{l}) as an open subscheme and this gives us again by a gluing process a scheme

\begin{matrix}V (M̃) \\ ↓ \\ X \end{matrix}

which is locally over the V_{j} equal to Spec(Sym(M^{∨}(V_{j}). This scheme is called the vector bundle attached to M̃.

Let us consider the projective space ℙ^{n}/ Spec(k). We have seen that the set of k-valued points of the projective space is the set of lines in k^{n+1} passing through zero. We want to construct the tautological line bundle over ℙ^{n}/ Spec(k). A k-valued point should be a pair (x, ζ) where x ∈ ℙ^{n}(k) and ζ ∈ x. Of course it is very easy to describe this line bundle as a closed subscheme of ℙ^{n} ×_{Spec(k)} 𝔸_{k}^{n+1}.

It is left to the reader to verify that this bundle is O_{ℙ^{n}}(−1)

1.3.5 The tangent bundle of ℙ^{n}

It is of course obvious that the scheme

\begin{matrix}ℙ_{A}^{n} \\ ↓ \\ Spec(A) \end{matrix}

is always smooth. Locally on one of the U_{i} the module of differentials is the free module generated by the dt_{i0}, …, dt_{i,i−1}, dt_{i,i+1}, …, dt_{i,n}, of course dt_{ii} = 0.

We consider the n-th exterior power of this module

Λ^{n}Ω_{ℙ^{n}/A} = Ω_{ℙ^{n}/A}^{n}.

The restriction Ω_{ℙ^{n}/A}^{n} | U_{i} is the trivial line bundle generated by

dt_{i0} ∧ … dt_{i,i−1} ∧ dt_{i,i+1} … dt_{in},

and on U_{i} ∩ U_{j} we have

t_{iv} = \frac{1}{t_{ij}} t_{jv},

and hence

dt_{iv} = \frac{1}{t_{ij}} dt_{jv} − \frac{t_{jv}}{t_{ji}^{2}} · dt_{ji}.

Taking the highest exterior power we get

dt_{i0} ∧ … ∧ dt_{i,i−1} ∧ dt_{i,i+1} ∧ … ∧ dt_{in} = (\frac{dt_{j0}}{t_{ji}} − \frac{t_{j0}}{t_{ji}^{2}} dt_{ji}) ∧ … ∧ (\frac{dt_{jn}}{t_{ji}} − \frac{t_{jn}}{t_{ji}^{2}} dt_{ji}),

where on the right hand side we h ave to leave out the factor with index ν = i, and where we observe that the factor ν = j simplifies to

−\frac{1}{t_{ji}^{2}} dt_{ji}.

Hence we see that the right hand side is

(−1)^{i−j} \frac{1}{t_{ji}^{n+1}} dt_{j0} ∧ … ∧ dt_{j,j−1} ∧ dt_{j,j+1} … ∧ dt_{jn}.

Therefore we see that the line bundle Ω_{ℙ_{A}^{n}/A} is obtained from the cocycle

(−1)^{i−j}t_{ij}^{n+1} = g_{ij}

in the sense of 1.3.1. But we may of course change our generator of Ω_{ℙ_{A}^{n}/A}^{n} over U_{i} by the

sign (−1)^{i} and the cocycle modifies into g_{ij} = t_{ij}^{n+1} which then implies

Ω_{ℙ_{A}^{n}/A}^{n} ≃ O_{ℙ_{A}^{n}/A}(−n − 1).

If we consider the dual sheaf of Ω_{ℙ^{n}/A} then we get the sheav of tangent vectors. We can pass to the associated vector bundle, and we get the tangent bundle

T ℙ^{n} ⟶ ℙ^{n}
↘ ↙
Spec(A).

This tangent bundle has now a very simple geometric description. In 1.3.4 we consider the trivial bundle

ℙ^{n} ×_{Spec(A)} 𝔸_{Spec(A)}^{n+1},

and we described the tautological line bundle in it. This means that we have

V (O_{ℙ^{n}}(−1)) ⟶ ℙ^{n} × 𝔸^{n+1}
↘ ↙
ℙ^{n},

and we can devide the trivial bundle by the tautological bundle. It is rather clear that the result is the tangent bundle T ℙ^{n} (This has to be verified but from the geometric intuition it is clear.) Hence we get an exact sequence of vectorbundles

0 → (O_{ℙ^{n}}(−1)) → ℙ^{n} × 𝔸^{n+1} → T ℙ^{n} → 0

over ℙ^{n}. If we now take the (n + 1)-th exterior power of the trivial bundle

1.4 Seperated and proper morphisms:

The property of a morphism to be projective is a global property in contrast to the properties smooth and flat which can be checked locally.

There are two other global properties of morphism which are global in nature namely a morphism X → S can have the property to be separated and it can be proper and I think here is the right place to discuss them.

If we have a scheme π : X → S then we can form the fibered product X ×_{S} X/S and the identity id : X → X provides an element in (id, id) ∈ Hom_{S}(X, X) × Hom_{S}(X, X). By the universal property this is nothing else than an element Δ_{X} → X ×_{S} X.

The morphism π is called separated if Δ_{X} is a closed embedding.

It is not too hard to see that this property is local in the base, hence if we discuss this notion we may assume that S = Spec(A). Then

Lemma :A morphism π : X → Spec(A) = S is separated if for any two affine open subsets U, V ⊂ X the intersection U ∩ V is affine again and O_{X}(U ∩ V) is generated by the restriction of O_{X}(U)|U ∩ V and O_{X}(V)|U ∩ V.

This is rather clear because as I mentioned at the end of the section on fibered products we can cover X×_{S}X by open affine subsets U ×_{S}V. The morphism Δ is a closed embedding if for any such pair U ∩ V → U ×_{S} V is a closed embedding, But then U ∩ V is a closed subscheme of the affine scheme U ×_{S} V and its ring of regular functions O_{X}(U ∩ V) is a quotient of O_{X}(U) ⊗_{A} O_{X}(V).

We should notice that a morphism π : X → Spec(A) = S is separated if we can find some covering 𝔘 = {U_{i}}_{i∈I} by affine subschemes such that for any pair of indices i, j the affine schemes U_{i}, U_{j} satisfy the condition in the lemma above.

We have given an explicit construction of ℙ_{S}^{n} ×_{S} ℙ_{S}^{m} as a projective subscheme of some ℙ_{S}^{N}. We apply this to the case n = m. In this case we can either verify that the system of affine sets {U_{i}}_{i=0,...,n} satisfies the condition in the Lemma because the the elements in O_{ℙ^{n}}(U_{i}) which we have to invert to get O_{ℙ^{n}}(U_{i} ∩ U_{j}) lie in O_{ℙ^{n}}(U_{j}). Hence we see that that ℙ^{n} → S is separated. We could also argue that we can describe the diagonal as a closed subscheme of the product just by adding the polynomials Z_{ij} − Z_{ji} to the ideal which describes the product as a subscheme of ℙ_{S}^{(n+1)(m+1)−1}.

An open subscheme y ⊂ X/S of a projective scheme X/S is called quasiprojective over S. It is now clear that quasiprojective schemes y/S are also separated.

Finally I want to introduce the notion of a proper morphism. A morphism π : X → S is called proper if it is separated and if it is universally closed. This means that for any base change S′ → S and any closed subscheme Z ⊂ X ×_{S} S′ the image of Z under the projection π ×_{S} S′ is closed.

1.4.1 Theorem: A projective morphism

π : X ⟶ S

is always proper.

Proof: We have just seen that it is separated. We recall the definition of a projective morphism. It means that we have a diagram

X ↪^{i} P_{S}^{n}
π ↘ ↓
S

where i is a closed embedding. We have to show that for any base change S′ → S and any closed subset y ⊂ X ×_{S} S′ the image of π is closed. Since this closed subset is also a closed subset in ℙ_{S′}^{n} it suffices to show that for any closed subset Z ⊂ ℙ_{S}^{n} its image in S is closed.

The question is local in the base, we assume that S = Spec(A). Now we know that we can describe Z as the set of zeroes of a homogeneous ideal Ĩ ⊂ A[X_{0}, X_{1}, …, X_{n}] (see 1.1.2). Now we pick a point s ∈ S which is not in the image of Z. We localize A at s, let 𝔪_{s} be the maximal ideal of this local ring A_{𝔪_{s}}. Now we have Z_{s} = ∅ and our Lemma 1.1.2.1 tells us that for a sufficiently large d >> 0 we have

Ĩ(d) ⊗ A_{𝔪_{s}}/𝔪_{s} ≃ A_{𝔪_{s}}/𝔪_{s}[X_{0}, X_{1}, …, X_{n}](d)

or in other words

A_{𝔪_{s}}[X_{0}, X_{1}, …, X_{n}](d)/Ĩ(d) ⊗ A_{𝔪_{s}}/𝔪_{s} = (0).

By the lemma of Nakayama it follows that

A_{𝔪_{s}}[X_{0}, …, X_{n}](d)/Ĩ(d) = 0.

We are basically through but since we passed to the localization at s, we need still a little finiteness argument. We know that any monomial X_{0}^{r_{0}} … X_{n}^{r_{n}} of degree d can be written in the form

X^{r_{0}} … X_{n}^{r_{n}} = ∑ G_{𝔲} · F_{𝔲}

where F_{𝔲} ∈ Ĩ and G_{𝔲} ∈ A[X_{0}, …, X_{n}]. The local ring A_{𝔪_{s}} is obtained from our original A by localization. But to write down the G_{𝔲} we need only finitely many denominators which means that we can replace A by a localization A_{f} with some f with f(s) ≠ 0. Hence we see that already

Ĩ(d) ⊗ A_{f} ≃ A_{f}[X_{0}, X_{1} … X_{n}](d)

which proves that the image of Z does not intersect with the open neighborhood Spec(A_{f}) of s. This proves the theorem.

1.5 The valuative criteria

I want to state criteria for separatedness and properness which are extremely important, and which give a very intuitive idea of this notion. I will not give the proofs here, I refer to Grothendieck’s book EGA, II, §7. This section of the book is relatively selfcontained, so it can be read directly.

Since we know that the question whether a morphism π : X → y is separated (resp. proper) is local in the base, we consider the following situation. Let y = Spec(A) where A is noetherian, let π : X → y be of finite type, i.e. we can cover X by affines U_{α} = Spec(B_{α}) where B_{α} is a finitely generated A-algebra.

Now we have:

Theorem 1.5.1: Under our conditions above the morphism π : X → Spec(A) is

a) separated if for any discrete valuation ring C with quotient field K and any morphism

f : Spec(C) ⟶ Spec(A)

two Spec(C) valued points

Spec(C)  \overrightarrow{→}  X
f ↘  ↓
Spec(A)

which become equal if we restrict them to Spec(K) are already equal,

b) proper if for any such C and any f a Spec(K) valued point extends (uniquely) to a Spec(C) valued point.

For the proof see EGA, II, §7, 7.2.3 and Remarks 7.2.4 and 7.3.8.

1.5.2. The valuative criterion for the projective space:

The following property of a projective space is an algebraic substitute for the fact that the complex projective space ℙ^{(}ℂ) is compact.

Let us start from a discrete valuation ring R. We want to study the S = Spec(R) valued points of ℙ_{S}^{n} → S. Let π be a uniformizing element of R and let K be its quotient field. We have a diagram

ℙ^{n}
↓
Spec(R) ← Spec(K)

The valuative criterion asserts that the K valued points and the R valued points are the same. But this is almost obvious. We have seen that (1.1.3.)

ℙ_{S}^{n}(Spec(K)) = ℙ_{S}^{n}(K) = (K^{n+1} \ {0})/K^{∗}.

But for any x = (x_{0}, x_{1} …, x_{n}) ∈ K^{n+1} \ {0} we write x_{i} = u_{i}π^{n_{i}} and pick the index i_{0} for which n_{i_{0}} is minimal. Then (π^{−n_{i_{0}}} x_{0}, π^{−n_{i_{0}}} x_{1} …, π^{−n_{i_{0}}} x_{n}) represents the same point, but now the coordinates are in R and one of the coordinates is a unit. Therefore this is a R-valued point and have have shown

If R is a discrete valuation ring with quotient field K then ℙ_{Spec(R)}^{n}(R) = ℙ_{Spec(R)}^{n}(K) This is the valuative criterion for the projective space. This of course extends immediately

to projective schemes over R.

This expresses in algebraic terms the ”compactness” of the projective space. We should look at Spec(R) as a small disc, the generic point (0) ∈ Spec(R) corresponds to the disc minus the origin and the closed point (π) corresponds to the origin. The analogous object in function theory is a disc D = {z||z| < 1} in the complex plane. Then we know from function theory that a meromorphic map f : D\{0} → U_{i}(ℂ) ⊂ ℙ^{n}(ℂ) extends in a unique way to a holomorphic map f : D → ℙ^{n}(ℂ).

1.6. The construction Proj(R)

We have a construction of projective schemes starting from a graded algebra. Let A be an arbitrary ring and let R be a graded A-algebra. This means that R is an A-algebra and we have a direct sum decomposition

R = R_{0} ⊕ R_{1} + … ⊕ R_{n} ⊕ …

such that the R_{i} are A-modules and R_{i}R_{j} ⊂ R_{i+j}. The identity element is in R_{0} and the algebra homomorphism from A to R factor through R_{0}. We assume that A → R_{0} is surjective and we assume that R_{1} is a finitely generated A-module which generates the A-algebra.

We define the set Proj(R) (the projective spectrum) to be the set of homogeneous prime ideals of R which do not contain R_{1}. If we have such a prime ideal 𝔭 we pick on f ∈ R_{1} such that f ∉ 𝔭 and we define

Proj(R)_{f} = {𝔮 ∈ Proj(R)|f ∉ 𝔮}.

We form the ring

R_{f}^{(0)} = \{\frac{g}{f^{n}} |g ∈ R_{n}\}

i.e. the ring of elements of degree zero in the quotient ring R_{f}. It is very easy to see that

Proj(R)_{f} = Spec(R_{f}^{(0)}).

We use this to define a topology on Proj(R) and a structure of a ringed space. The open sets V ⊂ Proj(R) are those for which V ∩ Proj(R)_{f} is open for all f.

We define the sheaf of regular function so that its restriction to the Proj(R)_{f} is simply

the sheaf R̃_{g}^{(0)} on Spec(R_{g}^{(0)}). Hence we have defined a scheme (Proj(R), Õ) for any such graded A-algebra.

If we take for instance the polynomial graded algebra

A[X_{0}, …, X_{n}]

then Proj(A[X_{0}, …, X_{n}], O) = (ℙ^{n}, O_{ℙ^{n}}). If our A-algebra R is generated by elements x_{0}, …, x_{n} we have a homomorphism of graded A-algebras

A[X_{0}, …, X_{n}] ⟶ R_{0}[x_{0}, …, x_{n}]
↖  ↗
A

and we see that Proj(R) is a closed subscheme

Proj(R) ⟶ ℙ_{A}^{n} ↘ ↙ Spec(A) .

This generalizes easily to the case where we replace Spec(A) by an arbitrary scheme S and where R is a graded sheaf of O_{S}-algebras.

1.6.1 The assumption that R is generated by homogeneous elements in degree one is not essential. Let us assume that R is finitely generated by elements x_{0}, x_{1}, …, x_{n} which are of degree d_{0}, d ≥ 1 respectibely. Then we can define

R_{x_{i}}^{(0)} = \{\frac{f}{x_{i}} | deg(f) = d_{i}m\} ⊂ R_{x_{i}},

and we can consider Spec(R_{x_{i}}^{(0)}. Now we define Proj(R) as the space of homogeneous prime ideals 𝔭 which do not contain all the x_{i}. Then we have

Proj(R) = ⋃_{i=1}^{n} Spec(R_{x_{i}}^{(0)}

and we can proceed as before.

1.7. Ample and very ample sheaves

Let S = Spec(A) be a noetherian scheme. We want to discuss certain constructions which allow us to show that a scheme X → S is projective or which provide projective embeddings of this scheme.

Let us go back briefly to the case of affine schemes. If we have an arbitrary scheme X and if we want to show that this scheme is affine we have only one chance: We consider the ring of global sections B = Γ(X, O_{X}) and we try to prove that X→^{∼} Spec(B).

This may not be so easy. Of course we always have a morphism

π : X → Spec(B)

which sends a point x ∈ X to the prime ideal 𝔭_{x} of functions which vanish at x and then we get an inclusion h_{x} : B_{𝔭_{x}} ⟶O_{X,x}. (See V.1.6.) Both maps have to be bijective. It is clear what kind of information we need if we want to be successful. We need that we have enough regular functions on X. So for instance we certainly need the information that the regular functions separate points , this means: If we have two point x, y ∈ X and if y is not in the closure of {x} (???) then we can find an f ∈ B such that f(x) = 0 and f(y) ≠ 0. This would tell us that π is injective but this is by far not good enough. For instance the example V.2.1.1 tells us that π does not need to be surjective. Hence we have to assume the surjectivity of π or to make some assumptions which allow to conclude that π is surjective.

I want to discuss some instances where our strategy will be successful.

First I want to discuss an affine situation. Let X → Spec(A) be a separated scheme. I assume it can be covered by finitely many affine subschemes.

Let us assume that we have a A-subalgebra B ⊂ Γ(X, O_{X}) so that we get a diagram

X ⟶^{π} Spec(B) = y
↘ ↙
Spec(A) = S

Finally I assume that π is closed and any fibre π^{−1}(y) is contained in an open affine subset of X.

Under these assumptions I claim: the scheme X is affine.

Proof: We pick a point y ∈ Y and we choose an affine subset U which conyains the fibre π^{−1}(y). We consider the complement of U in X and by our assumption we know that the image of this complement is a closed subset Z ⊂ y which of course does not contain y. Hence we find a regular function g ∈ B such that g(y) ≠ 0 and g|Z = 0. Then we see that π^{−1}(y_{g}) ⊂ U and this is an affine scheme because π^{−1}(y_{g}) = U_{g}. We conclude that any point y ∈ y has an open neighborhood such that the inverse image of this neighborhood is affine. We can cover Y by affine subsets Y_{g_{i}} such that the π^{−1}(Y_{g_{i}}) = X_{i} are affine. We get homomorphisms

O_{y}(y_{i}) ⟶ O_{X}(X_{i}).

The algebras O_{y}(y_{i}) are localizations B_{g_{i}} and since B → Γ(X, O_{X}) is injective, we get inclusions

B_{g_{i}} → Γ(X, O_{X})_{g_{i}} ⟶ O_{X}(X_{i}).

We need that the last arrow is an isomorphism. This follows from a little

Lemma: Let X be a separated scheme which can be covered by a finite set of affine schemes. Let g ∈ Γ(X, O_{X}) and let h ∈ O_{X}(X_{g}). Then we can find an integer n > 0 such that g^{n}h is the restriction of an element in Γ(X, O_{X}) to X_{g}.

Proof: Let us write X = ⋃_{i∈E} U_{i} with U_{i} affine and E finite. We consider the restriction h_{i} of h to X_{g} ∩ U_{i} = U_{i,g}. Now it is clear by construction that h_{i} = \frac{f_{i}}{g^{n_{i}}} for some f_{i} ∈ Γ(U_{i}, O_{X}) and hence we can find an index n, and functions F_{i} ∈ O_{X}(U_{i}) such that

g^{n}h = F_{i} | U_{i} ∩ X_{g}.

Now we compare F_{i} and F_{j} on U_{i} ∩ U_{j}. We have

F_{i} | U_{i} ∩ U_{j} ∩ X_{g} = F_{j} | U_{i} ∩ U_{j} ∩ X_{g}.

Since we assumed that X is separated, we can conclude that we can find an m such that

g^{m}F_{i} | U_{i} ∩ U_{j} = g^{m}F_{j} | U_{i} ∩ U_{j}

and then the g^{m}F_{i} are restrictions of a function F ∈ Γ(X, O_{X}).

(If we want to avoid to assume that X is separated, we can assume instead that any open set has a finite covering by affines.)

Now it is rather clear that

X ⟶ Spec(Γ(X, O_{X}))

must be an isomorphism. It is obvious that Γ(X, O_{X}) separated points because B separates the points in y and the Γ(X_{i}, O_{X}) separate the points on X_{i}. It is also clear that for any x ∈ X the map

h : Γ(X, O_{X})_{𝔭_{x}} ⟶ O_{X,x}

is an isomorphism.

It rests to prove the surjectivity. Again we exploit the assumption that π is closed. From this we get easily that π is surjective. If x ∈ Spec(Γ(X, O_{X})) we consider its image in y and observe that x ∈ X_{i} = Spec(Γ(X_{i}, O_{X})).

We have seen that the assumption that π : X → Spec(B) = y is surjective is vital and it is not so clear how we can verify in a given case that it is fulfilled.

If we have a projective scheme over X → Spec(A) this strategy cannot work since we do not have enough regular functions. The fundamental principle is to replace the regular functions by the sections in the positive powers of a suitable line bundle L. If we have a section s ∈ H^{0}(X, L) then we can trivialize L on the open set X_{s} where s does not vanish and hence we can view elements in H^{0}(X, L^{⊗n}) as regular functions on X_{s} and therefore a line bundle provides a tool to construct regular functions on certain open sets. For a certain class of line bundles - the so called very ample bundles - this method can be used to construct projective embeddings and I will explain this construction.

If we start from X → S and if we have an embedding

X  ↪^{i}  ℙ_{A}^{n}
π ↘ ↙
Spec(A)

then we have the line bundle O_{ℙ_{A}^{n}}(1) on ℙ_{A}^{n} and we may consider its pullback L = i^{∗}(O_{ℙ^{n}}(1)). Starting from this line we can reconstruct an embedding from X into a projective space.

Now our starting point will be a scheme X → S of finite type and let L be a line bundle on X. We consider the direct image sheaf R_{1} = π_{∗}(L). Since we assume that S is affine R_{1} = π_{∗}(L) is an A-module.

We put R_{0} = A and form the graded A-algebra

R′ = R_{0} ⊕ R_{1} ⊕ ⊕_{n≥2}π_{∗}(L^{⊗n})

and in this algebra we consider the subalgebra

R = R_{0} ⊕ R_{1} ⊕ R_{2} ⊕ …

which is generated by R_{1}.

Now we make the assumption that our line bundle has no base point. This means that for any point s ∈ S we can find an affine neighborhood V such that the sections

π_{∗}(L)(V) = L(π^{−1}(V))

generate the stalk of the line bundle at any point x ∈ π^{−1}(s). This is the same as saying that for any x we can find a section t ∈ H^{0}(π^{−1}(V), L) which does not vanish at x, i.e. it does not go to zero in L ⊗_{O_{x}} k(x).

The assumption that the line bundle L over X has no base point gives us a morphism

j : X ⟶ Proj(R)
↘ ↙
Spec(A).

To see this morphism, we pick a point x and a section f ∈ H^{0}(π^{−1}(V), L) with f(x) ≠ 0. Then the line bundle L and and its powers L^{⊗n} become trivial over X_{f}, hence any section t ∈ H^{0}(π^{−1}(V), L^{⊗n}) can be written as

t = h · f^{n}

with a unique h ∈ O_{S}(X_{f}).

This provides an injective homomorphism

R_{f}^{(0)} → O_{X}(X_{f})

of A-algebras and hence a morphism

X_{f} → Proj(R_{f}^{(0)}).

These morphism match on the intersections X_{f} ∩X_{g} = X_{fg} and hence we get the morphism

X ⟶ Proj(R) ↘ ↙ Spec(A) .

It seems to me clear that this morphism must be surjective because we assumed that X → S should be projective.

Locally on the base S the sheaves π_{∗}(L) are finitely generated modules hence we may write them as quotients of free modules. We have seen that this gives us a morphism of graded algebras

A[X_{0}, …, X_{n}] → R

and a diagram

X ⟶ Proj(R) ↪ ℙ_{S}^{n}
↘ ↙
Spec S

hence we see that our base point free sheaf provides a morphism

\begin{aligned}X & ⟶ & ℙ_{S}^{n} \\ & ↘ ↙ \\ & S \end{aligned} .

We may ask under which circumstances this morphism has nice properties, for instance when is it an embedding?

If for instance we go back to the case where we start from an embedding

i : X  ↪  ℙ_{S}^{n}
↘ ↙
S

and take for L the sheaf L = i_{∗}(O_{ℙ_{S}^{n}}(1)) then our construction gives us back the given embedding.

If we start from a sheaf L on X/S and if our construction provides a projective embedding

X ↪ Proj(R) = Proj(A ⊕ π_{∗}(L) ⊕ …)

then the sheaf L is called relatively very ample. If we have a line bundle L for which a high power L^{⊗n} provides such an embedding then it is called relatively ample.

We consider a proper scheme π : X → where S is locally noetherian, actually we aasume without loss of generality that S = Spec(A) where A is a noetherian ring.

We make the assumption that for any line bundle L on X the direct image is a coherent O_{S}-module. This is not really an assumption, it follows from a general finiteness theorem by Grothendieck (see VIII, ???). This theorem will not be proved in full generality in this book, but in some applications where we need some implications of this result we can circumvent the use of this general theorem.

We start from a line bundle L on X. then π_{∗}(L) is coherent and hence by our assumption a finitely generated A-module R_{1} = H^{0}(X, L). We consider the graduated ring

R = A ⊕ R_{1} ⊕ R_{2} ⊕ …

where R_{i} ⊂ H^{0}(X, L^{⊗i}) is the submodule generated by the products of elements in R_{1}. We assume that R_{1} has no base point, and we consider the morphism

r_{L} : X ⟶  Proj(R)
↘ ↙
S

If we pick a point x ∈ X and an s ∈ R_{1} such that s(x) ≠ 0, then we can consider the localization

r_{L,s} : X_{s} ⟶ Spec(R_{s}^{(0)}).

Now we can make several assumptions: b1) For any choice of (x, s) this morphism has finite fibres.

b2) The algebra R_{s}^{(0)} seperates points on X_{s}.

(This assumption is equivalent to the assumption that H^{0}(X, L) separates points.)

The strongest assumption is

b3) The algebra R_{s}^{(0)} separetes points and for any x ∈ X_{s} the homomorphism

R_{s,y_{x}}^{(0)} ⟶ O_{X,x}/𝔪_{X,x}^{2}

is surjective.

Finally to make a technical assumption

c) Any finite subset in X lies in an affine open subscheme.

Theorem: (i) Under the assumptions b1) and c) or under the assumption b2) the X_{s} are affine and the scheme π : X → S is projective and the morphism r_{L} : X → Proj(R) is finite.

(ii) If we assume b3), then the morphism r_{L} is an isomorphism.

Proof: Since X → S is proper, this morphism is universally closed. If we consider the fibered product X ×_{S} Proj(R), then any closed subscheme Z ⊂ X ×_{S} Proj(R) has a closed image under the projection to Proj(R). We apply this to the graph Γ_{r_{L}} and conclude that the image of X in Proj(R) is closed. Since by construction the map between the rings

R_{s}^{(0)} ⟶ Γ(X_{s}, O_{X_{s}})

is injective. We conclude that r_{L} must be surjective.

Then it is clear that our previous considerations imply that the X_{s} must be affine, this holds for the two cases. Hence we also know that X_{s} = Spec(Γ(X_{s}, O_{X})).

The direct image r_{L,∗}(O_{X}) (???, sehr dubioses Argument) is a coherent O_{Proj(R)} module because π_{∗}(O_{X}) is a coherent O_{S}-module. Hence we see that the algebras Γ(X_{s}, O_{X_{s}}) are finite over R_{s_{i}}^{(0)}.

Now we assume b3). Since the algebra R_{s}^{(0)} separates points, we get that

X_{s} ⟶ Spec(R_{s}^{(0)}

is always bijective. We have to show that (see ???)

R_{s,𝔭_{x}}^{(0)} ⟶ O_{X_{s,x}}

is always an isomorphism. We localize at 𝔭_{x} and get the algebra

Γ(X_{s}, O_{X}) ⊗ R_{s,𝔭_{x}}^{(0)}

which is finite over R_{s,𝔭_{x}}^{(0)} and hence its maximal ideals lie over 𝔭_{x} (see Chap. CA, ???) and our assumption b3) implies that there is only one such maximal ideal. We get

Γ(X_{s}, O_{X}) ⊗ R_{s,𝔭_{x}}^{(0)} ⊂ O_{X,x}.

We reduce modulo the maximal ideal 𝔪_{𝔭_{x}} ⊂ R_{s.𝔭_{x}}^{(0)} and consider the algebra

Γ(X_{s}, O_{X}) ⊗ R_{s,𝔭_{x}}^{(0)}/𝔪_{𝔭_{x}},

this is a local finite algebra over the residue field. Its maximal \overline{𝔪}_{x} ideal is the image of 𝔪_{x} and if the rank of this algebra over the residue field is > 1, then we must have \overline{𝔪}_{x} ≠ \overline{𝔪}_{x}^{2} and this contradicts the surjectivity of 𝔪_{𝔭_{x}} → 𝔪_{x}/𝔪_{x}^{2}. We conclude that the rank must be one. It follows that

Γ(X_{s}, O_{X}) ⊗ R_{s,𝔭_{x}}^{(0)}/𝔪_{𝔭_{x}} = O_{X,x}/𝔪_{x}

and Nakayama’s lemma gives the desired result.

Now we assume b1) and c) or b2). We begin with the observation that we have a finite number of sections s_{i} such that

⋃ X_{s_{i}} = X.

The algebras Γ(X_{s_{i}}, O_{X}) are finite over R_{s_{i}}^{(0)} and ?????

Our problem is that L does not have enough sections. The sections of L produce the affine algebras R_{s}^{(0)} which are too small, but the algebras Γ(X_{s}, O_{X}) are finite over R_{s}^{(0)}. Hence we can chose a set of generators s_{0}, …, s_{n} of H^{0}(X, L), and we can choose finite sets f_{i,0} … f_{i,m_{i}} of generators of the R_{s_{i}}^{(0)}-module Γ(X_{s_{i}}, O_{X}).

Our strategy will be to construct a line bunde H on Y = Proj(R) which contains the trivial bundle O_{Y}. We take the pullback of this bundle r_{L}^{∗}(H) on X. Then we get L ↪ L ⊗ r_{L}^{∗}(H) on X and H^{0}(X, L) ↪ H^{0}(X, L ⊗ r_{L}^{∗}(H)). Now we can consider functions t/s_{i} with t ∈ H^{0}(X, L ⊗ r_{L}^{∗}(H)), and we want that H is so big that this module of functions on X_{s_{i}} contains all the generators. If we succeed in doing this we know that the elements t/s_{i} with t ∈ H^{0}(X, L ⊗ r_{L}^{∗}(H)) generate Γ(X_{s_{i}}, O_{X}).

To construct this bunde X, we write the ring R as quotient of the graded polznomial ring A[X_{0}, …, X_{n}] we send x_{i} → s_{i}. This gives us an embedding

Y ⟶^{i} ℙ_{A}^{n}
↘ ↙
Spec(A)

(see ???), and we recall that i^{∗}(O_{ℙ_{A}^{n}}(1)) ≃ r_{L_{x}}(L).

On ℙ_{A}^{n} we have the sheaf

O_{ℙ_{A}^{n}}(−X_{0}) ⊗ … ⊗ O_{ℙ_{A}^{n}}(−X_{n}) = H_{0}

(see ???), and the restriction of a high positive power of this sheaf to Y will be our H.

We consider an element f ∈ Γ(X_{s_{i}}, O_{X}). If we restrict this element to an open subset X_{s_{i}} ∩ S_{s_{j}}, then we can find an element g_{j} ∈ Γ(X_{s_{j}}, O_{X}) such that

f | X_{i} ∩ X_{j} = \frac{g_{j}}{s_{i}^{d_{j}}} .

We can assume that all these d_{j} are equal to a fixed number d and now we can niew \frac{1}{s_{j}^{d}} as a section in

r_{L}^{∗}(i^{∗}(O_{ℙ_{A}^{n}}(−dX_{j}))

(see ???). Hence we conclude that the section f can be extended to a section in

H^{0}(X_{s_{i}} ∪ X_{s_{j}}, L ⊗ H_{0}^{⊗d}).

But if we compare two such extensions on the intersections, they become equal, maz be we have to pass to a still higher power of the bundle. But finally we conclude that f extends to a section in H^{0}(X, L ⊗ H^{⊗d}).

Čech cohomology of quasicoherent sheaves

We want to apply Čzech cohomology to the following situation. Let A be a ring and let us put S = Spec(A). We consider a separated scheme

\begin{matrix}X \\ ↓ & π \\ S \end{matrix} ,

recall that this means that for any two affine open subschemes U, V ⊂ X we have that U ∩ V is again affine and O_{X}(U ∩ V) is generated by the images of O_{X}(U) and O_{X}(V) in O_{X}(U ∩ V). We have seen in the section on projective spaces that ℙ_{S}^{n} → S is separated and from this we concluded that any quasiprojective scheme

\begin{aligned}X & ↪ & ℙ_{s}^{n} \\ ↘ & & ↙ \\ & S \end{aligned}

is separated too.

We consider quasicoherent sheaves F′, F, … on X. We choose a covering of our scheme X by affine subschemes: 𝔘 = {U_{i}}_{i∈I}. We have defined the cohomology groups

H^{q}(X, 𝔘, F) q = 0, 1, …

using the Čzech-complex. We have seen that H^{0}(X, 𝔘, F) = F(X).

These cohomology groups are A- modules. It is also clear that these cohomology groups depend functorially on F, if we have an O_{X}-module homomorphism of quasicoherent sheaves φ : F → F′ then it induces a morphism on the cohomology groups.

We want to show

Theorem:(1) The cohomology groups do not depend on the choice of the covering: Given another covering 𝔙 we can construct a functorial isomorphism

H^{•}(X, 𝔘, F) ⟶^{∼} H^{•}(X, 𝔙, F).

(2) Any short exact sequence of quasicoherent sheaves

0 → F′ → F → F′′ → 0

gives rise to a long exact sequence of cohomology groups

0 → H^{0}(X, 𝔘, F) → H^{0}(X, 𝔘, F) → H^{0}(X, 𝔘, F′′) →
→ H^{1}(X, 𝔘, F′) → H^{1}(X, 𝔘, F) → H^{1}(X, 𝔘, F′′) →
→ H^{2}(X, 𝔘, F′) → H^{2}(X, 𝔘, F) → H^{2}(X, 𝔘, F′′) →

which depends functorially on the sequence.

(3) If X is an affine scheme then

H^{q}(X, 𝔘, F) = \{ \begin{matrix}F(X) & for q = 0 \\ 0 & for q > 1. \end{matrix}

Before I come to the proof I want to make comments:

It is clear that (3) is actually a consequence of (1) since we can use the covering X = ∪_{{i}}U_{i} where U_{i} = X. For this covering (3) is clear if we use the alternating Čzech complex. But it is important for the understanding that one sees that (3) should be true for philosophical reasons. If we have X = Spec(B) then the quasicoherent sheaves on X are the sheaves M̃ obtained from the B-modules M. We have seen that we may recover M from M̃ just as the modules of global sections: M = M̃(X) (see section on affine schemes). Now it is clear that an exact sequence of B-modules gives rise to an exact sequence of sheaves. This is the flatness of the localization. But if we have an exact sequence of quasicoherent sheaves

0 → M̃′ → M̃ → M̃′′ → 0

then the sequence

0 → M′ → M → M′′ → 0

must also be exact. If it were not exact we would get an exact sequence

0 → M′ → M → M′′ → N → 0

with N = Ñ(X) and hence M̃ → M̃′′ would not be surjective.

The reason for introducing the cohomology groups is that they should control the deviation from exactness of the functor F → H^{0}(X, F). (That is also the reason why these functors are called the right derived functors of H^{0}). But for X affine the functor F → H^{0}(X, F) is exact on quasicoherent sheaves, we do not need the higher cohomology groups.

Now I come to the proof of the theorem. We start discussing (2). An exact sequence of quasicoherent sheaves 0 → F′ → F → F′′ → 0 gives rise to a sequence of complexes

0  0  0
↓  ↓  ↓
0 → C^{0}(X, 𝔘, F′) → C^{1}(X, 𝔘, F′) → C^{2}(X, 𝔘, F′) →
↓  ↓  ↓
0 → C^{0}(X, 𝔘, F) → C^{1}(X, 𝔘, F) → C^{2}(X, 𝔘, F) →
↓  ↓  ↓
0 → C^{0}(X, 𝔘, F′′) → C^{1}(X, 𝔘, F′′) → C^{2}(X, 𝔘, F′′) →
↓  ↓  ↓
0  0  0

The first thing we check is that the vertical complexes are exact. To see this we recall that for any of the three sheaves

C^{q}(X, 𝔘, G) = ∏_{(i_{0},...,i_{q})∈I^{q+1}} G(U_{i_{0},...,i_{q}})

and hence we have to check the exactness of (we turn by 90^{∘})

0 → F′(U_{i_{0},...,i_{q}}) → F(U_{i_{0},...,i_{q}}) → F′′(U_{i_{0},...,i_{q}}) → 0

for any (i_{0}, …, i_{q}) ∈ I^{q+1}. But since our scheme is separated we know that U_{i_{0},...,i_{q}} = U_{i_{0}} ∩ … ∩ U_{i_{q}} is affine and now it follows from the comment above that the sequence is exact.

The horizontal lines compute the cohomology and the long exact cohomology sequence is now obtained by the standard argument in homological algebra.

Now it is clear that we can prove (3) if we are able to prove the effaceability of our functor. his means that for any quasicoherent sheaf F on an affine scheme X we can find an embedding

F ↪ G

into a quasicoherent sheaf G such that

H^{q}(X, 𝔘, F) → H^{q}(X, 𝔘, G)

is the zero map for all q ≥ 1.

If we succeed in proving the existence of such a G for any F then we get an exact sequence

0 → H^{0}(X, 𝔘, F) → H^{0}(X, 𝔘, G) → H^{0}(X, 𝔘, G/F) → H^{1}(X, 𝔘, F) → 0.

The first three terms are modules of global sections of a quasicoherent sheaf on an affine scheme and this gives us that H^{0}(X, 𝔘, G) → H^{0}(X, 𝔘, G/F) is surjective and hence H^{1}(X, 𝔘, F) = 0 for any F. Of course we can apply this to G/F and we get also H^{1}(X, 𝔘, G/F) = 0. This implies H^{2}(X, 𝔘, F) = 0 for any F and so on.

To construct such a G we start with a little

Lemma: The scheme X is still affine and let us assume that in our covering 𝔘 = {U_{i}}_{i∈I} there is a member y ∈ I with U_{y} = X. Then we have H^{q}(X, 𝔘, F) = 0 for all q ≥ 1.

Proof: Let us assume we have a cocycle

c = (c_{i_{0},...,i_{q}})_{(i_{0},...,i_{q})} ∈ Z^{q}(X, 𝔘, F).

We construct a b ∈ C^{q−1}(X, 𝔘, F) by

b_{i_{0},...,i_{q−1}} = c_{y,i_{0},...,1_{q−1}}.

We have to observe that

U_{i_{0},...,i_{q−1}} = U_{y,i_{0},...,i_{q−1}}.

Then

\begin{matrix}(db)_{i_{0},...,i_{q}} = Σ(−1)^{ν}b_{i_{0},...,î_{ν},...,i_{q}} \\ = Σ(−1)^{ν}c_{y,i_{0},...,î_{ν},...,i_{q}} = −(dc)_{y,i_{0},...,i_{q}} + c_{i_{0},...,i_{q}} = c_{i_{0},...,i_{q}}. \end{matrix}

We come back to our problem to construct G. For any of the open sets U_{i} ⊂ X we construct a sheaf

F_{U_{i}} : V → F(U_{i} ∩ V)

on X. This sheaf is quasicoherent because if V is affine then it is obtained from the O_{X}(V)-module F(U_{i} ∩ V). We consider the cohomology groups

H^{q}(X, 𝔘, F_{U_{i}})

and to do this we look at the Čzech-complex.

C^{q}(X, 𝔘, F_{U_{i}}) = ∏_{(i_{0},...,i_{q})} F_{U_{i}}(U_{i_{0}} ∩ … ∩ U_{i_{q}}).

But then it is clear that this is isomorphic to the Čzech complex

C^{q}(U_{i}, 𝔘 ∩ U_{i}, F)

which computes the cohomology

H^{q}(U_{i}, 𝔘 ∩ U_{i}, F).

But now U_{i} itself is a member of the covering hence our lemma implies

H^{q}(U_{i}, 𝔘 ∩ U_{i}, F) = 0

for q ≥ 1.

We have a map

F → F_{U_{i}}

induced by the restriction

F(V) → F_{U_{i}}(V) = F(U_{i} ∩ V)

and hence we get an injection

F → F_{𝔘} = ∏_{i∈I} F_{U_{i}}

and it is clear that G = F_{𝔘} does the job.

Hence we have proved (3). But then it is also clear how we can prove (1).

I want to investigate the cohomology of the sheaves O_{ℙ^{n}}(r) on ℙ^{n}/S where S = Spec(A). Our main tool will be the exact sequences obtained from

\begin{aligned}0 & → O_{ℙ^{n}}(X_{0}) & → O_{ℙ^{n}} & → O_{ℙ^{n−1}} & → 0 \\ & ∥ \\ & O_{ℙ^{n}}(−1) \end{aligned} .

tensoring this with O_{ℙ^{n}}(r) we get

0 → O_{ℙ^{n}}(r − 1) → O_{ℙ^{n}}(r) → O_{ℙ^{(n−1)}}(r) → 0

for all integers r.

In the section on locally free sheaves we had the chain of inclusions

O_{ℙ^{n}} ↪ O_{ℙ^{n}}(1) ↪ O_{ℙ^{n}}(2) ↪ … O_{ℙ^{n}}(d) …

We consider the limit

O_{ℙ^{n}}(∞) = lim_{d→∞} O_{ℙ^{n}}(d).

We get

O_{ℙ^{n}}(∞)(ℙ^{n}) = O_{ℙ^{n}}(U_{0})

and hence we have for any open set V ⊂ ℙ^{n}

O_{ℙ^{n}}(∞)(V) = O_{ℙ^{n}}(U_{0} ∩ V).

This implies that

H^{i}(ℙ^{n}, O_{ℙ^{n}}(∞)) = 0 for all i > 0

simply because this cohomology is the same as H^{i}(U_{0}, O_{ℙ^{n}}). (This argument is the same as the one to prove the effaceability of the Čzech cohomology).

We proceed by analyzing the information provided by the long exact cohomology sequence

\begin{aligned}0 & → H^{0}(ℙ^{n}, O_{ℙ^{n}}(r − 1)) & → H^{0}(ℙ^{n}, O_{ℙ^{n}}(r)) & → H^{0}(ℙ^{n−1}, O_{ℙ^{n−1}}(r)) & ⟶^{δ} \\ & H^{1}(ℙ^{n}, O_{ℙ^{n}}(r − 1)) & → H^{1}(ℙ^{n}, O_{ℙ^{n}}(r)) & → H^{1}(ℙ^{n−1}, O_{ℙ^{n−1}}(r)) & ⟶^{δ} \end{aligned}

We know that

H^{0}(ℙ^{n}, O_{ℙ^{n}}(r)) → H^{0}(ℙ^{n−1}, O_{ℙ^{n−1}}(r))

is surjective unless we have n = 1 and r < 0. This follows from the Exercise (1) above. The map between the modules of homogeneous polynomials is given by putting X_{0} = 0. Hence we get for n = 1

0 → H^{0}(ℙ^{1}, O_{ℙ^{1}}(r − 1)) → H^{0}(ℙ^{1}, O_{ℙ^{1}}(r)) → H^{0}(ℙ^{0}, O_{ℙ^{0}}(r)) →
H^{1}(ℙ^{1}, O_{ℙ^{1}}(r − 1)) → H^{1}(ℙ^{1}, O_{ℙ^{1}}(r))  → 0  →
H^{2}(ℙ^{1}, O_{ℙ^{1}}(r − 1)) → H^{2}(ℙ^{1}, O_{ℙ^{1}}(r))  → 0  → .

We turn our attention to H^{i} with i ≥ 2. We see that for any r the map

H^{i}(ℙ^{1}, O_{ℙ^{1}}(r − 1)) → H^{i}(ℙ^{1}, O_{ℙ^{1}}(r))

is injective for i ≥ 2. Since H^{i}(ℙ^{n}, O_{ℙ^{n}}(∞)) = 0 for i ≥ 1 a simple argument shows that this implies

H^{2}(ℙ^{1}, O_{ℙ^{1}}(r)) = H^{3}(ℙ^{1}, O_{ℙ^{1}}(r)) = … = 0.

(Look at čzech cohomology and show that a class in H^{i}(ℙ^{1}, O_{ℙ^{1}}(r)) which vanishes in H^{i}(ℙ^{1}, O_{ℙ^{n}}(∞)) must already vanish in some H^{i}(ℙ^{1}, O_{ℙ^{1}}(r + m)) with m >> 0).

The same argument applies for H^{1}(ℙ^{1}, O_{ℙ^{1}}(r)) as long as we have surjectivity in the first line, i.e. r ≥ 0. Hence we conclude

H^{1}(ℙ^{1}, O_{ℙ^{1}}(r)) = 0

for all r ≥ 0. For r = −1 our sequence becomes

0 → 0 → 0 → H^{0}(ℙ^{0}, O_{ℙ^{0}}(−1)) → H^{1}(ℙ^{1}, O_{ℙ^{1}}(−2)) → 0
∥
A

and hence

H^{1}(ℙ^{1}, O_{ℙ^{1}}(−2)) ≃ A.

For r ≤ −2 we get a short exact sequence

A
∥
0 → H^{0}(ℙ^{0}, O_{ℙ^{0}}(r)) ⟶^{δ} H^{1}(ℙ^{1}, O_{ℙ^{1}}(r − 1)) → H^{0}(ℙ^{1}, O_{ℙ^{1}}(r)) → 0

and hence: For r ≤ −2

H^{1}(ℙ^{1}, O_{ℙ^{1}}(r)) = A^{(−r−1)}.

Now we state the theorem.

Theorem:For n ≥ 1 we have

H^{0}(ℙ^{n}, O_{ℙ^{n}}(r)) = Module of homogeneous polynomials of degree r in A[X_{0}, …, X_{n}].

Especially if it is zero for r < 0

H^{i}(ℙ^{n}, O_{ℙ^{n}}(r)) = 0 for 0 < i < n or i > n.

H^{n}(ℙ^{n}, O_{ℙ^{n}}(−r)) = \{ \begin{matrix}0 & for r < n + 1 \\ A & for r = n + 1 \\ A^{m} & for r ≥ n + 1 \end{matrix}

where m = rank of H^{0}(ℙ^{n}, O_{ℙ^{n}}(−n − 1 + r)).

We have proved this for n = 1. We get easily by induction that

H^{i}(ℙ^{n}, O_{ℙ^{n}}(r − 1)) → H^{i}(ℙ^{n}, O_{ℙ^{n}}(r))

is injective for all 0 < i ≤ n − 1 and i > n. Hence by the same argument as above we find H^{i}(ℙ^{n}, O_{ℙ^{n}}(r)) = 0 for all i ≠ 0, n. For i = n we get

0 → H^{n−1}(ℙ^{n−1}, O_{ℙ^{n−1}}(r)) → H^{n}(ℙ^{n}, O_{ℙ^{n}}(r − 1)) → H^{n}(ℙ^{n}, O_{ℙ^{n}}(r)) → 0.

For r > −n the first term is zero hence we get an isomorphism between the second and third term. This implies again by the limit argument

H^{n}(ℙ^{n}, O_{ℙ^{n}}(r)) = 0

for r > −n. For r = −n we get an isomorphism

H^{n−1}(ℙ^{n−1}, O_{ℙ^{n−1}}(−n)) ⟶^{∼} H^{n}(ℙ^{n}, O_{ℙ^{n}}(−n − 1−))

and for still smaller r we get

\begin{matrix}0 & → H^{n−1}(ℙ^{n−1}, O_{ℙ^{n−1}}(−n − s)) → H^{n}(ℙ^{n}, O_{ℙ^{n}}(−n − 1 − s)) → \\ H^{n}(ℙ^{n}, O_{ℙ^{n}}(−n − s)) → 0 \end{matrix}

and one checks easily that the recursion for the rank as a function of s is the same as for H^{0}(ℙ^{n}, O_{ℙ^{n}}(s)).

Cohomology of coherent sheaves.

From now on we assume that our groundring A is noetherian. A quasicoherent sheaf F on ℙ^{n} is coherent if one of the following conditions is fulfilled:

(1) For all U_{i} the O_{ℙ^{n}}(U_{i})-module F(U_{i}) is finitely generated.

(2) For any affine open set V ⊂ ℙ^{n} the O_{ℙ^{n}}(V)- module F(V) is finitely generated.

(3) The stalks F_{𝔭} are finitely generated O_{ℙ^{n},𝔭}-modules for all points 𝔭 ∈ ℙ^{n}.

We leave it as an exercise to prove that these conditions are equivalent.

In this section I want to prove the celebrated coherence theorem.

Theorem: Let A be a noetherian ring and let F be a coherent sheaf on ℙ^{n}/ Spec(A) = ℙ^{n}/S. Then the A-modules H^{i}(ℙ^{n}, F) are finitely generated. They are zero for i > n.

The proof requires a series of steps. Again we start from the exact sequence

0 → O_{ℙ^{n}}(r − 1) → O_{ℙ^{n}}(r) → O_{ℙ^{n−1}}(r) → 0.

We will consider the tensor products F(r) = F ⊗ O_{ℙ^{n}}(r), but we have to be aware of a minor technical complication: Since the tensor product is not an exact functor the sequence

0 → O_{ℙ^{n}}(X_{0}) → O_{ℙ^{n}} → O_{ℙ^{n−1}} → 0

yields only an exact sequence

0 → G → O_{ℙ^{n}}(X_{0}) ⊗ F → O_{ℙ^{n}} ⊗ F → O_{ℙ^{n−1}} ⊗ F → 0

where G is the kernel of the next arrow so the exactness is true by definition. Since O_{ℙ^{n}}(r) is locally free, we get for all r ∈ ℤan exact sequence

0 → G(r) → O_{ℙ^{n}}(r − 1) ⊗ F → O_{ℙ^{n}}(r) ⊗ F → O_{ℙ^{n−1}} ⊗ F → 0.

The first step in the proof of the coherence theorem will be the

Theorem: Under the assumptions of the theorem above we can find an r_{0} > 0 (depending on F) such that for all r ≥ r_{0} H^{0}(ℙ^{n}, F(r)) is a finitely generated A-module and H^{i}(ℙ^{n}, F(r)) = 0 for all i > 0.

The proof of this second theorem will basically be obtained by induction on n. We examine our sequence above. The first thing we have to do is to find out how the sheaf G looks like. Here we observe that locally we are in the following situation: We have an A-algebra B an element f ∈ B and we consider the sequence of B modules

0 → fB → B → B/fB → 0.

This will be tensorized by a B-module N (the local F) and we get

fB ⊗ N → N → N/fN → 0

and the first arrow is f ⊗ n ↦ fn and hence the kernel is exactly the annihilator of f in N. This means that we get

0 → Ann_{N}(f) → Bf ⊗ N → N → N/fN → 0.

Now it is clear that Ann_{N}(f) is an A/fA-module (which is of course finitely generated).

This implies for our sequence above that the kernel G is in fact a O_{ℙ^{n−1}}-module sheaf, hence it is a coherent sheaf on the hyperplane at infinity.

We break the four term exact sequences into two pieces

\begin{aligned}0 & → G(r) → O_{ℙ^{n}}(r − 1) ⊗ F → F′(r − 1) → 0 \\ 0 & → F′(r − 1) → O_{ℙ^{n}}(r) ⊗ F → O_{ℙ^{n−1}}(r) ⊗ F → 0. \end{aligned} .

Assuming that we have proved our theorem of ℙ^{n−1} we get

H^{i}(ℙ^{n}, G(r)) = H^{i}(ℙ^{n−1}, G(r)) = 0 for all r >> 0

and

H^{i}(ℙ^{n−1}, O_{ℙ^{n−1}}(r) ⊗ F) = 0 for all r >> 0, i > 0

Hence we get

H^{i}(ℙ^{n}, O_{ℙ^{n}}(r − 1) ⊗ F) = H^{i}(ℙ^{n}, F′(r − 1))

from the first exact sequence. We substitute this into the long exact sequence for the second short exact sequence of sheaves and get

\begin{matrix}H^{0}(ℙ^{n}, O_{ℙ^{n}}(r) ⊗ F) & → H^{0}(ℙ^{n}, O_{ℙ^{n−1}}(r) ⊗ F) & → \\ H^{1}(ℙ^{n}, O_{ℙ^{n}}(r − 1) ⊗ F) & → H^{1}(ℙ^{n}, O_{ℙ^{n}}(r) ⊗ F) & → 0 \\ · \\ · \\ · \\ → H^{i}(ℙ^{n}, O_{ℙ^{n}}(r − 1) ⊗ F) & ⟶^{∼} H^{i}(ℙ^{n}, O_{ℙ^{n}}(r) ⊗ F) & → \end{matrix} .

Now the limit argument saying that H^{i}(ℙ^{n}, O_{ℙ^{n}}(∞)⊗F) = 0 implies that H^{i}(ℙ^{n}, O_{ℙ^{n}}(r)⊗ F) = 0 for i ≥ 2. But for i = 1 we still are not finished because we do not know whether the arrow in degree 0 is surjective.

To get the case i = 1 we use a different method.

We need the following

Lemma: For any coherent sheaf F on ℙ^{n}/S we can find an integer r_{0} > 0 such that for all r ≥ r_{0} the sheaf F(r) = F ⊗O_{ℙ^{n}}(r) is generated by a finite number of global sections. In other words we can find s_{1}, …, s_{N} ∈ H^{0}(ℙ^{n}, F(r)) such that for any point 𝔭 ∈ ℙ^{n} the stalk F(r)_{𝔭} is generated by the restrictions of these sections to this stalk as an O_{ℙ^{n},𝔭}-module.

Proof : We consider the restriction of F to the open sets U_{i}, we know that F|U_{i} = F̃(U_{i}). The O_{ℙ^{n}}(U_{i})-modules F(U_{i}) are finitely generated. We write by s_{i1}, …, S_{iν_{i}} for these generators. They also generate the stalks in the points inside U_{i}. We have the embedding O_{ℙ^{n}} ⊂ O_{ℙ^{n}}(−dH_{i}) and this induces a morphism among sheaves F⊗O_{ℙ^{n}} → F⊗O_{ℙ^{n}}(−dH_{i}). This morphism is an isomorphism on the stalks inside U_{i}. If we pass to the limit we see that F ⊗O_{ℙ^{n}}(−∞H_{i})(ℙ^{n}) = F ⊗O_{ℙ^{n}}(U_{i}) and hence we see that all the sections s_{iν} extend to sections in the limit.

But then these extensions must already lie in some F ⊗ O_{ℙ^{n}}(−rX_{i}) and since we have O_{ℙ^{n}}(−rX_{i})→^{∼}F(r) = F ⊗ O_{ℙ^{n}}(r) we see that at least the stalks inside U_{i} can be generated by global sections in F(r). Since this is so for any i the lemma follows.

It follows from the Lemma that we get a morphism of sheaves

O_{ℙ^{n}}^{N} → F(r)

simply by sending (0, …, 1, …, 0) → s_{i}, . . .. This morphism is surjective and has a kernel G which again is coherent. We get an exact sequence which we still can twist by O_{ℙ^{n}}(s) for an arbitrary integer s. Hence we obtain

0 → G(s) → (O_{ℙ^{n}}(s))^{N} → F(r + s) → 0

which gives us for s >> 0

0 → H^{0}(ℙ^{n}, G(s)) → H^{0}(ℙ^{n}, O_{ℙ^{n}}^{N}(s)) → H^{0}(ℙ^{n}, F(r + s)) ⟶^{δ}
→ H^{1}(ℙ^{n}, G(s))  → 0  → H^{1}(ℙ^{n}, F(r + s)) ⟶^{δ}
H^{2}(ℙ^{n}, G(s))  → 0
·
·
·
0  H^{i}(ℙ^{n}, F(r + r)) ⟶^{δ}
→ H^{i+1}(ℙ^{n}, G(s))  ,

where all the boundary maps δ except the one in degree zero are isomorphisms.

We know already that H^{q}(ℙ^{n}, G(s)) = 0 for q ≥ 2 and s >> 0. Hence we get H^{i}(ℙ^{n}, F(r+ s)) = 0 for i ≥ 1 and r >> 0, s >> 0 which of course implies the second assertion in the theorem.

To get the first assertion we simply observe that in our exact sequence above also H^{1}(ℙ^{n}, G(s)) = 0 for s >> 0 and hence we get

H^{0}(ℙ^{n}, O_{ℙ^{n}}^{N}(s)) → H^{0}(ℙ^{n}, F(r + s))

will be surjective for r + s >> 0. Then the assertion concerning the finite generation follows from our computation of the cohomology of the sheaves O_{ℙ^{n}}(s).

Now the coherence theorem becomes easy. Given our coherent sheaf F we produce an acyclic resolution. We embed F → F(r) such that H^{i}(ℙ^{n}, F(r)) = 0 for i ≥ 0 and H^{0}(ℙ^{n}, F(r)) is a finitely generated A-module. We get an exact sequence

0 → F → F(r) → F_{1} → 0

and do the same thing to F_{1}. This yields an exact sequence of coherent sheaves

0 → F → F(r) → F_{1}(r_{1}) → F_{2}(r_{2}) →

which we write as

0 → F → F^{1} → F^{2} → … F^{m} → …

where the F^{i} are acyclic and H^{0}(ℙ^{n}, F^{i}) is a finitely generated A-module. Hence we have constructed an acyclic resolution of F by coherent sheaves. Taking global sections over ℙ^{n} we get a complex H^{0}(ℙ^{n}, F^{•}) of A-modules. I refer to the simple principle in the section on homological algebra. This principle tells us that this complex computes the cohomology groups. Hence

H^{•}(ℙ^{n}, F) = H^{•}(H^{0}(ℙ^{n}, F^{•}))

and since the individual members of the complex are finitely generated A-modules the theorem follows.

The reader should observe that the two parts of the proof have an overlap. In the first part we fail to say something in degree 1, in the second we cannot say what happens in large degrees (≥ n + 1). If we would allow to work with the alternating čzech complex, we would see H^{i}(ℙ^{n}, F) = 0 for i > n directly.

Nevertheless, the above discussion shows that after we have defined the cohomology by using Čzech complexes we can very successfully treat them if we only use their properties.

Base change:

If we have a projective scheme

\begin{aligned}X & ↪ & ℙ_{S}^{n} \\ ↘ & & ↓ \\ & S \end{aligned}

or more generally a coherent sheaf F on ℙ_{S}^{n} then the intuitive idea is that this should be viewed as a family of projective schemes parametrized by the points in S or a family of sheaves parametrized by the points in s ∈ S.

To be a little bit more precise, for any morphism of the spectrum of a field s : Spec(K) → S we have the base change

X ⟵ X ×_{S} Spec(K) = X_{s} ↓  ↓ S ⟵  Spec(K) : s .

The natural question is how certain data attached to the fibres X_{s} vary if we vary s. We may for instance ask how the dimensions of the irreducible components of the fibres X_{s} behave.

I give an example which shows that this behavior may be rather irregular. We consider the affine space over a field k, i.e. S = Spec(k[Y_{1}, …, Y_{n}]). In the projective space ℙ^{n−1}/S we define a subscheme Z by the system of homogeneous (even linear equations)

Y_{i}X_{j} − Y_{j}X_{i} = 0.

(The X_{1}, …, X_{n} are the homogeneous variables for ℙ^{n−1}). Now it is clear that for any s : Spec(K) → S, i.e. a homomorphism φ : k[Y_{1}, …, Y_{n}] → K, the fibre consists of a single point if (φ(Y_{1}), …, φ(Y_{n})) ≠ (0), because in this case the solutions are given by the line determined by (φ(Y_{1}), …, φ(Y_{n})). But if (φ(Y_{1}), …, φ(Y_{n})) = 0 then suddenly the system of equations degenerates to 0X_{i} − 0X_{j} = 0 and the fibre is the full ℙ^{n−1}/ Spec(K).

General educational remark: This process is called ”blowing up a point” and is of considerable importance in algebraic geometry. The intuitive meaning is that the point is replaced by the projective space attached to its tangent space. To see how useful this may be we give two examples.

(1) Let us assume we have the meromorphic function

f(X, Y) = \frac{X + Y}{X − Y}

in the function field of the affine plane S = Spec(k[X, Y]) where k is a field and char(k) ≠ 2. It is regular outside the diagonal X = Y and provides a morphism

f : Spec(k[X, Y])\Δ → Spec(k[T]) = 𝔸_{k}^{1}.

We can extend this to a morphism into ℙ_{k}^{1} if we send the points on the diagonal Δ to infinity as long as we avoid (X, Y) = (0, 0). But in the origin we do not know what to do.

But if we pull back the map to Z then a point in the fibre over (0, 0) tells us in ”which direction” we approach (0, 0) and we can take a ”limit”. This allows us to extend f to a regular morphism f̃: Z → ℙ^{1}.

(2) If we have a plane curve C ⊂ Spec(k[X, Y]), for example something like

f(X, Y) = XY + X^{7} + Y^{28} + … = 0

then it has a so-called ordinary double point at O = (0, 0). The curve has two tangents at O = (0, 0). If we blow up the origin then the inverse image of our curve has two irreducible components, one is the ”proper transform” C̃ and the other is the fibre over O. The curve C̃ meets the fibre in two points (the two tangents) and is now smooth in these two points. We have resolved the singularity.

End of the remark.

We also look at more general base changes which in our situation amounts to choosing an A-algebra A′, i.e. A → A′. Then we can consider

ℙ^{n} ×_{Spec(A)} Spec(A′) = ℙ_{S′}^{n}

and we get a sheaf F′ on ℙ_{S′}^{n} which is the pull-back, in other words F′(U′_{i}) = F(U_{i}) ⊗_{A} A′. This tells us that the Čzech-complex which computes H^{•}(ℙ_{S′}^{n}, F′) is the tensor product of C^{•}(ℙ^{n}, 𝔘, F) ⊗_{A} A′ and hence we get a map between the complexes

C^{•}(ℙ_{S}^{n}, 𝔘, F) → C^{•}(ℙ_{S′}^{n}, 𝔘, F′)

and therefore a map between the cohomology groups

H^{•}(ℙ_{S}^{n}, F) → H^{•}(ℙ_{S′}^{n}, F′).

Here we consider both sides as abelian groups, the left hand side carries the structure of A-modules, the right hand side cohomology groups are A′-modules. It is obvious that we get from the homomorphism above an A′-module homomorphism

H^{•}(ℙ_{S}^{n}, F) ⊗_{A} A′ → H^{•}(ℙ_{S′}^{n}, F).

This map between the cohomology groups is the base change homomorphism. In general we cannot say very much about this base change homomorphism.

We have to make some assumptions, and here we have two options (a) The base change A → A′ is flat.

(b) The sheaf F is flat over A (we say it is A-flat or S = Spec(A)-flat).

We consider the case (a) first. It is much easier. Here we can start from a more general situation, we can drop the assumption that A noetherian and our sheaf F needs only to be quasi-coherent.

Actually we can even assume that we have a quasiprojective scheme

X  ↪  ℙ_{S}^{n}
↘ π  ↙
S = Spec(A)

and a quasicoherent sheaf F on X.

In this case we have

Theorem: If S ← S′ is flat, i.e. if A → A′ is flat then the base change homomorphism is an isomorphism.

To see this we have to show if

K^{•} : 0 → K^{0} → K^{1} → …

is a complex of A-modules and if A → A′ is flat then

H^{•}(K^{•} ⊗_{A} A′) = H^{•}(K^{•}) ⊗_{A} A′.

This is clear since in a given degree i we have

\begin{aligned}0 → Z^{i−1} → K^{i−1} → & B^{i} & → 0 \\ & ↓ \\ & Z^{i} & → 0 \\ & ↓ \\ & H^{i}(K^{•}) \\ & ↓ \\ & 0 \end{aligned} .

Now since Z^{i} is the kernel of d : K^{i} → K^{i+1} we see that

Z^{i} ⊗_{A} A′ = ker(K^{i} ⊗_{A} A′ → K^{i+1} ⊗_{A} A′)

and

B^{i} ⊗_{A} A′ = im(K^{i+1} ⊗_{A} A′ → K^{i} ⊗_{A} A′)

because A → A′ is flat. Hence

H^{i}(K^{•} ⊗ A′) = Z^{i}(K^{•} ⊗_{A} A′)/B^{i}(K^{•} ⊗ A′) = Z^{i} ⊗_{A} A′/B^{i} ⊗_{A} A′

and the last quotient is equal to H^{i}(K^{•} ⊗_{A} A′) which is clear if we tensorize the above vertical sequence by A′ over A.

This rather simple fact has an important consequence. We may consider the special case that A′ = A_{f} is a localization, in that case

S′ = Spec(A_{f}) → S = Spec(A)

is an affine open subscheme of S. We just have seen that

H^{i}(X ×_{S} Spec(A_{f}), F) ⊗ A_{f} = H^{i}(ℙ_{Spec(A))}^{n}F) ⊗ A_{f}

and this means that we can view these cohomology groups as sections of a sheaf

H̃^{i}(ℙ_{Spec(A)}^{n}, F)

on Spec(A). This allows us to drop the assumption that the base S is an affine scheme. For any scheme S we may consider sheaves F on quasicoherent

\begin{matrix}X \\ ↓ π \\ S \end{matrix}

where X over S is quasiprojective. We can cover S by open affine subschemes S = ∪U_{ν} and consider the cohomology groups

H^{i}(X ×_{S} U_{ν}, F × U_{ν})

and from the sheaves H̃^{i}(X ×_{S} U_{ν}, F × U_{ν}) over U_{ν}. The remark above yields that the sheaves restricted to affine open subsets in U_{ν} ∩ U_{μ} are obtained by localization and hence they form a sheaf on S.

These sheaves get a new name. They are called the higher direct images of F under π and are denoted by

R^{i}π_{∗}(F).

Of course R^{0}π_{∗}(F) = π_{∗}(F).

Now we come to the case b). Here we are interested in situations where the fibres or the sheaves vary in a rather gentle manner. The technical concept is that of flatness. We will consider coherent sheaves F on ℙ^{n}/S which are flat over S. I recall that this means that for all U_{i} the A-module F(U_{i}) is flat. We call it S-flat or A-flat. I want to prove

Theorem: If S = Spec(A) and A noetherian and if F is an A-flat coherent sheaf on ℙ^{n}, then we can find an r >> 0 such that

(1) H^{i}(ℙ^{n} ×_{A} A′, F ×S′(r)) = 0 for all i > 0 and all schemes S′ = Spec(A′) → S = Spec(A).

(2) The A module H^{0}(ℙ^{n}, F(r)) is locally free.

(3) For any A → A′ the base change homomorphism

H^{0}(ℙ^{n} ×_{A} A′, F′(r)) ⟶^{∼} H^{0}(ℙ^{n}, F(r)) ⊗_{A} A′

is an isomorphism.

To prove the theorem we use the same ideas as in the section before. We write F(r) for r >> 0 as a quotient of O_{ℙ^{n}}^{N} and we get an exact sequence

0 → G → O_{ℙ^{n}}^{N} → F(r) → 0.

Since F is an A-flat sheaf and since O_{ℙ^{n}} is even free we see that also G must be an A-flat sheaf. Hence we see that this sequence stays exact after any base change

Spec(A) = S ← Spec(A′) = S′

so we can write

0 → G′ → O_{ℙ/S′}^{N} → F′(r) → 0

for any S ← S′. This yields a long exact sequence in cohomology and since we have

H^{i}(ℙ_{S′}^{n}, O_{ℙ_{S}^{n}}^{N}) = 0 for all i > 0

we get isomorphisms

H^{i}(ℙ_{S′}^{n}, F′(r)) ≃ H^{i+1}(ℙ_{S′}^{n}G′).

Since we may also tensorize the sequence still by O_{ℙ^{n}}(r_{1}) we get

\begin{aligned}0 & → H^{0}(ℙ_{S′}^{n}, G′(r_{1})) & → H^{0}(ℙ_{S′}^{n}, O_{ℙ^{n}/S′}^{N}(r_{1})) & → H^{0}(ℙ_{S′}^{n}F′(r + r_{1})) \\ & → H^{1}ℙ_{S′}^{n}G′(r_{1})) & → 0 & → H^{1}(ℙ_{S′}^{n}, F′(r + r_{1})) \\ & → H^{2}ℙ_{S′}^{n}, G′(r_{1})) & → 0 & → \end{aligned} .

Since H^{i}(ℙ_{S′}^{n}, G′(r_{1})) = 0 for i > n we conclude that

H^{i}(ℙ_{S′}^{n}, F′(r + r_{1})) = 0 for all i ≥ n

provided r, r_{1} >> 0 and any S′. But since G is also flat we may apply this to G and get that for all S ← S′

H^{i}(ℙ_{S′}^{n}, G(r_{1} + r_{2})) = 0 for all i ≥ n

provided r_{1}, r_{2} >> 0 and this in turn gives

H^{i}(ℙ_{S′}^{n}, F′(r + r_{1} + r_{2})) = 0 for all i ≥ n − 1

if r, r_{1}, r_{2} >> 0. Eventually we conclude by downwards induction

H^{i}(ℙ_{S′}^{n}, F′(r)) = 0 for all i > 0 and all S ← S′

provided r is sufficiently large. Again we apply this to G and we get for all S ← S′ an exact sequence

0 → H^{0}(ℙ_{S′}^{n}, G′(r_{1})) → H^{0}(ℙ_{S′}^{n}, O_{ℙ_{S′}^{n}} (r_{1})) → H^{0}(ℙ_{S′}^{n}, F(r + r_{1})) → 0

provided r_{1}, r are large enough.

Now we want to prove the assertion (2). To do this we may assume that A is a local ring with maximal ideal 𝔪 and we consider the base change A → A/𝔪. S′ = Spec(A/𝔪).

We consider the following diagram

\begin{aligned}& & A^{M} \\ & & ∥ \\ 0 & → H^{0}(ℙ_{S′}^{n}, G(r_{1})) & → H^{0}(ℙ_{S}, O_{ℙ_{S}^{n}}^{N}(r_{1})) & → H^{0}(ℙ_{S}^{n}, F(r + r_{1})) & → 0 \\ & ↓ & ↓ & ↓ \\ & H^{0}(ℙ_{S}^{n}, G(r_{1})) ⊗ A/𝔪 & → (A/𝔪)^{M} & → H^{0}(ℙ_{S}^{n}, F(r + r_{1})) ⊗ A/𝔪 & → 0 \\ & ↓ ρ_{G} & ∥ & ↓ ρ_{F} \\ 0 & → H^{0}(ℙ_{S′}^{n}, G′(r_{1})) & → H^{0}(ℙ_{S′}^{n}, O_{ℙ_{S′}^{n}} (r_{1}))^{N} & → H^{0}(ℙ_{S′}^{n}, F(r + r_{1})) & → 0 \end{aligned} .

The map ρ_{F} in the lower right corner is surjective. This is true for any A-flat coherent sheaf provided r + r_{1} is large enough. But then we conclude that also ρ_{G} in the lower left corner must be surjective at least if we enlarge r_{1} still further. But then it follows that ρ_{F} is even bijective and hence also ρ_{G}. (This proves assertion (3) for this special base change).

We summarize that for r_{1}, r large enough the two vertical arrows ρ_{G} and ρ_{F} are bijections. But if ρ_{G} is bijective we can conclude the supplemented sequence in the middle is exact

0 → H^{0}(ℙ_{S′}^{n}G′(r_{1})) ⊗ A/𝔪 → (A/𝔪)^{M} → H^{0}(ℙ_{S}^{n}, F(r + r_{1})) ⊗ A/𝔪 → 0

is exact. We pick a basis of (A/𝔪)^{M}

< y_{1}, …, y_{a}, x_{1}, …, x_{b} >

such that the y’s form a basis for

H^{0}(ℙ_{S′}^{n}G′(r_{1})) ⊗ A/𝔪

and the x’s map to a basis of H^{0}(ℙ_{S′}^{n}F′(r + r_{1})) ⊗ A/𝔪. Lifting this to a H^{0}(ℙ_{S}^{n}, O_{ℙ^{n}}^{N}(r_{1})) yields by Nakayama’s lemma a system of a + b = M generators and hence we get a basis of H^{0}(ℙ_{S}^{n}, O_{ℙ^{n}}^{N}(r_{1})). This proves that the modules of sections H^{0}(ℙ_{S}^{n}, F(r)) are locally free A-modules and therefore we proved (2). To prove the assertion (3) we work with the same diagram as above but we replace A′ = A/𝔪 by an arbitrary A-algebra A → A′. But now we know that the horizontal sequence in the middle is exact even if we supplement it by 0 → on the left. This is so because H^{0}(ℙ_{S}^{n}, F(r)) is locally free. Picking the same kind of basis as above for (A′)^{M}, we see that the images of the x_{i} in H^{0}(ℙ_{S}^{n}, F(r_{1} + r)) must form a basis.

The theorem has important consequences. We stick to our assumption that S = Spec(A) is local.

We look at the hyperplane H_{0} at infinity and have the morphism of O_{ℙ^{n}}-sheaves

F → F ⊗ O_{ℙ^{n}}(−rH_{0}).

This needs not to be injective because the tensor product is taken over O_{ℙ^{n}} and our sheaf F is only A-flat. But if we restrict this morphism to U_{0} it becomes an isomorphism hence it is injective on U_{0}. Therefore we get an embedding

F → ⊕_{i=0}^{n}F × O_{ℙ^{n}}(−rH_{i}) = F^{0}

and we know for r >> 0 that a) H^{i}(ℙ^{n} ×_{S} S′, F_{x}^{0}) = 0 for all i > 0 and all S ← S′.

b) H^{0}(ℙ^{n} × S, F^{0}) is a free A-module of finite rank.

c) H^{0}(ℙ^{n}, F^{0}) ⊗ A′ = H^{0}(ℙ^{n} ×_{Spec(A)} Spec(A′), F^{0} ⊗ A′).

Then we get an exact sequence of sheaves

0 → F → F^{0} → H → 0

where H is again flat. We apply the same process to H and get a resolution

0 → F → F^{0} → F^{1} → …

where all the sheaves F^{ν} satisfy a) b) c) above.

Remark: I want to point out that we used the same principle twice. In the section on homological algebra we explained that for the computation of the derived functors of a function we should start from an acyclic resolution. In general such resolutions are very big and for instance injective modules or sheaves are very big and their modules of sections don’t have a chance to be finite in some sense. But in the proof of the coherence theorem we showed that a coherent sheaf F has an acyclic resolution

0 → F → F^{0} → F^{1} →

by coherent sheaves and we showed that the modules of global sections are finitely generated A-modules.

Here we showed that for a coherent flat sheaf F we have an acyclic resolution where the F^{i} satisfy even more requirements.

The appropriate way of formulating this is provided by the language of derived categories.

The derived category here will be the category of complexes of quasicoherent sheaves. We rewrite our resolutions

0 ↓

0 → F → 0 → 0 →
↓ ↓ ↓
0 → F^{0} → F^{1} → F^{2} →

we have here a morphism between the two horizontal complexes. These complexes of sheaves have cohomology groups which are again sheaves we compute the cohomology stalkwise.

The fact that we have a resolution is equivalent to saying that this map between complexes induces an isomorphism in cohomology. But then this map is by the construction of the derived category on isomorphisms, this is built into the construction. Hence we can say:

In the derived category of quasicoherent sheaves a coherent (resp. coherent and A-flat) sheaf is isomorphic to

0 → F^{0} → F^{1} → …

where the F^{i} are acyclic (resp. acyclic after any base change) and have some further properties which I stated above.

Let us call this a flat, universally acyclic resolution. We know that for any S ← S′ the complex

0 → H^{0}(ℙ_{S′}^{n}, F^{′0}) → H^{0}(ℙ_{S′}, F^{′1}) → …

computes the cohomology groups H^{i}(ℙ_{S′}^{n}F′), and we also know that

H^{0}(ℙ_{S′}^{n}, F′) = H^{0}(ℙ_{S}^{n}, F) ⊗_{A} A′.

Hence we see that the above flat universally acyclic resolution gives us a complex of finitely generated free A-modules

K^{•} = 0 → H^{0}(ℙ_{S}^{n}, F^{0}) → H^{0}(ℙ_{S}^{n}, F^{1}) → …

such that for any S ← S′, i.e. A → A′ the complex

K^{•} ⊗_{A} A′

computes the cohomology groups

H^{i}(ℙ_{S′}^{n}, F′).

From this we can obtain all the information about the behavior of the cohomology if we vary the fibre by simple linear algebra.

Let us assume that A is local with maximal ideal 𝔪. Let 𝔭 be another prime ideal in A let A_{𝔭} be localization at 𝔭 and 𝔪_{𝔭} the maximal ideal. We have

A ⟶ A_{𝔭}
↓
A/𝔪  .

We assume that we have a flat sheaf F on ℙ_{Spec(A)}^{n} and we have the complex

0 → H^{0}(ℙ_{Spec(A)}^{n}, F^{0}) → H^{0}(ℙ_{Spec(A)}^{n}, F^{1}) ….

If we tensorise this complex by A/𝔪 we get the cohomology of

H^{i}(ℙ_{Spec(A/𝔪)}^{n}, F ⊗ A/𝔪)

and if we tensorise by A_{𝔭}/𝔪_{𝔭} we get the cohomology H^{i}(ℙ_{Spec(A_{𝔭})/𝔪_{𝔭}}^{n}F ⊗ A_{𝔭}/𝔪_{𝔭}).

Now we have to remember some linear algebra. If we have a map

φ : A^{M_{1}} → A^{M_{2}}

and reduce it mod 𝔪 then we can find a basis

x_{1}, …, x_{a}, y_{1}, …, y_{b}

of (A/𝔪)^{M_{1}} such that the φ(x_{i}) go to zero and the φ(y_{i}) are linearly independent in (A/𝔪)^{M_{2}}.

We lift this to a basis

x̃_{1}, …, x̃_{a}, ỹ_{1}, …, ỹ_{b}

in A^{M_{1}} and the φ(ỹ_{1}), …, φ(ỹ_{b}) will form a part of a basis in A^{M_{2}}. With respect to these two bases we write the matrix for φ and get

C = Mat(φ) = ( \begin{matrix}1 \\ . & . & . \\ & & 1 \\ ∗ & ∗ & ∗ & B \end{matrix} )

where B is a matrix with coefficients in 𝔪, the ∗ are in 𝔪, and all other entries are zero.

Now this matrix has rank equal to b if we map it to A/𝔪 but its rank may become bigger if we map it to A_{𝔭/𝔪}.

Actually we have for any homomorphism of A into a field K

Ψ : A → K

Rank(Ψ(C)) = b + Rank(Ψ(B)).

This implies

dim_{A_{𝔭}/𝔪_{𝔭}} ker((A_{𝔭}/𝔪_{𝔭})^{M_{1}} → (A_{𝔭}/𝔪_{𝔭})^{M_{2}})) ≤ dim_{A/𝔪} ker((A/𝔪)^{M_{1}} → (A/𝔪)^{M_{2}}) dim_{A_{𝔭}/𝔪_{𝔭}} Im((A_{𝔭}/𝔪_{𝔭})^{M_{1}} → (A_{𝔭}/𝔪_{𝔭})^{M_{2}}) ≥ dim_{A/𝔪} Im((A/𝔪)^{M_{1}} → (A/𝔪)^{M_{2}}) .

This applied to our complex yields that the dimensions of the boundaries drop and the dimensions of the cycles go up if we specialize 𝔭 → 𝔪. (See Chap. on aff. schemes, sect. Zariski topology.)

Hence we get

Semicontinuity:

If F is a coherent A-flat sheaf on ℙ^{n}(Spec(A)) and if 𝔪 ⊃ 𝔭 are two prime ideals then

dim_{A/𝔪}H^{i}(ℙ_{Spec(A/𝔪)}^{n}, F ⊗ A/𝔪) ≥ dim_{A_{𝔭}/𝔪_{𝔭}}H^{i}(ℙ_{Spec(A_{𝔭}/𝔪_{𝔭})}^{n}, F ⊗ A_{𝔭}/𝔪_{𝔭}).

Usually one finds this in a slightly different form in the literature. We consider an arbitrary noetherian base scheme S and a S-flat coherent sheaf F on a projective scheme π : X → S. Then we consider the sheaves R^{i}π_{∗}(F) on S. We consider the functions s → dim_{k(s)}H^{i}(X_{s}, F_{s}). Then this function is upper semicontinous as a function in s with respect to the Zariski topology on S. I find it easier to remember that the function may jump up if we specialize from a point s → s′. But this formulation is obviously the same as the one above because the assertion is local on S.

Invariance of the Euler characteristic

Let π : X → S be a projective scheme over a noetherian base scheme S and let F be a S flat coherent sheaf. Then the function

s ↦ χ(X_{s}, F) = ∑_{i=0}^{d}(−1)^{i}dim_{k(s)}H^{i}(X_{s}, F_{s})

is constant. Here d is large enough such that the cohomology groups vanish in degrees beyond d.

Local constancy:

We say that the direct image R^{i}π_{∗}(F) is locally constant at s ∈ S if this sheaf is locally free at s and if furthermore the base change homomorphism R^{i}π_{∗}(F)_{s} ⊗_{O_{X,s}} k(s)→^{∼}H^{i}(X_{s}, F_{s}) is an isomorphism. To see under which conditions this will be the case we switch to our previous notations, we assume that S = Spec(A) where A is noetherian local and s = 𝔪. Of course we should assume that F is flat over A then we are in our previous situation since also assume that π is projective.

It is quite clear what we have to do if we want to prove local constancy. We have to look at the complex

H^{i−1}(X, F^{i−1}) →^{d^{i−1}} H^{i}(X, F^{i}) →^{d^{i}} H^{i+1}(X, F^{i+1})

which is obtained from an universally acyclic resolution. We apply our linear algebra arguments and we see that we get local constancy if for the boundary maps d^{i−1}, d^{i} the 𝔪 valued matrices B_{i−1}, B_{i} are actually equal to zero. If we assume that the ring A is reduced we can conclude that these matrices must be zero if we have

dim_{A/𝔪}H^{i}(ℙ_{Spec(A/𝔪)}^{n}, F ⊗ A/𝔪) = dim_{A_{𝔭}/𝔪_{𝔭}}H^{i}(ℙ_{Spec(A_{𝔭}/𝔪_{𝔭})}^{n}, F ⊗ A_{𝔭}/𝔪_{𝔭})

for any minimal prime ideal 𝔭. This is clear because if one of these B_{j} is not zero then it is not zero in some residue field A_{𝔭}/𝔪_{𝔭} for a minimal prime ideal 𝔭 and the the dimension of the cohomology group in the fibre over this 𝔭 must necessarily drop because either the rank of the modules of boundaries will go up or the rank of the group of cycles drops.

Hence we see that we have local constancy for R^{i}π_{∗}(F) in a point s = 𝔪 if the local ring at s is reduced and if the rank of the cohomology stays constant in an open neighborhood of s.

Chapter IX

Curves and the theorem of Riemann Roch

A curve over field k is a scheme C/k of finite type over k all whose irreducible components are of dimension 1. In other words our scheme has a finite covering by affine schemes U_{i} = Spec(O_{C}(U_{i})) where the O_{C}(U_{i}) are finitely generated k-algebras and dim O_{C}(U_{i}) = 1.

To give simple examples we can take a non zero homogeneous polynomial of degree d say

f(x, y, z) = ∑ a_{ν_{1}ν_{2}ν_{2}}x^{ν_{1}}y^{ν_{2}}z^{ν_{3}} a_{ν_{1}ν_{2}ν_{3}} ∈ k, ν_{1} + ν_{2} + ν_{3} = d.

Then the ideal (f) defines a curve

C = V (f)  ↪  ℙ^{2}
↘  ↙
Spec(k)

To see that this defines a curve we restrict to one of the affine planes which cover ℙ^{2}. This means we put one of the variables equal to 1 and divide the polynomial ring in the remaining 2 variables by the resulting polynomial. We have to show that the irreducible components of C intersected with this plane are of dimension one. This intersection is non empty if and only if this resulting polynomial is not constant. But then the irreducible components correspond to the minimal prime ideals containing the polynomial and these are of height one (Krull Hauptidealsatz) and hence the quotient ring by this ideal is of dimension one

Polynomial rings over fields are factorial (See ???). Therefore a principal ideal defined by an irreducible polynomial F (X_{1}, …, X_{n}) ∈ k[X_{1}, …, X_{n}] is a prime ideal. In our case here we see that an irreducible polynomial f defines an irreducible curve C. It may happen that f is irreducible but can be factorized over a bigger field. If for instance we take k = ℚ then (x + y − \sqrt{2}z)(x + y + \sqrt{2}z) = (x + y)^{2} − 2z^{2} is irreducible over ℚ but factors over ℚ[\sqrt{2}]. If this occurs the curve is irreducible but not absolutely irreducible. In our example the curve C ⊗_{ℚ} ℚ̄ is the union of two lines which are interchanged by the Galois group. They intersect in one point, which is not smooth.

It is also possible that a curve C/k is smooth, irreducible but not absolutely irreducible. This happens if we start from an absolutely irreducible curve C/k. Let k_{0} be a subfield of k such that k/k_{0} is a finite separable non trivial extension. Then we have C → Spec(k) → Spec(k_{0}) and we can view C/k_{0} as a curve over k_{0}. This curve will still be smooth, but it is not absolutely irreducible. To see this we simply look at C ×_{k_{0}} k̄ and verify that this is a disjoint union of [k : k_{0}] smooth curves.

There is a slightly different way of looking at this phenomenon. If we have a smooth irreducible curve C/k and an element f ∈ O_{C}(U) which is not in k it can happen that this element is still algebraic over k. It generates a finite extension L/k contained in O_{C}(U) and now O_{C}(U) ⊗_{k} k̄ must have zero divisors and hence our curve is not absolutely irreducible.

We call such elements f which are algebraic over k the constant elements or simply the constants (???). They form a finite extension of the ground field. A smooth irreducible curve is absolutely irreducible if and only if the field of constants is equal to k.

Exercise:

1) We return to the case of a curve defined by a homogeneous polynomial f as above. Use the results in the previous section (???) to compute the cohomology groups H^{•}(C, O_{C}) and show especially that H^{0}(C, O_{C}) is a k-vector space of dimension one. Conclude that C cannot decompose into disjoint closed subschemes and show that C must be absolutely irreducible if it is smooth.

Remark: We can construct a universal curve of degree d. If we consider forms of a fixed degree and we remove the trivial form, then we see that we can view the coefficients (a_{ν_{1}ν_{2}ν_{3}}) as the k-valued point of a projective scheme S_{d} = Proj(ℤ[…, a_{ν_{1}ν_{2}ν_{3}}, …]) where we consider the a_{ν_{1}ν_{2}ν_{3}} as independent variables in degree one. We can define the universal curve of degree d which is a subscheme of

C  ↪  S_{d} ×_{Spec(ℤ)} ℙ^{2}
↘  ↙
S_{d}

Exercise 2:

a) Prove that there is a closed subscheme S_{d}^{sing} ⊂ S_{d} such that the universal curve restricted to the complement S_{d}^{smooth} := S_{d} \ S_{d}^{sing} is smooth and that all the fibres over S^{sing} are singular.

b) Compute S_{d}^{smooth}(ℤ) for d = 1, d = 2. ( It is a deep theorem that S_{d}^{smooth}(ℤ) = ∅ for d ≥ 3)

Proposition: For a smooth curve C/k the local rings O_{C,𝔭} at a closed point 𝔭 is a discrete valuation ring. For any non empty affine open set U ⊂ C the ring O_{C}(U) is a Dedekind ring.

Proof: Let P be a geometric point, let 𝔭 be the corresponding point on C. We can choose a local parameter f at P. Recall that this is an element in f ∈ O_{C,𝔭} such that the differential df generates Ω_{C/k}^{1} at 𝔭. We have seen in the section on smooth points (???) that we get an embedding of the polynomial ring

\begin{matrix}k[X] ↪ O_{C,𝔭} \\ X ↦ f. \end{matrix}

If 𝔭_{0} = k[X] ∩ 𝔭 then the embedding of local rings

k[X]_{𝔭_{0}} ↪ O_{C,𝔭}

is étale (???):

(i) The maximal ideal 𝔭_{0} ⊂ k[X]_{𝔭_{0}} generates the maximal ideal 𝔭 ⊂ O_{C,𝔭}.

(ii) The extension

k[X]_{𝔭_{0}}/𝔭_{0} ↪ O_{C,𝔭}/𝔭

is a finite separable extension. (This is explained at the end of the section on smooth points, just before the section on flat morphisms).

Since the ring k[X] is principal, we have 𝔭_{0} = (p(X)) with an irreducible polynomial p(X). But the maximal ideal of the local ring O_{C,𝔭} is also generated by p(X) and the proposition follows. Since we had the habit to denote a uniformizing element of the maximal ideal 𝔭 of a discrete valuation ring by π_{𝔭} we can choose π_{𝔭} = π_{𝔭_{0}} = p(X).

The above étale morphism provides in a certain sense a good approximation of O_{C,𝔭} by k[X]_{𝔭_{0}}. If we assume for instance that k(𝔭_{0}) = k(𝔭) then it is clear that for any n we have k[X]/𝔭_{0}^{n} = O_{C,𝔭}/𝔭^{n} and hence we get an isomorphism between the completions

k̂[X]_{𝔭_{0}} = lim_{←} k[X]/𝔭_{0}^{n}→^{∼} lim_{←} O_{C,𝔭}/𝔭^{n}O_{C,𝔭} = Ô_{C,𝔭}.

Under our assumptions the extension k(𝔭)/k(𝔭_{0}) is separable, but the first step k(𝔭_{0})/k can be inseparable. But

If 𝔭 is a closed point on the smooth curve C/k and if the extension of residue fields k(𝔭)/k is separable, and if π_{𝔭} is a uniformizing element at 𝔭, then dπ_{𝔭} generates the module of differentials at 𝔭.

We choose our X as above. Then our assumption implies that p′(X) is a unit in k[X]_{𝔭_{0}} and then dπ_{𝔭_{0}} = dπ_{𝔭} = p′(X)dX is a generator for the module Ω_{C,𝔭}^{1} of differentials.

The structure of Ô_{C,𝔭}:

We need a little bit of information concerning the structure of Ô_{C,𝔭}. We have the diagram

Ô_{C,𝔭} ⟶^{Ψ} k(𝔭)
↑
↗
k.

If the extension of residue fields k(𝔭)/k is separable then we have a unique section s : k(𝔭) → Ô_{C,𝔭} with s|k = Id_{k} and Ψ ∘ s = Id_{k(𝔭)}. To get this section we write k(𝔭) = k(θ) where θ is the zero of an irreducible and separable polynomial f(X) ∈ k[X]. We lift θ to an element θ̃ ∈ Ô_{C,𝔭}. Then F (θ) ≡ 0 mod 𝔭. Let π_{𝔭} be a uniformizing element. We modify θ̃ into θ̃ + απ_{𝔭} where α ∈ Ô_{C,𝔭}. We evaluate F at this new argument and get F (θ̃ + απ_{𝔭}) = F′(θ̃) + F′(θ̃)απ_{𝔭}. We know that F′(θ̃) is a unit in Ô_{C,𝔭} and hence we see that we can choose α in such a way that

F (θ̃ + απ_{𝔭}) ≡ 0 mod 𝔭^{2}

. We modify again by adding a βπ_{𝔭}^{2} and improve our solution to a solution mod 𝔭^{3}. This yields a sequence which converges to an exact solution. This argument is called the Lemma of Hensel and it is the p-adic version of Newtons method.

We identify s(k(𝔭)) to k(𝔭) and our diagram above becomes

k(𝔭)[[π_{𝔭}]] ⟶ k(𝔭)
↑
↗
k

in other words our ring Ô_{C,𝔭} is the power series ring in one variable π_{𝔭} over the residue field. The quotient field is k(𝔭)[[π_{𝔭}]][1/π_{𝔭}], it is sometimes called the field of Laurent expansions at 𝔭.

If the residue field extension is not separable then the structure of Ô_{C,𝔭} is not so nice and will cause us some trouble.

If the extension k(𝔭)/k purely inseparable then it is clear that the L-algebra k(𝔭) ⊗_{k} L is local and hence we have only one prime ideal 𝔭′ in the fibre over 𝔭.

Base change:

We have to investigate systematically what happens if we extend the field of constants k, i.e. we choose an extension L/k and we consider the curve C ×_{Spec(k)} Spec(L) or simply C ×_{k} L. We have a morphism (the base change morphism)

C ← C ×_{k} L.

We considered this at the end of the section on affine schemes for the case L = k̄ but it is clear that the considerations carry over to this case. The base change morphism induces a map on the closed points and the fibre over a closed point 𝔭 is the set of prime ideals of the finite L-algebra k(𝔭) ⊗_{k} L . If we consider a closed point 𝔭 and an affine neighborhood U of 𝔭 then our base change morphism corresponds to the inclusion

A(U) → A(U) ⊗_{k} L.

Let us assume that L/k is finite. Then the prime ideal 𝔭 ⊂ A(U) decomposes in A(U)⊗_{k} L, i.e.

𝔭(A(U) ⊗_{k} L) = 𝔭_{1}^{e_{1}} · . . . · 𝔭_{s}^{e_{s}}

or

(A(U) ⊗ L)/𝔭A(U) ⊗ L ≃ A(U) ⊗_{k} L/𝔭_{1}^{e_{1}} ⊕ … ⊕ (A ⊗_{k} L)/𝔭_{s}^{e_{s}}.

The e_{ν} are called the ramification indices. There are some special cases

(1) If the residue field k(𝔭) = k, then it is clear that 𝔭 stays prime. (This tells us also that our assumption that L/k is finite is inessential. After passing to a suitable finite extension nothing essential happens after that (if we stick to one 𝔭)).

(2) If the extension k(𝔭)/k is separable then the ramification exponents e_{ν} are one, the algebra (A(U) ⊗ L)/𝔭A(U) ⊗ L = k(𝔭) ⊗_{k} L has no nilpotent elements.

(3) But if for instance k(𝔭) = k(^{p}\sqrt{a}) where p = char(k) and a ∉ k^{p} and A(U) = k[X]/(X^{p} − a) and if L = k(^{p}\sqrt{a}) then

k[X] ⊗_{k} k(^{p}\sqrt{a})/_{(X^{p}−a)} = k(^{p}\sqrt{a})[X]/(X − ^{p} \sqrt{a})

and hence we get nilpotent elements.

(4) If more generally k(𝔭)/k is purely inseparable we choose L big enough so that it can contain the k-algebra k(𝔭). Then the ramification index e = [k(𝔭) : k] in other words in the group of fractional ideals of A(U) ⊗_{k} L we have 𝔭 = 𝔭′^{[k(𝔭):k]}. The completion O_{C×_{k}L,𝔭′} is again a power series ring.

If we pass to the completions, i.e. the infinitesimal neighborhoods of 𝔭 then we get

Ô_{C,𝔭} ⊗_{k} L = (lim_{←} A(U)/𝔭^{n} ⊗ L) ≃ ˆ=O_{C×_{k}L,𝔭_{1}} ⊕ … ⊕ Ô_{C×_{k}L,𝔭_{s}}.

If the extension k(𝔭)/k is separable then we get

Ô_{C,𝔭} ⊗_{k} L = k(𝔭)[[π_{𝔭}]] ⊗_{k} L ≃ (k(𝔭) ⊗_{k} L)[[π_{𝔭}]]

where now the points above 𝔭 correspond to the points of Spec(k(𝔭) ⊗_{k} L).

Curves and their function fields

For an irreducible curve C/k we consider the function field k(C) of meromorphic functions. This is simply the stalk O_{C,η} of the sheaf O_{C} in the generic point η. For any affine non-empty open subset U ⊂ C the ring O_{C}(U) is a one-dimensional, integral ring and k(C) is its quotient field. We pick an f ∈ O_{C}(U), which is not constant. By definition this element cannot be algebraic over k. It yields an embedding of the polynomial ring

k[f] ↪ O_{C}(U)

and an embedding of fields

k(f) ↪ k(C).

I claim

The field k(C) is a finite extension of k(f) and hence of transcendence degree one.

This is rather clear, if we had an g ∈ k(C) which is not algebraic over k(f) then k[f, g] would be a polynomial ring in two variables sitting in some Dedekind ring O_{C}(V) which is absurd.

Let us denote by D_{f} ⊃ U the set of points 𝔭 where f is regular, i.e. we have f ∈ O_{C,𝔭} Since O_{C} is a sheaf then we have f ∈ O_{C}(D_{f}). Now we consider the integral closure A of k[f] in k(C). In the section on Dedekind rings we indicated that A is again a finitely generated k-algebra and a Dedekind ring. (The extension k(C)/k(f) needs not to be separable anymore). It is clear that the elements of A are integral at all points of D_{f} in other words we have A ↪ O_{C}(D_{f}). We get a diagram

D_{f} → Spec(A)
↘
C

Now we assume in addition that our curve C/k is projective. Then it is clear that the morphism D_{f} ↪ C extends uniquely to Spec(A) → C. Our diagram can be completed by a vertical arrow:

D_{f} → Spec(A)
↘  ↓
C.

If 𝔭 is a closed point in the image of Spec(A) then we have A ⊂ O_{C,𝔭} hence we have f ∈ O_{C,𝔭} and this implies 𝔭 ∈ D_{f}. Hence our diagram becomes

D_{f} → Spec(A)
↘ Id  ↓
D_{f}  ⊂ C.

Looking at the resulting diagrams for rings we see that the inclusion A ↪ O_{C}(D_{f}) must be an isomorphism.

We summarize

Let C/k be a projective smooth curve. If we pick a non constant f ∈ k(C) and consider the set D_{f} ⊂ C where f is regular. Then O_{C}(D_{f}) is the integral closure of k[f] in k(C).

We may also consider k[f^{−1}] and by the same procedure we get k[f^{−1}] ⊂ O_{C}(D_{f^{−1}}). We have seen how to glue Spec(k[f])and Spec(k[f^{−1}]) over Spec(k[f, f^{−1}]) and construct ℙ^{1}/k. Hence we see that our element f defines a morphism

Φ_{f} : C → ℙ^{1}.

This morphism is finite, i.e. on any open affine Y ⊂ ℙ^{1} the morphism O_{ℙ^{1}}(Y) ↪ O_{C}(Φ_{f}^{−1}(Y)) is finite.

This tells us that for a smooth, projective and absolutely irreducible curve we can reconstruct the curve from its field of meromorphic functions, i.e. from O_{C,η} = k(C). The set of closed points of C can be identified to the set V of all discrete valuation rings V in k(C) which contain the field of constants k and have quotient field k(C). To see this we have to show that any such discrete valuation ring V ⊂ k(C) is the stalk of O_{C} at a closed point. We pick an f ∈ V which is not in k then V ⊃ k[f]. Then we must have V ⊃ O_{C}(D_{f}). Hence V is the discrete valuation ring at a point of D_{f}. We put a topology onto the set V: The non empty open sets are the sets V_{f} = {V |f ∈ V }. Then we see that the bijection V→^{∼} closed points of C becomes a homeomorphism. Eventually we define a sheaf O(V_{f}) = ⋂_{V ∈V_{f}} V, we get a ringed space and if we add a generic point we get an isomorphism of ringed spaces.

If we have two smooth, absolutely irreducible projective curves C_{1}, C_{2} then we can consider morphisms ϕ : C_{1} → C_{2}. Such a morphism is not constant (i.e. it does not map C_{1} to a point) then it maps the generic point to the generic point. Hence it induces a map between the function fields

k(C_{2}) →^{^{t}ϕ} k(C_{1})
↘  ↙
k

and it is not difficult to see, that we can recover ϕ from ^{t}ϕ. Hence we see that non constant morphisms

φ : C_{1} ⟶ C_{2}

between two smooth, absolutely irreducible projective curves are in one to one correspondence to morphisms

tf : k(C_{2}) ⟶ k(C_{1})

between the function fields which are the identity on the constants.

This is an exceptional phenomenon in dimension one.

We may even start from a field K/k of trancendence degree one. If we assume in addition a) we can find an x ∈ K which is transcendental over k and which is such that K/k(x)

is a finite separable extension.

b) The field k is algebraically closed in K which means that for any α ∈ K which is algebraic over k we have α ∈ k,

then we can construct a smooth, absolutely irreducible, projective curve C/k with k(C) = K. The set of closed points will be the set of discrete valuation rings in K which contain k. The non empty open sets will be the sets of closed points containing a given f ∈ K plus the generic point and the ring of holomorphic functions on such an open set will be the integral closure of k[f] in K.

Let φ : C_{1} → C_{2} be a separable, finite morphism between two smooth and absolutely irreducible projective curves, this means that

tφ : k(C_{2}) ↪ k(C_{1})

is a finite separable extension. For any affine open subset U ⊂ C_{2} we know that

B = O_{C_{1}}(φ^{−1}(U)) ⊃ O_{C_{2}}(U) = A

and B is the integral closure of A in k(C_{1}). We introduced the concept of ramification in (???), we said that B is ramified at a point 𝔭 ∈ Spec(A) if and only if the A/𝔭-algebra B/𝔭B = B ⊗_{A} A/𝔭 has nilpotent elements. This is obviously equivalent to the assertion that the trace pairing

tr_{(B/𝔭B)/(A/𝔭)} (xy)

is degenerate.

This leads to the definition of the fractional ideal

𝔇_{B/A}^{−1} = {x ∈ K_{1} | tr_{K_{1}/K_{2}}(xA) ⊂ A} .

Since K_{1}/K_{2} is separable we see that

\begin{aligned}tr & : & K_{1} × K_{1} & ⟶ & K_{2} \\ tr & : & (x, y) & ⟶ & tr_{K_{1}/K_{2}}(xy) \end{aligned}

is non degenerate and we conclude that we have B ⊂ 𝔇_{B/A}^{−1} and that 𝔇_{B/A}^{−1} is indeed the inverse of a non zero ideal 𝔇_{B/A} ⊂ B.

This ideal can be written as

𝔇_{B/A} = ∏_{i=1}^{t} 𝔭_{i}^{m_{i}},

and it gives us the ramified primes and it also measures the ramification.

By definition we have a homomorphism

ψ : 𝔇_{B/A}^{−1} → Hom_{A}(B, A)

x ↦ {b ↦ tr_{B/A}(xb)}

and I claim that this map is an isomorphism.

To see this we pick any non zero prime ideal 𝔭 in A and localize at this prime ideal. If now x ∈ 𝔇_{B/A}^{−1} \ 𝔇_{B/A}^{−1}𝔭 then we can find a b ∈ B s.t. tr_{B/A}(xb) ≢ 0 mod 𝔭. Otherwise we would have x/π_{𝔭} ∈ 𝔇_{B/A}^{−1} in contradiction to our assumption on x.Hence we see that

𝔇_{B/A}^{−1} ⊗ A/𝔭⟶ Hom_{A}(B, A/𝔭)

and since these two vector spaces have the same dimension the claim follows.

If we vary the open set U we can put these different ideals together and get a sheaf

𝔇_{C_{1}/C_{2}}

which measures the ramification of ψ : C_{1} → C_{2}.

Now we consider the sheaf of differentials Ω_{C_{1}/k}^{1} and Ω_{C_{2}/k}^{1}. Both of them are line bundles because we made the assumption that C_{1}, C_{2} are smooth.

If we pull back the sheaf Ω_{C_{2}/k}^{1} via φ to the sheaf φ^{∗}(Ω_{C_{2}/k}), then we have an obvious inclusion between the two line bundles

φ^{∗}(Ω_{C_{2}/k}^{1}) ↪ Ω_{C_{1}/k}^{1},

and we have the Riemann-Hurwitz formula. This inclusion extends to an isomorphism

φ^{∗}(Ω_{C_{2}/k}^{1}) ⊗ 𝔇_{C_{1}/C_{2}}^{−1} ˜ ⟶Ω_{C_{1}/k}^{1}.

This is a local formula: If we choose an affine open set U ⊂ C_{2} such that Ω_{A/k}^{1}, Ω_{B/k}^{1} and 𝔇_{B/A} become free modules, and if ω_{A} is generator of Ω_{A/k}^{1} and F a generator of 𝔇_{B/A}, then F^{−1} · ω = ω′ is a generator of Ω_{B/k}^{1}.

We have seen for the sheaves of differentials that they behave well under extensions of the ground field. Hence we perform the base change Spec(k) ← Spec(\overline{k}), then

\begin{aligned}Ω_{A⊗_{k}\overline{k}/\overline{k}}^{1} & = & Ω_{A/k}^{1} ⊗_{k} \overline{k} \\ Ω_{B⊗_{k}\overline{k}/\overline{k}}^{1} & = & Ω_{B/k}^{1} ⊗_{k} \overline{k} \end{aligned}

Now we look at the discriminant 𝔇_{A/B}. Our assumption is that A ⊗_{k} \overline{k}, B ⊗_{k} \overline{k} are still fields.

We showed that ψ : 𝔇_{B/A}^{−1} → Hom_{A}(B, A) is an isomorphism, hence

𝔇_{B/A}^{−1} ⊗_{k} k′ ⟶ Hom_{A⊗_{k}k′}(B ⊗_{k} k′, A ⊗_{k} k′)

is an isomorphism. This shows

𝔇_{B/A}^{−1} ⊗_{k} k′ = 𝔇_{B⊗_{k}k′/A⊗_{k}k′}^{−1}.

Hence it suffices to prove the Riemann-Hurwitz formula. We can assume that our ground field is algebraically closed.

Since we may also pass to the completion, we are reduced to the case

Ô_{C_{2},𝔭} = k[[x]] ⟶ Ô_{C_{1},𝔭} = k[[y]]

where the extension is finite.

We can write x as a power series in y

x = Q(y) = y^{m} + a_{m+1}y^{m+1} + ⋯ = y^{m}(1 + a_{m+1}y + ⋯)

which implies that ord_{(y)}(x) = m. Then it is obvious that the elements 1, y, ⋯ y^{m−1} form a basis of the k[[x]]-module k[[y]], and we have an equation

y^{m} + a_{1}(x)y^{m−1} + ⋯ a_{m}(x) = 0

where the a_{i}(x) ∈ k[[x]] and a_{i}(x) ≡ mod(x). Looking at the order of vanishing yields a_{m}(x) = α_{0}x + α_{1}x^{2} ⋯ with α_{0} ≠ 0. If P (Y) = Y^{,} + a_{1}(x)Y^{m−1} ⋯ a_{m}(x) ∈ k[[x]][Y], we find a relation for the differentials

\frac{∂P}{∂Y} (y)dy + (∑ \frac{∂a_{v}}{∂x} y^{m−v}) dx = 0.

Since \frac{∂a_{m}}{∂x} = α_{0} ≠ 0 we see that (∑ \frac{∂a_{v}}{∂x} y^{m−v}) dx generates Ω_{A/k}^{1} ⊗ B, we see that Ω_{B/k}^{1} is generated by

\frac{dx}{\frac{∂P}{∂Y} (y)}.

Now everything boils down to show that

𝔇_{B/A} = (\frac{∂P}{∂Y} (y))

which is very easy.

Kommt noch (???)

One of the important consequences of our considerations is that we can define a trace map

tr : f_{∗}(Ω_{C_{1}/k}^{1}) ⟶ Ω_{C_{2}/k}^{1}.

This is clear. Locally we can write a differential ω = fdx where dx is a generator of Ω_{C_{2}/k}^{1} and f ∈ 𝔇_{C_{1}/C_{2}}^{−1}. Then

tr_{C_{1}/C_{2}}(ω) = tr_{C_{1}/C_{2}}(f)dx

and tr_{C_{1}/C_{2}}(f) is regular by definition of the different

Line bundles and Divisors

The following considerations are valid in a more general framework. Let X be any scheme. A line bundle on X is simply a locally free O_{X}-module of rank one. The structure sheaf itself is a line bundle. It is called the trivial bundle. If we have two line bundles L_{1}, L_{2}, we can form the tensor product L_{1} ⊗ L_{2}. We can form the line bundle L^{−1} = Hom_{O_{X}} (L, O_{X}) and we have L ⊗ L^{−1} ≅ O_{X}. Hence it is rather clear that the isomorphism classes of line bundles form a group under the multiplication. It is the so-called Picard group Pic(X). To define this group we do not need any assumption on X.

Under certain assumptions we can identify this group to the so called divisor class group. At first we define the group of divisors as the free group generated by the irreducible

subschemes 𝔭 of codimension one. If we want to do this we need the concepts from dimension theory and hence we assume that our scheme is covered by affine noetherian schemes (See???). I even want to assume that it is noetherian itself, i.e. it can be covered by a finite number of affine schemes. We are mainly interested in the case where X is covered by a finite number of affine schemes over a field k. For this special case we discussed the relevant results in dimension theory in section (???).

We decompose X into irreducibles X = ⋃ X_{i} let η_{i} be the generic point of X_{i}. Then a meromorphic function f on X is simply an element in the stalk O_{X_{i},η_{i}}. We want to attach a divisor Div(f) to the function f. Since everything is happening on X_{i} I assume that X is already irreducible.

We can find a covering X = ∪U_{i} by affine integral schemes such that on U_{i} we can write

f = \frac{g_{i}}{h_{i}}

with f_{i}, h_{i} ∈ O_{X}(U_{i}). We consider the prime ideals which contain (f_{i}) resp (g_{i}) and amoung those we we consider those which are minimal. Then the Hauptidealsatz of Krull asserts that there is a finite number of such minimal prime ideals 𝔭_{1}, …, 𝔭_{s} ⊃ (f_{i}) and 𝔮_{1}, …, 𝔮_{r} ⊃ (g_{i}) and these have height one ( ) [Ei], , ???). But this is not enough to attach a divisor to f, we need the multiplicities of the zeroes and poles.

To define these multiplicities we make the additional assumption that the scheme X should be normal: this means that the affine rings O_{X}(U_{i}) are integrally closed in there field of meromorphic functions. This implies that for all prime ideals 𝔭 of height one the local rings O_{X,𝔭} are discrete valuation rings. (I gave a reference and an indication of the proof of this fact in the section on low dimensional rings.)

If now 𝔭_{ν} ⊃ (f_{i}) ( resp. 𝔮_{μ} ⊃ (g_{i}) then we can write the principal ideals local rings (f_{i}) ⊂ O_{X,𝔭_{ν}} (resp. (g_{i}) ⊂ O_{X,𝔮_{ν}}) as

(f_{i}) = 𝔭_{ν}^{ord_{𝔭ν} (f_{i})}, (g_{i}) = 𝔮_{μ}^{ord_{𝔮μ} (g_{i})}.

Now we attach a divisor to the restriction of f to U_{i} namely

∑ ord_{𝔭_{ν}} (f_{i})𝔭_{ν} − ∑ ord_{𝔮_{μ}}(g_{i})𝔮_{μ}.

Comparing these divisors on the intersections U_{i} ∩ U_{j} gives us a divisor on X.

Of course it can happen that we have some cancellations, i.e. that the same 𝔭 occurs amoung the 𝔭_{ν} and the 𝔮_{μ}. We say that the function f has a zero (resp. pole) of order n ≥ 0 at an irreducible subscheme 𝔭 of codimension 1 if it occurs amoung the 𝔭_{ν}, 𝔮_{μ} and if n = ord_{𝔭}(f_{i}) − ord_{𝔭}(g_{i}) resp. n = ord_{𝔭}(g_{i}) − ord_{𝔭}(f_{i}) (for some i).

Hence we see that we can define the group of principal divisors on a noetherian, integral normal scheme as the group of divisors of meromorphic functions.

We define the divisor class group as the group of divisors modulo principal divisors (See ???)

Under certain conditions we have an isomorphism between Pic(X) and the divisor class group. To get this isomorphism we have to use some results from the commutative algebra of noetherian rings, which are not in our script. In the special case of rings of dimension one they are proved in the section on ”Low dimensional rings”.

Assume again that X is noetherian, integral and normal. Let L be a line bundle on X. Then the stalk L at the generic point η is a one dimensional O_{X,η}- vector space. Let s ∈ L_{η} be a generator. It is a meromorphic section in the line bundle. We want to attach a divisor to this meromorphic section.

We can find a covering X = ∪U_{i} by affine integral schemes such that L|U_{i} becomes trivial, i.e.

L(U_{i}) = (O_{X}|U_{i}) · t_{i}

where t_{i} ∈ H^{0}(U_{i}, L). Then we have

s = g_{i}t_{i}

where g_{i} is an element in the field of meromorphic functions O_{X,η}. We can write

g_{i} = \frac{f_{i}}{h_{i}}

with f_{i}, h_{i} ∈ O_{X}(U_{i}). As before we see that there is a finite number of minimal prime ideals 𝔭_{1}, …, 𝔭_{s} ⊃ (f_{i}) and 𝔮_{1}, …, 𝔮_{r} ⊃ (g_{i}) and these have height one.

Again we use the fact that the local rings O_{X,𝔭_{ν}} , O_{X,𝔮_{ν}} are discrete valuation rings. Hence we can write in these local rings

(f_{i}) = 𝔭_{ν}^{ord_{𝔭ν} (f_{i})}, (g_{i}) = 𝔮_{ν}^{ord_{𝔮ν} (g_{i})}.

The prime ideals 𝔭_{ν}, 𝔮_{ν} define irreducible subschemes of codimension one and we attach to (L, s, t_{i}) a divisor on U_{i} namely

∑ ord_{𝔭_{ν}} (f_{i})𝔭_{ν} − ∑ ord_{𝔮_{μ}}(g_{i})𝔮_{μ}.

If we compare D_{i} and D_{j} on the intersection U_{i} ∩ U_{j} we see that they must coincide and hence we see that we can attach a divisor D on X to our data (L, s, t_{i}). If we change then t_{i} then we modify g_{i} by a unit and D_{i} stays the same. If we modify the meromorphic section s, i.e. we multiply s by a meromorphic function g then the divisor is changed by the divisor of a meromorphic function.

Hence we get a homomorphism

Pic (X) ⟶ Divisors modulo principal divisors.

It can be shown that under our assumptions this homomorphism is injective: If we can choose the sections s and t_{i} in such a way that the divisor becomes zero, then s = g_{i}t_{i} where the divisor of g_{i} is zero. We conclude that g_{i} and g_{i}^{−1} lie in the intersection of all discrete valuation rings O_{X}(U_{i})_{𝔭} where 𝔭 runs over the prime ideals of height one. Now we know that this intersection is equal to O_{X}(U_{i}) (See [Ei], Cor. 4.11, in dimension one also in ”Low dimensional rings”) Hence g_{i} is a unit. But then t_{i} = g_{i}^{−1}s and the meromorphic section s is an element in H^{0}(X, L) which generates the free O_{X}-module L.

This homomorphism is not surjective in general. If we want this we need to assume that X is locally factorial. This means that for any point x ∈ X and any prime ideal 𝔭 ⊂ O_{X,x} of height one we find an f_{𝔭} ∈ O_{X,x} such that 𝔭 = (f_{𝔭}).

Under this assumption the homomorphism becomes bijective. To see this we start from a divisor D = ∑_{𝔭} n_{𝔭}𝔭. For any point x ∈ X we can find an open neighborhood U_{x} and an element f_{x} ∈ O_{X}(U_{x}) such that Div(f_{x})|U_{x} = D|X. Now we define the line bundle O_{X}(−D): Its sections over U_{x} are

O_{X}(U_{x}) = {h ∈ O_{X,η}|f_{x}h ∈ O_{X}(U_{x})}.

Then it is clear that this defines a line bundle because locally at x the function 1/f_{x} trivializes the bundle. The corresponding divisor class is the class of D (Our global meromorphic section s is simply the constant function 1 and for the t_{i} we choose f_{x}^{−1}. The the g_{i} become the f_{x}.)

If we have a line bundle L on X which has a non zero global section s ∈ H^{0}(X, L) then we can define the scheme V (s) of zeroes of s. We simply observe that locally L is trivial, hence locally s is nothing else than a non zero regular function.

Our divisor is called effective if all the multiplicities are ≥ 0. If D is effective then O_{X}(−D) is the sheaf of germs of meromorphic functions which may have poles of order ≤ n_{𝔭} along the 𝔭. In this case we have 1 ∈ H^{0}(X, O_{X}(−D)).

It is a theorem in commutative algebra that X is X is locally factorial if X/k is smooth (See ???). If X is of dimension one this follows from (???).

Divisors on curves :

Now we come back to the case where X is a smooth, projective and absolutely irreducible curve. The irreducible subschemes of codimension 1 are the closed points.

The group of divisors Div(C) is the free abelian group generated by the closed points. We write

D = ∑ n_{𝔭}𝔭.

We define the degree of a divisor as

deg(D) = ∑ n_{𝔭}[k(𝔭) : k]

where [k(𝔭) : k] is the degree of the extension of the residue field.

If we pick a non zero element f ∈ k(C) then we know that for any closed point 𝔭 we may write f = ϵπ_{𝔭}^{ν_{𝔭}(f)} in the local ring O_{C,𝔭}. Hence we can attach a divisor to our element f namely Div(f) = ∑_{𝔭} ord_{𝔭}(f)𝔭. A divisor which has such a presentation is called principal. The principal divisors form a subgroup of the group of all divisors.

Theorem: If C/k is a smooth, absolutely irreducible projective curve, then a principal divisor has degree zero.

Proof: This follows from the results which we explained in the section on Dedekind rings. We assume that f is not constant. We constructed the morphism (Dies muß noch erklärt werden!)

Φ_{f} : C → ℙ^{1}

and it is clear from the construction that f ⊂ k(ℙ^{1}). As an element in the function field of the projective line its divisor is (0) − (∞) where (0) is the closed point defined by (f) ⊂ k[f] and (∞) is the closed point defined by (f^{−1}) ⊂ k[f^{−1}]. This divisor has degree zero. Now we consider the divisor of f on C/k. We decompose the divisor into the divisor of zeroes and the divisor of poles:

Div(f) = Div_{0}(f) + Div_{∞}(f) = ∑_{𝔭,ord_{𝔭}(f)>0} ord_{𝔭}(f)𝔭 + ∑_{𝔭,ord_{𝔭}(f)<0} ord_{𝔭}(f)𝔭.

In the section on Dedekind rings we studied the behavior of prime ideals under extension of Dedekind rings. Our formula there says

deg(Div_{0}(f)) = ∑_{𝔭,ord_{𝔭}(f)>0} ord_{𝔭}(f)[k(𝔭) : k] = [k(C) : k(f)]

(Our (f) is the prime ideal 𝔭 there and the 𝔓 their correspond to the 𝔭 here.)

The same holds for the pole divisor and the theorem is clear.

If we have a base change C ← C ⊗_{k} L then this induces a homomorphism on the group of divisors. To see this we have to check what happens for prime divisors. We discussed what happens in the above section on base change and using the notation there we send 𝔭 to ∑ e_{i}𝔭_{i}. This extends to the group of divisors and we see that this homomorphism preserves the degree.

All our assumptions are valid in the case of smooth, absolutely irreducible curves hence we define as Pic(C/k) the group of line bundles of our curve C/k and we identify it to the group of divisors modulo linear equivalence. Especially we can now define the degree of a line bundle, it is simply the degree of the corresponding divisor class. The degree defines a homomorphism

deg : Pic(C/k) → ℤ.

The kernel of this homomorphism is denoted by Pic^{0}(C/k). It is a aim of these notes to give a proof, that this group is actually the group of rational points of a so called abelian variety of dimension g over k. An abelian variety is a connected projective variety which in addition has a structure of a group.

In the case of curves over ℂ this goes back to Abel, Riemann and Jacobi.

properties of the degree

Let C/k be a smooth curve, let us consider an effective divisor D = ∑ n_{𝔭}𝔭. In this casethe sections of O_{C}(−D) over an affine open set V ⊂ C are the meromorphic functions on U whose order of poles at 𝔭 is estimated by ≤ n_{𝔭}.

We notice that we have O_{C} ⊂ O_{C}(−D) and the quotient sheaf O_{C}(−D)/O_{C} has non zero stalk only at the points 𝔭 with n_{𝔭} > 0 this set is called the support of the divisor D and sometimes denoted by |D|. If V is an affine open set containing the support of D then O_{C}(V) is a Dedekind ring and we can interprete D as a fractional ideal for this Dedekind ring, it is clear from the definition (see section on Dedekind rings) that O_{C}(−D)(V) = ∏_{𝔭∈V} 𝔭^{−n_{𝔭}}. Now it is clear that for the global sections

O_{C}(−D)/O_{C})(C) = (O_{C}(−D))/O_{C})(V) = ∑ 𝔭^{−n_{𝔭}}/O_{C,𝔭}.

This direct sum ∑ 𝔭^{−n_{𝔭}}O_{C,𝔭}/O_{C,𝔭} is a finite dimensional vector space over k and the dimension of this vector space is ∑ n_{𝔭}[k(𝔭) : k] = deg(D), we summarize

dim_{k}(H^{0}(C, O_{C}(−D)/O_{C}) = deg(D)

This can be generalized to arbitrary line bundles. If we have a line bundle L and an effective divisor D, then we can define L(−D) = L ⊗ O_{C}(−D) and we have L ⊂ L(−D). On the other hand if we have two line bundles one of them contained in the other L ⊂ L_{1} then there is a unique divisor D such that L_{1} = L(−D). In these cases we have the formula

deg(L_{1}) = deg L(−D) = deg(L) + deg(D) = deg(L) + dim_{k}((L_{1}/L)(C)).

Vector bundles over a curve

A locally free coherent O_{C}−module E is called a vector bundle. Let n be its rank. It is an easy exercise in algebra to prove the following: If we have a subspace V ⊂ E_{η} in the O_{C,η} vector space E_{η} then we can find a submodule F ⊂ E such that F_{η} = V and the quotient E/F is again locally free. This subbundle is unique one could denote it by E ∩ V. This implies that our vector bundle E admits a complete flag (0) = L_{0} ⊂ L_{1} ⊂ L_{2} … L_{n−1} ⊂ E of subbundles such that the quotient of two successive bundles is a line bundle. We can define the n-th exterior power, this is a line bundle det(E) = Λ^{n}(E). We define the degree by deg(E) = deg(det(E)) and it is clear that we can express the degree in terms of a flag as deg(E) = ∑ deg(L_{i}/L_{i−1}).

For any vector bundle we can define the dual bundle as Hom_{O_{C}} (E, O_{C}) = E^{∨}. Again it is easy to see that we have deg(E) + deg(E^{∨}) = 0. Again can derive it easily from the case of line bundles.

Our formula above generalizes to vector bundles. If we have two vector bundles E ⊂ E_{1} of the same rank, then E_{1}/E is a finitely generated torsion sheaf and

deg(E_{1}) = deg(E) + dim_{k}((E_{1}/E)(C)

To see this we trivialize both bundles on a suitably small non empty affine open set V. Then the matrices which transform the two bases into each other have non zero determinants in O_{C,η} which are units over a still smaller but still non empty affine open set V_{1} ⊂ V. This means that E|V_{1} = E_{1}|V_{1}. This shows that the stalks of E_{1}/E are non zero only in the finite set 𝔭 ∈ C \ V_{1}. But for these points it is clear that we can find m_{𝔭} such that 𝔭^{m_{𝔭}}E_{1,𝔭} ⊂ E_{𝔭} and hence it is clear that the quotient is a finitely generated torsion sheaf. To see the assertion concerning the degree we may proceed as follows. We choose an affine set V containing the support of E_{1}/E. If we take this set to be sufficiently small then we may assume that both bundles are trivial and the theorem on elementary divisors tells us that we can find a basis e_{1}, e_{2} … e_{n} for E_{1}(V) and non zero elements a_{1}, a_{2}, … a_{n} in O_{C}(V) such that a_{1}e_{1}, a_{2}e_{2} … a_{n}e_{n} form a basis for E(V). Now these bases define complete flags in both vector bundles. These are the flags which are induced by the subspaces V_{i} = O_{C,η}e_{1} ⊕ … ⊕ O_{C,η}e_{i} of the generic fibre. We used these flags to define the degree: The degree was the sum of the degrees of the successive quotient line bundles induced by the flag. But if we compare these quotient line bundles for both vector bundles then we see the following: If M_{i} resp. M_{1,i} is such a quotient line bundle obtained by the flag in E resp. E_{1} then the stalks (M_{i})_{𝔮} = (M_{1,i})_{𝔮} for all 𝔮 not in the support of E_{1}/E. For the points 𝔭 in the support of E_{1}/E and even for the 𝔭 ∈ V we have a_{i}^{−1}M_{i,𝔭} = M_{1,i,𝔭}. Hence we see deg(M_{1,i}) = deg(M_{i}) + dim_{k}(a_{i}^{−1}O_{C}(V)/O_{C}(V)). Hence we get deg(E_{1}) = deg(E) + ∑_{i} dim_{k}(a_{i}^{−1}O_{C}(V)/O_{C}(V)). On the other hand we see that

E_{1}/E(C) = E_{1}/E(V) = ⊕_{i}a_{i}^{−1}O_{C}(V)/O_{C}(V,

this implies our formula.

We consider vector bundles over the projective line ℙ_{k}^{1}. In this case we have

Theorem: Any vector bundle E/ℙ_{k}^{1} is a direct sum of line bundles, i.e.

E ≃ O_{ℙ^{1}}(d_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(d_{n})

with some integers d_{1} ⋯ d_{n}. The integers are well defined up to order.

Proof: We consider the case rank E = 2. In this case we may tensorize by a line bundle L ≃ O_{ℙ^{1}}(r) such that deg(E) = 0 or 1. Now it follows from the theorem of Riemann-Roch that

dim H^{0}(ℙ^{1}, E) − dim H^{1}(ℙ^{1}, E) = deg(E) + 2 > 0,

and we find a non zero section s ∈ H^{0}(ℙ^{1}, E). The restriction of s to the generic point defines a subspace V ⊂ E_{η}, and this subspace extends to a unique line bundle L ≃ O_{ℙ^{1}}(m) such that E/L is torsion free. We must have deg(L) ≥ 0 and hence if

0 ⟶ L ⟶ E ⟶ E/L ⟶ 0

we see that deg((E/L)^{∨} ⊗ L) ≥ −1. If we tensorize by (E/L)^{∨}, then we get

0 ⟶ L ⊗ (E/L)^{∨} ⟶ E ⊗ (E/L)^{∨} ⟶ O_{ℙ^{1}} ⟶ 0,

and we have the section 1 ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}) which is every where ≠ 0. Since deg((E/L)^{∨} ⊗ L) ≥ −1 we see that

H^{1}(ℙ^{1}, (E/L)^{∨} ⊗ L) = 0

and the section 1 lifts to a section s_{0} ∈ H^{0}(ℙ^{1}, (E/L)^{∨} ⊗ L) which does not have any zero. We conclude that we have an inclusion O_{ℙ^{1}} ↪ E ⊗ (E/L)^{∨} such that its composition with the map to O_{ℙ^{1}} is the identity and therefore

E ⊗ (E/L)^{∨} = L ⊗ (E/L)^{∨} ⊕ O_{ℙ^{1}}.

Since our assertion is insensitive to tensorization by line bundles, the claim follows.

The general case follows by induction. Let rank (E) = r. We consider a line subbundle L ⊂ E, s.t. E/L is a vector bundle and such that deg(L) = d_{1} is maximal.

Then we have

0 ⟶ L ⟶ E ⟶ E/L ⟶ 0

and

\begin{aligned}E/L & ≃ & ⨁_{ν=2}^{r} O_{ℙ^{1}}(d_{ν}) \\ L & ≃ & O_{ℙ^{1}}(d_{1}). \end{aligned}

I claim that d_{1} ≥ d_{ν} for all ν ≥ 2. If we had an index ν_{0} s.t. d_{ν_{0}} > d_{1}, then we could consider the rank 2 bundle E′ ⊂ E which is the inverse image of the line subbundle O_{ℙ^{1}}(d_{ν_{0}}) in E/L. This bundle would decompose

E′ = O_{ℙ^{1}}(a) ⊕ O_{ℙ^{1}}(b)

where a + b = d_{1} + d_{ν_{0}} = deg(E′). One of the summands has to map non trivially to E′/L = O_{ℙ^{1}}(d_{ν_{0}}). If this is O_{ℙ^{1}}(a), then we conclude a ≤ d_{ν_{0}}. We cannot have equality because then deg O_{ℙ^{1}}(a) = d_{ν_{0}}, and this contradicts the choice of L. But then b > d_{1} and this is again a contradiction.

Now we show that the sequence

0 ⟶ L ⟶ E ⟶ E/L ⟶ 0

must split. Basically we argue as in the rank 2 case but we modify our argument slightly. We can cover ℙ^{1} by affines U_{1}, U_{2} such that on these affines we have sections s_{i} : E/L → E. These local sections may differ by a homomorphism

φ_{12} : E/L | U_{1} ∩ U_{2} ⟶ L

and φ_{12} gives us a class in

H^{1}(ℙ^{1}, Hom(E/L, L)).

But Hom(E/L) = ⨁_{v=2}^{r} O_{ℙ^{1}}(d_{1} − d_{ν}) and hence this class vanishes and we can bound the cocycle.

We add the following observation. Once we have

E = ⨁_{ν=1}^{r} O_{ℙ^{1}}(d_{ν})

we can tensorize by O_{ℙ^{1}}(−d_{ν_{0}}) where now d_{v_{0}} is maximal. Then we get

E ⊗ O_{ℙ^{1}}(−d_{ν_{0}}) = O_{ℙ^{1}}^{r_{0}} ⊕ ⨁_{ν=ν_{0}+1}^{r} O_{ℙ^{1}}(d_{ν} − d_{ν_{0}})

where we assumed that d_{1} = ⋯ = d_{ν_{0}} = d_{ν_{0}}, and the other d_{v} are smaller. Then we conclude that

H^{0}(ℙ^{1}, E ⊗ O_{ℙ^{1}}(−d_{ν_{0}})) = k^{r_{0}}

and that these sections generate the subbundle O_{ℙ^{1}}^{r_{0}}. This implies that d_{ν_{0}} is determined by E and that the subbundle

⨁_{ν=1}^{r_{0}} O_{ℙ^{1}}(d_{ν}) d_{ν} = d_{ν_{0}}

is unique.

This shows: If we order these numbers

d_{1} = ⋯ = d_{r_{0}} > d_{r_{0}+1} = ⋯ = d_{r_{1}} > d_{r_{1}+1} ⋯,

then the resulting sequence of numbers is determined by E and that the flag

⨁_{ν=1}^{r_{0}} O_{ℙ^{1}}(d_{ν}) ⊂ ⨁_{ν=1}^{r_{1}} O_{ℙ^{1}}(d_{ν}) ⊂ ⋯

is also determined by E.

The theorem above and the consequences are called Grothendieck’s theorem but it has been known to Dedekind (see ???).

The Theorem of Riemann-Roch

We consider line bundles on a smooth, projective, absolutely irreducible curve C/k. Such a line bundle L has cohomology group H^{0}(C, L), H^{1}(C, L), … which are finite dimensional k-vector spaces. We shall see that H^{i}(C, L) = 0 for i ≥ 2 and we will give a formula for

χ(C, L) = dim_{k}H^{0}(C, L) − dim_{k}H^{1}(C, L).

The first observation is

dim_{k}H^{0}(C, O_{C}) = 1.

If f ∈ H^{0}(C, O_{C}) then we have f^{2}, f^{3}, … ∈ H^{0}(C, O_{C}) and since this is a finite dimension vector space we see that f must be algebraic over k. If f ∉ k then we find that

k(C) ⊗_{k} k̄

has zero divisors and this contradicts the assumption that C/k should be absolutely irreducible.

We define the genus of the curve as

g = dim_{k}H^{1}(C, O_{C}).

Now we can state Theorem of Riemann-Roch (first version) For any line bundle L on C we have

χ(C, L) = dim_{k}H^{0}(C, L) − dim_{k}H^{1}(C, L) = deg(L) + 1 − g.

The proof is not too difficult. (The main difficulty is to prove that dim_{k}H^{1}(C, L) is finite dimensional but this follows from our general results in the previous chapter). The theorem is true for L = O_{C} by definition.

If we have a closed point 𝔭 then we have an exact sequence of sheaves

0 → O_{C} → O_{C}(−𝔭) → O_{C}(−𝔭)/O_{C} → 0.

The quotient is a skyscraper sheaf, its only non-zero stalk is the stalk at 𝔭 and there it is a one dimensional k(𝔭)-vector space.

Since L is locally free we get an exact sequence by tensoring by L

0 → L → L ⊗ O_{C}(−𝔭) → O_{C}(−𝔭)/O_{C} → 0.

We observe that

dim_{k}(H^{0}(C, O_{C}(−𝔭)/O_{C})) = [k(𝔭) : k]

and

H^{i}(C, O_{C}(−𝔭)/O_{C})) = 0 for i ≥ 1.

We write the long exact sequence in cohomology

\begin{aligned}0 & → H^{0}(C, L) → H^{0}(C, L ⊗ O_{C}(−𝔭)) → H^{0}(C, O_{C}(−𝔭)/O_{C}) → \\ & → H^{1}(C, L) → H^{1}(C, L ⊗ O_{C}(−𝔭)) → 0 \end{aligned}

and and higher degrees we get

H^{i}(C, L) ⟶^{∼} H^{i}(C, L ⊗ O_{C}(−𝔭)) for all i ≥ 2.

From this we get easily that our assertions are true for L ⊗ O_{C}(−𝔭) if and only if they are true for L itself. The rest is more or less clear. We have seen that H^{i}(C, O_{C}(−∞𝔭)) = 0 for i ≥ 2 in the section on cohomology of coherent sheaves (see exercise below). Since we know that our line bundle is isomorphic to some O_{C}(−D) the theorem follows.

Now it is an easy argument in homological algebra that we have a Riemann-Roch formula for vector bundles:

χ(C, E) = dim_{k}H^{0}(C, E) − dim_{k}H^{1}(C, E) = deg(E) + n(1 − g).

(Use the flag to prove it by induction)

Exercise:

1) I cheated a a little bit when I said that H^{i}(C, O_{C}(−∞𝔭)) = 0 for i ≥ 2. To see this we need to know that C \ 𝔭 is affine. Derive this from the theorem of Riemann-Roch. Why is this not a circulus vitiosus?

Differentials and Residues

On our curve C/k we have two privileged line bundles (???). The first one is the structure sheaf O_{C} and the other one is the sheaf Ω_{C/k}^{1} of differentials. It is called the canonical bundle. We know the degree of O_{C} and we computed the cohomology groups H^{0}(C, O_{C}) = k and dim_{k}H^{1}(C, O_{C}) = g where the second assertion is tautological.

Our next aim is the study of Ω_{C/k}^{1}. We will show that

deg(Ω_{C/k}^{1}) = 2g − 2

and that we have a canonical isomorphism

Res : H^{1}(C, Ω_{C/k}^{1}) → k.

Here we mean by canonical that this map is consistent with the trace map which we defined for separable morphisms. If we have C_{1}/k, C_{2}/k and if f : C_{1} → C_{2} is a finite separable morphism, then we defined the tr_{C_{1}/C_{2}} : f_{∗}(Ω_{C_{1}/k}^{1}) ⟶ Ω_{C_{2}/k}^{1}. and we require that the resulting k-linear map yields a commutative diagram

H^{1}(C_{1}, Ω_{C_{1}/k}^{1})  ⟶^{tr} H^{1}(C_{2}, Ω_{C_{2}/k}^{1})
Res ↘  ↙ Res
k  .

Of course we also want compatibility with base change in the obvious sense.

These assertions are much more difficult to prove.

If we accept that they are true then we can take any line bundle L and define a pairing

H^{0}(C, L) ⊗ H^{1}(C, L^{−1} ⊗ Ω_{C/k}^{1}) → H^{1}(C, Ω_{C/k}^{1}) = k

which turns out to be non degenate. As a special case we get that

H^{0}(C, Ω_{C/k}^{1}) = H^{1}(C, O_{C/k})^{∨}

and

dim_{k}(H^{0}(C, Ω_{C/k}^{1}) = g.

To understand the pairing we recall how we computed H^{1}(C, O_{C}). We looked at effective divisors

D = ∑ n_{𝔭}𝔭

with n_{𝔭} ≥ 0, and at the resulting exact sequence

0 → O_{C} → O_{C}(−D) → O_{C}(−D)/O_{C} → 0.

The quotient sheaf O_{C}(−D)/O_{C} is a collection of Laurent expansions at the points in the support of D. We introduce the notation 𝕃(−D) for it. For any closed point 𝔭 in |D| the stalk of 𝕃(−D) at 𝔭 is

{f ∈ k(C)| ord_{𝔭}(f) ≥ −n_{𝔭}} = π^{−n_{𝔭}}O_{C,𝔭}/O_{C,𝔭}

where π_{𝔭} is a generator of the maximal ideal 𝔭 ⊂ O_{C,𝔭}. We may (or may not) pass to the direct limit

lim_{\overrightarrow{D}} O_{C}(−D)/O_{C} = 𝕃

where 𝕃 is the sheaf of all Laurent expansions: For an open set U its sections are given by

𝕃(U) = ⨁_{𝔭∈U, 𝔭 closed} K/O_{C,𝔭}.

We introduce the completions of the local rings lim_{←}(O_{C,𝔭}/𝔭^{n}) = Ô_{C,𝔭} and the quotient

fields of these completions K̂_{𝔭}. We define the module of differentials Ω̂_{C_{1},𝔭}^{1} = Ω_{C,𝔭}^{1} ⊗ Ô_{C,𝔭}. Of course this is again a free module of rank one over Ô_{C,𝔭} and it is not difficult to see that it is a universal separated module for continuous differentials. We may also introduce the infinitesimal meromorphic differentials Ω̂_{C,𝔭}^{1} = Ω_{C,𝔭}^{1} ⊗ K̂_{𝔭}

Now we choose an effective divisor with deg(D) >> 0 and we consider the long exact sequence in cohomology. We have seen that H^{1}(C, O_{C}(−D)) = 0 provided deg(D) >> 0 and hence we get

0 → H^{0}(C, O_{C}) → H^{0}(C, O_{C}(−D)) → H^{0}(C, O_{C}(−D)/O_{C}) → H^{1}(C, O_{C}) → 0

if deg(D) >> 0, we could as well look at the same sequence where we passed to the limit over all D.

Hence we can interprete H^{1}(C, O_{C}) as a quotient of a space of Laurent expansions modulo the space of those Laurent expansions which come from a meromorphic function. This means that have to understand the obstruction for a collection of Laurent expansions

ξ ∈ ⊕_{𝔭}π_{𝔭}^{−n_{𝔭}}O_{C,𝔭}/O_{C,𝔭}

to come from a meromorphic function. The point is that the holomorphic differentials produce such obstructions.

We introduce the sheaf 𝕃(−D) = O_{C}(−D)/O_{C}, and we identify it with its space of global sections, i.e. we write

H^{0}(C, 𝕃(−D)) ≃ 𝕃(−D).

For any 𝔭 ∈ |D| we have

𝕃(−D)_{𝔭} = π_{𝔭}^{−n_{𝔭}}O_{C,𝔭}/O_{C,𝔭},

and we will define a residue map

res_{𝔭} : lim_{→} π_{𝔭}^{−n}O_{C,𝔭}/O_{C,𝔭} ⊗ Ω_{C,𝔭}^{1} = lim_{→} π^{−n}Ω_{C,𝔭}^{1}/Ω_{C,𝔭}^{1} ⟶^{res_{𝔭}} k.

The definition of this residue map requires some work especially if our ground field is of positive characteristic.

Once we have this map, then we can define

\begin{matrix}𝕃(−D)⊗ & H^{0}(C, Ω_{C}^{1}) ⟶ k \\ \underline{ξ}⊗ & ω ⟶ ∑_{𝔭∈|D|} res_{𝔭}(ξ_{𝔭} ⊗ ω) \end{matrix}

and we will show:

An element \underline{ξ} ∈ 𝕃(−D) is the Laurent expansion of an element f ∈ H^{0}(C, O_{C}(−D)) if and only if

∑_{𝔭} res_{𝔭}(fω) = 0.

It is clear that this says that we get a nondegenerate pairing

H^{1}(C, O_{C}) × H^{0}(C, Ω_{C}^{1}) ⟶ k.

The elements

ω′ = fω ∈ H^{0}(C, Ω_{C}^{1}(−D))

are called meromorphic differentials and part of the assertion above is that

∑_{𝔭} res_{𝔭}(ω′) = 0

for any meromorphic differential.

We come to the definition of the residue map

res_{𝔭} : lim_{→} π_{𝔭}^{−n}O_{C,𝔭}/O_{C,𝔭} ⊗ Ω_{C,𝔭}^{1} ⟶ k.

We notice that we can replace O_{C,𝔭} by the completion because

π_{𝔭}^{−n}O_{C,𝔭}/O_{C,𝔭} = π_{𝔭}^{−n}Ô_{C,𝔭}/Ô_{C,𝔭}.

At first we define this residue map only for rational points, i.e. those for which k(𝔭) = k. We change the notation slightly and denote the local parameter by X, then we have(???)

Ô_{C,𝔭} = k[[X]],

the quotient field is

k((X)) = k[[X]] [\frac{1}{X} ] ,

and our differentials are of the form

ω = f(X)dX = (\frac{a_{−n}}{X^{n}} + \frac{a_{n−1}}{X^{n−1}} + ⋯ + \frac{a_{−1}}{X} + ⋯) dX,

and we define

res_{𝔭}(ω) = res_{𝔭}(f ⊗ dX) = a_{−1}.

We have to show that this definition does not depend on the choice of the local parameter X. If we replace X by another local parameter

Y = tX + u_{2}X^{2} + ⋯ + u_{n}X^{n} + ⋯

where t ≠ 0 and otherwise P (X) is an arbitrary power series, then

ω = (\frac{a′_{−n}}{Y^{n}} + ⋯ + \frac{a′_{−1}}{Y′} ⋯) dY,

and we want (invariance of the residue)

a_{−1} = a′_{−1}.

This is indeed the case, but it is not so easy to prove.

If we want to prove the invariance of the residue, we have to write X as a power series in Y

X = τY + v_{2}Y^{2} + ⋯ + v_{m}Y^{m} + ⋯ = Q(Y)

where τ = t^{−1} and the v_{μ}/τ are polynomial in the u_{ν}/t.

Then we have to expand

\frac{dX}{X^{m}} = \frac{τ + 2v_{2}Y + ⋯ mv_{m}Y^{m−1} + ⋯}{τ^{m}Y^{m}(1 + \frac{v_{2}}{τ} Y + \frac{v_{m}}{τ} ⋯ Y^{m−1})^{m}} dY = (⋯ \frac{a′_{−1}}{Y} ⋯) dY.

We have to show that in this expansion the coefficient a′_{−1} of \frac{dY}{Y} is (i) equal to one if m = 1 and (ii) it is zero if m ≥ 2. The first assertion is clear because the factor τ cancels. But the second assertion is not

so clear. (I recommend to do the calculation for some small values of m.)

It is clear that our coefficient is a polynomial

P_{m} (\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} )

times τ^{1−m}, hence we have to show that this polynomial is identically zero if m ≥ 2. (We can view the \frac{v_{ν}}{τ} as indeterminates.)

Now we can prove the invariance of the residue by an argument which goes back to H. Hasse.

If the characteristic of k is equal to zero, then we can write for m ≥ 2

\frac{dX}{X^{m}} = −\frac{1}{m − 1}dX^{1−m}

but then

(\frac{a′_{−m}}{Y^{m}} + ⋯ \frac{a′_{−1}}{Y} ⋯) dY = −\frac{1}{m − 1}dQ(Y)^{1−m},

and this is only possible if a′_{−1} = 0.

This argument fails if char(k) = p > 0. For instance

\frac{dX}{X^{p+1}}

is not of the form dG(X). But we have to show that the polynomial P_{m} (\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ) vanishes identically. If we start from the case k = ℚ, then we see easily that P_{m} (\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ) has integer coefficients, hence it lies in

ℤ [\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ] .

Moreover it is obvious that the corresponding polynomial in arbitrary characteristic is obtained by reducing the coefficients mod p, it lies in

𝔽_{p} [\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ] ⊂ k [\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ] .

Since the polynomial in ℤ [\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ] must become the zero polynomial in ℚ [\frac{v_{1}}{τ} , ⋯, \frac{v_{m}}{τ} ], it must be identically zero itself. This proves (ii) in general.

We observe that our problem to construct res_{𝔭} is purely local, it concerns the structure of the space of meromorphic differentials in an infinitesimally small neighborhood of 𝔭.

We keep in mind that so far we constructed the residue map only for rational points. In the next section we will give a second proof of the invariance of the residue which uses

global arguments and which I like better. In this second proof we will use the fact that for differentials which have only a first order pole the invariance of the residue is obvious. (Assertion (i) above)

The other assertions which we made in the beginning of this section are global in nature. First we will prove for the special case ℙ^{1}/k. In the course of this proof we will see that the well definedness of res_{𝔭} follows from global information.

Hence we concentrate on the special case C = ℙ^{1}. We have seen H^{1}(ℙ^{1}, O_{ℙ^{1}}) = 0 hence g = 0, we know that

Ω_{ℙ^{1}}^{1} ≃ O_{ℙ^{1}}(−2)

(see ???) and hence we have

deg(Ω_{ℙ^{1}}^{1}) = 0 − 2 = −2.

We want to construct the canonical isomorphism

Res : H^{1}(ℙ^{1}, Ω_{ℙ^{1}}^{1}) ⟶ k.

To do this we apply a principle which also works in the general situation. So let us return to the case of an arbitrary smooth, projective and absolutely irreducible curve C = C/k. We will show later:

Proposition xxx:For any non zero effective divisor D we have H^{1}(C, Ω_{C}^{1}) = 0 and especially for any k rational point a ∈ C(k) we have an isomorphism

δ : H^{0}(C, 𝕃(−a)→^{∼}H^{1}(C, Ω_{C}^{1}).

The map δ is of course the boundary morphism.

This proposition is of course clear for the case C = ℙ^{1} because Ω_{ℙ^{1}}^{1}, (−a) ≃ O_{ℙ^{1}}(−1) and

H^{0}(ℙ^{1}, O_{ℙ^{1}}(−1)) = H^{1}(ℙ^{1}, O_{ℙ^{1}}(−1)) = 0.

So we pick a point a ∈ ℙ^{1}(k) and consider the sequence

0 → Ω_{ℙ^{1}}^{1} → Ω_{ℙ^{1}}^{1}(−a) → 𝕃(−a) ⊗ Ω_{ℙ^{1}}^{1} → 0

which gives us an isomorphism

δ_{a} : 𝕃(−a) ⊗ Ω_{ℙ^{1}}^{1} ˜ ⟶H^{1}(ℙ^{1}, Ω_{ℙ^{1}}^{1})

Now we noticed that we can define the map

res_{a} : 𝕃(−a) ⊗ Ω_{ℙ^{1}}^{1} ⟶ k,

and we put

Res = res_{a} ∘δ_{a}^{−1}.

Of course something would be wrong if we did not have the problem of well definedness again. Let us pick a point b ∈ ℙ^{1}(k) which is different from a. Then we get a diagram

𝕃(−a) ⊗ Ω_{ℙ^{1}}^{1}
↘^{i_{a}}

𝕃(−a − b) ⊗ Ω_{ℙ^{1}}^{1} ⟶^{δ} H^{1}(ℙ^{1}, Ω_{ℙ}^{1})

i_{b} ↗

𝕃(−b) ⊗ Ω_{ℙ^{1}}^{1}

Now the map δ has a nontrivial kernel and it is clear that this kernel is spanned by the form

ω_{a,b} = \frac{dx}{x − a} − \frac{dx}{x − b}

which is a generator of H^{0}(ℙ^{1}, Ω_{ℙ^{1}}^{1}(−a−b)). Since res_{a}(ω_{a,b})+res_{b}(ω_{a,b}) = 0. But δ_{a} = δ∘i_{a} and δ_{b} = δ ∘ i_{b} and it is clear that

res_{a} ∘ δ_{a}^{−1} = res_{b} ∘ δ_{b}^{−1}.

Now we consider the special point ∞ ∈ ℙ^{1}(k). We have

Spec(k[X]) ∪ Spec(k[Y]) = ℙ^{1}

where XY = 1 and ∞ is given by 0 ∈ Spec(k[Y]). We consider the exact sequence

0 → Ω_{ℙ^{1}}^{1} → Ω_{ℙ^{1}}^{1}(−n∞) → 𝕃(−n∞) ⊗ Ω_{ℙ^{1}}^{1} → 0

and get

δ : 𝕃(−n∞) ⊗ Ω_{ℙ^{1}}^{1} → H^{1}(ℙ^{1}, Ω_{ℙ^{1}}^{1}) → 0.

Hence we try a new definition

res′_{∞} = Res ∘δ

for the local residue map.

We want to give a local formula for res′_{∞}, this is easy. We observe that the elements of 𝕃(−n∞) ⊗ Ω_{ℙ^{1}}^{1} can be written as

ω = + (\frac{a_{1}}{Y} + \frac{a_{2}}{Y^{2}} ⋯ \frac{a_{n}}{Y^{n}} ) ⊗ dY = −(a_{1}X + ⋯ a_{n}X^{n}) ⊗ \frac{dX}{X^{2}} .

By definition we have res′_{∞}(ω) = 0 if and only if the element ω is the image of

H^{0}(ℙ^{1}, Ω_{ℙ^{1}}^{1}(−n∞)) ⟶ 𝕃(−na) ⊗ Ω_{ℙ^{1}}^{1},

and this is the case if and only if a_{1} = 0. We can also say that the form

−(a_{1}X + ⋯ + a_{n}X^{n})\frac{dX}{X^{2}}

is holomorphic on Spec(k[X]) \ {0} and has a pole of order ≤ 1 at zero with residue −a_{1} at 0. Hence it is clear that

res′_{∞}(ω) = a_{1},

and this is our old definition of the residue with respect to the parameter Y at infinity.

Now we give a second proof for the independence of the local parameter of the residue map.

We have seen that it suffices to show that the differentials

\frac{dY}{Y^{m}} ∈ 𝕃(−n∞) ⊗ Ω_{ℙ^{1}}^{1}

which have residue zero for m > 1 still have residue zero if we make a change of local parameters

Y = tU + v_{2}U^{2} ⋯ + v_{r}U^{r} + ⋯

where t ≠ 0.

But we know how to characterize the elements in 𝕃(−n∞) ⊗ Ω_{ℙ^{1}}^{1} which have residue zero. These are the elements which lie on the image of

H^{0}(ℙ^{1}, Ω_{ℙ^{1}}^{1}(−n∞)) ⟶ 𝕃(−n∞) ⊗ Ω_{ℙ^{1}}^{1}.

We approximate the power series which defines the change of parameters by a polynomial of degree r ≥ n, only the first in coefficients are relevant. Since the degree can be chosen to be larger, we may assume that the polynomial

F (U) = tU + v_{2}U^{2} + ⋯ + v_{r}U^{r}

is separable.

It provides an inclusion k[X] ⊂ k[U] and a morphism

Φ_{F} : ℙ^{1} ⟶ ℙ^{1}

where the first ℙ^{1} is Spec(k[U]) ⊂ Spec k[U^{−1}] and the second one is Spec(k[X]) ⊂ Spec k[X^{−1}]. We can consider the pull back Φ_{F}^{∗} (\frac{dX}{X^{m}}) = ω, this form has poles of order m in the point U = 0 and in the other r − 1 points u_{2} ⋯ u_{r} where F vanishes. (We assume that F is separable.)

In the point u = 0 its Laurent expansion is

\frac{F′(U)dU}{F (U)^{m}} ,

and we want to show that the residue at u = 0 with respect to the local parameter U is zero.

But it has some more poles at u_{2} ⋯ u_{r}. The total order of the pole is rm. Hence ω must have a zero divisor Z of degree rm − 2 because the degree of Ω^{1} is −2. We consider the line bundle

Ω_{ℙ^{1}}^{1}(−Z + ma_{2} + ⋯ + ma_{r}).

If we multiply ω by a section f ∈ H^{0}(ℙ^{1}, Ω_{ℙ^{1}}^{1}(−Z + ma_{2} ⋯ + ma_{r})), then we get a meromorphic form fω which now has only one pole at U = 0, the other poles are cancelled.

Of course we may have changed the expansion at U = 0. But if we map f to O_{ℙ^{1},0}/(U)^{m−1} O_{ℙ_{1},0}, then we get an exact sequence

0 → O_{ℙ^{1}}(−Z+ma_{2}+⋯+ma_{r}+(m−1)(0)) → O_{ℙ^{1}}(−Z+ma_{2}+⋯+ma_{r}) → O_{ℙ^{1}}/(U)^{m−1}O_{ℙ^{1}} → 0, and the line bundle on the left has degree −1, hence it has no cohomology. This yields that

H^{0}(ℙ^{1}, O_{ℙ^{1}}(−Z + ma_{2} + ⋯ + ma_{r}) ˜ ⟶O_{ℙ^{1}}/(U)^{m−1}O_{ℙ^{1}}

is an isomorphism. This means that we find an f ∈ H^{0}(ℙ^{1}, O_{ℙ^{1}}(−Z + ma_{2} + ⋯ + ma_{r})) such that its expansion at U = 0 with respect to U is

f(U) = 1 + a_{m}U^{m} + · · · .

Now fω has a pole only at U = 0 and its polar expansion is the same as the polar expansion of ω. Since this polar expansion is the expansion of fω the residue must be zero.

Historical and heuristical remark:

If our ground field is the field of complex numbers then the set of complex points S = C(ℂ) carries the structure of a compact Riemann surface. (Complex manifold in dimension one it has a natural orientation!) . In this case everything is much easier. A meromorphic differential ω which has a pole at a point P can be written locally as

Ω_{C/k}^{1} = f(z)dz

where z is a uniformizing element at P and f is meromorphic. In this case we look at a small disc around P and we know that

\frac{1}{2πi} ∮ ω = res_{p}(ω)

where the left hand side is defined intrinsically as the integral over the boundary of the disc taken counterclockwise and where the right hand side is computed in terms of the Laurent expansion of f(z). Hence it is clear that the right hand side does not depend on the choice of the local parameter z.

A meromorphic differential has a finite number of poles and we can compute the sum of the residues at these poles as the sum of the integrals over the boundary of these discs. Then we consider the complement U of the open discs. This is a compact manifold with boundary ∂U= union of the boundaries of the discs. We can apply Stokes theorem which says the the integral over this boundary is equal to the integral of dω over U. But ω is holomorphic on U hence we have dω = 0, and we see that the sum over the residues is zero.

It is interesting to look into the original paper by Riemann and to find out what Riemann and Roch actually proved.

For them cohomology did not exist, hence they could not define H^{1}(S, O_{S}) and g = dim H^{1}(S, O_{S}).

On the other hand they could of course study the cokernel of

H^{0}(S, O_{S}(−D)) ⟶ 𝕃(−D)

which is this cohomology if deg(D) >> 0.

Of course it was clear that the holomorphic 1-forms produced linear forms on this cokernel. They could define g = dim H^{0}(S, Ω_{S}^{1}) and it was clear to them that 2g is the the first Betti- number. It was also proved that the common kernel of linear forms produced by the differentials described the image of the map above. This is the content of Serre duality, which is much deeper. Therefore it i clear that they proved more that just the first version of Riemann-Roch.

We have to understand the sheaf Ω_{C/k}^{1}. To do this we consider a finite morphism

π : C ⟶ ℙ^{1}

which is induced by a choice of a function f ∈ k(C) for which k(C)/k(f) is separable.

In this case the functor π_{∗} which sends coherent sheaves on C to coherent sheaves on ℙ^{1} is acyclic because the fibres are finite. Hence we know that for any coherent sheaf F on C

H^{•}(C, F) ≃ H^{•}(ℙ^{1}, π_{∗}(F)).

For any line bundle L on C the bundle π_{∗}(L) is a vector bundle of rank d = degree (π) over ℙ^{1}.

We apply Grothendieck’s theorem to this vector bundle and write

π_{∗}(L) = O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{d})

with a_{1} ≥ a_{2} ⋯ ≥ a_{d}.

If L = O_{C} we have H^{0}(C, O_{C}) = k and this implies

a_{1} = 0  and  a_{ν} < 0  for all  ν ≥ 2.

Since by definition

dim H^{1}(C, O_{C}) = dim H^{1}(ℙ^{1}, π_{∗}(O_{C})) = g

we must have (see ???)

g = ∑_{ν=2}^{d} (−a_{ν} − 1) = − ∑_{ν=2}^{d} a_{ν} − (d − 1).

We invoke the Riemann-Hurwitz formula

Ω_{ℙ^{1}}^{1} ⊗ 𝔇_{C/ℙ^{1}}^{−1} →^{∼}Ω_{C}^{1}

By construction the vector bundle π_{∗}(𝔇_{C/ℙ^{1}}^{−1}) is dual to the bundle π_{∗}(O_{C}). We wrote

π_{∗}(O_{C}) = O_{ℙ^{1}} ⊕ ⨁_{ν=2}^{d} O_{ℙ^{1}}(a_{ν}),

hence we see that

π_{∗}(𝔇_{C/ℙ^{1}}^{−1}) = O_{ℙ^{1}} ⊕ ⨁_{ν=2}^{d} O_{ℙ^{1}}(−a_{ν})

and

π_{∗}(Ω_{C}^{1}) = O_{ℙ^{1}}(−2) ⊕ ⨁_{ν=2}^{d} O_{ℙ^{1}}(−a_{ν} − 2).

Since the a_{ν} < 0 we see −a_{ν} − 2 ≥ −1. We get

H^{1}(C, Ω_{C}^{1}) ≃ H^{1}(C, π_{∗}(Ω_{C}^{1})) = H^{1}(ℙ^{1}, O_{ℙ^{1}}(−2)) ≃ k

and

dim_{k}H^{0}(C, Ω_{C}^{1}) = ∑_{v=2}^{d} (−a_{ν} − 1) = g.

For any line bundle L on C and any effective divisor D we can consider the exact sequence

0 ⟶ L ⟶ L(−D) ⟶ 𝕃(−D) ⟶ 0

which yields the exact sequence

0 ⟶ π_{∗}(L) ⟶ π_{∗}(L(−D)) ⟶ π_{∗}(𝕃(−D)) ⟶ 0

and π_{∗}(𝕃(−D)) is a torsion sheaf on ℙ^{1}, its space of section has

dim_{k}π_{∗}(𝕃(−D)) = deg D.

Hence we see that (see our fourmula ???)

deg(π_{∗}(L(−D)) = deg(π_{∗}(L)) + deg D.

If we apply this to the exact sequence

0 ⟶ O_{C} ⟶ 𝔇^{−1} ⟶ 𝔇^{−1}/O_{C} → 00

we get

deg(π_{∗}(𝔇^{−1})) = deg(π_{∗}(O_{C})) + dim_{k}(𝔇^{−1}/O).

On the other hand

deg(π_{∗}(O_{C})) + deg(π_{∗}(𝔇^{−1})) = 0

because these bundlex are dual to eachother and hence

dim_{k}(𝔇^{−1}/O) = 2 deg(π_{∗}(𝔇^{−1})) = 2 (− ∑_{ν=2}^{d} a_{ν}) = 2(g + d − 1).

This can be read on C. We get

deg(𝔇^{−1}) = 2(g − d + 1).

On the other hand we know that deg(π^{∗}(Ω_{ℙ^{1}}^{1})) = −2d and therefore

deg(Ω_{C}^{1}) = −2d + 2(g + d − 1) = 2g − 2.

We want to construct an explicit linear map

Res : H^{1}(C, Ω_{C}^{1}) ⟶ k.

To get this map we proceed in the same way as in the case C = ℙ^{1}. We pick a point x ∈ C(k) and consider the line bundle Ω_{C}^{1}(−x). I claim that

H^{1}(C, Ω_{C}^{1}(−x)) = 0.

To see that this is so we choose a meromorphic f such that df generates the differential at x and we consider the map induced by f

π_{f} = π : C ⟶ ℙ^{1},

it is étale at the point a. It is clear that the bundles

π_{∗}(O_{C}(x))  and  π_{∗}(𝔇^{−1}(−x))

are dual to each other. We have

π_{∗}(O_{C}(x)) ⊂ π_{∗}(O_{C}) = O_{ℙ^{1}} ⊕ ⨁_{v=2}^{d} O_{ℙ^{1}}(−a_{ν}),

and the degree of this subbundle drops by one. Since π_{∗}(O_{C}(x)) has no non trivial section, we conclude that

π_{∗}(O_{C}(x)) ≃ O_{ℙ^{1}}(−1) ⊕ ⨁_{v=2}^{d} O_{ℙ^{1}}(−a_{ν})

and hence because of duality

π_{∗}(𝔇^{−1}(−x)) = O_{ℙ^{1}}(1) ⊕ ⨁_{v=2}^{d} O_{ℙ^{1}}(a_{ν}).

But now

H^{1}(ℙ^{1}.π_{∗}(π^{∗}(Ω_{ℙ^{1}}^{1}) ⊗ 𝔇^{−1}(−x))) = H^{1}(C, Ω_{C}^{1}(−x)) = 0.

We have the exact sequence

0 ⟶ Ω_{C}^{1} ⟶ Ω_{C}^{1}(−x) ⟶ 𝕃(−x) ⊗ Ω_{C}^{1} ⟶ 0

and obtain the exact sequence in cohomology

H^{0}(C, Ω_{C}^{1}) → H^{0}(C, Ω_{C}^{1}(−x)) → H^{0}(C, 𝕃(−x) ⊗ Ω_{C}^{1}) →^{δ} H^{1}(C, Ω_{C}^{1}) → 0.

The map δ must be an isomorphism, we have

res_{x} : H^{0}(C, 𝕃(−x) ⊗ Ω_{C}^{1}) ˜ ⟶k

and define

Res = res_{x} ∘δ^{−1}.

To see that this morphism does not depend on the choice of a, we choose a second point b. Now we consider the exact sequence

0 → Ω_{C}^{1} → Ω_{C}^{1}(−a − b) → 𝕃(−a − b) ⊗ Ω_{C}^{1} → 0,

and find that there exists a 1-form ω′_{a,b} on C which has a simple pole at a and b and is holomorphic elswhere. We choose an f ∈ k(C) such that df generates Ω_{C}^{1} in the two points a, b and consider the resulting morphism

π = π_{F} : C ⟶ ℙ^{1}.

This morphism π induces isomorphisms between the completions Ô_{C,a} ≃ Ô_{ℙ^{1},π(a)}, Ô_{C,b} ≃ Ô_{ℙ^{1},π(a)}, and it is clear that

res_{a}(ω′_{a,b}) = res_{π(a)} (tr_{C/ℙ^{1}}(ω′_{a,b}))
res_{b}(ω′_{a,b}) = res_{π(b)} (tr_{C/ℙ^{1}}(ω′_{a,b})) .

Of course it is clear that tr_{C/ℙ^{1}}(ω′_{a,b}) is a non zero multiple of the 1-form ω_{π(a),π(b)} which we constructed on ℙ^{1}, hence we get

res_{a}(ω′_{a,b}) + res_{b}(ω′_{a,b}) = 0.

This argument shows that Res : H^{1}(C, Ω_{C}^{1}) → k is well defined but it also shows that it is compatible with a map

π : C ⟶ ℙ^{1}

as above: By construction we habe the commutative diagram

H^{0}(C, 𝕃(−a) ⊗ Ω_{C}^{1}) ⟶ H^{1}(C, Ω_{C}^{1})
↓ tr_{C/ℙ^{1}}  ↓ tr_{C/ℙ^{1}}
H^{0}(ℙ^{1}, 𝕃(−π(a)) ⊗ Ω_{ℙ^{1}}^{1}) ⟶ H^{1}(ℙ^{1}, Ω_{ℙ^{1}}^{1})

Which then implies the commutativity

H^{1}(C, Ω_{C}^{1})
↘^{Res}
↓ tr_{C/ℙ^{1}}  k
↗_{Res}
H^{1}(ℙ^{1}, Ω_{ℙ^{1}}^{1})

and then it is easy to derive the general compatibility for arbitrary separable d : C_{1} → C_{2}.

It is also clear that in the diagram above we can replace 𝕃(−a) by 𝕃(−na) with any n > 0, then we get a commutative diagram

H^{0}(C, 𝕃(−na)) ⟶^{δ} H^{1}(C, Ω_{C}^{1})
↘^{res^{a}}  ↙^{Res}
k  .

Hence we see that under the assumption that k is algebraically closed, we have for any effective divisor D ≠ 0 that

H^{0}(C, ω_{C}^{1}(−D)) → H^{0}(C, 𝕃(−D) ⊗ Ω_{C}^{1}) → H^{1}(C, Ω_{C}^{1}) →
∑_{a∈|D|} res_{a} ↘  ↙ Res
k

is an exact sequence and the triangle in the bottom is commutative.

Finally we have to remove our assumption that k is algebraically closed.

We needed a rational point a ∈ C(k) to construct Res : H^{1}(C, Ω_{C}^{1}) → k and so far we defined the residue map

res_{𝔭} : 𝕃(−∞𝔭) ⊗ Ω_{C,𝔭}^{1} ⟶ k

only fo rational points 𝔭 ∈ C(k). The decisive point is that we can extend the definition easily to “separable” points, i.e. points for which k(𝔭)/k is a separable extension and in addition we always have many “separable” points on C.

How do we get separable points? We choose an f ∈ k(C) such that k(C)/k(f) becomes separable. Then we have the resulting π_{f} : C → ℙ^{1}. We have a non empty affine open set V ⊂ ℙ^{1} such that π^{−1}(V) = U → V is unramified.

Now if k is finite, then any point 𝔭 ∈ C is separable. Otherwise it is clear that V (k) is infinite and for any 𝔭_{0} = X_{0} ∈ V (k) the points 𝔶 ∈ C lying over 𝔭_{0} are separable.

(Vielleicht hier Ref. auf früher ???)

Now we want to define res_{𝔭} for a separable point 𝔭 ∈ C.

Text schon vorhanden!

Now we define Res : H^{1}(C, Ω_{C}^{1}) → k as before: We pick a separable point 𝔭 ∈ C and consider the exact sequence in cohomology

H^{0}(C, Ω_{C}^{1}(−𝔭)) → H^{0}(C, 𝕃(−𝔭) ⊗ Ω_{C}^{1}) →^{δ} H^{1}(C, Ω_{C}^{1}) → 0.

Here the map δ is not necessarily an isomorphism. But if we extend our base field to the algebraic closure, then see that the kernel of our map

res : H^{0}(C, 𝕃(−𝔭) ⊗ Ω_{C}^{1}) ⟶ k

becomes the kernel of the sum of the residues, hence ker(δ) = ker(res_{𝔶}). Hence we can define Res by the diagram

H^{0}(C, 𝕃(−𝔭) ⊗ Ω_{C}^{1}) ⟶ H^{1}(C, ω_{C}^{1})
res_{𝔭} ↘  ↙ Res
k

But once we defined Res we can define res_{𝔭} for arbitrary points using the same diagram.

It is clear that our definition of Res has the right functoriality principles with respect to separable morphisms and extension of the ground field.

Especially we know now that for any effective divisor D the image of

H^{0}(C, Ω_{C}^{1}(−D)) ⟶ H^{0}(C, 𝕃(−D) ⊗ Ω_{C}^{1})

consists of the elements \underline{ξ} ⊗ ω for which

∑_{𝔭} res_{𝔭}(ξ_{𝔭} ⊗ ω) = 0.

Finally we can state the Serre duality theorem

Theorem: For any line bundle L on C we have a perfect pairing

H^{0}(C, L) × H^{1}(C, L^{−1} ⊗ Ω_{C}^{1}) ⟶ k

which is given by the cup product. Of course that means that a global section s ∈ H^{0}(C, L) defines by multiplication a map

⟨s, ⟩ : H^{1}(C, L^{−1} ⊗ Ω_{C}^{1}) ⟶ H^{1}(C, Ω_{C}^{1}) ⟶^{Res} k.

Proof:

Now we can formulate the final version of the Riemann-Roch theorem. We start from a line bundle L on C and we pick a point 𝔭 (or an effective divisor). We compute the cohomology H^{1}(C, L) and to do this we start from the sequence

0 → L → L(−∞𝔭) → L(−∞𝔭)/L → 0.

We have the cohomology sequence

H^{0}(C, L(−∞𝔭)) → H^{0}(C, L(−∞𝔭)/L) → H^{1}(C, L) → 0.

Now we consider the sheaf of differentials with coefficients in L^{−1} namely L^{−1} ⊗ Ω. If 𝔭 ∈ H^{0}(C, L(−∞𝔭)) and ω ∈ H^{0}(L^{−1} ⊗ Ω_{C/k}^{1}) then ξ_{𝔭}ω is a Laurentexpansion of a meromorphic differential at 𝔭 and res_{𝔭}(ξ_{𝔭}ω) is defined. If ξ𝔭 comes from a meromorphic section s ∈ H^{0}(C, L(−∞𝔭)) then sω is a meromorphic differential which is holomorphic outside 𝔭. Hence res_{𝔭}(sω) = 0 and we get again a pairing

Res : H^{1}(C, L) × H^{0}(C, L^{−1} ⊗ Ω_{C/k}^{1}) → k.

This generalizes the pairing we had for L = O_{C}.

Theorem (Serre Duality or final version of Riemann-Roch):

The pairing

Res : H^{1}(C, L) × H^{0}(C, L^{−1} ⊗ Ω_{C/k}^{1}) → k

is non-degenerated.

Proof: It is quite clear that a non zero element α ∈ H^{0}(C, L^{−1} ⊗ Ω_{C/k}^{1}) will induce a non trivial linear form on H^{1}(C, L). Hence we conclude dim_{k}H^{1}(C, L) ≥ dim_{k}H^{0}(C, L^{−1} ⊗ Ω_{C/k}^{1}). This last dimension is equal to (first version of Riemann-Roch)

dim_{k}H^{1}(C, L^{−1} ⊗ Ω_{C/k}^{1}) − deg(L) + 2g − 2 + 1 − g

We apply our argument a second time and get dim_{k}H^{1}(C, L^{−1} ⊗ Ω_{C/k}^{1}) ≥ dim_{k}H^{0}(C, L) Hence we see that

dim_{k}H^{1}(C, L) ≥ dim_{k}H^{0}(C, L) − deg(L) + g − 1

.

But the first version of Riemann-Roch tells us that we must have equality in this last inequality. Hence we see that all the inequalities in between where actually equalities. But if the dimensions are equal it follows that the pairing is non degenerate.

Finally we conclude: If we have a line bundle L with deg(L) < 0 then we have H^{0}(C, L) = 0, because if we had a section s ≠ 0 then deg(div(s)) = deg(L) and div(s) is effective.

Now we can say: If L is a line bundle with deg(L) > 2g−2 then H^{1}(C, L) = H^{0}(C, L^{−1}⊗ Ω_{C/k}^{1}) = 0 and

dim H^{0}(C, L) = deg(L) + 1 − g.

Comments on the first version of the theorem of Riemann-Roch.

I begin with a very general remark. If we have a projective scheme X → Spec(k) and a coherent sheaf F on X, then we have seen that the cohomology groups H^{i}(X, F) are finite dimensional k-vector spaces and the cohomology vanishes, if i >> 0. This allows us to define the Euler-characteristic

χ(X, F) = ∑_{i}(−1)^{i}dim_{k}H^{i}(X, F).

A short exact sequence of sheaves

0 → F′ → F → F” → 0

yields a long exact sequence in cohomology. Now it is an easy exercise in linear algebra to show that the long exact sequence provides the addivity of the Euler characteristic

χ(X, F) = χ(X, F′) + χ(X, F”).

Now we consider the special case where X/k = C/k is a smooth curve. In this case, we have seen that for any locally free sheaf E on C we can define rank(E) and deg(E). The rank is simply the dimension of the generic fibre as a vector space over O_{C,η} and the degree can be defined inductively: If 0 → E′ → E → E” → 0 is an exact sequence of locally free sheaves then

deg(E) = deg(E′) + deg(E”).

Then we show that any locally free sheaf admits a filtration (complete flag):

(0) = F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ F_{n−1} ⊂ F_{n} = E

such that F_{i+1}/F_{i} = M_{i} is a line bundle. We reduced the definition of the degree to the case of line bundles. One has to check that if n = rank(E)

Λ^{n}E ≃ ⊗F_{i}/F_{i−1} = ⊗M_{i}.

The theorem of Riemann-Roch gives as a formula for χ(C, E) in terms of rank(E), deg(E) namely

χ(C, E) = deg(E) + rank(E)(1 − g).

To prove this we observe that both sides behave additively under exact sequences and then the existence of flags reduces the problem to the case of line bundles.

But there is a different way to look at the theorem of Riemann - Roch. We could consider all coherent sheaves on C and ask for a formula for χ(C, F) in terms of deg(F), rank(F). The first problem is that we do not have a notion of deg(F) and rank(F) for arbitrary coherent sheaves.

But let us have a closer look at the coherent sheaves on C. Locally they are finitely generated modules over Dedekind rings. If A is a Dedekind ring and M a finitely generated A-module then we have an exact sequence

0 → M_{tors} → M → M/M_{tors} → 0

where M_{tors} is the module of torsion elements and M/M_{tors} is locally free. Hence we see that any coherent sheaf on C sits in such a sequence

0 → E_{tors} → E → E/E_{tors} → 0

where E/E_{tors} is locally free. Now it is clear that a torsion module over a discrete valuation ring O_{C,𝔭}M = ⊕O_{C,𝔭}/𝔭^{m_{i}} and this implies that any torsion module M on C can be written as quotient of a vector bundle by a subbundle of the same rank. Hence it sits in an exact sequence

0 → E′ → E → M → 0.

This suggests the definition

deg(M) = deg(E) − deg(E′) = dim_{k}H^{0}(C, M)
rank(M) = rank(E) − rank(E′) = 0.

The first formula has been verified earlier. We define the degree and the rank for arbitrary coherent sheaves by

deg(E) = deg(E/E_{tors}) + deg(E_{tors})
rank(E) = rank(E/E_{tors})

and this gives a more general Riemann-Roch formula.

For any coherent sheaf F on C we have

χ(C, F) = deg(F) + (1 − g) rank(F).

The proof is almost obvious but I write it down in a slight sophisticated form. I introduce the group K′(C). This group is generated by the isomorphism classes of coherent sheaves. For such a sheaf E let [E] be its class in K′(C). We introduce the relations: For any exact sequence

0 → E′ → E → E” → 0

we have

[E] = [E′] + [E”].

The Euler characteristic, the degree and the rank provide homomorphisms

χ : K′(C) → ℤ
deg : K′(C) → ℤ
rank : K′(C) → ℤ

and the theorem of Riemann-Roch says

χ = deg +(1 − g) · rank .

ur previous considerations make it clear that K′(C) is generated by line bundles and torsion sheaves. But since any line bundle is of the form L = O_{C}(−D) we see that the group K′(C) is actually generated the structure sheaf and the torsion sheaves. But now it is clear that

χ(O_{C}) = deg(O_{C}) + (1 − g) rank(O_{C}) = 1 − g

and for torsion sheaves

χ(M) = deg(M)

and this proves the formula.

The structure of K′(C)

The structure of this group is rather complicated. We have the surjective homomorphism (deg, rank) : K′(C) → ℤ ⊕ ℤ which means that the group has a very simple quotient. But the kernel of this group is very complicated. We leave it as an exercise to prove that the kernel is the group Pic^{0}(C) which is the group of line bundles of degree zero.

Applications of the Riemann-Roch Theorem

I want to begin by discussing the cases of curves with low genus. Let C/k be an absolutely irreducible, smooth and projective curve. Let us assume the genus of this curve is 0. If this curve has a rational point P ∈ C(k), then we can consider the line bundle L = O_{C}(−P) and as in VII 3.1, we can consider the morphism

r_{L} : C ⟶ Proj (⨁_{n=0}^{∞} H^{0}(C, L^{⊗n})) ,

The vector space H^{0}(C, L) has rank 2, it is generated by the constant function 1 ∈ H^{0}(C, L) which we call X_{0} and another function X_{1} which has a first order pole at P. Then it is clear that H^{0}(C, L^{⊗n}) is spanned by the homogeneous polynomials of degree n in X_{0}, X_{1}, and these form a basis. Hence we see that

⨁_{n=0}^{∞} H^{0}(C, L^{⊗n}) = k[X_{0}, X_{1}].

I leave it as an exercise to the reader to show that r_{L} provides an isomorphism

r_{L} : C ⟶^{∼} ℙ^{1}

(see VII ...).

If the curve does not have a rational point, then we can say at least the following. We have the sheaf Ω_{C/k}^{1} and this is a line bundle of degree −2. Hence the dual of this line bundle has degree 2 and therefore we can find a non zero section t ∈ H^{0}(C, (Ω_{ℙ^{1}}^{1})^{∨}). This section must have zeroes. The divisor of zeroes has degree 2, hence it must be a point P of degree 2 or of the form P_{1} + P_{2} with two rational points P_{1} and P_{2}. In the second case we see that we have a rational point. In the first case the point P has a residue field k(P) which is of degree 2 over k. Hence we conclude that we can always find a quadratic extension over which C will split.

Of course the first case can happen if k is not algebraically closed. If we consider a quadratic form over a field of characteristic ≠ 2

ax^{2} + by^{2} + cz^{2},

then it defines a curve of genus zero if it is not reducible (Exercise 1 ???). Now it is clear that this curve does not have a rational point if this form does not represent zero (i.e. we can not find x_{0}, y_{0}, z_{0}) ∈ k^{3}\{0} such that ax_{0}^{2} + by_{0}^{2} + cz_{0}^{2} = 0).

It is not difficult to see that the line bundle L = (Ω_{ℙ^{1}}^{1})^{∨} provides an embedding

r_{L} : C ⟶ ℙ^{2}

where the image is described by a quadratic form.

Now we consider curves of genus one. Again we assume that we have a point P ∈ C(k), and we consider the line bundle L = O_{C}(−P). We have the inclusions

O_{C} ⊂ O_{C}(−P) ⊂ O_{C}(−2P) ⊂ O(−3P).

Now we consider the graded ring

⨁_{n} H^{0}(C, L^{⊗n}).

In contrast to our precious situations this ring is not generated by elements in degree one, but we will see that this does not really matter.

The constant function 1 yields a section x_{0} ∈ H^{0}(C, L) and this section spans this space. Then we get from the Riemann-Roch theorem that we have a section x_{1} ∈ H^{0}(C, L^{⊗2}) which is independent of x_{0}^{2}, and we have a section x_{2} ∈ H^{0}(C, L^{⊗3}) which is independent of x_{0}^{3}, x_{0}^{2}x_{1}. Now it is not difficult to see that the element x_{0} (in degree one), x_{1} (in degree 2) and x_{2} (in degree 3) generate the graded ring. It follows from the Riemann-Roch theorem that the elements

{x_{0}^{6}, x_{0}^{4}x_{1}, x_{0}^{2}x_{1}^{2}, x_{1}^{3}, x_{0}^{3}x_{2}, x_{0}x_{1}x_{2}, x_{2}^{2}}

must be linearily dependent because the space H^{0}(C, L^{⊗6}) has dimension 6. Therefore we find a linear relation amoung them. Before we write it down we want to derive some information about it. Let π_{P} be a uniformizing element at ℙ, then

x_{2} = \frac{a}{π_{P}^{3}} + … a ∈ k^{∗}

and

x_{1} = \frac{b}{π_{P}^{2}} + … b ∈ k^{∗}.

It is clear that x_{2}^{2}, x_{1}^{3} are the only terms which have a 6-th order pole, hence our searched for relation has to cancel that pole.

On the other hand we can modify x_{1}, x_{2} by a scalar factor such that the above numbers a = b = 1. Then we can conclude that our relation must be of the form

x_{2}^{2} + a_{1}x_{0}x_{1}x_{2} + a_{3}x_{0}^{3}x_{2} = x_{1}^{3} + a_{2}x_{0}^{2}x_{1}^{2} + a_{4}x_{0}^{4}x_{1} + a_{6}x_{0}^{6}.

this is a relation among elements in H^{0}(C, L^{⊗6}) = H^{0}(C, O_{C}(−6P)). Since x_{0} is the constant function 1 viewed as element in H^{0}(C, O_{C}(−6P)) we see that the exponent in x_{0}^{∨} is irrelevant and we can modify our relation to a relation in variables of degree one

x_{0}x_{2}^{2} + a_{1}x_{0}x_{1}x_{2} + a_{3}x_{0}^{2}x_{2} + x_{1}^{3} + a_{2}x_{0}x_{1}^{2} + a_{4}x_{0}^{2}x_{1} + a_{6}x_{0}^{3},

and this is a homogeneous relation of degree three among elements in H^{0}(C, L^{⊗3}). (The point of this trick is that we see that we do not need all monomials of degree 3 in x_{0}, x_{1}, x_{2} ∈ H^{0}(C, L^{⊗3})). Hence we get the morphism

r_{L^{⊗3}} : C ⟶ ℙ^{2}
↘ ↙
Spec(k).

Again I leave it to the reader to show that this is a closed embedding.

If we do not have a point on C then we have difficulties. In this case we are not able to produce a point P ∈ C whose residue field has a degree which can be estimated independently of C.

The moduli space

At this point I want to give an outlook to more advanced topics.

We consider an elliptic curve C over a base scheme S which for simplicity I assume is affine, i.e. S = Spec(A). By this I mean that C is projective and smooth over S, I assume that all the fibres are absolutelz irreducible and of dimension one and also of genus one. Finally I assume that we have a section s

\begin{matrix}C \\ π ↓↑ s \\ S. \end{matrix}

As before, we view the image of S under the section s as a divisor P on C. It defines a line bundle L = O_{C}(−P) on C. If we evaluate at a point x ∈ S, then we get the curve

C× Spec(k(x))
↓
Spec(k(x)),

the point P gives us a k(x)-rational point, and we are in the previous situation.

Now we consider the direct images of our line bundle π_{∗}(L^{⊗n}). The Riemann-Roch theorem tells us that the k(x)-vector spaces

H^{0}(C ×_{S} k(x), L^{⊗n} | C ×_{S} k(x))

have dimension n if x varies. Then our semicontinouity implies that π_{∗}(L^{⊗n}) are locally free of rank n over A.

Let us make the further assumption that any locally free module over A is actually free. (There are interesting rings having this property.) Then we can proceed as before. We find sections

x_{0} ∈ H^{0}(C, L), x_{1} ∈ H^{0}(C, L^{⊗2}), x_{3} ∈ H^{0}(C, L^{⊗3})

such that they provide respectively a basis in

H^{0}(C, L) ⊂ H^{0}(C, L^{⊗2}) ⊂ H^{0}(C, L^{⊗3}).

If J ⊂ O_{C} is the ideal defining P the module J/J^{2} is free or rank one over A. We may choose an uniformizing element π_{P} which generates this A-module. The elements π_{P}^{2} x_{1} and π_{P}^{3} x_{2} are elements which are regular along P, we can evaluate at P, the result is a unit in A. Now we can require

π_{P}^{2} x − 2(P) = π_{P}^{3} x_{3}(P) = 1.

Then we conclude that we have a relation

x_{2}^{2} + a_{1}x_{0}x_{1}x_{2} + a_{3}x_{0}^{3}x_{2} = x_{1}^{3} + a_{2}x_{0}^{2}x_{1}^{2} + a_{4}x_{0}^{4}x_{1} = a_{6}x_{0}^{6}

with some coefficients in A. Again we perform the change of the bundle, we replace L bz L^{⊗3} = O_{C}(−3P) and we consider x_{0}, x_{1}, x_{2} as sections of this bundle, then get degree one and consider the relation

x_{0}x_{2}^{2} + a_{1}x_{0}x_{1}x_{2} + a_{3}x_{0}^{2}x_{2} = x_{1}^{3} + a_{2}x_{0}x_{1}^{2} + a_{4}x_{0}^{2}x_{1} + a_{6}x_{0}^{3}

and the homogeneous relation of degree 3 describes C as a closed subscheme of ℙ^{2} = Proj A[X_{0}, X_{1}, X_{2}]. Now we have to take into account that the choices of x_{0}, x_{1}, x_{2} as sections in H^{0}(C, O_{C}(−3P)) are not unique. Of course the choice of x_{0} as constant function 1 is canonical. If we stick to our choice of π_{P}, then we can modify

\begin{aligned}x_{1} & ⟶ & x_{1} + αx_{0} & = & x′_{1} \\ x_{2} & ⟶ & x_{2} + βx_{1} + γx_{0} & = & x′_{2}. \end{aligned}  (S)

Then we will get a new relation with new coefficients, in other words the coefficeints of the relationare by no means determined by the curve and the choice of π_{p}.

The following statements have to be varified by computations and a little bit of thinking.

Assume that 2 and 3 are invertible in A. Then we can find a unique substitution as above such that the new relation will be of the form

x_{0}x_{2}^{2} + x_{1}^{3} + a′_{4}x_{0}^{2}x_{1} + a′_{6}x_{0}^{3}  (∗ ∗ ∗)

where a′_{4} and a′_{6} are the following expressions in the a_{1}, a_{3}, a_{2}, a_{4}, a_{6}:

\begin{matrix}a′_{4} = \frac{a_{1}}{48} + \frac{a_{1}^{2}a_{2}}{6} + \frac{a_{2}^{2}}{3} − \frac{a_{1}a_{3}}{2} − a_{4} \\ a′_{6} = \frac{a_{1}^{6}}{864} + \frac{a_{1}^{4}a_{2}^{2}}{72} + \frac{a_{1}^{2}a_{2}^{2}}{18} + \frac{2a_{2}^{2}}{27} − \frac{a_{1}^{3}a_{3}}{24} − \frac{a_{1}a_{2}a_{3}}{6} + \frac{a_{3}^{2}}{4} − \frac{a_{1}^{2}a_{4}}{12} − \frac{a_{2}a_{4}}{3} + a_{6} \end{matrix}

Now it is quite clear that the curve defined by the equation

x_{0}x_{2}^{2} = x_{1}^{3} + a′_{4}x_{1}x_{0}^{2} + a′_{6}x_{0}^{3}

is smooth if and only if the discriminant

−4(a′_{4})^{3} − 27(a′_{6})^{2}

is a unit in A. If we write down this expression, then we get an expression which is a sum of monomials in the a_{1}, a_{3}, a_{2}, a_{4}, a_{6} which is homogeneous of degree 12 and where the coefficients are rational numbers which have only powers of 2 in their denominator and where the largest denominator is 16. We define

Δ(a_{1}, a_{3}, a_{2}, a_{4}, a_{6}) = 16 · (−4(a′_{4})^{3} − 27(a′_{6})^{2}).

This is again a sum of such monomials where now the coefficients are integral and where we have some monomials with coefficient ±1.

Finally we can apply the same process to a′_{4} and a′_{6} and put

C_{4} = 48 · a′_{4} , C_{6} = 862 · a′_{6},

here the monomials have integral coefficients.

We have 48 = 4 · 12 and 862 = 12^{3}/2. Then it becomes clear that

△(a_{1}, a_{3}, a_{2}, a_{4}, a_{6}) = \frac{1}{12^{3}} · (−C_{4}^{3} − C_{6}^{2}),

and with the help of a computer I found

Δ(a_{1}, a_{3}, a_{2}, a_{4}, a_{6}) = −a_{1}^{4}a_{2}a_{3}^{2} − 8a_{1}^{2}a_{2}^{2}a_{3}^{2} − 16a_{2}^{3}a_{3}^{2} + a_{1}^{3}a_{3}^{3} + 36a_{1}a_{2}a_{3}^{3}−

27a_{3}^{4} + a_{1}^{5}a_{3}a_{4} + 8a_{1}^{3}a_{2}a_{3}a_{4} + 16a_{1}a_{2}^{2}a_{3}a_{4} − 30a_{1}^{2}a_{3}^{2}a_{4} + 72a_{2}a_{3}^{2}a_{4} + a_{1}^{4}a_{4}^{2} + 8a_{1}^{2}a_{2}a_{4}^{2}+

16a_{2}^{2}a_{4}^{2} − 96a_{1}a_{3}a_{4}^{2} − 64a_{4}^{3} − a_{1}^{6}a_{6} − 12a_{1}^{4}a_{2}a_{6} − 48a_{1}^{2}a_{2}^{2}a_{6} − 64a_{2}^{3}a_{6} + 36a_{1}^{3}a_{3}a_{6}+

144a_{1}a_{2}a_{3}a_{6} − 216a_{3}^{2}a_{6} + 72a_{1}^{2}a_{4}a_{6} + 288a_{2}a_{4}a_{6} − 432a_{6}^{2}.

Since we have aunique substitution which transforms a general relation into the specail form (***), we conclude that a′_{4}, a′_{6} are invariant expressions under our substitutions.

Now we observe that the expressions C_{4}, C_{6} and Δ can be written down without the assumption that \frac{1}{2}, 1|over3 ∈ A. And it is clear that:

The expressions for C_{4}, C_{6} and Δ are invariant under the substitutions induced by substitutions of the form (S).

The following theorem is almost clear from our considerations above¿

Theorem: The equation

x_{0}x_{2}^{2} + a_{1}x_{0}x_{1}x_{2} + a_{3}x_{2}x_{0}^{2} = x_{1}^{3}x_{0} + a_{4}x_{1}x_{0}^{2} + a_{6}x_{0}^{3}

defines a smooth projection curve over Spec(A) if and only if Δ(a_{1}, a_{3}, a_{2}, a_{4}, a_{6}) ∈ A^{∗}. It contains the point

P = (0, 0, 1).

The following considerations are more geometric in nature, hence we assume at this point that C/k is a smooth, projective and irreducible curve over an algebraically closed field k. Let g be the genus of our curve.

We pick g points P_{1}, …, P_{g} on our curve. We form the divisor D = ∑ P_{i} and ask Is there a non-constant meromorphic function f ∈ H^{0}(C, O_{C}(−D)) ?

a) What does the Riemann-Roch theorem tell us?

We see that the simple version of the Riemann-Roch is not good enough.

b) What do we need to know about the behavior of the holomorphic differentials at D if we want to answer our question?

c) Prove that we can find g points P_{1}, …, P_{g} such that there is no a non-zero differential ω ∈ H^{0}(C, Ω_{C}) which vanishes in all these points.

d) If we have such a g-tupel, as in c) what do we know about our original question?

We want to discuss this kind of questions in a more sytematic way. We consider the product of our curve by itself C ×_{k} C, then we may consider the diagonal Δ ⊂ C ×_{k} C. This diagonal can locally be described by one equation and hence we can look at the line bundle O_{C×C}(−Δ). (See section on divisors).

We now form the product

C × C^{g} = C × \underbrace{C × C × … × C}_{g−times} = X.

On this variety we have the line bundles O_{X}(−Δ_{i}) which is obtained by the diagonal in the factor C in front and the i-th factor in the product.

This gives us a line bundle L = ⊗_{i=1}^{g}O_{X}(−Δ_{i}) on X.

Now we look at the projection to the second factor

\begin{aligned}C & × & C^{g} \\ & & ↓π \\ & & C^{g} \end{aligned} .

The sheaf L is flat over C^{g}. If we pick a point P = (P_{1}, …, P_{g}) ∈ C^{g} and restrict L to the fibre π^{−1}(P) ≃ C then this restriction is exactly the line bundle O_{C}(−P_{1} … − P_{g}) on C.

e) Prove that we have a non-empty Zariski open subset U ⊂ C^{g} such that

dim_{k}H^{0}(C, O_{C}(−P_{1} … − P_{g})) = 1 if only if P = (P_{1}, …, P_{g}) ∈ U.

f) Prove that U ≠ C^{g} and show that the complement is of codimension one.

The next exercise is difficult, thats the reason why it has two stars.

g**) Show that if g ≥ 2 we can find a point P for which we can find a holomorphic differential form ω ≠ 0 which has a zero of order g in P.

(This is at least plausible, because the assertion means that the intersection of the total diagonal in C^{g} and the complement of U have an non empty intersection. But there is no theorem that guarantees non emptyness of intersections just from dimensional reasons.)

Hint: Consider C × C. Let Ω^{(1)} be the pullback of the differentials under a projection to the first factor. Consider the sheaf

L = Ω^{(1)} ⊗ O_{C×C}((g − 1)Δ).

What happens if we restrict this sheaf to a fibre C × {P }. How can we describe the resulting sheaf?

Prove that the Riemann-Roch theorem shows that

dim_{k}H^{0}(C, Ω_{C}((g − 1)P)) ≥ 1,

this means we can always find a one form ≠ 0 which has a (g − 1)-th order zero at a given point. Show that our claim follows if we can find a point P_{0} where

dim_{k}H^{0}(C, Ω_{C}((g − 1)P_{0})) ≥ 2.

Hence to prove our theorem, we may assume that

dim_{k}H^{0}(C, Ω_{C}((g − 1)P)) = 1

for all P ∈ C(k).

Now we exploit the semi-continuity. Let p_{2} be the projection to the second factor. The sheaf p_{2∗}(O_{C×C}((g − 1)Δ)) is a line bundle on the curve. Hence wecan find for any P_{0} ∈ C(k) an open neighborhood V and a section ω ∈ p_{2∗}(O_{C×C}((g − 1)Δ))(V) such that ω_{P} generates H^{0}(C, Ω^{1}C((g − 1)P) at all points P in V.

Then ω_{P} has 2g − 2 zeroes, hence it has g − 1 zeroes apart from the (g − 1)-th order zero at P. We consider the subset Z ⊂ C(k) × C(k) which consists of pairs

(Q, P) ∈ C × C

where Q is a zero of the section ω_{P} ∈ H^{0}(C, Ω_{C}^{1}((g − 1)P)). This is by construction a divisor on C ×_{k} C. Now we must prove that Z has a non empty intersection with the diagonal! Again we are in the same situation as above, we cannot conclude this from dimensional reasons.

To proceed we need the concept of the intersection number L_{1} · L_{2} of two line bundles on C ×_{k} C which I will not discuss in detail. The definition of the intersection number is easy in special cases. If for instance the line bundle L_{1} has a non zero global section s_{1} ∈ H^{0}(C ×_{k} C, L_{1}) such that the scheme V (s) of zeroes (see section on divisors) is a smooth curve then we define L_{1} · L_{2} as the degree of the line bundle L_{2} restricted to V (s).

Of course we have to verify that this number does not depend on the choice of s. Here we aply this to the line bundle O_{C×C}(−Z) on C × C. We consider the intersection number O_{C×C}(−Z) · O_{C×C}(−Δ) = Z · Δ. It is clear that this restriction is trivial if Z ∩ Δ = ∅. Hence we can conclude that the intersection is not empty if we show that the intersection number is non zero. ( This argument is valid independently of whether we have shown that the intersection number is well defined)

Actually much more is true:

If Z, Δ have no component in common then the intersection number is equal to the number of points in Z ∩ Δ counted with the right multiplicities.

( For details on intersection numbers I refer to Hartshornes book. From now I will use on the concept of intersection numbers. )

But how can we compute the intersection number O_{C×C}(−Z) · O_{C×C}(−Δ)??

To do this we compute the line bundle p_{2∗}(Ω^{(1)} ⊗ O_{C×C}((g − 1)Δ) (of course still under the above assumption). This can be done inductively by applying the direct image functor to the exact sequences (0 ≤ a < g)

0 → Ω^{(1)} ⊗ O_{C×C}(aΔ) → Ω^{(1)} ⊗ O_{C×C}((a − 1)Δ) → \frac{Ω^{(1)} ⊗ O_{C×C}((a − 1)Δ)}{Ω^{(1)} ⊗ O_{C×C}(aΔ)} → 0.

The term on the right is a sheaf concentrated on the diagonal and can be computed ( see also the discussion of the Grothendieck-Riemann-Roch theorem following later). From this one can compute O_{C×C}(−Z) by exploiting the definition of Z. Then it is easy to compute the intersection number. ( It is a cubic polynomial in g which has three integral zeroes, one of them does not occur, the other two are necessary because our therem is false for these values.)

End of the exercise g**)

If we pick a point P ∈ C(k) and consider the spaces H^{0}(C, O_{C}(−aP)) for a = 1, …, … then we get from the Riemann-Roch formula

dim_{k}H^{0}(C, O_{C}(−aP)) − dim_{k}H^{1}(C, O_{C}(−aP)) = a + 1 − g.

We know that 1 ∈ H^{0}(C, O_{C}(−aP)) hence dim_{k}H^{0}(C, O_{C}(−aP)) ≥ 1.

On the other hand it is clear that

dim_{k}H^{0}(C, O_{C}(−(a + 1)P)) ≤ dim_{k}H^{0}(C, O_{C}(−aP)) + 1

the dimension may jump by one if we go from a to a + 1 or it may stay constant. Finally we have

dimH^{0}(C, O_{C}(−(2g − 1)P)) = g.

This means that if we go up from a = 0 to a = 2g − 1 the dimension will jump exactly g − 1 times. It is at least plausible that for P in some non-empty open set U ⊂ C we have the first jump if we go from a = g to g + 1.

h) Show that for a point P for which H^{0}(C, Ω_{C}(gP)) = 0 we have

dim_{k}H^{0}(C, O_{C}(−aP)) = \{ \begin{matrix}1 & for 0 ≤ a ≤ g \\ a + 1 − g & for g + 1 ≤ a \end{matrix} .

Find out - perhaps by research in the literature - whether there is always a point P for which

H^{0}(C, Ω_{C}(gP)) = 0.

Show that we have always points P for which

dim_{k}H^{0}(C, O_{C}(gP)) ≥ 2.

i) Of course we know that a non-zero differential cannot have a zero of order ≥ 2g − 1. But I do not know what the record is. We could ask several questions

1) For a given g can we find a curve such that H^{0}(C, Ω_{C}((g + 1)P)) = 0 for all points P ∈ C(k)?

2) Which is the maximal number a_{g}(g ≤ a_{g} < 2g − 1) for which we can find a curve of genus g such that

dim_{k}H^{0}(C, Ω_{C}(a_{g}g)) ≥ 1.

3) How early can it happen that

dim_{k}H^{0}(C, Ω_{C}(aP)) ≥ g − a + 1,

which is the minimal a for which this happens for a special curve or the generic curve.

We consider line bundles of degree g + 1, g + 2, … and ask whether they are base point free or provide an embedding of the curve into projective space.

j) Any line bundle L of degree d ≥ g + 1 is of the form

L = O(−P_{1} … − P_{d}).

What does it mean that the bundle L is base point free?

Prove that there is a non-empty subset U ⊂ C^{g+1} such that O(−P_{1} … − P_{g+1}) is a base point free for (P_{1} …, P_{g+1}) ∈ U.

k) Try to prove: For a non-open set U ⊂ C^{g+2} the line bundle O(−P_{1} …−P_{g+2}) provides a closed immersion

i : C → ℙ^{2},

this means that on a non-empty open set the morphism i is injective and its differential is always non-zero. (Show that not all curves can be realized as plane projective curves). l) Prove that for a non-empty open set U ⊂ C^{g+3} the line bundle

L = O(−P_{1} … − P_{g+3})

provides a closed embedding of C into ℙ^{3}.

A special case of the Grothendieck-Riemann-Roch Thereom

We form the product of our curve by itself, we put X = C ×_{k} C we have the two projections p_{1}, p_{2} : X \overrightarrow{→} C. Of course we may again ask for a formula for the Euler characteristic

χ(X, L) = ∑ dim_{k}(−1)^{ν}H^{ν}(X, L)

in terms of certain data which are attached to the line bundle L. This is the classical Riemann-Roch problem for surfaces. It is treated in Hartshornes book in Chapter IV for arbitrary smooth projective surfaces X. Here I want to discuss the special case where X is the product of a curve by itself, but I want to discuss a stronger theorem which is a special case of a Riemann- Roch theorem in the sense of Grothendieck.

In the supplementary section on the properties of the degree function I introduced the group K′(C) and I showed that the formula of Riemann-Roch gave an expression for the Euler characteristic of an element in K^{(}C) in terms of its degree and its rank. Here we recall that we interpreted the Euler chacteristic, the degree and the rank as homomorphisms

χ_{C} : K′(C) → ℤ
rank_{C} : K′(C) → ℤ
deg_{C} : K′(C) → ℤ.

We may ask the following question: Let L be a line bundle on X then we can consider the coherent sheaves R^{ν}p_{2∗}(L) on the curve and we may consider the Euler characteristic

χ(X, R^{•}p_{2∗}L) = ∑(−1)^{ν}[R^{ν}p_{2∗}(L)]

as an element in K′(C). If we are able to ”compute” this element then we can also compute χ(X, L).

This follows from the spectral sequence

χ_{C}(χ(X, R^{•}p_{2∗}L)) = χ(X, L) = ∑ dim_{k}(−1)^{ν}H^{ν}(X, L)

Hence it is clear that we have a better theorem if we can prove a formula for χ(X, R^{•}p_{2∗}(L)) such a formula will imply the classical Riemann-Roch formula for C ×_{k} C.

Again we should be much more consequent and ask for a formula for χ(X, R^{•}p_{2∗}(F)) for any coherent sheaf on X. We can define the group K′(X) and F → χ(X, R^{•}p_{2∗}(F)) is a homomorphism χ : K′(X) → K′(C).

I will explain how we can get such a formula a special situation. Let us assume that our line bundle is of the form L = O_{X}(−Z) where Z is a smooth irreducible curve in X which is not one of the fibres. Then we proceed in exactly the same way as in the case of curves. We write the exact sequence

0 → O_{X} → O_{X}(−Z) → O_{X}(−Z)/O_{X} → 0.

This gives us the formula

χ(X, R^{•}p_{2∗}(L)) = χ(X, R^{•}p_{2∗}(O_{X})) + χ(R^{•}p_{2∗}(O_{X}(−Z)/O_{X}))

But now the situation is similar (but much more complicated) as in the case of curves. The sheaf O_{X}(−Z)/O_{X}) is supported on Z and the morphism p_{2∗} : Z → C is finite. Actually the sheaf O_{X}(−Z)/O_{X}) is a line bundle on Z and since p_{2∗} : Z → C is finite the higher direct images vanish. Therefore we have χ(R^{•}p_{2∗}(O_{X}(−Z)/O_{X}) = [p_{2∗}(O_{X}(−Z)/O_{X})] and since it is clear that χ(X, R^{•}p_{2∗}(O_{X})) = −[O_{C}^{g−1}] our formula simplifies

χ(X, R^{•}p_{2∗}(L)) = −[O_{C}^{g−1}] + [p_{2∗}(O_{X}(−Z)/O_{X})].

Hence we are left with the ”computation” of [p_{2∗}(O_{X}(−Z)/O_{X})].

Here we have to stop for a second. What does it mean to compute this element in K′(C)? This group is much to complicated to identify this individual object. But remember what we actually want. Eventually we want to compute the Euler characteristic χ_{C}(χ(X, R^{•}p_{2∗}L)) and to do this we only need to know the degree and the rank of χ(X, R^{•}p_{2∗}L). Therefore we will be content if we can compute the two numbers

deg p_{2∗}(O_{X}(−Z)/O_{X}) and rank p_{2∗}(O_{X}(−Z)/O_{X})

.

In principle we learned how to do that. We did this in the special case when we discussed the degree of the module of differentials. It is quite clear that the rank rank p_{2∗}(O_{X}(−Z)/O_{X}) is degree d_{2}(Z) of the morphism p_{2∗} : Z → C. The computation of the degree is more subtle.

Some geometric considerations

These considerations are valid for arbitrary smooth surfaces X and a smooth curve Z ⊂ X. The O_{X} - module sheaf N = O_{X}(−Z)/O_{X} is a line bundle when we restrict it to Z. To see this we recall that we can write Z locally by one equation and hence the ideal I(Z) which defines Z is a line bundle, it is the line bundle O_{X}(Z). Then I(Z) ⊗ O_{Z} = I(Z)/I(Z)^{2} is a line bundle on Z, it is called the conormal bundle. The bundle N = O_{X}(−Z)/O_{X} is obviously dual to I(Z)/I(Z)^{2}.

If we restrict the tangent bundle T_{X} to Z then we get an exact sequence

0 → T_{Z} → T_{X}|Z → T_{X}|_{Z}/T_{Z} → 0

But know one sees that I(Z)/I(Z)^{2} is the dual bundle to the quotient T_{X}|Z/T_{Z}. This is so because the differentials df for f ∈ I(Z) can be evaluated on the tangent vectors t_{z} ∈ T_{X,z} they vanish on the subspace T_{Z,z} and this gives the pairing. Hence it is clear that N = T_{X}|_{Z}/T_{Z}, the bundle N is the normal bundle.

We get a formula

Λ^{2}(T_{X}|_{Z}) = T_{Z} ⊗ N

which is called the adjunction formula. Very often is is written in dual form. The second exterior power Λ(T_{X}) is the dual bundle to the so called canonical bundle K_{X} = Λ(Ω_{X}^{1}) and then we get

K_{X}|_{Z} = Ω_{Z}^{1} ⊗ I(Z)/I(Z)^{2}

. This yields in terms of degrees

deg(K_{X}|_{Z}) = deg(Ω_{Z}^{1}) + deg(I(Z)/I(Z)^{2})

and this can be interpreted in terms of intersection numbers as (see the considerations following below)

K_{X} · Z = 2g_{Z} − 2 − Z · Z

.

We return to our discussion of the Grothendieck-Riemann-Roch theorem, we assume again that X = C ×_{k} C. The degree of the line bundle N on Z is called the selfintersection Z · Z = O_{X}(−Z) · O_{X}(−Z). Hence we have in K′(Z) that [N] = [O_{Z}] + [M], where M is a virtual torsion sheaf of degree Z · Z. Hence we get

p_{2∗}(O_{X}(−Z)/O_{X}) = p_{2∗}(N) = [p_{2∗}(O_{Z})] + [p_{2∗}M]

The second term is torsion and therefore we have that its rank is zero and the degree is Z · Z. We are not yet at the end we need to compute p_{2∗}(O_{Z}). This can be done by the same method which we applied when we computed the degree of the sheaf of differentials. Again we introduce the ”different”-module D_{Z/C} using the same definition as before, we replace ℙ^{1} by C and C by Z. We have the perfect duality of O_{C}-modules

p_{2∗}(O_{Z}) × p_{2∗}(D_{Z/C}^{−1}) → O_{C}

which implies that the degrees of these two modules add up to zero. Again we have the Hurwitz formula

Ω_{Z}^{1} = p_{2}^{∗}(Ω_{C}^{1}) ⊗ D_{Z/C}^{−1}

This gives a formula for the degree of D_{Z/C}^{−1} and this formula yields

deg(p_{2∗}(O_{Z})) = −\frac{1}{2} deg(D_{Z/C}^{−1}) = d_{2}(Z)(g − 1) − (g_{Z} − 1)

This formula is still not completely satisfactory, we want to compute the genus g_{Z} of the curve Z. To get this we recall that we have seen that T_{Z} ⊗ N = Λ^{2}(T_{X}|_{Z}) which implies 2g_{Z} − 2 = Z · Z − deg(Λ^{2}(T_{X}|_{Z})

But still it is not clear what this means, because the result will depend on Z and we need a way to produce numbers from the divisor Z. Collecting all the terms gives us

(rank, deg)(O_{X}(−Z)/O_{X}) = (d_{2}(Z), d_{2}(Z)(g − 1) + \frac{1}{2}(Z · Z + deg(Λ^{2}(T_{X}|_{Z})))

The tangent bundle T_{X} is the direct sum of the two pullbacks of the tangent bundle on C hence it is clear that deg(Λ^{2}(T_{X}|_{Z})) = (2−2g)(d_{1}(Z)+d_{2}(Z)) and our formula simplfies to

(rank, deg)(O_{X}(−Z)/O_{X}) = (d_{2}(Z), −d_{1}(Z)(g − 1) + \frac{1}{2}Z · Z)

To get the final formula we remember that O_{X}(−Z)/O_{X} was only one term in our exact sequence. For our original line bundle L = O_{X}(−Z) we get

(rank, deg)(R^{•}p_{2∗}(O_{X}(−Z))) = (−(g − 1) + d_{2}(Z), −d_{1}(Z)(g − 1) + \frac{1}{2}Z · Z)

At this point it is clear how to get a formula for arbitrary line bundles on X. On the product X we have the two special divisors H_{2} = x_{0} × C, H_{1} = C × x_{0} where x_{0} is just an arbitrary point. Then it is clear that the degrees d_{1}(Z) = Z · H_{1}, d_{2}(Z) = Z · H_{2}. Hence the general formula should be

(rank, deg)(L) = (−(g − 1) + L · H_{2}, −(g − 1)L · H_{1} + \frac{1}{2}L · L)

To prove it we consider exact sequences (a > 0)

0 → L → L(−aH_{1}) → L(−aH_{1})/L → 0

0 → L → L(−aH_{2}) → L(−aH_{2})/L → 0

and we apply R^{•}p_{2∗} to both sequences. It is not difficult to show that in both cases the resulting long exact sequence shows that the formula above is true for L if and only if it is true for L(−aH_{1}) resp. L(−aH_{2}).

Then it is clear that it suffices to prove the Riemann-Roch formula for L(−a_{1}H_{1} −a_{2}H_{2}) where a_{1}, a_{2} >> 0. But then the bundle L(−a_{1}H_{1} − a_{2}H_{2}) will be very ample and provide a projective embedding of C ×_{k} C. Then we get L(−a_{1}H_{1} − a_{2}H_{2}) = O_{X}(−Z) where Z is a section with a hyperplane. Now we invest the theorem of Bertini which says that we can choose the hyperlane so that Z is smooth and now we are in the case which we treated above.

Of course we can now easily derive the formula for the Euler characteristic of H^{•}(X, L):

∑(−1)^{ν}dim_{k}H^{ν}(X, L) = (g − 1)^{2} − (g − 1)L · (H_{1} + H_{2}) + \frac{1}{2}L · L

It is quite clear that our approach has a serious defect. In our argument we used that we have a Riemann -Roch formula for arbitrary coherent sheaves, but on our surface we proved it only for line bundles. This is not a consistent way to proceed. We should do something for arbitrary coherent sheaves on X. I will give a scetchy description how we can proceed. Again we introduce the group K′(X). But we need a way to produce numbers. To get these we introduce the Chow ring

A^{•}(X) = A^{0}(X) ⊕ A^{1}(X) ⊕ A^{2}(X)

of our surface. The group A^{ν}(X) is generated by irreducible subvarieties of codimension ν. The elements Z = ∑ n_{i}Y_{i} are called codimension ν cycles. But we have to introduce an equivalence relation

Z = 0 if and only if Z · Z′ = 0 for all codimension 2 − ν cycles

.

This Chow ring can be introduced for arbitrary smooth projective schemes. It has a very intuitive geometric meaning, but it is very difficult to understand. It is technically a little bit difficult to define the intersection of cycles. Remember that we have an equivalence on the linear combinations of cycles and the intersection has to be compatible with that.

Here the situation is a little bit simpler than in general: We have A^{0}(X) = ℤ, A^{2}(X) = ℤ and the only complicated group is A^{1}(X).

Now we define a homomorphism (the Chern character)

ch : K′(X) → A^{•}(X)

The construction of this homomorphism requires the theory of Chern classes.

To any vector bundle E on X we can attach Chern classes

(c_{1}(E), c_{2}(E), …) ∈ A^{1}(X) ⊕ A^{2}(X) + …

(Here X could also be of greater dimension.) We encode these Chern classes by writing the Chern polynomial

P (E, t) = 1 + c_{1}(E)t + c_{2}(E)t^{2} + …

The fundamental properties of these Chern classes are:

(1) If we have an exact sequence of vector bundles

0 → E′ → E → E” → 0

then

P (E, t) = P (E′, t)P (E”, t),

here we use the ring structure of A(X).

(2) If L = O_{X}(−D) is a line bundle given by a divisor D = ∑ n_{i}ℤ_{i} where Z_{i} are irreducible of codimension one then

P (L, t) = 1 + (∑ n_{i}Z_{i})t

where we now consider D as an element in A^{1}(X).

Now it is possible to show that for a smooth variety the group K′(X) is isomorphic to K(X) which is the group generated by vector bundles and relations coming from exact sequences. (We have proved this in the case of curves).

Now we define ch(E) ∈ A(X) for a vector bundle E by the following rule.

If rank (E) = d then

1 + c_{1}(E)t + … c_{d}(E)t^{d} = (1 − λ_{1}t) … (1 − λ_{d}t)

and

ch(E) = (∑_{i=1}^{d} λ_{i}^{0}, ∑_{i=1}^{d} λ_{i}, \frac{1}{2!} ∑_{i=1}^{d} λ_{i}^{2}, …, \frac{1}{n!} ∑_{i=1}^{d} λ_{i}^{n}) ∈ A^{•}(X)

in other words

ch(E) = (rank(E), c_{1}(E), \frac{1}{2}c_{1}(E)^{2} − c_{2}(E), …).

It is clear that we get for an exact sequence of vector bundles

0 → E′ → E → E” → 0

that the property 1) translates into

ch(E) = ch(E′) + ch(E”).

If we have a line bundle L = O_{X}(−D) then

ch(L) = (1, D, \frac{1}{2}D · D, …, \frac{1}{n!}D^{n}, …).

If D is effective then we had the sequence

0 → O_{X} → O_{X}(−D) → O_{X}(−D)/O_{X} → 0

and we find

ch(O_{X}(−D)/O) = (0, D, \frac{1}{2}D · D, …).

We return to the case that X is a surface. Let us assume that we have two irreducible subvarieties Z_{1}, Z_{2} ⊂ X which intersect in smooth points. If J_{1}, J_{2} are the ideals defining these subvarieties then we get an exact sequence of sheaves

0 → O_{X}/O_{X}(Z_{1} + Z_{2}) → O_{X}/O_{X}(Z_{1}) ⊕ O/O_{X}(Z_{2}) → J → 0

where J is a torsion sheaf whose stalk is non zero only at the intersection points of Z_{1}, Z_{2}. At these points we have

J_{X} = O_{X,x}/(I_{1}, I_{2}).

Hence we see that

dim_{k}H^{0}(X, J) = Z_{1} · Z_{2}

and

ch(J) = (0, 0, Z_{1} · Z_{2}).

Therefore we must have

ch(O_{X}/O_{X}(Z_{1} + Z_{2})) + ch(J) = ch(O_{X}/I_{1}) + ch(O_{X}/I_{2})

and this just turns out to be right

(0, −Z_{1} − Z_{2}, \frac{(Z_{2} + Z_{2})^{2}}{2} ) + (0, 0, Z_{1} · Z_{2}) = (0, −Z_{1}, \frac{Z_{1}}{2}^{2}) + (0, −Z_{2}, \frac{Z_{2}^{2}}{2} ).

We have a map (remember X = C × C)

p2^{•} : A^{•}(X) → A^{•}(C)

which simply is given by the intersection

Z → (x_{0} × C) · Z

and hence we get a diagram

\begin{matrix}K′(X) & ⟶^{ch} & A^{•}(X) \\ R^{•}p_{∗} ↓ & & p2^{•} ↓ \\ K′(C) & ⟶^{ch} & A^{•}(C) \end{matrix} .

This diagram does not commute, but we have the Grothendieck-Riemann-Roch formula

p2^{•}(ch(F) · (T (T_{X})) = ch ∘Rp_{2∗}(F) · T (T_{C})).

Here T (T_{X}) resp T (T_{C}) is the so called Todd-genus of the tangent bundle T_{X} resp T_{C} It is defined as follows: We write as above

1 + c_{1}(T_{X})t + … c_{d}(T_{X})t^{d} = (1 − λ_{1}t) … (1 − λ_{d}t)

and then

T (T_{X}) = ∏ \frac{λ_{i}}{(1 − e^{−λ_{i}})}

This is now a general formulation of the Riemann-Roch for a morphism

p_{2} : X → C

between any two projective smooth varieties over k.

Actually we have seen a special case of this formula before. Let us assume that we have a separable finite morphism f : C_{1} → C_{2} between two smooth curves. Then we have

K′(C_{1}) ⟶^{ch} A^{•}(C_{1})
f_{∗} ↓  f^{•} ↓
K′(C_{2}) ⟶^{ch} A^{•}(C_{2})

and

(ch ∘f_{∗}(F)) · (T (T_{C_{2}})) = f^{•}(ch(F) · (T (T_{C_{1}}))

and this is the Hurwitz formula.

I want to discuss an application of the Riemann-Roch theorem. By definition we have a non degenerate pairing

A(X) ⊗ ℚ × A(X) ⊗ ℚ → ℚ

which is given by the intersection pairing. In A(X) we have the two classes given by H_{1}, H_{2}, it is clear that H_{1} · H_{1} = H_{2} · H_{2} = 0 and H_{1} · H_{2} = 1. Hence ℚH_{1} ⊕ ℚH_{2} is a hyperbolic plane in A(X) ⊗ ℚ. Hence it has an orthogonal complement A_{0}^{1}(X) and

A(X) ⊗ ℚ = ℚH_{1} ⊕ ℚH_{2} ⊕ A_{0}^{1}(X)

Now he have

Hodge index theorem:

The intersection form is negative definite on A_{0}^{1}(X)

Proof: Let L be a line bundle which corresponds to a class ξ ∈ A_{0}^{1}(X). Then we get from the Riemann-Roch theorem that

(rank, deg)(R^{•}p_{2∗}(L)) = (−(g − 1), \frac{1}{2}L · L)

It is clear that p_{2∗}(L) has no torsion and since the restriction of L to the fibre of p_{2} has degree zero we get that p_{2∗}(L) has at most rank one. We may apply the same argument to R^{1}p_{2∗}(L). In a fibre we have dim_{k}(R^{1}p_{2∗}(L)_{y}) ≤ g and henc we see that the rank of R^{1}p_{2∗}(L) is bounded. What can happen is that R^{1}p_{2∗}(L) has a large torsion subgroup. Now we replace L by L^{n}. Our previous considerations still apply. I claim that we cannot have L · L > 0. Otherwise we would get that L^{⊗n} · L^{⊗n} = n^{2}L · L would become a large positive number. But the only thing on the left hand side that can become large is the torsion subgroup in R^{1}p_{2∗}(L^{⊗n}) but it contributes by a negative term since it sits in degree one. Hence L · L ≤ 0 and the form is negative. Then it must be negative definite because the form is non degenerate.

Curves over finite fields.

We consider the special situation of a projective, smooth and absolutely irreducible curve C/𝔽_{q}, where 𝔽_{q} is the field with q elements. This special case is historically the origin for the theory of curves over arbitrary fields. The classical geometers always studied the case of curves over ℂ and the theory of Riemann surfaces.

As I explained, we can consider the function field K = 𝔽_{1}(C), this is a finite separable expension of a rational function field 𝔽_{q}[f] (see ???). The condition that C/𝔽_{q} says that the field of constants is equal to 𝔽_{q}. We can recover the curve C/𝔽_{q} from the field K as we explained in ???.

These function fields attracted the attention of number theorists since they are analogous to number fields. The closed points 𝔭 of the curve C are in one to one correspondence to the discrete valuation ring O_{𝔭} ⊂ K. If we remove one point ∞ from C, then C \ {∞} is the spectrum of the Dedekind ring O_{C}(C \ {∞}), and this has to be seen in analogy to the number field case where we have Spec(O) where O is the ring of integers.

We can attach a ζ-function to our curve which we define as

ζ_{K}(s) = ζ_{C}(s) = ∏_{𝔭:closed point} \frac{1}{1 − \frac{1}{N𝔭^{s}}}

where N𝔭 = #(O_{𝔭}/𝔭) is the number of elements in the residue field. This ζ-function is analogous to the Dedekind ζ-function of a number field, and it is easy to see that the product converges for Re(s) > 1.

We will show that the Riemann-Roch theorem implies that this ζ-function is a rational function in the variable t = q^{−s} and more precisely we have

ζ_{C}(s) = \frac{P_{1}(q^{−s})}{(1 − q · q^{−s})(1 − q^{−s})}

where P_{1}(q^{−s}) = 1 + a_{1}q^{−s} ⋯ + q^{g} · q^{−2gs} is a polynomial of degree 2g in q^{−s} with integer coefficients. We will see that the theorem of Riemann-Roch implies that we have a functional equation

ζ(s) = q^{(2−g2)s}q^{1−g}ζ(1 − s).

This is in a perfect analogy to the situation of a number field.

Now it has been observed by Artin and Hasse that we can formulate the analogon of the Riemann hypothesis which would say that the zeroes of ζ(s) have imaginary part Re(s) = \frac{1}{2}. This assertion can be formulated in terms of the polynomial P_{1}(q^{−s}) = P_{1}(t). We may factor it over \overline{ℚ} ⊂ ℂ and get

P_{1}(t) = ∏_{i=1}^{2g}(1 − ω_{i}t)

where the ω_{i} are algebraic integers. It is easy to see that the functional equation allows a grouping of these numbers ω_{i} so that we have

ω_{1} ⋯ ω_{g} , ω_{g+1} ⋯ ω_{2g}

and ω_{ν} · ω_{v+g} = q.

Then the Riemann hypothesis is equivalent to the assertion that all the ω_{i} have absolute value

|ω_{i}| = q^{\frac{1}{2}} .

This has been conjectured by Artin in his thesis (???), and he verified it in several cases. But actually Gauss knew it as special cases in a somewhat disguised form and also Artin’s thesis advisor Herglotz had proved in it in a special case.

This special case occurs in Gauss’ Tagebuch, where he states it as a fact which was found experimentally but he does not claim that he proved it.

The Riemann hypothesis was then proved by Hasse (1934) for curves of genus one and A. Weil announced the proof in the general case in 1941. The final proof appeared in 19.., and it is based on the theory of the Jacobian of curves.

Elementary properties of the ζ-function.

In this section I call elementary properties onf the ζ-function those which follow from the Riemann-Roch theorem for the curve.

As in the case of the Riemann ζ-function we can expand the product

∏_{𝔭} \frac{1}{1 − \frac{1}{N𝔭^{s}}} = ∑_{𝔞} \frac{1}{N(𝔞)^{s}}

where 𝔞 runs over the effective divisors

𝔞 = ∑ n_{i}𝔭_{i} n_{i} ≤ 0,

where N(𝔞) = ∏(N𝔭_{i})^{n_{i}} = q^{deg(𝔞)}. For the following discussion we have to assume that our curve has a divisor of degree one. This assumption is equivalent to the assumption that the degrees f_{𝔭} of the prime divisors are coprime. It is a theorem of F. K. Schmidt that this is always the case.

If we make this assumtion (or if we believe F. K. Schmidt’s theorem), then it is clear that the divisor classes of degree n for any n form a principal homogeneous space under Pic^{0}(C)(𝔽_{q}), this is the group of divisor classes of degree 0. The number h of these divisor classes is finite and it is called the class number. We get

ζ_{C}(s) = ∑_{n=0}^{∞} \frac{c(n)}{q^{ns}}

where c(n) = # of effective divisors of degree n. (This number is clearly finite.) For n > 2g − 2 we have a formula for c(n): If 𝔞 is an effective divisor of degree n, then we can consider the line bundle L = O_{C}(−𝔞), and we have

dim H^{0}(C, O_{C}(𝔞)) = n + 1 − g

(see ???). A non zero section s ∈ H^{0}(C, L) defines the divisor 𝔟 = div(s). and we see that the divisors 𝔟 which we get, if s varies are exactly those divisors which are linearly equivalent to 𝔞. Then

c(n) = h \frac{q^{n+1−g} − 1}{q − 1}.

(As a byproduct we proved the finiteness of the class number.)

We make the substitution q^{−s} = t, and write ζ_{C}(s) = Z_{C}(t). Then we define a new function Z_{C}^{∗}(t) by

Z_{C}(t) = Z_{C}^{∗}(t) + \frac{h}{q − 1} · (q^{g} \frac{t^{2g−1}}{1 − qt} − \frac{1}{1 − t}) ,

and

Z_{C}^{∗}(t) = ∑_{n=0}^{2g−1} c^{∗}(n)t^{n}.

The coefficiants c^{∗}(n) are equal to zero for n ≥ 2g − 1, and we have

c(n) = c^{∗}(n) − \frac{h}{q − 1} for 0 ≤ n ≤ 2g − 2.

The correcting term

R(t) = \frac{h}{q − 1} (q^{g} \frac{t^{2g−1}}{1 − qt} − \frac{1}{1 − t})

satisfies the functional equation

R(t) = t^{2g−2}q^{1−g}R(\frac{1}{qt}),

and hence we have to show that Z_{C}^{∗}(t) satisfies this functional equation.

Now we observe that we have a precise formula for c(n). We sum over the classes of bundles of degree n and count the effective divisors in a class. If L is a line bundle of degree n, then the number of effective divisors in this class is

\frac{1}{q − 1} (q^{dim H^{0}(C,L)} − 1) ,

and hence

c(n) = \frac{1}{q − 1} ∑_{L:deg(L)=n} (q^{dim H^{0}(C,L)} − 1) .

Now we assume that for all n this sum is not empty. This follows from the theorem of F. K. Schmidt that we can always find a divisor of degree one. Then we can conclude that the number of terms in the sum is equal to h and hence

c^{∗}(n) = ∑_{L:deg(L)=n} q^{dim H^{0}(C,L)}.

Our problem is that we do not know the dimension of the space of sections. Now we have in involution n, the set of divisor classes of degree between 0 and 2g − 2.

L ⟶ L^{−1} ⊗ Ω = L′,

and then deg(L) → 2g − 2 − deg(L). We see that

c^{∗}(2g − 2 − n) = ∑_{L′:deg(L′)=2g−2−n} q^{dim H^{0}(C,L′)}.

If now L′ = L^{−1} ⊗ Ω, then

dim H^{0}(C, L) − dim H^{0}(C, L′) = n + 1 − g,

hence this difference depends only on the degree and not on the class. This implies

c^{∗}(n) = q^{n+1−g}c^{∗}(2g − 2 − n),

and this implies the functional equation for Z_{C}^{∗}(t).

Now we can conclude that

Z_{C}(t) = \frac{1 + a_{1}t ⋯ + g^{g}t^{2g}}{(1 − t)(1 − qt)} .

The coefficients a_{i} must be integers, since the c(n) are integers.

We can write

P_{1}(t) = ∏_{i=1}^{2g}(1 − ω_{i}t),

the ω_{i} ∈ ℂ are algebraic and the functional equation implies that the collection

{ω_{1} ⋯ ω_{2g}}

is invariant under the substitution

ω_{i} ⟶ ω_{i}^{−1}q.

At this moment it is not yet clear that they are algebraic integers and that we have the Riemann hypothesis

|ω_{i}| = q^{\frac{1}{2}} for i = 1 ⋯ 2g.

These so called elementary results are all due to F. K. Schmidt. By the way F. K. Schmidt was clever enough to remove the assumption that we have a divisor of degree one, he just proved that this is always the case.

Can you also prove it?

The Riemann hypothesis.

We recall the multiplicative definition of the ζ-function, and we remember that

N𝔭^{−s} = q^{−f_{𝔭}s} = t^{f_{𝔭}}

where [𝔽_{q}(𝔭) : 𝔽_{q}] = q^{f_{𝔭}}, i.e. f_{𝔭} is the degree of the residue field at 𝔭.

Now we consider the expression

t · \frac{Z′_{C}(t)}{Z_{C}(t)} = ∑_{𝔭} \frac{f_{𝔭}t^{f_{𝔭}}}{1 − t^{f_{𝔭}}} .

If we expand the expression on the right, we get an infinite sum ∑ a_{n}t^{n} and clearly

a_{n} = ∑_{f_{𝔭}|n} f_{𝔭}.

But f_{𝔭} is exactly the number of geometric points over 𝔭, and these geometric points lie in

C(𝔽_{q^{f𝔭}}) ⊂ C(𝔽_{q^{n}}),

and hence we see that the right hand side must be

t · \frac{Z′_{C}(t)}{Z_{C}(t)} = ∑ #C(𝔽_{q^{n}})t^{n}.

Going back to our expression for Z_{c}(t) as a rational function we find

t · \frac{Z′_{C}(t)}{Z_{C}(t)} = \frac{qt}{1 − qt} + \frac{t}{1 − t} − ∑_{i=1}^{2g} \frac{ω_{i}t}{1 − ω_{i}t},

and from here we get

#C(𝔽_{q^{n}}) = q^{n} + 1 − ∑_{i=1}^{2g} ω_{i}^{n}

for all n.

(This implies that ∑_{i=1}^{2g} ω_{i}^{n} is an integer for all numbers n, and this implies that the ω_{i} must be integers themselves.)

The following proof of the Riemann hypothesis is due to A. Grothendieck ([ ]) and is based on intersection theory and the Hodge index theorem.

Our curve C/𝔽_{q} is defined over 𝔽_{q}, and this allows us to define the Frobenius morphism

Φ_{q} : C ⟶ C.

To define this morphism we consider an affine open subset U ⊂ C, then the ring of regular function O_{c}(U) is an 𝔽_{q}-algebra and the map

Ψ_{q} : O_{c}(U) ⟶ O_{c}(U)

which sends f ∈ O_{c}(U) to f^{q} is an 𝔽_{q}-algebra homomorphism. This defines the restriction of Φ_{q} to U, and since we can cover C by such open affine sets, we have defined it every where.

If we have an embedding of our curve

i : C ⟶ ℙ_{Spec(𝔽_{q})}^{n}
↘ ↙
Spec(𝔽_{q}).

Then the action of the Frobenius Φ_{q} on the geometric points is given by

\underline{x} = (x_{0}, ⋯, x_{n}) ⟶ x^{q} = x^{q} = (x_{0}^{q}, ⋯, x_{n}^{q})

for any point \underline{x} ∈ BP^{n}(𝔽_{p}).

Now we pass to a geometric situation, we replace C by \overline{C} = C×_{𝔽_{q}} \overline{𝔽}_{q} and our information that \overline{C} comes from the curve C/𝔽_{q} is encoded in the datum of the Fronenius morphism

Φ_{q} : \overline{C} ⟶ \overline{C}

which also can be viewed as given by its graph

Γ_{Φ} ⊂ \overline{C} × \overline{C} = \overline{X}.

Then it is clear by definition that

C(𝔽_{q}) = Γ_{Φ} ∩ Δ ⊂ C(\overline{𝔽}_{q}) × C(\overline{𝔽}_{q}).

The tangent space of \overline{X} in any geometric point p = (x, y) is given by

T_{\overline{X},p} = T_{\overline{C},x} ⊕ T_{\overline{C},y},

and the tangent space of Γ_{Φ} at p = (x, Φ(x)) consist of vectors in

T_{Γ_{Φ},p} = T_{\overline{C},x}.

The tangent space of Δ is an point (x, x) is the diagonal, hence we see that Γ_{Φ} and Δ intersect transversally, and this gives us

#C(𝔽_{q}) = Γ_{Φ} · Δ.

If we pick any two points x_{0} ∈ C(\overline{𝔽}_{q}), y_{0} ∈ C(\overline{𝔽}_{q}), then we may consider the divisors

H_{1} = \{x_{0}\} × \overline{C} , \overline{C} × \{y_{0}\} = H_{2}

on \overline{C} × \overline{C}. We get an orthogonal decomposition of the Chow group A^{1}(\overline{X}) = Ã^{1}(\overline{X}) ⊕ ℤ[H_{1}] ⊕ ℤ[H_{2}].

Since H_{1} + H_{2} is ample on \overline{X}, we can conclude that the intersection form restricted to Ã^{1}(\overline{X}) is negative definite.

We write in A^{1}(\overline{X})

[Δ] = [A′] + [H_{1}] + [H_{2}],

then [Δ′] ∈ ỹ^{1}(\overline{X}).

More generally we can say: If Γ ⊂ \overline{C} × \overline{C} is the graph of a morphism f : \overline{C} → \overline{C}, then

\begin{aligned}Γ · (\{x_{0}\} × \overline{C}) & = & Γ · H_{2} & = & 1 \\ Γ · (\overline{C} × \{y_{0}\}) & = & Γ · H_{1} & = & deg(f) \end{aligned}

and

[Γ′] = [Γ] − (deg f)[H_{2}] − [H_{1}] ∈ Ã(\overline{X}).

We need the selfintersection Γ · Γ. To get this number we recall the adjunction formula. The tangent bundle T\overline{X} ≃ p_{1}^{∗}(T\overline{C}) + p_{2}^{∗}(T\overline{C}) and hence

Λ^{2}T\overline{X} = p_{1}^{∗}(T\overline{C}) ⊗ p_{2}^{∗}(T\overline{C}).

Now we have

Γ · Λ^{2}T\overline{X} = Γ · p_{1}^{∗}(T\overline{C}) + Γ · p_{2}^{∗}(T\overline{C}) = (2g − 2) · (deg(f) + 1).

Then the adjunction formula

Γ · Λ^{2}T\overline{X} = 2g_{Γ} − 2 − Γ · Γ,

and g_{Γ} = g because the projection p_{1} provides an isomorphism p_{1} : Γ → \overline{C}. Hence we get

Γ · Γ = (2 − 2g) · deg(f)

and this yields

Γ′ · Γ′ = −2g deg(f).

We look at the case Γ = Γ_{Φ}. We have

Γ_{Φ} · Δ = Γ_{Φ} · ([Δ′] + [H_{1}] + [H_{2}]) = Γ_{Φ}Δ′ + q + 1,

and hence we get

− ∑_{i=1}^{2g} ω_{i} = Γ_{Φ} · Δ′ = Γ′_{Φ} · Δ′.

We know that

(nΓ′_{Φ} + mΔ′) · (nΓ′_{Φ} + mΔ′) ≥ 0

for all integers n.m, and this says that

−n^{2} · 2gq + 2nmΓ′_{Φ} · Δ′ − m^{2} · 2g ≥ 0

for all n.m. This is equivalent to the inequality

(Γ′_{Φ} · Δ′)^{2} = (∑_{i=1}^{2g} ω_{i})^{2} ≤ 4g^{2}q,

and hence we get the estimate

| ∑_{i=1}^{2g} ω_{i}| ≤ 2g\sqrt{q}.

This is in principle the Riemann hypothesis. The only thing we have to say is that we can replace 𝔽_{q} by any field 𝔽_{q^{r}}, i.e. we get the inequality

| ∑_{i=1}^{2g} ω_{i}^{r}| ≤ 2g\sqrt{q^{r}}

for all r = 1, 2, · · ·. Since we have

∏_{i=1}^{2g} ω_{i} = q^{g},

this can only be the case, if we have

|ω_{i}| = \sqrt{q}

for all i = 1 ⋯ 2g.

Chapter X

The Picard functor for curves

Introduction: In the previous chapter we constructed the Jacobian for compact Riemann surfaces and we heavily relied on transcendental methods. We got an complex torus and gave indications why it is in fact algebraic.

The goal of this chapter is the construction of a Jacobian J_{C}/k = J/k for any smooth, projective and absolutely irreducible curve C/k where k is an arbitrary field. Of course we want that J is the ”parameter space” for the line bundles of degree zero on C/k but we have to make it precise what we mean by that.

Once we constructed J/k we also want to investigate the group of line bundles on J/k, we want to construct the dual J^{∨} of J/k and we want to show that we have a canonical identification J → J^{∨}.

At the end of the last chapter we formulated a universality property for the line bundle N on J × J and S × J, which we did not prove. This universality property will become the basic principle for the construction of J/k.

1. The construction of the Jacobian

1.1 Generalities and heuristics : If we consider any scheme X over a base S, we can define the Picard functor

Pic_{X/S} : Schemes S ⟶ Abelian groups
T  ⟶ Pic(X ×_{S} T)

where Pic(X ×_{S} T) = Pic_{X/S}(T) is the group of isomorphism classes of line bundles over X ×_{S} T. We may ask whether this functor is representable? (See ???). I recall what this means: We can find a group scheme Pic(X)/S and a ”universal” line bundle

L on X ×_{S} Pic(X)

such that for any scheme T → S and any line bundle L_{1} over X ×_{S} T we have a unique

ψ : T → Pic(X)

such that we can find an isomorphism

η : L_{1} ⟶^{∼} (Id ×ψ)^{∗}(L).

The answer to this question is simply “No”.

The reason for this negative answer is that line bundles have non trivial automorphisms. This has the effect that we do not have a canonical way to identify two isomorphic line bundles L_{1}, L_{2} on a scheme Y. Hence we cannot say that they are equal, in other words: The line bundles on X ×_{S} T form a category but not a set.

We explain this difficulty in a different way. Let us assume that our base scheme is S = Spec(k) and k is an algebraically closed field. Then the intuitive meaning of the ”universal” bundle L on X×_{S}Pic(X) is the following: For any p ∈ Pic(X)(k) the restriction of L to X × {p} gives us a line bundle L_{p} on X. We want that Pic(X) parameterizes the line bundles on X/ Spec(k) and by this we mean that p → [L_{p}] is a “natural” bijection between Pic_{X/S}(k) and the isomorphism classes of line bundles on X. By “natural” we mean that this parametrization should at least have the following property: If we have a scheme T/k of finite type over k, and if we have a line bundle L_{1} on X ×_{k} T then a point t ∈ T (k) provides a line bundle L_{1,t} on X/k. To this line bundle corresponds a point p = ψ_{0}(t) on Pic(X)(k) such that L_{1,t} ≃ L_{p}. At this point ψ_{0} is just a map ψ_{0} : T (k) → Pic(X)(k) between the two sets of geometric points. We want that this map between the sets of geometric points is induced by a morphism ψ : T → Pic(X). But still this cannot be enough since in general this morphism will not be determined by its effect on the geometric points if T is not reduced. (See ???).

1.1.1. We still want more. Let us now assume in addition that X/k is projective, reduced and connected. If we have such a ψ as above, then the line bundles L_{1} and (Id ×ψ)^{∗}(L) are “isomorphic point by point” on X × T. We want an isomorphism η : L_{1}→^{∼}(Id ×ψ)^{∗}(L) between the bundles on X × T. But now it is clear that we cannot have such a property: If we tensorize L_{1} by any line bundle M on X × T which is the pullback of a line bundle on T, we get isomorphic fibres (L_{1} ⊗ M)_{t} ≃ L_{1,t} if t varies in T (k). This means that the morphism ψ does not change. But we cannot expect that L_{1} is isomorphic to L_{1} ⊗ M.

But we can hope that a slightly weaker property is satisfied. We can form the tensor product L_{1}⊗((Id ×ψ)^{∗}(L))^{−1} which is a line bundle on X×_{k}T. It is trivial at any geometric point t ∈ T (k). Since we assumed that X/k is projective reduced and connected we have

dim_{k} H^{0}(X × {t}, L_{1} ⊗ ((Id ×ψ)^{∗}(L))_{t}^{−1}) = 1

for all the closed points t ∈ T (k). ( See ???) The semi-continuity theorem ( See ???) implies that the direct image p_{2,∗}(L_{1}⊗((Id ×ψ)^{∗}(L))^{−1} for the projection p_{2} is a line bundle of rank one on T (we should assume that T is reduced). Hence we can find a covering T = ∪T_{α} by open subschemes such that L_{1} ⊗ ((Id ×ψ)^{∗}(L))^{−1} is trivial on X × T_{α}. So we extended our point by point isomorphisms locally into some neighborhood of any point t_{0}.

1.1.1.1. In general we will call a line bundle L on X × T locally trivial in T if we can cover T by open sets T_{α} such that L|X × T_{α} becomes trivial. We call two line bundles L_{1}, L_{2} on X × T locally isomorphic in T if we can cover T by open sets T_{α} such that

L_{1}|X × T_{α}→^{∼}L_{2}|X × T_{α}

This means that the best we can hope for is to show that our functor is locally representable.

This means that we want to construct a scheme Pic(X) over k and a line bundle P on X × Pic(X) such that for any bundle L on X × T with T of finite type over k we find a unique ψ : T → Pic(X) such that

(Id ×ψ)^{∗}(P) ∼_{T} L,

and this means that the bundles are locally isomorphic in T.

If X is reduced, projective and connected and let p_{2} be the projection to the second factor T. Then it is clear that

p_{2,∗}((Id ×ψ)^{∗}(P) ⊗ L^{−1}) = M

is locally free of rank one and we have

(Id ×ψ)^{∗}(P)→^{∼}L ⊗ p_{2}^{∗}(M)

1.1.1.2 Once we have achieved this goal we will say that Pic_{X/k} is locally represented by the pair (Pic(X), P).

Of course it seems to be reasonable to put some finiteness conditions on T we will need them. But the condition for T/k to be of finite type is perhaps to strong, we should also allow localizations of such schemes.

1.1.1.3. We can state the results of our considerations by saying the Picard functor is not a sheaf for the Zariski topology (See ???). If we have a covering of T by open subsets then we cannot expect that the sequence

Pic_{X/S}(T) → ∏ Pic_{X/S}(T_{α}) \overrightarrow{→} ∏ Pic_{X/S}(T_{α} ∩ T_{β})

is exact.

On the other hand we have seen in (???) that a necessary condition for representability of a functor is, that it is a sheaf for the faithfully flat topology.

1.1.2. The situation becomes better, if we assume that X is projective, absolutely reduced and connected over a field k and it comes with a distinguished point P ∈ X(k). Now we define a modified functor

P̃ic_{X/k}(T) = \{(L, s)| \begin{matrix}L line bundle on X × T, \\ s ∈ H^{0}(\{P \} × T, L) which \\ trivializes L|\{P \} × T \end{matrix} \} .

This means we look at the group of isomorphism classes of line bundles on X × T which are trivial on {P } × T and which are equipped with

η_{P} : L|{P } × T ⟶^{∼} O_{{P }×T}

where we require that η_{P} (s) = 1.

If we have two such pairs (L_{1}, s_{1}) and (L_{2}, s_{2}) with isomorphic line bundles then we have exactly one isomorphism which sends s_{1} into s_{2}. Hence we can say that the two pairs (L_{1}, s_{1}), (L_{2}, s_{2}) are not only isomorphic, they are even equal. Here we used of course that X/ Spec(k) is projective, reduced and connected.

1.1.2.1. Now it is also not difficult to show that the modified functor P̃ic_{X/k} is in fact a sheaf for the faithfully flat topology.

We see that our chances to represent the functor P̃ic_{X/k} are much better. We will prove representability of P̃ic_{C/k} for curves C/k. The case of general smooth projective schemes X/k can be reduced to the case of curves.

But if we compare the two functors P̃ic_{X/k} and Pic_{X/k} then we see that the difference is not so dramatic. We have a homomorphism between abelian groups

P̃ic_{X/k}(T) ⟶ Pic_{X/k}(T).

The image consists of those isomorphism classes of line bundles which are trivial over {P } × T. The homomorphism is injective since if we have two pairs (L_{1}, s_{1}), (L_{2}, s_{2}) such that the bundles L_{1} and L_{2} are isomorphic, then we can find an unique isomorphism which sends s_{1} to s_{2}.

I recall that representability of P̃ic_{X/k} gives us a universal triplet

(X × P̃ic_{X/k}, P, s)

where is P̃ic_{X/k} a scheme, P a line bundle on

X × P̃ic_{X/k},

and s_{0} ∈ H^{0}({P } × P̃ic_{X/k}, P) a section which is everywhere non zero. I do not formulate the universality property, it is now obvious.

But if we consider P on X × P̃ic_{X/k}, and if we forget the section s, then we see easily, that this pair gives us local representability of the Picard functor. If we have a line bundle L on X ×_{k} T (T over Spec(k) of finite type), then we can find a covering T = ∪T_{α} such that L | {P } × T_{α} is trivial. We choose sections s_{α} ∈ H^{0}({P } × T_{α}, L) which trivializes this restriction, and we find unique

ψ_{α} : T_{α} ⟶ P̃ic_{X/k}

such that

(Id ×ψ_{α})^{∗}(P) | X × T_{α} ≃ L | X × T_{α}.

Then these ψ_{α} coincide on the intersections T_{α} ∩ T_{β} and give us a ψ : T → P̃ic_{X/k}.

Hence we see that representability of P̃ic_{X/k} implies that the Picard functor is locally representable by a pair Pic(X) and line bundle P on X × Pic(X). The scheme Pic(X) is unique up to a canonical isomorphism: If we have another pair (Pic_{X/k}^{∗}, P^{∗}), then we have a unique

ψ : Pic_{X/k}^{∗} ⟶ Pic_{X/k}

such that (Id ×ψ)^{∗}(P) ∼_{Pic_{X^{∗}/k}} P and vice versa. Hence we see that Pic_{X/k} is unique, but P is not.

I remind you that the definition of P̃ic_{X/k} depends on the choice of a point P ∈ X(k) where the definition of the Picard functor does not.

Once we have local representability for the Picard functor, i.e. we have Pic(X) and the bundle P on X × Pic(X), and if we also selected a P ∈ X(k) (if this is possible), then we see that we have representability for P̃ic_{X/k}. We simply modify

P ⟶ P ⊗ p_{2}^{∗}(P | {P } × Pic(X))^{−1} = P′

where p_{2} : X × Pic(X) → Pic(X) is the projection. Then P′ is trivial on {P } × Pic(X) by construction, and we choose a trivializing section.

1.2. General properties of the functor Pic

1.2.1. : We consider a reduced, projective and connected scheme X/k and line bundles on X × T where T should be of finite type. In the following considerations it is always possible to replace these schemes T by the affine schemes Spec(A) since our questions will be local in T. We will also allow ourselves to pass to local rings at points in T.

We need some finiteness condition for T because we have to apply the finiteness theorems from (???). Since the Picard functor will be of finite type our condition above seems to be natural. But the passage to local rings does not hurt.

Lemma 1.2.2: Let X/k be an irreducible, reduced projective scheme T → Spec(k) of finite type. Let L be a line bundle on X ×_{Spec(k)} T. Then there exists a ”largest” closed subscheme T_{1} ⊂ T such that L_{1} = L|S ×_{Spec(k)} T_{1} is locally trivial in T_{1} and ”largest” means for any closed subscheme T′ ⊂ T for which L|X ×_{k} T′ is locally trivial in T′, the inclusion T′ ↪ T factors over T_{1}.

This lemma would of course be clear, if we had representability. Let T′ ⊂ T be a subscheme such that L_{T′} = L|X ×_{k} T′ is locally trivial. Our assumptions on X imply that the projection p : X ×_{k} T′ → T′ yields sheaves

p_{∗}(L_{T′})  and  p_{∗}(L_{T′}^{−1})

which are locally constant of rank one.

But if in turn these two sheaves are locally constant of rank one, then L_{T′}, L_{T′}^{−1} are locally trivial in T′. To see this, we can consider two such local generators

s ∈ H^{0}(V_{t}, p_{∗}(L_{T′})) , s_{−} ∈ H^{0}(V_{t}, p_{∗}(L_{T′}^{−1})).

(Here V_{t} is an open neighborhood of a point t ∈ T′) and their product gives an element

ss_{−} ∈ H^{0}(X × V_{t}, O_{X×V_{t}}),

By definition of local constancy (see ???) we know that the restrictions of these sections to the special fibre over t generate the spaces of sections H^{0}(X×{t}, L_{t}) and H^{0}(X×{t}, L_{t}^{−1}) which are both of dimension one. Then the product ss_{−} restricted to the fibre is not zero, but since X ×{t} is projective and connected, it follows that they never vanish. The closed subset in X × V_{t} where ss_{−} vanishes does not meet the fibre X × {t}. Its projection to V_{t} is closed in V_{t} (???) and does not contain t. Hence we find an open neighborhood W_{t} of t such that s, s_{−} are nowhere zero on X × W_{t} and hence L_{T′} and L_{T′}^{−1} are trivial on X × W_{t}.

We assume that we have a point t ∈ T such that L|X × {t} is trivial. Let A be the local ring O_{T,t}. We restrict our sheaf to X × Spec(A), let 𝔪 be the maximal ideal. We want to show that there is a smallest ideal I_{0} such that our sheaf restricted to X × Spec(A/I_{0}) becomes trivial.

In the section (semicontinuity) we have seen that we can construct a resolution of L on X ×_{k} T

0 → L → E_{0} → E_{1} → ⋯,

such that the E_{i} are flat, acyclic for p_{∗} and such that the p_{∗}(E_{i}) = M_{i} are locally free on T. We may assume that T = Spec(A), such that the M_{i} are actually free, and we get

0 → p_{∗}(L) → M_{0} → M_{1} → · · · .

We have M_{0} = A^{N_{0}}, M_{1} = A^{N_{1}} and the map α_{0} : M_{0} → M_{1} is given by a N_{0}×N_{1}-matrix.

Now we choose an element m_{1} ∈ A^{N_{0}} which reduces to a basis element of the kernel

(A/𝔪)^{N_{0}} ⟶ (A/𝔪)^{N_{1}}.

We can assume that m_{1} is the first element of a basis of A^{N_{0}}. We consider m_{1} as a column vector

m_{1} = ( 1 0 ⋮ 0 ) .

The linear map A^{N_{0}} → A^{N_{1}} is given by a matrix M(α_{0}). Our matrix operates by multiplication from the left, then it will be of the form

M(α_{0}) = ( \begin{matrix}m_{11} & ∗ & ⋯ & ∗ \\ m_{12} \\ ⋮ & & B \\ m_{1,N} \end{matrix} )

where the m_{1,ν} are in the maximal ideal 𝔪. We reduce mod 𝔪. The matrix B mod 𝔪 maps the space of vectors

( 0 ; x_{2} ; ⋮ ; x_{N_{0}}  )  with  x_{i} ∈ A/𝔪

injectively into (A/𝔪)^{N_{1}}. This implies that B contains an invertible (N_{0} − 1) × (N_{0} − 1) submatrix. This allows us to modify the basis in the target such that the vectors

( 0 1 ⋮ ⋮ 0 ) ⋯ ( 0 ⋮ 1 0 0 )

are the images of the (N_{0} − 1)-vectors

( 0 1 ⋮ 0 ) , ⋯, ( 0 0 ⋮ 1 )

and then our matrix Bis of the form

( \begin{matrix}m_{11} & 0 & ⋯ & 0 \\ m_{12} & 1 & 0 \\ ⋮ & 0 & ... & 1 \\ m_{1,N_{1}} \end{matrix} ) .

Now we take an ideal I ⊂ A and ask: when is the kernel

(A/I)^{N_{0}} ⟶ (A/I)^{N_{1}}

free of rank one and surjects to the kernel of

(A/𝔪)^{N_{0}} ⟶ (A/𝔪)^{N_{1}}?

Clearly this is the case if and only if this kernel is spanned by the element

( 1 ; −m_{12} ; ⋮ ; −m_{1,N_{0}}  ) .

The first condition to be satisfied is that m_{11} ∈ I. Then the second, third up to N_{0}-th component of the image is zero. Now the vanishing of the N_{0} + 1-th up to the N_{1}-th entry means that the m_{1,μ} with μ = N_{0} + 1 … to N_{1} satisfy

m_{1,μ} − ∑_{j=2}^{j=N_{0}} b_{μj}m_{1,j} = 0.

This gives us a collection of elements which must be in I and in turn if I contains these elements, then the kernel of (A/I)^{N_{0}} → (A/I)^{N_{1}} is free of rank one.

Hence we see that we can take for I_{0} the ideal generated by these elements

{m_{11}, m_{1,μ} − ∑ b_{μj}m_{1,j}} = I_{0}

and Spec(A/I_{0}) is the largest subscheme of Spec(A) on which L|X × Spec(A/I_{0}) becomes trivial.

1.2.3. Infinitesimal properties: In our chapter on Riemann surfaces we exploited the exact sequence of sheaves

0 → ℤ → O_{S} → O_{S}^{∗} → 1,

which give rise to the sequence in cohomology

0 → H^{1}(S, ℤ) → H^{1}(S, O_{S}) → H^{1}(S, O_{S}^{∗}) → H^{2}(S, ℤ) →

(actually we can do this over any compact complex algebraic variety). We got from this sequence that H^{1}(S, O_{S}^{∗}) is a complex analytic group with tangent space H^{1}(S, O_{S}). We showed that H^{1}(S, ℤ) is a lattice in H^{1}(S, ℤ) and H^{1}(S, O_{S})/H^{1}(S, ℤ) = Pic^{0}(S) is a compact complex analytic group. The homomorphism

H^{1}(S, O_{S}) → H^{1}(S, O_{S})/H^{1}(S, ℤ) = Pic^{0}(S)

should be viewed as the exponential map from the tangent space to this complex analytic group.

We want to save some parts of this argument to our abstract situation. For instance we want to make it clear that the “additive” coherent cohomology group H^{1}(X, O_{X}) is the tangent space of our functor Pic_{X/k}.

To understand the infinitesimal properties we assume that T = Spec(A) where A is a local artinian k-algebra, let 𝔪 be its maximal ideal and hence A = k⊕𝔪. We want to consider the line bundles on X ×_{k} Spec(A) which are trivial on the special fibre X = X ×_{k} A/𝔪_{0}. This means that we want to understand the kernel Pic(X ×_{k} Spec(A))_{e} = H^{1}(X ×_{k} A, O_{X×A}^{∗})_{e} of the homomorphism

H^{1}(X ×_{k} A, O_{X×A}^{∗}) ⟶ H^{1}(X, O_{X}^{∗}).

For any integer n we put A_{n} = A/𝔪^{n} and hence k = A_{1}. We put X_{n} = X ×_{k} Spec(A_{n}), then our assumption implies that H^{0}(X_{n}, O_{X_{m}}^{∗}) = (A/𝔪^{n})^{∗}.

For any pair of integers n > n_{1} we can consider the embedding

X_{n_{1}} ↪ X_{n}

as a closed subscheme, it is an isomorphism on the topological spaces and we have the restriction

(O_{X_{n}})^{∗} ⟶ (O_{X_{n_{1}}} )^{∗}

which is surjective. For n = n_{1} + 1 we can see easily that the kernel is isomorphic to O_{X} ⊗ 𝔪^{n}/𝔪^{n+1} and since H^{0}(X_{n}, O_{X_{n}}^{∗}) = (A/𝔪^{n})^{∗} → H^{0}(X_{n−1}, O_{X_{n−1}}^{∗}) = (A/𝔪^{n−1})^{∗} is surjective we get an exact sequence

0 → H^{1}(X, O_{X} ⊗ 𝔪^{n−1}/𝔪^{n}) → H^{1}(X_{n}, O_{X_{n}}^{∗}) → H^{1}(X_{n−1}, O_{X_{n−1}}^{∗})

We have

H^{1}(X, O_{X} ⊗ 𝔪^{n−1}/𝔪^{n})→^{∼}H^{1}(X, O_{X}) ⊗ 𝔪^{n−1}/𝔪^{n}

and since we know that H^{1}(X, O_{X}) is finite dimensional, the we can conclude that

1.2.3.1. The abelian group Pic(X ×_{k} Spec(A))_{e} is an extension of finite dimensional kvector spaces.

The sum of the dimensions which I want to call the size of Pic(X ×_{k} Spec(A))_{e} can be estimated by H^{1}(X, O_{X}) and the structure of A. Especially if A = k[ϵ] is the k algebra of dual numbers then

Pic(X(k[ϵ])_{e}→^{∼}H^{1}(X, O_{X}).

Of course in general we do not know whether

H^{1}(X_{n}, O_{X_{n}}^{∗}) ⟶ H^{1}(X_{n−1}, O_{X_{n−1}}^{∗})

is surjective. We could say that the functor is smooth if this is the case.

1.3. The theorem of the cube.

We combine the above results and Lemma 1.2.2. Again we consider a line bundle L on X × Spec(A) where A is still local. Let I_{1} ⊂ A be the ideal which defines the maximal subscheme Spec(A/I_{1}) such that L|X × Spec(A/I_{1}) is trivial. Let 𝔪 ⊂ A be the maximal ideal, then I_{1}/𝔪I_{1} is a finite dimensional A/𝔪-vector space. Any non-zero linear form Ψ : I_{1}/𝔪I_{1} → A/𝔪 defines an ideal I_{Ψ} {⊂ \atop ≠}I_{1} and L|X × Spec(A/I_{Ψ}) is not trivial anymore.

We want to compute the obstruction to trivialize L|X × Spec(A/I_{Ψ}), i.e. we try to extend a trivialization to this larger scheme, and we will get a non zero element ξ_{Ψ} ∈ H^{1}(X × Spec(A/𝔪), O_{X}) which tells us that this is impossible.

Let I be any ideal such that L|X × Spec(A/I) is trivial. Let I′ = I𝔪. We have the inclusion

X × Spec(A/I) ↪ X × Spec(A/I′),

and our line bundle L|X × Spec(A/I) has a section s ∈ H^{0}(X × Spec(A/I), L) which is a generator at all points. We can cover X × Spec(A/I′) by affine open sets U_{i} such that the section s extends to a section s_{i} ∈ H^{0}(U_{i}, L) which then will be a generator at all points. We have

s_{i} = g_{ij}s_{j}

with g_{ij} ∈ H^{0}(U_{i} ∩ U_{j}, O_{U_{i}∩U_{j}}^{∗}). These g_{ij} are of the form

g_{ij} = 1 + h_{ij}

where h_{ij} ∈ I. Since we compute modulo I′, we have I/I′ ≃ k(t_{0}) and {h_{ij}} defines a class in H^{1}(X × k(t_{0}), O_{X×k(t_{0})}). This makes it clear that for any point t_{0} ∈ Spec(A/I) we get a linear map

δ_{L} : I/𝔪_{t_{0}}I ⟶ H^{1}(X × k(t_{0}), O).

We may for instance take as our ideal I simply the maximal ideal attached to a point t_{0} ∈ Spec(A) and get a linear map

δ_{L} : 𝔪_{t_{0}}/𝔪_{t_{0}}^{2} ⟶ H^{1}(X × k(t_{0}), O)

which can be interpreted as the map from the tangent space of Spec(A) in the point t_{0} to the “tangent space” of Pic(X)(k(t_{0})) in the “point” L.

It is clear: If I ⊂ A is the maximal ideal such that L|X × Spec(A/I) is trivial, then the map

δ_{L} : I/𝔪_{t_{0}}I ⟶ H^{1}(X × k(t_{0}), O_{X})

is injective for all t_{0} ∈ Spec(A/I). This has the following consequence

1.3.1. Theorem of the cube: Let X, Y be projective schemes over k. Let T be a connected scheme of finite type over k, and let L be a line bundle on

X × Y × T.

Let us assume that we have points x_{0} ∈ X(k), y_{0} ∈ Y (k), t_{0} ∈ T (k) such that L restricted to

x_{0} × Y × T X × y_{0} × T , X × Y × t_{0}

becomes trivial. Then the bundle itself is trivial.

Proof: We assume T = Spec(A), where A is local with maximal ideal t_{0}. Let us consider the maximal subscheme Spec(A/I) such that

L|X × Y × Spec(A/I)

is trivial. We want to show I = 0. If not, then I/𝔪_{t_{0}}I ≠ 0, and we get an injective map

δ_{L} : I/𝔪_{t_{0}}I ⟶ H^{1}(X × Y × k(t_{0}), O_{X×Y}).

But now we have the Künneth formula which tells us

\begin{aligned}H^{1}(X × Y × k(t_{0}), O_{X×Y}) & = H^{1}(X × k(t_{0}), O_{X}) ⊕ H^{1}(Y × k(t_{0}), O_{Y}) \\ & = H^{1}(X × y_{0} × k(t_{0}), O_{X}) ⊕ H^{1}(X × y_{0} × k(t_{0}), O_{Y}). \end{aligned}

But L|X × y_{0} × T and L|x_{0} × Y × T are assumed to be trivial. This means δ_{L} = 0. This is a contradiction.

Now we have seen that we can find an open neighborhood V of t_{0} such that L | X ×Y ×V is trivial. On the other hand we know (Lemma 1.2.1) that we have a maximal closed subscheme T_{1} ⊂ T, t_{0} ∈ T_{1} such that L | X × Y × T_{1} is locally closed in T_{1}. We must have T_{1} = T.

The Picard functor has two natural subfunctors. We say that a line bundle L on our projective scheme X/k is numerically equivalent to zero, if its restriction to any curve C ↪ X/k has degree zero.

We say that L is algebraically equivalent to zero, if we can find a connected scheme of finite type T/k with two points t_{0}, t_{1} ∈ T (k) on it and a line bundle L̃ over X ×_{k} T such that

L̃|X × t_{0} is trivial
L̃|X × t_{1} is isomorphic to L.

The intuitive meaning of this concept is: A line bundle is algebraically equivalent to zero, if it can be “deformed” into the trivial bundle.

1.3.2 Proposition: If a line bundle L on X is algebraically equivalent to zero, then it is numerically equivalent to zero.

Proof: Let us consider the case of a non singular projective curve C/k first. If we have a connected family L̃ on C × T with L̃_{t_{0}} ≃ O_{c}, L̃_{t_{1}} ≃ L then the function

t ⟶ dim_{k(t)}H^{0}(C × k(t), L̃) − dim_{k(t)}H^{1}(C × k(t), L̃)

is constant on T (???). But the right hand side is equal to deg(L̃_{t}) + 1 − g, therefore the degree is constant and equal to zero because deg(L̃_{t_{0}}) = 0.

We define the subfunctors. We put

Pic^{0}(X ×_{k} T) =Isomorphism classes of line bundles on X ×_{k} T

such that L_{t} is numerically equivalent to zero for all t ∈ T.

The second functor is defined on the category of connected schemes T/k of finite type, equipped with a base point t_{0} ∈ T (k) and the morphisms respecting the base point. Then

Pic^{00}(X ×_{k} T) =Isomorphism classes of line bundles L on X ×_{k} T
for which L|X × {t_{0}) ≃ O_{X}

It is a central point is that this last functor is “linear”:

1.3.3. For a product X × Y of two projective schemes which are also equipped with base points x_{0} ∈ X(k) and y_{0} ∈ Y (k) we have

Pic^{00}(X × Y × T) ≃ Pic^{00}(X × T) ⊕ Pic^{00}(Y × T).

This is a direct consequence of the theorem of the cube. If we have a line bundle L over X × Y × T, then the restrictions

\begin{aligned}L_{X} & = & L|X × y_{0} × T \\ L_{Y} & = & L|x_{0} × Y × T \end{aligned}

provide line bundles on X × T and Y × T. Now we consider the two projections

\begin{aligned}p_{X} & : & X × Y × T & → & X × T \\ p_{Y} & : & X × Y × T & → & Y × T, \end{aligned}

and we consider

L ⊗ p_{X}^{∗}(L_{X})^{−1} ⊗ p_{Y}^{∗} (L_{Y})^{−1}.

This line bundle on X × Y × T is trivial by the theorem of the cube.

We could think of Pic^{00}(X) as being the ”connected component ” of the identity of our functor, the quotient Pic(X)/Pic^{00}(X) gives us discrete invariants. They can viewed as elements in the second cohomology ( Chern classes See ???)) and the second cohomology of a product is bigger than the sum of the second cohomology groups and hence Pic(X)/Pic^{00}(X) should not be linear but quadratic.

1.3.4. The basic principle of the construction

The principal aim of this chapter is the construction of Pic(C) for a smooth, projective and absolutely irreducible curve C/k.

It is of course clear that this functor is a “disjoint union”

Pic(C) = ⨆ Pic^{r}(C),

where Pic^{r}(C) are the line bundles of degree r on C.

It will turn out that Pic^{0}(C) is an abelian variety, it is called J = J_{C}, the Jacobian of the curve. We will show that J and the Pic^{r}(C) are smooth projective and irreducible. In the second section of this chapter we will also discuss the Picard functor of J.

The basic idea of the construction is simple and goes back to Jacobi. The first object which we will construct is Pic^{g}(C).

Let us assume that k = \overline{k} is algebraically closed. Let us consider line bundles L of degree g on C. The Riemann-Roch theorem says

dim H^{0}(C, L) = g + 1 − g + dim H^{1}(C, L) = 1 + dim H^{1}(C, L).

We introduce the notion of a ”generic” line bundle of degree g. A bundle of degree g is generic if dim H^{1}(C, L) = 0 which implies that dim H^{0}(C, L) = 1.

The leading principle will be that for a ”generic” line bundle L have a non zero global section s ∈ dim H^{0}(C, L) which is unique up to a non zero scalar. The set of zeroes of s is a divisor (ref ???)

Div(s) = P_{1} + ⋯ + P_{g},

where P_{i} ∈ C(k). This collection of g points is unique up to permutations. Therefore we can view it as a point in C^{g}/Σ_{g}(k) where C^{g}/Σ_{g} is the quotient of C^{g} divided by the symmetric group Σ_{g}.

We will discuss the construction of the quotient C^{g}/Σ_{g} in section 1.4. and then we will show:

1.3.5. A non empty open set in C^{g}/Σ_{g} is the “parameter space” for the “generic” line bundles of degree g on C.

Hence we have to do the following:

a) We have to define and construct C^{g}/Σ_{g} and investigate its properties. (section 1.4.)

b) We have to make precise in what sense a suitable open subset U ⊂ C^{g}/Σ_{g} parameterizes bundles of degree g. (section ???)

c) We have to define some “translates” of these open sets U and glue them together to the parameter scheme Pic^{g}(C).

This is a long way to go.

1.4 Symmetric powers

Let k be a field and let X/k be a quasi-projective scheme. For any integer r we can form the r-fold product X^{r} = X ×_{k} X ⋯ × X (r factors) and on this product we have an action of the symmetric group Σ_{r}. We will construct a quotient X^{r}/Σ_{r} together with a projection

π : X^{r} ⟶ X^{r}/Σ_{r}

such that we have the obvious universal property: For any scheme T and any morphism h : X^{r} → T which commutes with the action of Σ_{r} (i.e. h ∘ σ = h for all σ ∈ Σ_{r}) we have a unique morphism h̄ : X^{r}/Σ_{r} → T such that h = h̄ ∘ π.

The construction of this quotient is easy. If X = Spec(A), where A is an affine k-algebra, we have X^{r} = Spec(A^{⊗r}) and

X^{r}/Σ_{r} = Spec ((A^{⊗r})^{Σ_{r}}) .

If X/k is quasi-projective, then we write

X  ↪  ℙ^{n}
↘  ↙
Spec(k)

and we observe that a point P ∈ X^{r} gives us an r-tuple of points (P_{1}, ⋯, P_{r}) with P_{i} ∈ X. If k is infinite we can find a hyperplane H ⊂ ℙ^{n} such that P_{i} ∉ H for all i. This implies that we can find an open affine subset U ⊂ X such that all our points lie in U. If k is finite we can find a non zero section s ∈ H^{0}(ℙ^{n}, O_{ℙ^{n}}(m)) for some m >> 0 such that non of our points lies on the zero set of s and the same argument works. Hence it is clear that for any point P ∈ X^{r} we can find an affine open subset U ⊂ X such that P_{1}, ⋯, P_{r} ∈ U. and we can cover X^{r} by a finite number of open affine subsets U_{ν} × … U_{ν}. We can form the quotients U_{ν}^{r}/Σ_{r} and now it is obvious how to glue (see ...??) these pieces together to get X^{r}/Σ_{r}.

I claim that this construction has the following properties:

(i) It commutes with extensions of the field of scalars, i.e. for any k → L we have

(X ×_{k} L)^{r}/Σ_{r} = X^{r}/Σ_{r} ×_{k} L.

(ii) The scheme X^{r}/Σ_{r} is again quasi-projective, and it is affine if X/k is affine and it is projective if X/k is projective.

To see the first assertion we assume that X = Spec(A) where A is an affine k-algebra. This algebra is free as a k-vector space, let e_{o}, e_{1}, ⋯, e_{m}, ⋯ be a basis. The elements

e\underline{i} = e_{i_{1}} ⊗ ⋯ e_{i_{r}} for (i_{1}, ⋯, i_{r}) ∈ ℕ^{r}

form a basis of the k-vector space A^{⊗r}.

The group Σ_{r} acts upon ℕ^{r}, let Σ_{r,i} be the stabilizer of \underline{i} in Σ_{r}. We put

E\underline{i} = ∑_{σ∈Σ_{r}/Σ_{r,\underline{i}}} e_{σ(\underline{i})},

and these elements form a basis of the k-vector space (A^{⊗r})^{Σ_{r}}. Since the e_{0}, ⋯, e_{m}, ⋯ also form a basis of A ⊗_{k} L over L, our assertion (i) is clear for affine k-algebras. But then it is obviously also true for any quasi-projective X/k. The assertion (ii) is obvious for affine schemes. It suffices to prove it for projective schemes. But now we may write for instance

X = Proj(k[x_{0}, ⋯, x_{n}])

where k[x_{0}, ⋯, x_{n}] is a graded k-algebra and where the x_{i} have degree one. We have seen that we can write

X^{r} = Proj(k[⋯ u\underline{α} ⋯])

where \underline{α} = (α_{1}, ⋯, α_{r}) ∈ [0, n]^{r} and

u\underline{α} = x_{α_{1}} ⊗ x_{α_{2}} ⋯ ⊗ x_{α_{r}}.

Then we get

X^{r}/Σ_{r} = Proj ((k[⋯ x_{α} ⋯])^{Σ_{r}}) .

Proposition 1.4.1. : If A/k is an affine k-algebra, then the algebra (A^{⊗r})^{Σ_{r}}/k is generated by the elementary symmetric functions

σ_{μ}(f) = ∑ 1 ⊗ ⋯ f ⊗ 1 ⋯ ⊗ f ⋯ ⊗ 1,

where we sum over all the possible placements of μ(≤ r) factors f in the above tensor product.

To see this we start from our basis e_{0}, e_{1}, ⋯ e_{m} ⋯ above where now in addition e_{0} = 1. We consider a basis element E\underline{i} as above, and we want to show that it is in the algebra generated by the elementary symmetric functions. We proceed by downward induction on the number of times 0 occurs in \underline{i} = (i_{1}, ⋯, i_{r}). If 0 occurs r-times the assertion is clear. Now we write

\underline{i} = ( \underbrace{0, ⋯, 0}_{ν_{0}−indices}, \underbrace{i, ⋯, i}_{ν_{1}−indices}, ⋯, \underbrace{j, ⋯, j}_{ν_{m}−indices})

where 0, i, ⋯, j are pairwise different.

Then

e\underline{i} = (1 ⊗ ⋯ e_{i} ⊗ ⋯ ⊗ e_{i} ⊗ 1 ⋯ 1) · (1 ⊗ · ⊗ e_{i} ⊗ e_{i} ⋯ e_{i} ⊗ 1 ⋯ ⊗ 1)(1 ⊗ 1 ⋯ 1 ⊗ e_{j} ⊗ ⋯ e_{j}).

Consider

E\underline{i} − σ_{ν_{1}}(e_{i}) ⋯ σ_{ν_{m}}(e_{j})

which is in (A^{⊗r})^{Σr}. If we look carefully at

(+ ⋯ 1 ⊗ ⋯ ⊗ e_{i} ⊗ ⋯ 1 ⊗ e_{i} ⊗ 1 ⋯) ⋯ (⋯ 1 ⊗ e_{j} ⊗ ⋯ 1 ⊗ e_{j} ⋯ ⊗ 1 ⋯),

and expand, then we see that E\underline{i} is exactly that part of the expansion where we multiply tensors which have the e_{i} ⋯ e_{j} at different places. So this part cancels. But for the remaining terms we see that we get some

⋯ ⊗ e_{ν}e_{μ} ⊗ ⋯

in the product. This e_{ν}e_{μ} can be expressed by linear combinations of the e_{i} ⋯ e_{j} and hence the number of 1’s in the tensor product goes up in these terms. This proves the proposition.

It is clear that the morphism

π : X^{r} ⟶ X^{r}/Σ_{r}

is finite. This implies that we have a surjective map on the set of geometric points

X^{r}(\overline{k}) ⟶ X^{r}/Σ_{r}(\overline{k}).

It is obvious that points which are equivalent under the action of the symmetric group map to the same point in X^{r}/Σ_{r}(k̄) hence the map factorizes

X^{r}(\overline{k})/Σ_{r} → X^{r}/Σ_{r}(\overline{k})

Exercise 1.4.2. Show that the above map is a bijection.

But over an arbitrary field the map X^{r}(k) → X^{r}/Σ_{r}(k) may not be surjective. A point Q ∈ X^{r}/Σ_{r}(k) can be lifted to a point P = (P_{1}, ⋯, P_{r}) ∈ X^{r}(\overline{k}) but then we have to answer the question

When does a geometric point P = (P_{1}, ⋯, P_{r}) ∈ X^{r}(\overline{k}) give a point π(P) ∈ X^{r}/Σ_{r}(k)?

To answer this question we may assume that X = Spec(A). Then our point P = (P_{1} ⋯ P_{r}) = (ψ_{1}, ⋯, ψ_{r}) where

\begin{matrix}B \\ ↑ & ↘^{ψ_{i}} \\ k & ⟶ & \overline{k} \end{matrix}

this is the same thing as ⊗ψ_{i} : B^{⊗r} → \overline{k}. The image of (ψ_{1}, ⋯, ψ_{r}) lies in X^{r}/Σ_{r}(k) if and only if the restriction ⊗ψ_{i} : (B^{⊗r})^{Σ_{r}} → k. Now it follows from proposition 1.4.1

Proposition 1.4.3. We have ⊗ψ_{i} : (B^{⊗r})^{Σ_{r}} → k if and only if for all f ∈ A the polynomial

(Y − ψ_{1}(f)) ⋯ (Y − ψ_{r}(f)) ∈ k[Y].

Now we want to assume that our scheme is a smooth projective curve C/k. A point P = (P_{1}, ⋯, P_{r}) ∈ C^{r}(\overline{k}) gives us a divisor

\overline{D} = ∑_{i=1}^{r} P_{i} = ∑_{P ∈C(\overline{k})} n_{P} P

on the curve C ×_{k} \overline{k}. We want to discuss the question whether this divisor is the base extension of a divisor on C/k

D = ∑_{𝔭} n_{𝔭}𝔭.

In this case we say that our divisor is rational over k. Of course we must have

deg(\overline{D}) = ∑ n_{P} = deg(D) = ∑ n_{𝔭}[k(𝔭) : k].

We have the action of the Galois group Gal(k̄/k) on the points in the support of our divisor. An orbit of this action consist of the points P which lie above one point 𝔭, i.e. of the points P → 𝔭. It is clear that we can assume that we have just one orbit which means that all the points P_{i} lie over one closed point 𝔭 on C/k.

Hence we get: The base extension of the divisor 𝔭 on C/k to the geometric curve is

n_{𝔭}^{(i)}( ∑_{P →𝔭} P),

where the Galois acts transitively on the P → 𝔭 and where n_{𝔭}^{(i)} is the degree of inseparability of the extension k(𝔭)/k. (Haben wir das nicht schon frueher gesehen???)

We conclude: If we look at our divisor \overline{D} = ∑_{P ∈C(\overline{k})} n_{P} P, and if we divide the points in its support under the action of the Galois group, then the orbits correspond to points 𝔭 on C/k. Our divisor

\overline{D} = ∑_{𝔭} n_{𝔭}( ∑_{P →𝔭} P),

is rational over k if and only if

n_{𝔭}^{(i)}|n_{𝔭}.

But now it is clear from our considerations in (p. 221 Chapter on curves Base change ??? Verweis auf obige proposition))) that the condition n_{𝔭}^{(i)}|n_{𝔭} is equivalent to the condition that

(⋯ \underbrace{P, ⋯, P}_{n_{𝔭} times} ⋯) ∈ C^{r}/Σ_{r}(k).

To see that this is the case we can assume that our points lie in an affine open subset U ⊂ C and U = Spec(A). Let k ⟶k̄ be an algebraic closure, let k_{s}, k_{i} be the maximal separable ( inseparable) subextension respectively. Our points are k-homomorphism ψ_{ν} : A → k̄, they form an orbit under the action of Gal(k̄/k). For any f ∈ A the polynomial

∏_{ν}(Y − ψ_{ν}(f)) ∈ k_{i}[Y]

because it is invariant under the action of the Galois group.

Lemma 1.4.4. :The degree of inseparability n_{𝔭}^{(i)} = p^{e} is the smallest power of p for which all the ψ_{ν}(f)^{p^{e}} ∈ k_{s}.

Let us assume this Lemma. Then we know that p^{e} is the smallest power of p such that for all f ∈ A the polynomial

∏_{ν}(Y − ψ_{ν}(f))^{p^{e}} ∈ k[Y],

and now it follows from proposition ??? that p^{e} is the smallest power of p such that

(⋯ \underbrace{P, ⋯, P}_{p^{e} times} ⋯)_{P →𝔭} ∈ C^{r}/Σ_{r}(k).

It remains to prove the Lemma. We choose an F ∈ A such that the differential dF generates the module of differentials at 𝔭. (See ???). Then we know that the extension k[F]⟶A is etale at 𝔭. This implies that ψ_{ν}(f)^{p^{e}} ∈ k_{s} for all f ∈ A if we have ψ_{ν}(F)^{p^{e}} ∈ k_{s} (Verweis???). But now 𝔭 ∩ k[F] = (p(F)) where p(F) ∈ k[F] is an irreducible polynomial. Now it is known from elementary algebra that

p(F) = F^{p^{e}n} + a_{1}F^{p^{e}(n−1)} + … a_{n−1}F^{p^{e}} + a_{n}

with some coefficients in k. This polynomial is equal to

∏_{ν}(F − ψ_{ν}(F))^{p^{e}} ∈ k[F].

If we know remember that this polynomial is irreducible we get the assertion of the Lemma.

Hence we come to the conclusion that the effective divisors on C/k of degree r are the points on C^{r}/Σ_{r}(k).

In our special case we have

Theorem 1.4.5. : For a smooth curve C/k the quotient C^{r}/Σ_{r} is again smooth.

We may assume that k is algebraically closed. We can pick a geometric point P ∈ C^{r}/Σ_{r}, we find a finite number of points Q_{1}, …, Q_{s} in C^{r}(k) lying above this point. We can write Q_{1} = (P_{1}, …, P_{r}) and the other Q_{i} are obtained by permuting the coordinates of Q_{1}. Now we have two possibilities to proceed. The first one is to pick a meromorphic function f on C which is regular at these points P_{1}, …, P_{r} and which has the property that df is a generator of the differentials in these points. We can find an affine scheme Spec(A) ⊂ C such that our points lie in Spec(A). Now we know that k[f]⟶A is etale (See ???) and from this we conclude that (k[f]^{⊗r})^{Σ_{r}} ⟶(A^{⊗r})^{Σ_{r}} (boring argument omitted). Hence we see that we have to prove our theorem for the special case C = Spec(k[f]). But now the theorem of the elementary symmetric functions says

(k[f]^{⊗r})^{Σ_{r}} →^{∼}k[X_{1}, …, X_{r}]^{Σ_{r}} = k[σ_{1}, …, σ_{r}]

where the σ_{i} are the elementary symmetric functions. This is a polynomial ring and hence smooth.

The second possibility is to investigate the completion of the local ring O_{C^{r}/Σ_{r},P} at the point P. We have two extreme possibilities. If all the coordinates ofP = (P_{1}, …, P_{r}) are pairwise different then it clear that for the completions

Ô_{C^{r}/Σ_{r},P} = Ô_{C^{r},(P_{1},...,P_{r})}

and smoothness becomes clear. The opposite case is that all the coordinates are equal say to P′. Then this point (P′, …, P′) is the only point in C^{r}(k) which lies over P and it is clear

(Ô_{C^{r},(P′,...,P′)})^{Σ_{r}} = Ô_{C^{r}/Σ_{r},P} .

But now the ring on the left is the power series ring in r variables and then the ring of invariants is the power series ring in the elementary symmetric functions. To treat the general case one has to mix the arguments.

1.5. The actual construction of the Jacobian of a curve.

Now want to prove the existence of the Jacobian. We assume that we have a smooth, projective and connected curve C/k which comes with a point P_{0} ∈ C(k)

Theorem I: The functor T → Pic_{P_{0}}^{0}(C ×_{k} T) is representable: We can construct a projective scheme J/k and an isomorphism of functors

J(T) → Pic_{P_{0}}^{0}(C ×_{k} T)

for all T → Spec(k) of finite type.

In our heuristical discussion in the beginning of this chapter we compared this representability to the local representability. To prove the above theorem we will prove

Theorem II:Let C/k be smooth, projective and connected and let g be the genus of C/k. The functor T → Pic^{g}(C ×_{k} T) is is locally representable.

Of course this implies that we construct a line bundle P on C × Pic^{g}(C) but this bundle cannot be unique, it can be multiplied by a bundle M which is the pullback of a bundle on Pic^{g}(C). (See 1.5.7.(

In exercise ... in section ... we constructed a line bundle on

C × C^{g}

which was the product

L′ = ⊗O_{C×C^{g}}(−Δ_{i}),

where Δ_{i} was the inverse image of the diagonal under the projection

pr_{0i} : C × C^{g} ⟶ C × C

to the zeroe’th and the i’th factor. The symmetric group Σ_{g} acts on the second factor and we have the quotient map

Π : C × C^{g} → C × C^{g}/Σ_{g}.

I want to construct a line bundle L on C × C^{g}/Σ_{g} whose inverse image Π^{∗}(L)→^{∼}L′. If P ∈ C × C^{g}/Σ_{g} is a point and if we lift this point to a point P̃ ∈ C × C^{g} then we can project it to the components and we get a g + 1 tuple (P_{0}, P_{1}, …, P_{g}). We can find a function in k(C) which is holomorphic at all the points P_{0}, …, P_{g} and which has the additional property that df generates the differentials at all these points. In a suitable neighborhood of P̃ the inverse image of the diagonal Δ ⊂ C × C under the projection pr_{0i} is defined by the ideal generated by

F_{i} = f ⊗ 1 … ⊗ 1 − 1 ⊗ 1 … ⊗ f … ⊗ 1

where we put the f into the i-th place (See ???). The inverse of this function trivializes the bundle O_{C×C^{g}}(−Δ_{i}) in our given neighborhood. This means that L′ is trivialized locally at P̃ by the product (∏_{i=1}^{i=g} F_{i})^{−1} and this is a meromorphic function on C × C^{g}/Σ_{g}. This system of trivializing sections defines L. (See ???)

We left it as an exercise (See ???) to the reader to show that the set

U′ ⊂ C^{g} = dim_{k(u)}H^{0}(C ×_{k} k(u), L_{u}) = 1.

is a non empty open subset in C^{g}. It is clearly invariant under the action of the symmetric group on C × U′ we can form the quotient by this action, we get an open subset U = U′/Σ_{g} ⊂ C^{g}/Σ_{g}.

Our line bundle L over C × C^{g}/Σ_{g}, has a specific properties: If we project π_{0} : C × C^{g}/Σ_{g} → C^{g}/Σ_{g}, and if we restrict this projection to C × U → U then the sheaf π_{0}^{∗}(L) will be locally free of rank one over U. (See ???). We want to denote the restriction of L to C × U by P′. Since our sheaf P′ contains the structure sheaf O_{C×C^{g}/Σ_{g}} we can even say that the O_{U} module π_{0}^{∗}(P′) is generated by the element 1 ∈ H^{0}(C × U, O_{C×U}).

The following proposition says that our line bundle P′ has a universality property.

Proposition 1.5.1. Let T be a scheme of finite type over k, let us assume that we have a line bundle L_{1} over C ×_{k} T, such that for any point t ∈ T we have deg L_{1,t} = deg L_{1}|C ×_{k} k(t) = g and

dim_{k(t)}H^{0}(C ×_{k} k(t), L_{1,t}) = 1.

Then there is a unique morphism

ψ : T ⟶ U,

such that for any point t ∈ T we can find an open subset V_{t} of t in T such that we have an isomorphism

η : (Id ×ψ)^{∗}(P′)|C × V_{t}→^{∼}L_{1}|C ×_{k} V_{t}

In other word we have

(Id ×ψ)^{∗}(P′)|C × T ∼_{T} L_{1}|C × T.

I want to comment on this proposition. It gives us already a large part of Theorem II, it says that the subfunctor of Pic^{g}(C) given by the generic bundles is locally represented by (C × U, P′) .

The second comment is that we proved the proposition in a special case. I recommend strongly to do the following

Exercise 1.5.2. Assume that our scheme T is integral then it has a field L = k(T) of meromorphic function which is the residue field of the generic point. We can restrict our line bundle to C ×Spec(L).

a) Show that our considerations in 1.4. imply that this restriction gives us an L valued point in

C^{g}/Σ_{g}(L).

b) Show that this L valued point gives us a morphism ψ′ : T′ → C^{g}/Σ_{g} from some non empty

open T′ ⊂ T such that (Id ×ψ′)^{∗}(P′)|C × T′→^{∼}L_{1}|C ×_{k} T′.

c) Let us assume we have a line bundle L_{0} on C/k which is generic, it yields a point P ∈ C^{g}/Σ_{g}(k). Assume that this point lifts to a point P = (P_{1}, P_{2}, ⋯, P_{g}) ∈ C^{g}(k) we all the components are pairwise different. We consider line bundles L on T = Spec(k[ϵ]) whose restriction to C × Spec(k) give L_{0}. We have seen in ??? that these L form a torseur under the group of line bundles on C × T which are trivial on C × Spec(k) and this group is isomorphic to to H^{1}(C, O). The tangent space to C^{g}/Σ_{g} at P is (see ???) the direct sum of tangent spaces T_{P_{i}} at the points P_{i} ∈ C(k). Hence our proposition tells us that we should have an isomorphism

H^{1}(C, O)→^{∼} ∑ T_{P_{i}}.

Write down this isomorphism!

(Es muss noch irgendwo im Buch der Zusammenhang zwischen dualen Zahlen und Tangentialräumen hergestellt werden!!!!!!)

We come to the proof of the proposition, it will occupy us for a while. Our assertion is local in the base scheme T, therefore we may assume that T = Spec(A) where A is an affine k−algebra. We pick a point t_{0} ∈ Spec(A). In the following considerations we have to replace A several times by a localization Spec(A_{f}) with f ∉ 𝔪_{0} and this new ring will again be called A. If we make such a replacement we will say ” by passing to a neighborhood of t_{0}”. (We could of course pass to the local ring at t_{0} directly, but then our algebra is not of finite type and formulating an extension of the theorem to this larger class of rings may also cause problems).

We consider the projection

p_{2} : C ×_{k} Spec(A) → Spec(A).

Our first aim is to show

1.5.3. Under our assumptions above the sheaf p_{_{2}∗}(L_{1}) is a locally free module of rank 1 over A.

Our assumption implies that the fibres π_{∗}(L_{1,t}) are one-dimensional vector spaces over k(t) for all t ∈ Spec(A). Now we assume that the k-algebra A is reduced. Then the semicontinuity theorem (See ???) implies that π_{∗}(L_{1}) is a locally free module of rank 1 over A. After passing to a neighborhood of t_{0} we may assume that it is free. Now I claim that we can drop the assumption that A is reduced. Let 𝔞 be the radical of A, let us put A_{0} = A/𝔞. We have the inclusion

X × Spec(A_{0})⟶X × Spec(A).

We can restrict the sheaf L_{1} to X × Spec(A_{0}) let us call the result L_{1,0}. We get an exact sequence of sheaves

0 → L_{1} ⊗ 𝔞 → L_{1} → L_{1,0} → 0.

We know already that H^{0}(X × Spec(A_{0}), L_{1,0}) is free of rank one over A_{0}. But now our assumption on L_{1} imply that H^{1}(X × k(t), L_{1,t}) = 0 for all points t ∈ T. Again we invoke the semicontinuity theorems (See ???, dies muss nochmal genau geprueft werden)) and we conclude R^{1}π_{∗}(L_{1}) = 0. We see that the generator s_{0} ∈ p_{∗}(L_{1})(Spec(A_{0})) lifts to an element s ∈ p_{∗}(L_{1})(Spec(A) and this clearly is a free generator of this module.

Let V (L_{1}) be the bundle of one-dimensional vector space over C × Spec(A) obtained from L_{1} (see PJ....) let p be the projection morphism. Then we have the zero section in this bundle

C ×_{k} Spec(A) × {0} ↪^{i_{0}} V (L_{1})
p ∘ i_{0} ↘  p ↓
C ×_{k} Spec(A)

and we use the isomorphism p_{0} ∘ i_{0} to identify the zero section and C ×_{k} Spec(A). Our section s defines a subscheme in V (L_{1}) which is defined locally by one equation. This subscheme intersected with the zero section defines via our identification a subscheme

[s = 0] ⊂ C ×_{k} Spec(A).

We know that this subscheme is of degree g in any fibre, this means that for any t ∈ Spec(A) the subscheme

[s_{t} = 0] := [s = 0] ×_{k} Spec(k(t)) ⊂ C ×_{k} Spec(k(t))

is finite and of degree g (see ???). But we have something much more precise

Lemma 1.5.4. : After passing to a neighborhood of t_{0} the scheme [s = 0] is flat over Spec(A) and therefore the A-algebra B_{0} = B ⊗ A/I is free of rank g (as an A-module).

Proof: For this proof we may and do replace A by the local ring at t_{0}. In a first step we show that [s = 0] is a finite affine scheme over Spec(A). We consider a finite subscheme F ⊂ C, F ≠ ∅ such that F ∩ [s_{t_{0}} = 0] = ∅. We put V = C \ F, this is affine and we consider

(F ×_{k} Spec(A)) ∩ [s = 0].

This must be empty, because if we project it to Spec(A) we get a closed subscheme which does not contain t_{0} (see ....). Since A is local, it follows that this closed subscheme is empty.

We see that the scheme [s = 0] is an affine scheme. But of course the restriction π_{0} : [s = 0] → Spec(A) is also projective we know that π_{0,∗}(O_{[s=0]}) is coherent and therefore [s = 0] is finite over Spec(A). We put B = O_{C}(V), then V = Spec(B) ⊂ C, and

[s = 0] = Spec(B ⊗ A/I),

where I is an ideal which is locally principal.

In the second step we prove the flatness. Assume we know that Tor_{1}^{A}(B⊗A/I, k(t_{0})) = 0. Then we choose a basis for the k(t_{0}) vector space (B ⊗ A/I) ⊗ k(t_{0}) and lift these basis elements to elements h_{1} ⋯, h_{g} in B ⊗ A/I, which then will be generators by Nakayama’s lemma. Hence we get a surjective homomorphism and an exact sequence

0 → R → A^{g} → A ⊗ B/I → 0,

where R is the A-module of relations. If we tensorize by A/𝔪_{0} = k(t_{0}) the sequence is still exact, because of the vanishing of the Tor^{1}. The arrow (A/𝔪_{0})^{g} → (B ⊗ A/I) ⊗ k(t_{0}) becomes an isomorphism and hence we conclude that R ⊗ A/𝔪_{0} = 0. Again we apply Nakayama and get the result.

Now we have to show the vanishing of the Tor_{1}. We consider the sequence

0 → I → B ⊗ A → B ⊗ A/I → 0,

and since B ⊗ A is free over A we get an exact sequence

0 → Tor_{1}^{A}(B ⊗ A/I, A/𝔪_{0}) → I ⊗ A/𝔪_{0} → B ⊗ A/𝔪_{0} → .

We are through, if we show that

I ⊗ A/𝔪_{0} → B ⊗ A/𝔪_{0}

is injective. We may assume that I is generated by an element f. (We may take our V above so small that L_{1} becomes trivial on an open neighborhood of V ×Spec(A/𝔪)). Then we have to consider elements

yf ⊗ 1 ∈ I ⊗ A/𝔪_{0},

whose image yf in B ⊗ A/𝔪_{0} is zero. But B ⊗ A/𝔪_{0} is integral and since f is non-zero in B ⊗ A/𝔪_{0}, we can conclude y goes to zero in B ⊗ A/𝔪_{0}. But then it follows that yf ⊗ 1 is zero and this is the injectivity.

1.5.4.1 At this point we introduce the notion of a relative divisor on C × Spec(A). By this we mean a subscheme Y ⊂ C × Spec(A) which is finite and flat over Spec(A) and where the sheaf of ideals defining Y is locally principal. The argument in the lemma above shows that such a subscheme is always contained in an open subscheme Spec(B)×Spec(A) where B is the algebra of regular functions on a suitable affine subset V ⊂ C, and hence

Y = Spec(B ⊗ A/I)

where the ideal I is locally principal.

The A/𝔪_{0} algebra (B ⊗ A/I) ⊗ (A/𝔪_{0}) has a rank r which is called the degree of the relative divisor. Our A-algebra (B ⊗ A)/I is free of rank r (Lemma 1.5.5.)

1.5.4.2. Our aim is to show that a relative divisor of degree r is nothing else than a Spec(A) valued point on C^{r}/Σ_{r}. I would like to point out that this is clear for r = 1 because in this special case our assumptions imply that B ⊗ A/I = A. Especially the following proposition is obvious for the case g = 1 and all the considerations in the next 12 pages are superflous if g = 1.

In the folling considerations our relative divisors have degree g this number could be replaced by an arbitrary r.

Our [s = 0] ⊂ C × Spec(A) is such a relative divisor, but we may also apply our construction to the line bundle L on C × C^{g}/Σ_{g} which provides the diagram

C × C^{g}/Σ_{g} ↩ [Δ = 0]
↓ ↙
C^{g}/Σ_{g}.

Here [Δ = 0] is of course the scheme ( divisor) of zeroes of the section 1 ∈ H^{0}(C × C^{g}/Σ_{g}, L).

Now we can our goal in 1.5.4.2. in a more precise form: We want to show that this diagram provides a universal relative divisor of degree g and this means that we can interprete [s = 0] as an A-valued point of C^{g}/Σ_{g}, i.e. a morphism ψ : Spec(A) → C^{g}/Σ_{g} over Spec(k). The following assertion (B) makes this precise and we will see in a minute that (B) implies proposition 1.5.1.

(B) There exists a unique morphism

C^{g}/Σ_{g} ⟵ Spec(A) : ψ,

such that in the diagram

C × C^{g}/Σ_{g} ⟵^{Id ×ψ} C × Spec(A)
∪  ∪
[Δ = 0]  [s = 0]
↓  ↓
C^{g}/Σ_{g} ⟵  Spec(A)

the scheme [s = 0] is the pull back of [Δ = 0] under Id ×ψ, i.e.

(Id ×ψ)^{−1}[Δ = 0] = [s = 0].

I want to introduce a terminology. If [s = 0] is given and if we found a morphism ψ which fulfills (Id ×ψ)^{−1}[Δ = 0] = [s = 0] then we say that ψ realizes [s = 0] . This formulation only concerns the existence. If we have also uniqueness then we could simply write [s = 0] = ψ

Before we show this assertion I want to explain that implies proposition 1.5.1. and conversely that a ψ as in proposition 1.5.1. must be a realization of [s = 0].

Assume we have a ψ as in proposition 1.5.1.. Then we have an isomorphism

L_{1} ⟶^{∼} (Id ×ψ)^{∗}(P′).

Let ψ(t_{0}) = x_{0}, then we may localize at x_{0}. The direct image π_{∗}(P′) is locally a free module generated by 1. The isomoprhism between our sheaves sends the section 1 to s up to a unit. Hence [s = 0] is the pullback of [Δ = 0] which is the subscheme defined by the section 1.

Conversely, if we have a realization ψ, we can easily construct an isomorphism between

L_{1} and (Id ×ψ)^{∗}(P′).

Outside of [s = 0] we identify by mapping s to 1. In a neighborhood of [s = 0] we know that the sections differ by a unit and this gives the identification over [s = 0].

Now we come to the proof of (B).

We have reduced our problem to a problem in commutative algebra: We have the affine algebra B of an open subset V of the curve C/k and the locally principal ideal I ⊂ B ⊗_{k} A such that

A ⟶ B ⊗_{k} A/I

is free of rank g. In terms of the algebras we have to show the following:

First of all it is clear from the definition of the subscheme [Δ = 0] that our morphism ψ must factor through

Spec ((B^{⊗g})^{Σ_{g}}) ⊂ C^{g}/Σ_{g}.

We translate our assertion (B) where we now introduce the convention that the homomorphism between affine algebras induced by ψ is denoted by ψ̃. Then we have to show:

We can find a unique homomorphism of k-algebras

ψ̃ : (B^{⊗g})^{Σ_{g}} ⟶ A,

such that for

(Id ×ψ̃) : B ⊗ (B^{⊗g})^{Σ_{g}} ⟶ B ⊗ A

image of I_{Δ} generates I, i.e.

(Id ×ψ̃)(I_{Δ}) · (B ⊗ A) = I.

We say that our relative divisor is very simple if the quotient algebra (B ⊗ A)/I is of the following simple form

(B ⊗ A)/I→^{∼}A[X]/(F (X))

where F (X) = X^{g} + a_{1}X^{g−1} + … a_{g} ∈ A[X] and where in addition the polynomial F (X) decomposes completely F (X) = (X − α_{1}) … (X − α_{g}) with α_{i} ∈ A. The reader should be aware that this decompositions is not necessarily unique because we allow that A has zero divisors.

1.5.5.1 The strategy: We will apply a localization technique.

1) For any relative divisor [s = 0] ⊂ C × T we will construct a faithfully flat T′ → T which is of finite type such that the pullback of our relative divisor becomes a very simple divisor. (This is the localization of our problem.)

2) We will prove our assertion for the special case of very simple divisors.

3) Then we will show that the localization does not destroy information. Let us asume have a relative divisor [s = 0] ⊂ C × T, a morphism ψ : T → C and a faithfully flat ϕ : T′ → T. Then we can take the pullback [s′ = 0] = (Id ×ϕ)^{−1}[s = 0] ⊂ C × T′ which is again a relative divisor. If then ψ′ = ψ ∘ ϕ realizes [s′ = 0] then already ψ realizes [s = 0].

This is actually almost obvious and follows from exercise ??? in ???

4) Finally we notice that the notion of a very simple divisor is obviously stable under pullback by morphisms p : T′′ → T′. Then we start from [s = 0] and construct a faithfully flat base change T′ → T such that the pullback [s′ = 0] is realized by ψ′ : T′ → C. Since at this point we know uniqueness we could also say [s′ = 0] = ψ′. Now we have the two projections p_{1}, p_{2} : T′ ×_{T} T′ → T′ and the two morphisms ψ′ ∘p_{1}, ψ′ ∘p_{2} which both realize

[s′′ = 0] = (p ∘ p_{1})^{−1}[s = 0] = (p ∘ p_{2})^{−1}[s = 0]

and since the realization is unique it follows

ψ′ ∘ p_{1} = ψ′ ∘ p_{2}

which then implies that we can find a ψ such that ψ′ = ψ ∘ p. Then it follows from 3) that [s = 0] = ψ.

I want to comment on this strategy. Here we are dealing with very general principles. First of all we learn that we can learn that we can solve problems by passing to localiza-

tions.

Secondly we learn that localization does not necessarily mean that we pass to small open subset in a toplogical space, there are more general concepts. (See also ???)

And finally we learn that what a fundamental role the uniqueness of the solution after localization plays. Our case here is in contrast to the situation in Chapter ??? where we applied the same strategy and where we saw that the non uniqueness created the cohomology groups.

Now we are ready for the hard work and tackle 2)

We still have the projection q : C^{g} → C^{g}/Σ_{g} and if we pull back our relative divisor [Δ = 0] to C × C^{g}, then we know that it is the product of g divisors

(Id ×q)^{∗}[Δ = 0] = ∏[Δ_{ν} = 0],

and where these [Δ_{ν} = 0] are the inverse images of the diagonal in C × C under the projection

C × C^{g} ⟶^{p_{0ν}} C × C.

These diagonals are relative divisors over C of degree one. We have the morphism

Ψ = (…, ψ_{ν}, …) : Spec(A) → C^{g}

such that (Id ×Ψ)^{∗}([Δ_{ν} = 0]) = (b̃−α_{ν}). We simply define ψ̃_{ν} : B → A as the composition

B ⊗ A → B ⊗ A/I→^{∼}A[X]/(F (X)) →^{X↦α_{ν}} A

Then ψ = q ∘ Ψ has the required property.

Of course it looks as if we have a problem with the uniqueness since our construction depends on the choice of a decomposition of F. This means that our morphism Ψ might be not unique. But the morphism Ψ is the same thing as Ψ̃ = (…, ψ̃_{ν}, …) : B^{⊗g} → A. We show that the restriction ψ of Ψ̃ to the subring of symmetric elements does not depend on this choice. An element b ∈ B is mapped to a polynomial g(X) = b_{m}X^{m} + … + b_{0} ∈ A[X]/(F (X)) under the composition B → B ⊗ A → B ⊗ A/I and ψ_{ν}(b) = g(α_{ν}). Now we refer to the theorem of elementary symmetric functions. The algebra (B^{⊗g})^{Σ_{g}} is generated by elements of the form σ_{μ}(b) and such an element is mapped to the corresponding coefficient in the polynomial (X − g(α_{1})) … (X − g(α_{g})). But this polynomial is independent of the choice of the α_{i}, its coefficients can be expressed in terms of the coefficients of the polynomial F (X) = (X − α_{1}) … (X − α_{g}).

Still this argument does not prove uniqueness.

To get uniqueness we first assume that our ψ lifts to a Ψ : Spec(A) → C^{g} as above in terms of algebras and we assume that we have a diagram

B ⊗ B^{⊗g}
↑  ↘^{Id ×Ψ̃}
B ⊗ (B^{⊗g})^{Σ_{g}}  ⟶^{Id ×ψ̃} B ⊗ A
↑  ↑
(B^{⊗g})^{Σ_{g}}  ⟶^{ψ̃}  B

The ideal I′_{Δ} in B ⊗B^{⊗g} decomposes into a product of locally principal ideals I′_{Δ} = ∏ I_{Δ_{ν}}, the image (Id ×Ψ̃)(I′_{Δ}) generate I which then decomposes it to a locally principal ideal

I = ∏ I′_{ν} with I′_{ν} = (Id ×Ψ̃)(I_{Δ_{ν}}) · B ⊗ A.

Now we have (We still assume that our divisor is very simple.)

B → B ⊗ A → B ⊗ A/I  = A[X]/(F (X))
↓  ↙ h_{ν}
A = B ⊗ A/I_{ν}

and A → B ⊗ A/I_{ν} must be an isomorphism since I_{ν} is of degree one. This implies that the morphism A[X]/(F (X)) ⟶^{h_{ν}} A must send X to a zero of F, and hence we see that I_{ν} = (X − β_{ν}) such that

F (X) = ∏_{ν=1}^{g}(X − β_{ν})^{.}

But now our previous remark tells us that this implies that ψ is unique.

We have to remove the hypothesis that ψ lifts to a morphism Ψ. To do this we observe: If we constructed a ψ : T → C^{g}/Σ_{g}, then we may consider Π_{2} : C^{g} → C^{g}/Σ_{g} and T via

ψ as schemes over C^{g}/Σ_{g}. The morphism Π_{2} is flat (See ???). We can form the fibered product

T′ = C^{g} ×_{C^{g}/Σ_{g}} T

and get a diagram

C × C^{g} ⟵^{Id ×ψ′} C × T′
↓ Id ×q_{0}  ↓ Id ×q
C × C^{g}/Σ_{g} ⊃ [Δ = 0] ⟵^{Id ×ψ} C × T ⊃ [s = 0]
↓  ↓
C^{g}/Σ^{g} ⟵^{ψ} T

where ψ′ is the projection.

Now the uniqueness of ψ follows from the argument 4) in 1.5.5.

Now we know that we can prove (B) for very simple relative divisors and it is clear that he notion of a very simple relative divisor is stable under base change.

We will prove

Lemma 1.5.5.1.: For any relative divisor we can find a finitely presented faithfully flat A-algebra

A ⟶ A′ such that

Spec(B ⊗ A′/I′) ⊃ [s = 0]
↓
Spec(A′)

is very simple.

Once we proved this we get by our strategy above the assertion (B) in general.

Now we prove that any relative divisor

B ⊗ A ⟶ B ⊗ A/I
↑
A

becomes very simple after a suitable faithfully flat base change of finite type.

Our problem is local in A. Let us pick a closed point t_{0} ∈ Spec(A), then the residue field k(t_{0}) is finite over k. We look at the fibre [s_{t_{0}} = 0] ⊂ Spec(B ⊗_{k} k(t_{0})). This scheme is finite, and we can find a finite normal extension L ⊃ k(t_{0}) ⊃ k such that all the residue fields k(𝔭) of the points 𝔭 ∈ [s_{t_{0}} = 0] can be embedded into L.

We now consider as a first step the base change A → A ⊗_{k} L = A′. We have finitely many closed points t′_{0} ∈ Spec(A′) which lie above t_{0}. This means that after a suitable base change we can assume that k(t_{0}) = k and that all points in [s_{t_{0}} = 0] have a residue field k, i.e. they are rational over k. Now we can find an element ȳ ∈ B ⊗_{k} k(t_{0}) such that dȳ ≠ 0 at all points 𝔭 ∈ [s_{t_{0}} = 0] and such that the values ȳ(𝔭) are different at the different points 𝔭 ∈ [s_{t_{0}} = 0]. Now we know that ȳ − ȳ(𝔭) is a uniformizing element at 𝔭, and it is clear that

B ⊗_{k} k(t_{0})/(f_{t_{0}}) ≃ k(t_{0})[X]/(∏_{𝔭}(X − ȳ(𝔭))^{m_{𝔭}}

(see CA ....). We lift ȳ to an element y ∈ B. Then we know that after a suitable localization in the base Spec(A) around t_{0} the images of the element 1, y, ⋯, y^{g−1} yield B ⊗ A/I of this A-module and we have

y^{g} = −a_{0} − a_{1}y − ⋯ − a_{g−1}y^{g−1}.

This means that after a localization in Spec(A) we have

B ⊗ A/I ≃ A[X]/(F (X))

where F (X) = X^{g} + a_{g−1}X^{g−1} + −a_{0} ∈ A[X]. The rest is clear: We can find a faithfully flat extension A → A′ of finite type such that

F (X) = (X − α_{1}) ⋯ (X − α_{g}) ∈ A′[X]

with α_{i} ∈ A′.

1.5.6 The gluing

Our goal was to construct a scheme Pic^{g}(C)/k and a line bundle P over C × Pic^{g}(C)/k such that for any line bundle L_{1} over a scheme C ×_{k} T for which deg L_{t} = g for all points t ∈ T we can find a

ψ : T ⟶ Pic^{g}(C),

such that (Id ×ψ)^{∗}(P) is locally on T isomorphic to L.

We reached this goal for a certain subfunctor: We considered only those families of bundles of degree g for which dim H^{0}(C × k(t), L_{t}) = 1.

So far we could work over our given ground field but now we have to extend the ground field k to an algebraically closed field. Then we have

1.5.6.1. Proposition: If we have a smooth projective curve C/k of genus g over an algebraically closed field k, then we can find a finite family of degree zero line bundles

F_{1}, ⋯, F_{r}

on C/k such that for any line bundle L′ on C ×_{k} L of degree g (here L is an arbitrary field extension of k), we can find an index i such that

dim H^{0}(C ×_{k} L, L′ ⊗ F_{i}) = 1

i.e. L′ ⊗ F_{i} is generic.

Proof: Our proof is based on a general principle namely that the line bundles of degree g on C/k form a bounded family [See Lit]. This means in our case that we have the bundle L on C × C^{g}/Σ_{g} and for any line bundle L_{1} on C/k we can find a point u ∈ C^{g}/Σ_{g}(k) such that L_{u}→^{∼}L_{1}. ( Every line bundle L_{1} ”occurs” (perhaps several times) in the family L on C × C^{g}/Σ_{g}). If such an L_{1} is given then we can find a line bundle F on C such that L_{1} ⊗ F is generic. Now we pick an u such that L_{1}→^{∼}L_{u}. Then we can find an open set V ⊂ C^{g}/Σ_{g} containing the point u such that for all v ∈ V the line bundle L_{v} ⊗ F is generic. This yields a covering of C^{g} by open sets V_{i} and bundles F_{i} such that L ⊗ F_{i} is generic on C × V_{i} and this covering has a finite subcovering and the finite list of F_{i}. For i = 1 we choose V_{1} = U and F_{1} = O_{C}.

Now we come back to our line bundle L′ on C ×L. Of course we can assume that L is the quotient field of a finitely generated k-algebra B, and that L′ is obtained by base change from a line bundle L^{∗} on C × Spec(B). The pullbacks of F_{i} by p_{1} are still called F_{i}. Now we consider a closed point t_{0} ∈ Hom_{k}(B, k) and consider the line bundle L_{t_{0}}^{∗} on C/k. We can find an index i such that L_{t_{0}}^{∗} ⊗ F_{i} is generic. Then it follows from semicontinuity that L ⊗ F_{i} is generic. This finishes the proof.

This makes it clear how to proceed with the construction of Pic^{g}(C). If we have line bundles L_{1} on C ×_{k} T, where T is of finite type, then we have a finite covering T = ∪ T_{i} by open sets such that L_{1} ⊗ F_{i} restricted to C × T_{i} is generic in all points t ∈ T_{i}. Then we find a

ψ_{i} : T_{i} ⟶ U

such that (Id ×ψ_{i})^{∗}(P′) is locally isomorphic to L_{1} ⊗ F_{i} or in other words

(Id ×ψ_{i})^{∗}(P′ ⊗ F_{i}^{−1})∼_{T_{i}}L_{1} | C × T_{i}.

Hence we do the following: We consider the open set U ⊂ C^{g}/Σ_{g} and the line bundle P′ on C × U. We consider r copies of C × U and on these copies we put the line bundles P′ ⊗ F_{i}^{−1}. Our previous consideration shows that C × U together with P′ ⊗ F_{i}^{−1} is a universal bundle for families of line bundles of degree g for which L_{1} ⊗ F_{i} is generic.

We form the disjoint union

⋃_{i=1}^{r} C × U = C × U × [1, …, r]

on which we consider the line bundle L_{i} = P′ ⊗ F_{i}^{−1} on the i’-th component. Remember that for i = 1 we put F_{1} = O_{C}. Let us put U_{i} = U × {i}.

For any of the components (C ×U_{i}, P′ ⊗F_{i}^{−1}) and for any other index j we may consider the open subset C × U_{ij} ⊂ C × U_{i} where u ∈ U_{ij} if and only if P′_{u} ⊗ F_{i}^{−1} ⊗ F_{j} is generic. Then it follows from our previous arguments that we have unique isomorphisms

ψ_{ij} : U_{ij}→^{∼}U_{ji} ⊂ U_{j}

such that we have

(Id ×ψ_{ij})^{∗}(P′ ⊗ F_{j}^{−1}) ∼_{U_{ij}} P′ ⊗ F_{i}^{−1}.

It is clear that this family of morphisms satisfies Ψ_{ik} ∘ ψ_{ij} = ψ_{ik} , and

ψ_{ji} ∘ ψ_{ij} = Id .

We get an equivalence relation on our disjoint union and this allows us to glue these copies of U via the identifications ψ_{ij} to a scheme Pic^{g}(C)/k (See ???).

By construction we have a covering of Pic^{g}(C) by open sets U_{i} and we have line bundles L_{i} on the products C × U_{i} We would also like to glue the bundles to a bundle P on C × C^{g}/Σ_{g} but before we can do this we have to work a little bit.

We have the following provisorial statement:

1.5.6.2. For any scheme T/k of finite type and for any line bundle L′ on C × T/k such that L′_{t} is of degree g for all t ∈ T we have a unique morphism

ψ : T ⟶ Pic^{g}(C)

such that for any t ∈ T we can find a neighborhood V_{t} of T such that ψ : V_{t} → Pic^{g}(C) factor through one of the U_{i}, and

η : (Id ×ψ)^{∗}(L_{i})→^{∼}L′ | C × V_{t}.

We say that ψ realizes L′ We can apply this to our bundle L on C × C^{g}/Σ_{g}. Hence we get a morphism

ψ : C^{g}/Σ_{g} ⟶ Pic^{g}(C)

which has the property above. Of course we can restrict Ψ to our open set U ⊂ C^{g}/Σ_{g} and on this set our map Ψ is the identity.

Now it is clear that we have an action of the group Pic^{0}(C)(k) on Pic^{g}(C): For any line bundle of degree zero F on C and any of our open sets U_{i} ⊂ Pic^{g}(C) we can define a new line bundle L_{i} ⊗ F on C × U_{i} which is of degree g in any point. Hence we get a unique morphism

T_{F} : U_{i} ⟶ Pic^{g}(C)

which realizes L_{i} ⊗F . It is clear that these morphisms are compatible on the intersections U_{i} ∩ U_{j} and therefore these morphisms fit together to an automorphism

T_{F} : Pic^{g}(C) ⟶ Pic^{g}(C),

and it is clear that T_{F} ∘ T_{G} = T_{F⊗G}.

Under this action a geometric point u ∈ Pic^{g}(C) which gives us an isomorphism class of line bundles L_{u} is mapped to T_{F}(u) and L_{u} ⊗ F = L_{T_{F} (u)}. Since this action is transitive we can conclude

1.5.6.3 The scheme Pic^{g}(C)/k is smooth and separated

Proof: We write Pic^{g}(C) = X. Of course the non empty open subscheme U ⊂ X is smooth and separated. The non smooth points form a closed subset in Z ⊂ X. If Z ≠ ∅ then Z(k)not = ∅. Since Pic^{0}(C)(k) acts transitively on X(k) and since Z(k) is invariant under this action it follows that Z(k) = ∅ because U(k)) ≠ ∅. A similar argument works for separatedness. We have to show that the diagonal Δ_{X} is closed in X ×X. Assume it is not, then we can find a geometric point (P, Q) ∈ Δ̄_{X}(k) ⊂ X(k) × X(k) with P ≠ Q. We have the diagonal action of Pic^{g}(C)(k) on X(k) × X(k) and clearly Δ_{X}(k) and Δ̄_{X}(k) are invariant under this action. We can find an element u ∈ Pic^{0}(C)(k) such that T_{u}(P, Q) ∈ U × U. But since U/k is separated we see that the diagonal is closed in U × U. This is a contradiction.

Now I claim

1.5.6.4. Our open set U ⊂ Pic^{g}(C) has a complement of codimension ≥ 2 and the line bundle P′ on C × U extends uniquely to a line bundle P on C × Pic^{g}(C). This line bundle P has the property that P|C × U_{i} ∼_{U_{i}} L_{i}

Proof : We consider the morphism ψ : C^{g}/Σ_{g} → Pic^{g}(C). The fibre of a point u ∈ Pic^{g}(C) consists of those points (P_{1} ⋯ P_{g}) ∈ C^{g}/Σ_{g}(k) for which

L_{u} ≃ O(−P_{1} − ⋯ − P_{g}),

and hence we see that the fibre has dimension

dim H^{0}(C × k(u), L_{u}) − 1.

The map ψ identifies the open set U ⊂ C^{g}/Σ_{g} with its image in Pic^{g}(C) by construction, hence we see that for points u ∈ Pic^{g}(C) \ U the dimension of the fibre is ≥ 1. (Ref ???) This implies the first assertion.

To prove the extendability we use a general principle which says that on smooth, irreducible schemes isomorphism between line bundles or sections in line bundles extend over closed subset of codimension ≥ 2.

To be more precise: Let V/k be a smooth. irreducible scheme of finite type, let L be a line bundle over V and let U ⊂ V be an open subset such that the complement has codimension ≥ 2. Let s ∈ H^{0}(U, L) be a section. Then this section extends to a unique section on V. To see that this is so we pick a point p ∈ V \ U and we choose a local section s_{p} ∈ H^{0}(V_{p}, L). Then we have s = hs_{p} over V_{p} ∩ U. Now it follows from theorem ??? in [Ei] that h extends uniquely to a regular function in O_{X}(V_{p}).

If we have two line bundles L_{1}, L_{2} over V which are isomorphic over U then this isomorphism is given by non zero sections in s_{1} ∈ H^{0}(U, L_{1} ⊗ L_{2}^{−1}), s_{2} ∈ H^{0}(U, L_{2} ⊗ L_{1}^{−1}) hose product is one. Our argument above shows that these sections extend to sections on V and their product is still one.

This implies that an extension from P′ on C × U to a line bundle P on C × Pic^{0}g(C) is unique.

Finally we extend P′ from C × U to a line bundle P on C × Pic^{g}(C). Let us assume that we extended P′ to a line bundle-still called P′- on C × U′ where U′ ⊃ U is open. Let a ∈ Pic^{g}(C)\U′. We find an index i such that a ∈ U_{i}. We consider the open subset U′_{i} ⊂ U′ where P′ ⊗ F_{i} is generic. By construction we have U′ ⊂ U_{i} and P′|C × U′_{i} ∼_{U′_{i}} L_{i}|C × U′_{i}. (The morphism ψ which realizes P′|C × U′_{i} is now the inclusion). But now we can say that we can find a line bundle M on U′_{i} such that

P′|C × U′_{i}→^{∼}L_{i}|C × U′_{i} ⊗ p_{2}^{∗}(M)

( see ???) and hence we can glue the bundles P′ on C × U′ and L_{i} ⊗ p_{2}^{∗}(M) on C × U_{i} over C × U′_{i} to a larger extension of P′. This process stops after a finite number of steps.

1.5.7. Hence have constructed the the scheme C × Pic^{g}(C) and a line bundle P which is the unique extension of P′ on C ×U such that we have the following universality property:

For any line bundle L′ on C×_{k}T where T/k is of finite type and where for all deg(L_{t}) = g t ∈ T, we have a unique morphism

Ψ : T ⟶ Pic^{g}(C)

such that

L′ ≃ (Id ×ψ)^{∗}(P) ⊗ M,

where M is a line bundle on T.

This means of course that we have local representability for Pic^{g}(C). We are aware that we needed to extend the ground field to k̄. Actually we can find a finite separable extension k_{1} of k such that our auxiliary bundles F_{i} are defined over k_{1} and then we can define Pic(C) over k_{1}. But it is clear that Pic^{g}(C)/k_{1} comes with a distinguished descend datum (see ???) and hence we can expect that we have a canonical descend to k. Before we can carry out this argument, we need more information on Pic^{g}(C) (for instance projectivity See ???) and these will be proved later.

If we keep our assumption that k = \overline{k}, or if we assume that we have a rational point P_{0} ∈ C(k). We want to show that we can replace g by any number r. We construct the line bundle

P_{r} = P ⊗ p_{1}^{∗}(O_{C}(−νP_{0}))  on  C × X,

and it is clear that this bundle has the universal property for bundles of degree rg − ν. If r is given we rebaptize X and call it Pic^{r}(C) and

P_{r}  on  C × Pic^{r}(C),

is the universal bundle for bundles of degree r.. Especially we can construct a line bundle P_{0} on C × Pic^{0}(C). Of course all these Pic^{r}(C) are isomorphic, but the identification depends on the choice of P_{0}.

If we consider the scheme

C × Pic^{r}(C) × Pic^{s}(C),

then we can consider the line bundle

p_{12}^{∗}(P_{r}) ⊗ p_{13}^{∗}(P_{s})

on it and this line bundle has degree r + s in any point y ∈ Pic^{r}(C) × Pic^{s}(C). Therefore we get a unique morphism

m : Pic^{r}(C) × Pic^{s}(C) ⟶ Pic^{r+s}(C)

such that

p_{12}^{∗}(P_{r}) ⊗ p_{13}^{∗}(P_{s}) ∼ (Id ×m)^{∗}(P_{r+s}).

It is clear that this defines a structure of a group scheme (See ???) on

Pic(C) = ⨆ Pic^{r}(C)

which is a scheme over k which is not of finite type.

As a special case this gives us a group scheme structure on Pic^{0}(C), and all the Pic^{r}(C) are principal homogeneous spaces under the action of Pic^{0}(C).

2. The Picard functor on X and on J

2.1. Some heuristical remarks

We want to get a better understanding of the scheme Pic(C). Since all its connected components are isomorphic, we can concentrate on the components. For us it is sometimes convenient to consider Pic^{g}(C) and Pic^{0}(C). Therefore I introduce the notations

\begin{aligned}X & = & Pic^{g}(C) \\ J & = & Pic^{0}(C) \end{aligned}

We have seen that X is a smooth projective scheme. Then J is also smooth projective, and it is a group scheme. Smooth projective group schemes are called abelian varieties.

It is already clear that Pic^{0}(C)/k is a proper scheme (see ???). To see this we recall that we have the surjective morphism

π : C^{g}/Σ_{g} ⟶ Pic^{g}(C),

and that we have identifications Pic^{0}(C) ≃ Pic^{g}(C). The morphism π is birational, because it induces an isomorphism on the open subset U.

As in the transcendental case our main object are the Picard schemes Pic(X) and Pic(J). The understanding of the structure of these Picard schemes is the key to many beautiful results.

2.1.1. I remind you that the fundamental tool for this study was the existence of the polarization and the resulting line bundles. (See ???)

We assume that k has rational points, we pick a point P_{0} ∈ C(k). Then we can restrict the bundle P to {P_{0}} × X ≃ X and this gives us a line bundle P_{P_{0}} on X. This bundle will play the role of - or is - the principal polarization. We have the action of J on X and this allows us to translate line bundle and divisors. For any x ∈ J(k) we have the translation T_{x} : X → X and we can consider the translated bundle T_{x}^{∗}(P_{P_{0}}) and we compare it to P_{P_{0}} by forming the quotient

T_{x}^{∗}(P_{P_{0}}) ⊗ P_{P_{0}}^{−1}.

Now we can view x as a variable or better we will construct a bundle

m^{∗}(P_{P_{0}}) ⊗ p_{2}^{∗}(P_{P_{0}}^{−1}) on X × J

(see ???) which evaluated at x gives the above bundle. Our main result will be that this bundle gives as a local representation of Pic^{00}(X) (see ???).

2.2. The homomorphisms φ_{M}:

Let us consider any line bundle M on X. We can consider the map

φ_{M} : J(k) = Pic^{0}(C)(k) → P^{00}(k)

defined by

φ_{M}(x) = T_{x}(M) ⊗ M^{−1},

and we want to show:

Proposition 2.2.1.: This map φ_{M} is a homomorphism, and if M is algebraically equivalent to zero, then this homomorphism is trivial.

Proof: We reduce the proof to the second assertion. Let us pick a point x, then

L = T_{x}^{∗}(M) ⊗ M^{−1}

is algebraically equivalent to zero. Then the second assertion says

T_{y}^{∗}(L) ≃ L

therefore

T_{y}^{∗}(T_{x}^{∗}(M) ⊗ M^{−1}) ≃ T_{x}^{∗}M ⊗ M^{−1}

and rearranging yields

T_{x+y}^{∗}(M) ⊗ M^{−1} ≃ T_{x}^{∗}(M) ⊗ M^{−1} ⊗ T_{y}^{∗}(M) ⊗ M^{−1}.

Now we consider the case of an M which is algebraically equivalent to zero. This means that we have a line bundle

M̃|X × Z

where Z is of finite type and connected over k and such that

\begin{aligned}M̃|X & × & z_{1} & ≃ & M \\ M̃|X & × & z_{0} & ≃ & O_{X}. \end{aligned}

for some points z_{1}, z_{0} ∈ Z. We can pick a point x_{0} ∈ X(k) and tensorize M̃ by p_{2}^{∗}(M̃|x_{0} × Z)^{−1}); then M̃|x_{0} × Z ≃ O_{Z}. We have two morphisms

\begin{aligned}p_{13} & : & X × J × Z & ⟶ & X × Z \\ m_{12} × Id_{Z} & : & X × J × Z & ⟶ & X × Z, \end{aligned}

the first one is the projection, the second one multiplication in the first to factors (times identity). We consider the bundle

(m_{12} × Id_{Z})^{∗}(M̃) ⊗ p_{13}^{∗}(M̃)^{−1} = N

which evaluated at a point x ∈ J(k) and z ∈ Z(k) gives us

T_{x}^{∗}(M̃_{z}) ⊗ M̃_{z}^{−1}

where M_{z} is of course M̃|X × {z}. Now we know that

N |x_{0} × J × Z  and  N |X × e × Z  and  N |X × J × z_{0}

are trivial. Now the theorem of the cube would imply that N is locally trivial in Z. But to apply it we need to know that X, J are projective. We don’t know this yet, even worse we need our proposition to prove the projectivity. Therefore we have to make a detour at this point.

We remember that we have a morphism

π × Id_{Z} : C^{g}/Σ_{g} × C^{g}/Σ_{g} × Z ⟶ X × J × Z

which is birational and where the fibres are product of projective spaces. We pull our line bundle back and apply the theorem of the cube upstairs. To do this we need a little

Lemma 2.2.2. For any line bundle L on X × J × Z we have (π × Id_{Z})_{∗}(π × Id_{Z})^{∗}(L)→^{∼}L

Proof: We consider the pullback of the bundle

L̃ = (π × Id_{Z})^{∗}(L) | C^{g}/Σ_{g} × C^{g}/Σ_{g} × Z.

It follows from the construction (See V. 2.1.5.) that we have a morphism

j : L → (π × Id_{Z})_{∗}((π × Id_{Z})^{∗}(L))

We want to show that this is an isomorphism. This is a local question on X × J × Z and therefore we can assume that L is the trivial bundle O_{X×J×Z}. Then the pullback is by definition the structure sheaf O_{C^{g}/Σ_{g}×C^{g}/Σ_{g}×Z}. We have to show that

(π × Id_{Z})_{∗}(O_{C^{g}/Σ_{g}×C^{g}/Σ_{g}×Z}) = O_{X×J×Z}

Of course this follows if we can show the corresponding assertion for the morphism

π : C^{g}/Σ_{g} × C^{g}/Σ_{g} ⟶ X × J

because the sheaves in question are simply the pullbacks via the projection p_{12} to the first and second factor. Now we have seen that the morphism π is projective and hence we can conclude that π_{∗}(O_{C^{g}/Σ_{g}×C^{g}/Σ_{g}}) is a coherent sheaf on X × J (See ???). Hence we see that for any open subset V ⊂ X × J the algebra π_{∗}(O_{C^{g}/Σ_{g}×C^{g}/Σ_{g}})(V) is finite over O_{X×J}(V). If we pass to the stalks at a point x then we have O_{X×J,x} ⊂ (O_{C^{g}/Σ_{g}×C^{g}/Σ_{g},x} the extension is finite and the two local rings have the same field of fractions, namely the field of meromorphic functions on X × J (??? vorher was dazu sagen). But since X × J is smooth we know that the local rings O_{X×J,x} are closed in their field of fraction ( See [Ei], ???). Hence we conclude that the local rings are equal and this proves the Lemma.

Now we know that the first to factors are projective. and if we take inverse images of our points x̃_{0} and ẽ, then we still have the triviality conditions. Hence we can conclude Ñ is locally trivial in Z, i.e. we can cover Z by open schemes Z_{α} s.t.

Ñ | C^{g}/Σ_{g} × C^{g}/Σ_{g} × Z_{α}

is trivial. But then the lemma above yields that already N is locally trivial in Z.

2.3. The projectivity of X and J

Now we are prepared to prove that the scheme X/k is projective. To do this we construct an ample line bundle on X.

We start from our line bundle P_{P_{0}}. We have seen that it can be written as

P_{P_{0}} = O_{X}(−D)

where D is an effective divisor (??? Ref. dafür, auch für effektiv?). Then we have

\begin{aligned}T_{x}^{∗}P_{P_{0}} & = & O_{x}(−T_{x}^{∗}D) \\ T_{−x}^{∗}P_{P_{0}} & = & O_{X}(−T_{−x}^{∗}D) \end{aligned}

and

T_{x}^{∗}P_{P_{0}} ⊗ T_{−x}^{∗}P_{P_{0}} ≃ P_{P_{0}} ⊗ L_{x} ⊗ P_{P_{0}} ⊗ L_{−x} ≃ P_{P_{0}}^{⊗2}.

For any point u ∈ X we find an x ∈ Pic^{0}(C) such that

u ∉ T_{x}^{∗}D ∪ T_{−x}^{∗}D,

and hence we see that u is not a base point.

Then we have seen that we get a morphism

r_{P_{P_{0}}^{⊗2}} : X ⟶ ℙ^{N} = ℙ(H^{0}(X, P_{P_{0}}^{⊗2})),

and I claim that this morphism is finite. First we assume that we have a fibre r_{P_{P_{0}}^{⊗2}}^{−1}(y)

which has positive dimension. This fibre contains an irreducible curve Z. We can find a section s ∈ H^{0}(X, P_{P_{0}}^{⊗2}) which does not vanish at a particular point x_{1} ∈ Z(k) since we do not have base points, but then this section will not vanish at any point of Z which implies that P_{P_{0}}^{⊗2} | Z is the trivial bundle.

Therefore we know that P_{P_{0}} and all its translates T_{x}^{∗}(P_{P_{0}}) have degree zero on Z. (Ref. ???) Since P_{P_{0}} = O_{X}(−D) we see that for any translate of D we have

T_{x}^{∗}(D) ∩ Z = \{ \begin{matrix}∅ \\ Z \end{matrix}

or in other words we have T_{x}^{∗}(Z) ⊂ |D| or T_{x}^{∗}(Z) ∩ |D| = ∅.

Now we show that for any two points z_{1}, z_{2} ∈ Z(k) we have

T_{z_{2}−z_{1}}^{∗}(|D|) = |D|.

Let d_{0} ∈ |D|(k) then d_{0} = T_{d_{0}−z_{1}}^{∗}(z_{1}) and it follows

T_{d_{0}−z_{1}}^{∗}(Z) ⊂ |D|.

Now T_{d_{o}−z_{1}}^{∗}(z_{2}) = T_{d_{0}−z_{1}}^{∗}(T_{z_{2}−z_{1}}^{∗}(z_{1})) = T_{z_{2}−z_{1}}^{∗}(T_{d_{0}−z_{1}}^{∗}(z_{1})) = T_{z_{2}−z_{1}}^{∗}(d_{0}) ∈ |D|.

But this is impossible since it would imply that T_{z_{2}−z_{1}}^{∗}(P_{P_{0}}) ≃ P_{P_{0}} for all z_{2}, z_{1} ∈ Z and this contradicts our theorem ??? and we proved that all the schemes Pic^{r}(C) are projective.

2.3.1 Now we want to show that the homomorphisms ϕ_{M} are not just homomorphisms on the groups of geometric points but also between the functors.

In 1.3.2. we defined the subfunktor Pic^{00}(X)(T). It is defined on connected schemes T/k of finite type which are equipped with a base point t_{0}. In the following considerations we evaluate this functor on local rings A = O_{Y,t_{0}} where Y is of finite type over k and the base point is t_{0}.

The action m : X ×J → X gives us for any T an action of the group J(T) on the scheme X × T and hence on Pic(X × T). If we denote again by T_{x} the automorphism induced by x ∈ J(A) on the scheme then we denote by T_{x}^{∗} the action on the line bundles. Again we pick a bundle M on X. For T = Spec(A) where A is local as above we consider

\begin{matrix}x → Pic(X)(A) \\ x ↦ T_{x}^{∗}(M) ⊗ M^{−1} \end{matrix}

Again this is a homomorphism. To see this we apply the theorem of the cube. We consider

X × J × J,

we have three morphisms to X × J, namely, the projections p_{12}, p_{13} and Id_{X} ×m_{23} where m_{23} is the multiplication in the second and third factor.

Then we have the two morphisms

X × J ⟶^{m} ; p_{1}^{⟶}X,

where m is the action of J on X. For any line bundle M on X we can consider the line bundle

\begin{aligned}N = (m ∘ m_{23})^{∗}(M) & ⊗ (m ∘ p_{12})^{∗}(M)^{−1} ⊗ (m ∘ p_{13})^{∗}(M)^{−1}⊗ \\ & ⊗ (p_{1} ⊗ m_{23})^{∗}(M)^{−1} ⊗ (p_{1} ∘ p_{12})^{∗}(M) ⊗ (p_{1} ∘ p_{13})^{∗}(M) \end{aligned}

on X × J × J.

We pick a point x_{0} ∈ X(k) and evaluate this bundle on

x_{0} × J × J  X × e × J  e × X × J,

and find that its restriction to the second and the this subscheme is trivial. Let N_{x_{0}} be its restriction to the first subscheme which we identify to J × J. Let

Ñ_{x_{0}} = p_{23}^{∗}(N_{x_{0}})

then we see that

N ⊗ p_{23}^{∗}(N_{x_{0}})^{−1}

is trivial on all three subschemes. Hence we see by the theorem of the cube that

N ≃ p_{23}^{∗}(N_{x_{0}}).

This can be formulated differently by saying N ∼_{J×J} O_{X×J×J}.

If now x, y ∈ J(A) and if x, y then (x, y) ∈ J × J(A)

T_{x+y}^{∗}(M) ⊗ T_{x}^{∗}(M)^{−1} ⊗ T_{y}^{∗}(M)^{−1} ⊗ M

on X ×_{k} T is trivial, because the contribution of N_{x_{0}} is trivial since A is local. This means that we get a homomorphism

ψ_{M} : J(A) ⟶ Pic(X)(A).

We will apply this to the special bundle P_{P_{0}}. We can now interprete this as a homomorphism of functors

ϕ_{P_{P_{0}}} : J → Pic^{00}(X)

2.4. Maps from the curve C to X

Now we construct certain morphisms from the curve into X and study the restriction (via these maps) of P_{P_{0}} and of other line bundles on X to C. We have seen in Ch. RS 6.3 (???) that such morphisms explain the selfduality of the Jacobian.

In this section we assume that k is algebraically closed.

2.4.1. I explain the morphisms naively: We choose points

(Q_{1}, ⋯, Q_{g}) ∈ C^{g}(k)

We still choose another point P_{0}, and we assume that the images of the points

(Q_{1}, ⋯, Q_{g})  and  (P_{0}, Q_{1}, ⋯, Q̂_{i}, ⋯, Q_{g})

in the quotient C^{g}/Σ_{g} lie always in U. From these data we construct a morphism

j_{P_{0},\underline{Q}} : C ⟶ X

which sends a point P ∈ C(k) to the line bundle O_{C}(P − P_{0} − Q_{1} − ⋯ − Q_{g}).

To give the correct definition of this morphism we consider the curve C × C with its two projections p_{1}, p_{2} to C. On this surface we have the line bundle

p_{1}^{∗}(O_{C}(−P_{0} − Q_{1} − ⋯ − Q_{g})) ⊗ O_{C×C}(Δ) = L,

where Δ ⊂ C × C is the diagonal. This gives us a family of line bundles on the first factor which is parameterized by the second factor. For a point P ∈ C(k) in the second factor the restriction of L to C × {P } is the bundle O_{C}(P − P_{0} − Q_{1} − ⋯ − Q_{g}). Hence we have a unique morphism - and this is our j_{P_{0},\underline{Q}} - such that locally in the second component

(Id ×j_{P_{0},\underline{Q}}(P)) ∼ L.

I claim that under our above assumptions the morphism j_{P_{0},\underline{Q}} : C → Pic^{g}(C) factors through the open subset U. To see this we have to show that

dim H^{0}(C, O_{C}(P − P_{0} − Q_{1} ⋯ − Q_{g})) = 1

for all points P ∈ C(k). This is of course clear if P is one of our points P_{0} or Q_{1} ⋯ Q_{g} because this is our assumption. If P is not equal to any of these points, and if we had

dim H^{0}(C, O_{C}(P − P_{0} − Q_{1} ⋯ − Q_{g})) ≥ 2

we could conclude that

dim H^{0}(C, O_{C}(−P_{0} − Q_{1} ⋯ − Q_{g})) ≥ 3,

because this space contains 1 and 1 is not zero at P. But then we could conclude that

dim H^{0}(C, O_{C}(−Q_{1} ⋯ − Q_{g})) ≥ 2

and this is again a contradiction to our assumption.

We want to compute the line bundle

j_{P_{0},\underline{Q}}^{∗}(P_{P_{0}}),

this is a line bundle on C. The morphism j_{P_{0},\underline{Q}} sends the point P to a point j_{P_{0},\underline{Q}}(P) = (Q_{1}(P), ⋯, Q_{g}(P)) where (Q_{1}(P) ⋯ Q_{g}(P)) ∈ C^{g}(\overline{k}) is well defined up to an element in the symmetric group. It is determined by the relation

O_{C}(P − P_{0} − Q_{1} ⋯ − Q_{g}) ≃ O_{C}(−Q_{1}(P) ⋯ − Q_{g}(P)).

Now we recall the definition of P and P_{P_{0}}. The restriction of P to {P_{0}} × U is the line bundle which is induced by the divisor

D′ = Σ C × ⋯ C × P_{0} × C ⋯ C

on C^{g}. This divisor descends to a divisor D on C^{g}/Σ_{g} and the line bundle P_{P_{0}}|U = O_{U}(−D).

Then it is clear that the curve j_{P_{0},\underline{Q}}(C) intersects D in the points j_{P_{0},\underline{Q}}(Q_{i}) with multiplicity one and this yields the innocent looking but fundamental

j_{P_{0},\underline{Q}}(P_{P_{0}}) ≃ O_{C}(−Q_{1} ⋯ − Q_{g}).  (2.4.2.)

2.4.3. Now we do what we did already very often, we consider the point Q ∈ U as a variable, more precisely we consider it as a point in X.

We consider the diagram

⟶^{p_{12}} C × X
C × X × C
⟶^{p_{13}} C × C ⟶^{p_{1}} C

and the line bundle

p_{12}^{∗}(P) ⊗ p_{13}^{∗}(p_{1}^{∗}(O_{C}(−P_{0})) ⊗ O_{C×C}(Δ)) = L̃

on the on C × X × C. If we evaluate at the point \underline{Q} ∈ U(k) ⊂ X(k), then we get our line bundle L above. We view this as a family of degree g bundles on the first factor which is parameterized be the product of the second and third factor. The universal property gives us a unique morphism

j_{P_{0}} : X × C ⟶ X

such that (Id_{C} ×j_{P_{0}})^{∗}(P) isomorphic locally in X × C to the line bundle L̃ .

Of course it is quite clear to describe this map on geometric points. If (u, P) is a geometric point on X × C then u is the isomorphism class of a line bundle L_{u} on C × k̄. Then L_{u} ⊗ O_{C}(P − P_{0}) is a line bundle of degree g on C, hence a geometric point on X and this point is the image.

This description yields a little piece of extra information

2.4.3.1. The morphism j_{P_{0}} commutes with the action of J on both sides

This map j_{P_{0}} gives us a line bundle

j_{P_{0}}^{∗}(P_{P_{0}})

on X × C. We consider this as a family of line bundles on C which now is parameterized by the first factor and find a unique morphism

ψ_{P_{0}} : X ⟶ X

such that (ψ_{P_{0}} × Id_{C})^{∗}(P) ∼_{X} j_{P_{0}}(P_{P_{0}}) where in this case ∼ means that the bundles are locally isomorphic in the first variable X.

But for a dense set of geometric points \underline{Q} ∈ U(k) we have shown that

ψ_{P_{0}}(\underline{Q}) = \underline{Q}

and since X is reduced we can conclude that ψ_{P_{0}} = Id_{X} (Ref ???). But we have of course (Id_{X} × Id_{C})^{∗}(P) ≃ P, and conclude

P ∼ j_{P_{0}}^{∗}(P_{P_{0}})  (2.4.4.)

locally in X on X × C.

2.4.5.: Now we have constructed the homomorphism

ϕ_{P_{P_{0}}} : Pic^{0}(C)(k) → Pic^{00}(X)(k)

and we have the resriction

j_{P_{0},\underline{Q}}^{∗} : Pic^{0}(C)(k) → Pic^{00}(X)(k) → Pic^{0}(C)(k)

and I claim that the composition of these to homomorphisms is the identity. To see that this is the case we have to show

j_{P_{0},\underline{Q}}^{∗} (T_{x}^{∗}(P_{P_{0}}) ⊗ P_{P_{0}}^{−1}) ≃ L_{x}

where L_{x} is a line bundle corresponding to x. For the computation of left hand side we use the invariance under translations (2.4.3.1.) and (2.4.2 ???): We get

T_{x}^{∗}(j_{P_{0},\underline{Q}}^{∗} (P_{P_{0}} ⊗ P_{P_{0}}^{−1}) →^{∼}T_{x}^{∗}(P) ⊗ P^{−1}→^{∼}L_{x}

and this is the claim.

We have the two morphisms

X × Pic^{0}(C)⟶^{m} ; p_{1}^{⟶}X,

and we can consider the bundle

m^{∗}(P_{P_{0}}) ⊗ p_{1}^{∗}(P_{P_{0}})^{−1}

on X × J. Then for any \underline{Q} the bundle

(j_{P_{0},\underline{Q}})^{∗}(m^{∗}(P_{P_{0}}) ⊗ p_{1}^{∗}(P_{P_{0}})^{−1})

is a line bundle on C × J which has degree zero at any closed point t ∈ J. We remember that we denoted a universal bundle on C × J by P_{0}. We see that we have a unique morphism

ψ_{P_{0},\underline{Q}} : J ⟶ J

such that

(Id ×ψ_{P_{0},\underline{Q}})^{∗}(P_{0}) ∼_{J} (j_{P_{0},\underline{Q}})^{∗}(m^{∗}(P_{P_{0}}) ⊗ p_{1}^{∗}(P_{P_{0}})^{−1}).

But if we evaluate at points x ∈ J(k) then the bundle on the left hand side and the bundle on the right are both isomorphic to L_{x}. For the bundle on the left this is the definition of a universal bundle, for the bundle on the right it is our formula above.

This says that the restriction of (m^{∗}(P_{P_{0}})⊗p_{1}^{∗}(P_{P_{0}})^{−1} to C×J by any j_{P_{0}.\underline{Q}} is a universal bundle on C × J = C × Pic^{0}(C).

We will show that the composition with j_{P_{0},\underline{Q}} : C → X does not depend on \underline{Q} and yields the identity on J(A).

On X × J we constructed the line bundle

m^{∗}(P_{P_{0}}) ⊗ p_{1}^{∗}(P_{P_{0}})^{−1} = P̃,

and I claim:

Theorem: The scheme X × J together with the line bundle P̃ represent the functor Pic^{00}(X), i.e. for any line bundle on

L | X × T, where T is connected and of finite type such that L | X × t_{0} ≃ O_{X} for some point t_{0} ∈ T. We have a unique morphism

ψ : T ⟶ J

such that L ∼ (Id ×ψ)^{∗}(P̃).

Proof: Our previous assertion says that for any j_{\underline{Q},P_{0}} : C → X the pullback of the line bundle P̃ to C × J is the universal bundle on C × J. Hence we can take the pullback of L on X × T via j_{P_{0},\underline{Q}} to C × T, and we see that we have a unique morphism

ψ : T ⟶ J

such that

(Id_{C} × ψ)^{∗}(P̃) ≃ (j_{P_{0},\underline{Q}} × Id_{T})^{∗}(L)

here we assume that T should be local. We have to show that already

(Id_{X} × ψ)^{∗}(P̃) ≃ L.

Let T = Spec(A) be the local ring at t_{0}. We have the two elements

(Id_{X} × ψ)^{∗}(P̃), L ∈ Pic^{00}(X)(A)

which are trivial in the special fibre and which become isomorphic if we restrict via j_{P_{0},\underline{Q}}. We have to show that they are actually equal. Let 𝔪 be the maximal ideal of A. It is clear that it suffices to prove that they are equal if we send them to Pic^{00}(X)(A/𝔪^{N}) for any N. For N = 1 both line bundles are trivial by assumption. We apply our infinitesimal considerations from ???. We put X_{N} = A ⊗ A/𝔪^{N}

We apply this sequence to the diagram

X ⟵ C^{g}/Σ_{g} ⟵ C^{g}

and put C^{g}/Σ_{g} = Y, C^{g} = Z. Then we get

\begin{aligned}0 & → & H^{1}(X, O_{X}) ⊗ 𝔪^{N}/𝔪^{N−1} & → & H^{1}(X_{N}, O_{X_{N}}^{∗})(0) & → & H^{1}(X × A_{N}, O_{X_{N−1}}^{∗})(0) → \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & H^{1}(Y, O_{Y}) ⊗ 𝔪^{N}/𝔪^{N−1} & → & H^{1}(Y_{N}, O_{Y_{N}}^{∗})(0) & → & H^{1}(Y × A_{N}, O_{Y_{N−1}}^{∗})(0) \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & H^{1}(Z, O_{Z})^{Σ_{g}}(0) ⊗ 𝔪^{N}/𝔪^{N−1} & → & H^{1}(Z_{N}, O_{Z_{N}}^{∗})^{Σ_{g}} & → & H^{1}(Z × A_{N}, O_{Z_{N−1}}^{∗})(0) \end{aligned}

Now we observe that the morphisms

H^{1}(X, O_{X}) ⟶ H^{1}(Y, O_{Y}) ⟶ H^{1}(Z, O_{Z})^{Σ_{g}}

are isomorphisms. To see this for the first morphism we recall that

Ψ : Y ⟶ X

is birational and the fibres are projective spaces. This implies that

Ψ_{∗}(O_{Y}) = O_{X},

and for any point x ∈ X we have

H^{1}(Ψ^{−1}(x), O_{Ψ^{−1}(x)}) = 0.

From this it follows that R^{1}Ψ_{∗}(O_{Y}) = 0 (???) and then the assertion follows from the spectral sequence.

The fact that the second map is an isomorphism follows from the Künneth formula. (???)

This implies that

H^{1}(X, O_{X}) ⟶^{∼} H^{1}(C, O_{C})

(see Chap. ???).

Now we have our diagram

J(A) ⟶ Pic^{00}(X)(A)
↘^{Id}  ↓
J(A)

and our infinitesimal considerations show that P ic^{00}(X)(A) and J(A) are of the same size. This implies that the vertical arrow must be an isomorphism.

2.4.4. Descend to the ground field.

We go back to 1.5.7. We have shown that for our given curve C/k the functor Pic^{r} is locally representable if we extend our ground field to the algebraic closure k̄. We want to discuss the question to what extend we can carry out such a construction over k.

The situation is easier for the functor P̃ic_{C/k,P_{0}}. This functor requires a k-rational point. We know that this functor is representable if we extend our groundfield to k̄ (See 1.5.7.)

But now we apply our considerations from ( V, ????), we have our functor defined over S = Spec(k) and it becones representable over S′ = Spec(k̄). We have a descend datum, which is effective since we know that the components of our representing scheme are projective ( See V, ??? and ???).

Actually we could also apply some less abstract arguments if we remember that we can also represent the functor already over a finite separable extension (See 1.5.7.) Then we can apply the arguments of Galois descend in Chap. ???.

Now we come to the functor Pic^{r}(C). We cannot apply the previous arguments since the functor is not representable. The situation is more delicate.

But I claim that we still have a canonical descend datum for the scheme Pic^{r}(C). In the following discussion I assume that we found a separable extension such that we have local representability by a scheme X′ and a line bundle P_{r} on (C × L) × X′.

To get the descend datum, we consider elements σ ∈ Gal(L/k) and form the scheme C × L × (X′)^{σ} and the line bundle P_{r}^{σ} on it. The universality property tells us that we can find a unique

f_{σ} : X′ ⟶ (X′)^{σ}

such that (Id ×f_{σ})^{∗}(P_{r}^{σ}) is locally in X′ isomorphic to P_{r}. Because these morphisms are always unique we can conclude that they have to satisfy the cocycle condition

f_{στ} = f_{τ}^{σ} ∘ f_{σ},

and therefore find that we can construct a projective scheme X/k together with an isomorphism

h : X ×_{k} L ⟶ X′

such that f_{σ} = h^{σ} ∘ h^{−1} (see ...). This scheme together with h is unique up to a canonical isomorphism.

But we want to do better, we want a line bundle P_{r} on C ×_{k} X such that (C × X, P_{r}) is representing our funcor locally. Here we have a problem because there is no way to get a descend datum which includes also the bundles. This is really not so good because now we cannot characterize X/k in terms of the functor.

We have such a bundle if r = g. In this case we have a bundle L on C × C^{g}/Σ_{g}, and we have shown that over the extension L we have the birational morphism

π′ : C^{g}/Σ_{g} ×_{k} L ⟶ X′

such that L_{L} is isomorphic to the bundle (Id ×π′)^{∗}(P′_{g}). We have the canonical descend of X′/L to a scheme X/k together with h : X ×_{k} L → X′. It is clear that we have π : C^{g}/Σ_{g} → X sucht that

π′ = h ∘ π.

Now we construct the bundle P_{g} on C × X in the same way as before (see ...). It is the unique bundle on C × X such that (Id ×π)^{∗}(P_{g}) ≃ L. I claim that this bundle has the universal property. If we extend k to L, then N will be isomorphic to P′_{g} on C_{L} × X′.

But now assume we have a scheme T/k of finite type and a bundle L on C ×_{k} T such that for any point t ∈ T the bundle L|C × {t} has degree g. We pass to the extension L/k, and we know that we have a unique

ψ′ : X′ ⟶ T ×_{k} L

such that (Id ×ψ)^{∗}(P′_{g}) is locally in T isomorphic to the extension of L to C × X × L ≃ C_{L} × X′ (via the maps h). It is clear that ψ′ is obtained from a ψ : X → T, i. e. ψ′ = h∘ψ ×_{k} L. Then we are in the situation that the bundles (Id ×ψ)^{∗}(P_{g}) and L became locally isomorphic in T after we extend the ground field from k to L. But then they must be already locally isomorphic in T on C × X.

This argument finishes the proof of the theorem ... which says that Pic^{g} is locally representable by a projective scheme.

Of course we would like to have the same assertion for all r. This is of course easy if our curve has a rational point. If not then this is not possible. To see this we simply consider a curve C/k of genus one, which does not have a rational point. Then we have the line bundle P_{1} which is given by the diagonal O_{C×C}(−Δ) on the product. But it is not difficult to see, that we can not have a line bundle Θ̃ = P_{0} on C × C.

2.5.The ℓ-adic modules T_{ℓ}(J) and the ring of endomorphisms End(J).

2.5.1 We start from a smooth projective curve C/k which has a point P ∈ C(k) and we consider its jacobian J. On the product C × J we have the universal bundle P_{0}. We restrict P_{0} to {P } × J and the resulting line bundle will now be called Θ it is the socalled Theta divisor.

For our Jacobian J we consider the endomorphism ring End(J). The understanding of the structure of this endomorphism ring is fundamental for the entire field of algebraic geometry.

I recomend to compare the following consideration to the analogous reasoning in the transcendental case. ( Chap. RS, 6.2.)

Any endomorphism φ : J → J induces an endomorphism

φ̃^{∗} : Pic(J)(k) ⟶ Pic(J)(k),

and we have to understand the function φ → φ̃^{∗}.

First of all it is clear that we can restrict φ̃^{∗} to the geometric points of the abelian variety Pic^{00}(J) = J^{∨}, this restriction is denoted by

φ^{∨} : J^{∨}(k) ⟶ J^{∨}(k).

We will show that - as in the transcendental case - this restriction is given by a endomorphism of J^{∨} = Pic^{00}(C) and it is linear, i.e. we will see

(φ + ψ)^{∨} = φ^{∨} + ψ^{∨}.

In analogy to our discussion in the complex analytic case we define

NS(J) = Pic(J)/Pic^{00}(J),

and φ̃^{∗} induces an endomorpism

φ^{∗} : NS(J) ⟶ NS(J).

In the case of Riemann surfaces the group End(J) could be considered as a subgroup of the endomorphisms of the first homology group (Chap. RS 6.2.1.) but these homology groups are ( not yet ) available, hence we have to look at this object as it is and get our insights from elsewhere.

We also should have the idea that NS(J) is related to the second cohomology, which in turn is related to the tensor product of the first cohomology groups (Chap RS , 6.3 ??). Having the idea that φ induces endomorphism in the cohomology in degree 1 and these should depend linearly on φ, we expect that φ → φ̃^{∗} is a quadratic function. This means that

˜ (φ + ψ)^{∗} − φ̃^{∗} − ψ̃^{∗} = ⟨φ, ψ⟩

where ⟨φ, ψ⟩ ∈ End(J) and depends bilinearily on φ, ψ.

2.5.2. We want to understand why φ^{∨} is a morphism. The key is the representability of Pic^{00}(J). We have the divisor Θ on J and the three morphisms

J × J ⟶^{p_{1}} ; ⟶^{m} ; ⟶^{p_{2}} J.

We have shown in the previous section that the bundle

Θ̃ = m^{∗}(Θ) ⊗ p_{1}^{∗}(Θ)^{−1} ⊗ p_{2}^{∗}(Θ)^{−1}

is universal:

The functor Pic^{00}(J) is locally represented by the line bundle Θ̃ on J × J Especially this implies that on the set of geometric points the homomorphism

φ_{Θ} : x ↦ T_{x}^{∗}(Θ) ⊗ Θ^{−1}

yields an isomorphism J(k) → Pic^{00}(J)(k).

If now φ : J → J is an endomorphism of J then φ^{∨} should be an endomorphism of Pic^{00}(J). Since this is given by J our φ^{∨} must be an endomorphism of J itself and now we explain how we get this endomorphism:

The endomorphism φ induces an endomorphism (φ × Id) : J × J → J × J and we can consider the pull back (φ×Id)^{∗}(Θ̃) on J ×J. This means that we can find an endomorphism tφ : J → J such that

(φ × Id)^{∗}(Θ̃)→^{∼}(Id ×^{t}φ)(Θ̃) ⊗ M

where M is the pullback of a line bundle on the second factor. But then we can evalute at e × J and find that M must be trivial. This gives us the defining formula for ^{t}φ

(φ × Id)^{∗}(Θ̃)→^{∼}(Id ×^{t}φ)(Θ̃)  (2.5.2.1)

On the geometric points this specializes to

φ^{∨}(T_{x}^{∗}(Θ) ⊗ Θ^{−1})→^{∼}T_{^{t}φ(x)}^{∗}(Θ) ⊗ Θ^{−1}

Theorem. 2.5.2.2 Let φ, ψ, η ∈ End(J) and let L be a line bundle on J. Then the bundle

\begin{aligned}(φ + ψ + η)^{∗}(L) & ⊗ (φ + ψ)^{∗}(L)^{−1} ⊗ (φ + η)^{∗}(L)^{−1} ⊗ (ψ + η)(L)^{−1}⊗ \\ & ⊗ φ^{∗}(L) ⊗ ψ^{∗}(L) ⊗ η^{∗}(L) ⊗ (O^{∗}L)^{−1} \end{aligned}

is trivial (here 0 : J → J is the zero homomorphism).

Proof. This is again a consequence of the theorem of the cube. We consider the threefold product J × J × J, and we consider the following 8 homomorphisms from J × J × J to J

m_{123} : J × J × J → J sum of all components
m_{23} ∘ p_{1}, m_{13} ∘ p_{2}, m_{12} ∘ p_{3}
p_{12}, p_{13}, p_{23}

and 0

We consider the bundle

\begin{matrix}N_{0} = & m_{123}^{∗}(L) ⊗ (m_{23} ∘ p_{1})^{∗}(L)^{−1} ⊗ (m_{12} ⊗ p_{3})^{∗}(L)^{−1} ⊗ (m_{13} ∘ p_{2})^{∗}(L)^{−1}⊗ \\ ⊗ p_{12}^{∗}(L) ⊗ p_{13}^{∗}(L) ⊗ p_{23}^{∗}(L) ⊗ (0^{∗}L)^{−1} \end{matrix}

If we restrict this bundle to one of the subvarieties

e × J × J , J × e × J , J × J × e ,

then in any case two of the 8 maps become equal and occur with opposite signs in the product. Hence the restriction becomes trivial.

Then the theorem of the cube tells us that the bundle is trivial.

To get our theorem we use φ, ψ, η to map

(φ, ψ, η) : J ⟶ J × J × J,

and the bundle in question is the pullback of N_{0}, hence trivial.

If our above bundle L is in Pic^{00}(J) = J^{∨}, then we need only two factors. In this case we know by definition that we can find a connected scheme of finite type and a line bundle L̃ on J × T such that

\begin{aligned}L̃ | J × \{t_{0}\} & ≃ & O_{J} \\ L̃ | J × \{t_{1}\} & ≃ & L \end{aligned}

for suitable points t_{0}, t_{1} ∈ T (k). Now we look at the four maps

J × J × T ⟶ J × T

given by summation in the first two summands, the two projections to the first two summands and finally the zero map. We consider

m_{12}^{∗}(L̃) ⊗ p_{1}^{∗}(L̃)^{−1} ⊗ p_{2}^{∗}(L̃)^{−1} ⊗ 0^{∗}(L).

Again we find that this bundle is trivial on

{e} × J × T , J × {e} × T and J × J × {t_{0}},

and again by the theorem of the cube we get that the bundle is trivial on J × J × {t_{1}}.

If we have a pair of endomorphisms φ, ψ

(φ, ψ) = (φ × ψ) ∘ Δ : J ⟶ J × J

then then composition with the multiplication gives φ + ψ and the composition with the projections gives φ and ψ. Hence

(φ + ψ)^{∗}(L) = φ^{∗}(L) ⊗ ψ^{∗}(L)

and this is the linearity

t(φ + ψ) = ^{t}φ +^{t}ψ.

For any pair of endomorphisms φ, ψ ∈ End(J) we can look at the difference

(φ + ψ)^{∗} − φ^{∗} − ψ^{∗} = ⟨φ, ψ⟩

where of course ⟨φ, ψ⟩ is again an endomorphism of Pic(J).

Perhaps this is a good place to summarize the properties of φ → φ^{∗}.

2.5.2.3. i) If we have two endomorphism φ, ψ then

(φ ∘ ψ)^{∗} = ψ^{∗} ∘ φ^{∗}.

ii) The pairing

End(J) × End(J) ⟶ End(Pic(J))
(φ, ψ)  ⟶  ⟨φ, ψ⟩

is bilinear in both variables.

iii) The endomorphism < φ, ψ > is trivial on Pic^{00}(J)

Proof: The assertion i) is obvious, iii) is the linearity of φ →^{t} φ it remains to prove ii). Let us replace φ by φ_{1} + φ_{2} and apply our formula above to the sum φ_{1} + φ_{2} + ψ. Then we see

\begin{matrix}(φ_{1} + φ_{2}+ & ψ)^{∗}(L) ⊗ (φ_{1} + φ_{2})^{∗}(L)^{−1} ⊗ ψ^{∗}(L)^{−1} = \\ (φ_{1} + ψ)^{∗}(L) ⊗ φ_{1}^{∗}(L)^{−1} ⊗ ψ(L)^{−1} ⊗ (φ_{2} + ψ)^{∗}(L) ⊗ φ_{2}^{∗}(L)^{−1} ⊗ ψ(L)^{−1} \end{matrix}

the term on the left hand side is

⟨φ_{1} + φ_{2}, ψ⟩(L),

and on the right hand side we get

⟨φ_{1}, ψ⟩(L) ⊗ ⟨φ_{2}, ψ⟩(L).

Since the expression ⟨φ, ψ⟩ is symmetric, the rest is clear.

The map φ →^{t} φ from End(J) into itself is called the Rosati involution . This involution depends on the choice of the line bundle Θ. In principle we could choose other ample bundles on J which then would define other involutions ( See [Mu], ???)

2.5.3. For an endomorphism φ : J → J we can define the kernel ker(φ) = φ^{−1}(0). By definition this is a subgroup scheme (See ???). If this kernel is a fintite group scheme then it is a finte group scheme over k and in this case we define the degree of φ as

deg(φ) = rank(φ^{−1}(0))

If the kernel is not finite then we put deg(φ) = 0.

An endomorphism φ : J → J with finite kernel is called an isogeny of J. The following is quite clear:

If φ : J → J is an isogeny then the morphism φ is finite and locally free of rank deg(φ). For any point y ∈ J the fibre φ^{−1}(y) is finte of rank deg(φ) over k(y).

We apply our formula for the degree: Let L be an ample bundle on J then

deg(φ)L^{g} = (φ^{∗}(L))^{g}

We want to compute (n Id)^{∗}. It is clear how to do this, we have

(n Id)^{∗} = ((n − 1) Id + Id)^{∗} = ((n − 1) Id)^{∗} + Id^{∗} + (n − 1)⟨Id, Id⟩

or since Id^{∗} = Id_{Pic(J)} = Id

(n Id)^{∗} − ((n − 1) Id)^{∗} = Id +(n − 1)⟨Id, Id⟩.

Since (O · Id)^{∗} = 0 we get

(n Id)^{∗} = n Id +\frac{n(n − 1)}{2}⟨Id, Id⟩.

We can also say something about ⟨Id, Id⟩. We have

Id +(− Id) = 0

as endomorphism on J. Then

0 = (Id +(− Id))^{∗} = Id +(− Id)^{∗} − ⟨Id, Id⟩,

and hence we see

⟨Id, Id⟩ = Id^{∗} +(− Id)^{∗}.

We want to evaluate this formula on a line bundle L. We put L_{−} = (− Id)^{∗}(L)and our formula yields

(n Id)^{∗}(L)→^{∼}L^{⊗n} ⊗ L^{⊗ \frac{n(n−1}{2}} ⊗ L_{−}^{⊗ \frac{n(n−1}{2}}

This allows us to compute the degree of n Id. As we explained. we may take any ample line bundle L on J, we may even assume it to be very ample. We may replace L by L ⊗ (− Id)^{∗}(L), since − Id is an automorphism, it is clear that (− Id)^{∗}(L) is very ample and hence the tensor product is so too.

Then we have the formula

L^{g} deg(n Id) = (n Id)^{∗}(L)^{g}

and (n Id)^{∗}(L) = L^{n} ⊗ L^{\frac{n^{2}−n}{2}} ⊗ L^{\frac{n^{2}−n}{2}} (because (− Id)^{∗}(L) = L) and hence

(n Id)^{∗}(L) = L^{⊗n^{2}}.

And now (L^{⊗n^{2}})^{g} = L^{g} · n^{2g} and it follows

deg(n Id) = n^{2g}.

Since we have seen that the group low J × J → J induces the addition on the tangent space T_{J,e} (see ???) we conclude that the multiplication by n on the tangent space. We conclude:

The kernel of the multiplication by n is a finite group scheme

J[n] ⟶ Spec(k)

of rank n^{2h}. If the characteristic p of k does not divide n, then this group scheme is etale. In this case

J[n](\overline{k}) ≃ (ℤ/nℤ)^{2g}.

We consider the function

n ⟶ deg(ψ + n Id),

we know how to express it in terms of intersection number. We choose an ample line bundle L on J and then we have

deg(ψ + n Id) · L^{g} = ((ψ + n Id)^{∗}(L))^{g}.

We can expand the right hand side and find

deg(ψ + n Id) · L^{g} = ⋯ n^{2g−1}L^{g−1} · ⟨ψ, Id⟩(L) + n^{2g} · L^{g}.

This expression looks pretty much like a characreristic polynomial of an endomorphism and in the next section we will see that this is indeed the case. In view of this expectation we define

tr(ψ) = \frac{L^{g−1}⟨ψ, Id⟩(L)}{L^{g}}.

In any case it is clear that deg(ψ + n Id) is a polynomial in n of degree 2g with rational coefficients and this polynomial takes integer values.

If we have two endomorphisms φ, ψ then we can look at the diagram

J ⟶^{φ×ψ∘Δ} J × J ⟶^{p_{1}} ; ⟶^{m} ; ⟶^{p_{2}} J

and for any line bundle L on J we have the formula

< ψ, φ > (L)→^{∼}m^{∗}(L) ⊗ p_{1}^{∗}(L)^{−1} ⊗ p_{2}^{∗}(L)^{−1}

by definition. If we apply this to Θ we get

< ψ, φ > (Θ)→^{∼}((φ × ψ) ∘ Δ))^{∗}(Θ̃) = Δ^{∗} ∘ (φ × ψ)^{∗}(Θ̃)

We have the defining relation for the transpose ( ???) and get

< ψ, φ > (Θ)→^{∼}Δ^{∗}∘(^{t}ψφ×Id)^{∗}(Θ̃)→^{∼}Δ^{∗}∘(Id ×^{t}φψ)^{∗}(Θ̃)→^{∼} <^{t} ψφ, Id > (Θ)→^{∼} <^{t} φψ, Id > (Θ)

If we take ψ = Id then we get

tr(^{t}φ) = tr(φ).

Now we are ready for the famous Theorem : Positivity of the Rosati involution The bilinear form

(φ, ψ) ↦ tr(^{t}φψ)

is a symmetric positive definite form

Proof: We have to prove the positivity. From the formula just above the statement of the theorem we get

< ψ, ψ > (Θ)→^{∼} <^{t} φφ, Id > (Θ).

We multiply by Θ^{g−1} and observing the definition of the trace we get

Θ^{g−1} < ψ, ψ > (Θ) = Θ^{g−1} <^{t} φφ, Id > (Θ) = tr(^{t}φφ)Θ^{g}

But obviously < ψ, ψ > (Θ)→^{∼}ψ^{∗}(Θ)^{⊗2} and hence we find

Θ^{g} tr(^{t}ψψ) = 2Θ^{g−1}ψ^{∗}(Θ)

Since Θ is ample this expression is strictly positive if ψ ≠ 0.( na ja).

The ℓ-adic modules

Now we can pick a prime ℓ which is different from the characteristic of k, and we define

T_{ℓ}(J) = lim_{\overrightarrow{α}} J[ℓ^{α}]

as before.

The group of geometric points is

T_{ℓ}(J)(\overline{k}) = ℤ_{ℓ}^{2g},

but we have to observe that the Galois group acts upon this module.

This is now the replacement for the cohomology groups which were available in the transcendental case. But these ℓ-adic cohomology groups have the defect that they are ℤ_{ℓ}-modules and not ℤ-modules.

(It has been pointed out by J.-P. Serre that cohomology groups which are free ℤ-modules of rank 2g cannot exist.)

Of course it is clear that we get a homomorphism End(J) ⟶ End(T_{ℓ}(J)), and hence we can define a trace

tr(φ) = tr(φ_{ℓ} | T_{ℓ}(J))

and a determinant

det(φ) = det(φ_{ℓ} | T_{ℓ}).

The problem is that these numbers are ℓ-adic numbers and what we want are integers. But

Theorem: The number tr(φ_{ℓ} | T_{ℓ}) and det(φ_{ℓ} | T_{ℓ}) are integers which do not depend on ℓ. We have

det(φ_{ℓ} | T_{ℓ}) = deg(φ).

Before we can prove this theorem we have to prove the following fundamental theorem:

Theorem: The ℤ-module End(J) is finitely generated and for any prime ℓ the natural map

End(J) ⊗ ℤ_{ℓ} ⟶ End(T_{ℓ}(J))

is an inclusion.

Proof: Let M ⊂ End(J) be any finitely generated submodule which is stable under the involution. The trace defines an integer valued pairing

\begin{aligned}tr & : & M × M ⟶ ℤ \\ tr & : & ⟨φ_{1}, φ_{2}⟩ ⟶ tr(φ_{1} ^{t}φ_{2}) \end{aligned}

which is positive definite. We see that this bilinear pairing is non degenerate over ℚ, i.e. if we take a basis φ_{1} ⋯ φ_{k} of the ℤ-module M we can conclude that

det(tr(φ_{i} ^{t}φ_{j}))

is a non zero integer.

Let us assume that M ⊗ℤ_{ℓ} does not embed into End(T_{ℓ}(J)), this means that we can find α_{1}, ⋯, α_{k} ∈ ℤ_{ℓ} which are not all congruent zero mod ℓ such that the linear combination ∑_{i=1}^{k} α_{i}φ_{i} is zero. If we approximate the α_{i} by integers n_{i} such that α_{i} ≡ n_{i} mod ℓ^{α}, then the element

ψ = ∑ n_{i}φ_{i} ∈ End(J)

is zero on the group of ℓ^{α} division points. this implies that ψ = ℓ^{α}ψ′ where ψ′ ∈ End(J).

(We have the diagram

\begin{aligned}J & ⟶^{ℓ^{α}} & J \\ & ψ ↘ \\ & & J \end{aligned}

and since ψ is zero on the kernel of ℓ^{α}, we can see easily that we can find a ψ′ completing the diagram.)

We get a system of linear equations for the n_{i}:

∑ n_{i} tr(φ_{i} ^{t}φ_{j}) = ℓ^{α} · tr(ψ′ ^{t}φ_{j})

where the tr(ψ′ ^{t}φ_{j}) are integers. We solve this system for the n_{i} using Cramers rule and find

n_{i} = ℓ^{α} \frac{A_{i}}{det(tr(φ_{i} ^{t}φ_{j}))}

where the A_{i} are integers. The maximal power of ℓ dividing the denominator does not depend on α. Hence we get a contradiction to the assumption that not all of the n_{i} are divisible by ℓ. For this we conclude that End(J) is a finitely generated ℤ-module.

The homomorphism End(J) ⊗ ℤ_{ℓ} → End(T_{ℓ}(J)) is called the ℓ-adic representation.

Of course it follows that End(J)_{ℚ} is a finite dimensional ℚ-algebra. The positivity of the trace implies

The algebra End(J)_{ℚ} is semisimple.

To see this, we consider the radical 𝔞 which is the maximal two-sided ideal consisting of nilpotent elements. Clearly, 𝔞 is stable under the Rosati involution, and hence we see: If 𝔞 ≠ 0, then we find non zero φ ∈ 𝔞 with ^{t}φ = ±φ. But then φ^{2} ≠ 0 since tr(φ^{2}) = ± tr(φ ^{t}φ), and this is non zero by the positivity of the Rosati involution. But then φ cannot be nilpotent, we have a contradiction to 𝔞 ≠ 0.

The structure theory of algebras implies that we can find a set of central elements e_{1} ⋯ e_{r} ∈ End(J)_{ℚ}, such that

\begin{aligned}1 & = & ∑ e_{i} \\ e_{i}e_{j} & = & \{ \begin{matrix}e_{i} & if i = j \\ 0 & else \end{matrix} \end{aligned}

and such that

End(J)_{ℚ} = ⨁ End(J)_{ℚ}e_{i} = ⨁_{i=1}^{r} A_{i}

where the A_{i} are fields.

The e_{i} are not endomorphisms of J but, if we multiply them by an integer d_{i} ≠ 0, then

E_{i} = d_{i} e_{i}

will be an endomorphism of J. Now we can consider the endomorphisms

Φ_{i} = ∑_{j≠i} E_{j}

and let J̃_{i} be the connected component of the kernel of Φ_{i}. Then we have of course that J̃_{i} is an abelian subvariety of J and it is clear that the map

∏ J̃_{i} ⟶ J

is surjective with finite kernel, in other words, this map is an isogeny. The algebra A_{i,ℤ} = A_{i} ∩ End(J) acts trivially on the J̃_{j} with j ≠ i and it injects into End(J̃_{i}) and is clearly of finite index in this ring of endomorphisms. It is clear that the Rosati involution is the identity on the e_{i} and hence it induces involutions on the A_{i} and on End(J̃_{i}).

Theorem: For any φ ∈ End(J) the numbers tr(φ_{ℓ}(T_{ℓ}(J)) and det(φ_{ℓ}) are independent of ℓ and we have

det(φ_{ℓ}) = deg(φ)
tr(φ_{ℓ}) = tr(φ).

Proof: We have the decomposition

End(J)_{ℚ} = ⨁ A_{i} = ⨁ End(J)_{ℚ}e_{i},

and it is clear that End(J̃_{i})_{ℚ} = A_{i}. In other words, for any φ ∈ End(J)_{ℚ} we find a non zero integer m such that

mφ = φ_{1} + ⋯ + φ_{r}

where φ_{i} ∈ End(J̃_{i}). This means that we have a diagram

∏ J̃_{i}  ⟶ J
↓ \underline{Ψ}  ↓ mφ
∏ J̃_{i}  ⟶ J

where \underline{Ψ} = (⋯ Ψ_{i} ⋯) and Ψ_{i} ∈ End(J̃_{i}). This implies that deg(mφ) = ∏ deg(Ψ_{i}). If we choose ℚ-independent generators φ_{ij} of the End(J̃_{i}) and consider

φ = ∑_{i,j} m_{ij} φ_{ij}

where ∑_{j} m_{ij}φ_{ij} = Ψ_{i} ∈ End(J̃_{i}), then we know that the function

φ ⟶ deg(φ) = deg(∑ m_{ij}φ_{ij})

is a homogeneous polynomial of degree 2g in the m_{ij}.

It is also clear (???) that the individual factor deg(∑_{j} m_{ij}φ_{ij}) are homogeneous polynomials in degree 2 dim J̃_{i}. This reduces our problem to the consideration of one of these factors.

We want to show that we have the formula

deg(φ) = det(φ_{ℓ} | T_{ℓ}(J)).

To prove this formula we look at integral linear combinations of endomorphisms and use the observation that

deg(n_{1}φ_{1} + ⋯ + n_{r}φ_{r})

and

det(n_{1}φ_{1,ℓ} + ⋯ n_{r}φ_{r,ℓ} | T_{ℓ}(J))

are both homogeneous polynomials of degree 2g in the variable n_{1} ⋯ n_{r}.

On the other hand we have just seen that our algebra

End(J) ⊗ ℚ = ⨁_{i=1}^{t} M_{n}(D_{i})

where the D_{i} are division algebras over their centre F_{i}/ℚ.

It is well known that we have homomorphisms

N_{i} : M_{n}(D_{i})^{∗} ⟶ ℚ^{∗}

which are compositions of the reduced norm

Norm red : M_{n}(D_{i})^{∗} ⟶ F_{i}^{∗}

and the norm

N_{F_{i}/ℚ} : F_{i}^{∗} ⟶ ℚ^{∗}

which have the property that

N_{i}(n_{1}α_{1} + ⋯ n_{r}α_{r})

is homogeneous of degree nd_{i} · [F_{i} : ℚ] = f_{i} where d_{i} = \sqrt{[D_{i} : F_{i}]}. Moreover we know that any homomorphism

N′_{i} : M_{n}(D_{i})^{∗} ⟶ ℚ^{∗}

which has the homogeneity property with some degree m_{i} is of the form

N′_{i} = N_{i}^{m_{i}/f_{i}}

i.e. we must have f_{i} | m_{i}.

Hence we can conclude that

deg : (End(J) ⊗ ℚ)^{∗} ⟶ ℚ^{∗}

is of the form

det = ∏_{i=1}^{t} N_{i}^{g_{i}}

where ∑ f_{i}g_{i} = 2g.

We apply the same consideration to

det : (End(J) ⊗ ℚ_{ℓ})^{∗} ⟶ ℚ_{ℓ}^{∗}.

We have to observe that now it can happen that some of our field F_{i} may decompose

F_{i} ⊗ ℚ_{ℓ} = ⨁_{μ=1}^{a_{i}} D_{iμ}

and

M_{n}(D_{i}) ⊗ ℚ_{ℓ} = ⨁_{μ=1}^{a_{i}} M_{n_{iμ}}(D_{iμ})

and we conclude

det = ∏_{i=1}^{t} N_{iμ}^{g_{iμ}}

where now

N_{iμ} : M_{n_{iμ}}(D_{iμ})^{∗} ⟶ ℚ_{ℓ}^{∗}.

We need to show that the g_{iμ} do not depend on μ, i.e. g_{iμ} = g′_{i} for all μ = 1 ⋯ a_{i} and that g′_{i} = g_{i} for all i.

This is now easy (na ja!) if we use an approximation argument and the fact that

| deg φ|_{ℓ} = | det(φ_{ℓ})|_{ℓ}

which is obvious from the definition of the degree.

The Weil Pairing

We consider a Jacobian J/k where we assume that k is algebraically closed. For any integer n > 0 which is not divisible by the characteristic p of our ground field we consider the kernel

J[n] = ker(J ⟶^{n} J).

We have seen that n = n Id is an étale morphism, the kernel is a finite étale group scheme and

J[n] ≃ (ℤ/nℤ)^{2g}.

We can also say that J[n] is the Galois group of the covering J →^{n} J, this means that any a ∈ J[n] = J[n](k) induces a translation

J
↘ n
↓ T_{a}  J
↗ n
J

and for any affine open subset V ⊂ J, the open set (n Id)^{−1}(V) = V′ ⊂ J is affine, and

O_{J}(V) = O_{J}(V′)^{J[n]},

i.e. the algebra downstairs is the algebra of invariants under J[n].

Our aim is the construction of a non degenerate alternating pairing

w_{0} : J[n] × J[n] → μ_{n} = group of n’th roots of unity.

(Compare the discussion in the transcendental context ???)

If we pick a point ξ ∈ J[n], then this gives us a line bundle

L_{ξ} = N | J × ξ.

This line bundle is algebraically equivalent to zero and satisfies L_{ξ}^{⊗n} = O_{J}.

We take the pull back of this line bundle under

n : J ⟶ J

and clearly we have n^{∗}(L_{ξ}) = L_{ξ}^{⊗n} = O.

Hence we see that n^{∗}(L_{ξ}) is trivial and we conclude that H^{0}(J, n^{∗}(L_{ξ})) is a one dimensional k-vector space.

Since n^{∗}(L_{ξ}) is a pull back of a bundle on J under n, we see that we have an action of J[n] (the Galois group) on n^{∗}(L_{ξ}) and hence an action of J[n] on H^{0}(J, n^{∗}(L_{ξ})) which then defines a character

α_{ξ} : J[n] ⟶ μ_{n} ⊂ k^{∗}.

One thing is clear: This character is trivial if and only if L_{ξ} is trivial in other words if ξ = 0.

We put

w_{0}(ξ, η) = α_{ξ}(η)

for (ξ, η) ∈ J[n] × J[n]. It is linear in η by definition. But if ξ = ξ_{1} + ξ_{2}, then we have

L_{ξ} = L_{ξ_{1}} ⊗ L_{ξ_{2}}

and this provides a non zero bilinear map

H^{0}(J, n^{∗}(L_{ξ_{1}})) × H^{0}(J, n^{∗}(L_{ξ_{2}})) → H^{0}(J, n^{∗}(L_{ξ}))

which commutes with the action of J[n]. Hence we see that

α_{ξ_{1}+ξ_{2}} = α_{ξ_{1}} · α_{ξ_{2}}

and this shows that w_{0} is bilinear.

The most difficult step is to show that w_{0}(ξ, ξ) = α_{ξ}(ξ) = 1. Once we have shown this, then it is clear that the map is alternating and non degenerate.

This last property follows from the symmetry of N. Since we know that the adjoint of n = n Id is again n we have an isomorphism

Φ : (n × Id)^{∗}(N) ⟶ (Id ×n)^{∗}(N).

We can restrict Φ to the diagonal, then it yields

Φ_{x,x} : N_{(nx,x)} ⟶ N_{(x,nx)}
∥  ∥
(n × Id)^{∗}(N)_{(x,x)} ⟶ (Id xn)^{∗}(N)_{(x,x)}

and we can compose this with

Δ_{(x,nx)} : N_{(x,nx)} ⟶ N_{(nx,x)}

to get

(Δ ∘ Φ)_{(x,x)} : N_{(nx,x)} ⟶ N_{(nx,x)},

this morphism is a number and because of our normalization this number is equal to one.

We normalize Φ such that

Φ : (n × Id)^{∗}(N)_{(e,e)} = N_{e,e} ⟶ (Id ×n)^{∗}(N)_{e,e} = N_{e,e}

is the identity.

Now we consider two n-division points (ξ, η) and consider the isomorphism

Φ_{ξ,η} : (n × Id)^{∗}(N)_{(ξ,η)} ⟶ (Id ×n)^{∗}(N_{(ξ,η)}
∥  ∥
N_{(e,η)} ⟶ N_{(ξ,e)}.

Since N | e × J and N | J × e are both trivial, we have

\begin{aligned}H^{0}(J, N | e × J) & ≃ & k \\ H^{0}(J, N | J × e) & ≃ & k \end{aligned}

and this gives us an identification

\begin{aligned}N_{(e,y)} & = & N_{(e,e)} \\ N_{(x,e)} & = & N_{(e,e)} \end{aligned}

and hence we get

N_{(e,η)} ⟶ N_{(ξ,e)}
∥  ∥
N_{(e,e)} = N_{(e,e)}.

The isomorphism in the top row gives us a number (the diagram is not necessarily commutative) and a little bit of thinking yields that this number is w_{0}(ξ, η) or it’s inverse (???)

The restriction to the diagonal gives the desired result.

I explained already that we always can define the subgroup Pic^{00}(X), and we can define the Neron-Severi groups as

NS(X) = Pic(X)/Pic^{00}(X).

If we consider X = J where J is a Jacobian together with the bundle P, then we constructed the homomorphism

φ_{P} : J ⟶ J^{∨} = Pic^{00}(J)

which on the points is given by

x ⟶ T_{x}(P) ⊗ P^{−1},

and we have shown that this is an isomorphism.

In fact we proved that for any line bundle L we have

φ_{L} : J ⟶ J^{∨} ⟶^{φ_{P}^{−1}} J,

and hence we get a homomorphism

Φ : NS(J) ⟶ End(J).

In the transcendental context we proved that Φ provides an isomorphism between NS(J) and the submodule of symmetric elements.

We want to investigate this map in the context of algebraic geometry, and we will show that:

The map is injective (modulo 2-torsion which actually does not exist), and the image lies in End_{sym}(J) and between 2 End_{sym}(J) and End_{sym}(J).

To get this we also consider the Neron-Severi group NS(J ×J). As in characteristic zero we have the pull backs

NS(J) \overrightarrow{→} NS(J × J)

given by the two projections and we have the two homomorphisms

NS(J × J) \overrightarrow{→} NS(J)

given by restriction to {e} × J and J × {e}.

This allows us to write

NS(J × J) = N(J) ⊕ NS′(J × J) ⊕ NS(J).

Let L be a line bundle on J × J whose Chern class (the image of L in the Neron-Severi group) lies in the summand NS′(J × J), then we know that

L | J × e and L | e × J

lie in Pic^{00}(J). After tensorization by bundles of the form p_{i}^{∗}(M_{i}) we may assume that these restrictions are even trivial.

Now we have seen that we can find a homomorphism

ψ : J ⟶ J

(Hier muß vielleicht früher was gesagt werden.) such that

L ≃ (Id ×ψ)^{∗}(N).

The map ψ ↦ (Id ×ψ)^{∗}(N) gives us a homomorphism (???)

End(J) ⟶ NS′(J × J)

which is surjective (as we just saw).

It is also injective: If (Id ×ψ)^{∗}(N) is zero in NS′(J × J), then this means that

(Id ×ψ)^{∗}(N) ≃ p_{1}^{∗}(M_{1}) ⊗ p_{2}^{∗}(M_{2}).

Evaluation at e × J and J × e shows that M_{1}, M_{2} must be trivial, hence

(Id ×ψ)^{∗}(N) ≃ O_{J×J}.

But is ψ ≠ 0, then

(Id ×ψ)^{∗}(N) | J × y = N | J × ψ(y)

which is not trivial.

Hence we proved

NS′(J × J) ⟶^{∼} End(J)

a result which in the transcendental context was reduced to linear algebra in Λ^{•}Γ.

Now we consider the diagonal embedding

Δ : J ⟶ J × J,

and we put

L_{ψ} = Δ^{∗} ∘ (Id ×ψ)^{∗}(N),

this yields a homomorphism

End(J) ⟶ NS′(J × J) ⟶ NS(J).

We have the following formula

Φ(L_{ψ}) = ψ +^{t}ψ

(??? wird später eingefügt). If on the other hand L is a bundle on J, then we may consider

J × J ⟶^{p_{1}} ; ⟶^{m} ; p_{2}^{⟶} J

and look at

L̃ = m^{∗}(L) ⊗ p_{1}^{∗}(L)^{−1} ⊗ p_{2}^{∗}(L)^{−1}.

Again it is clear that L̃ | J × e ≃ O_{J} and L̂ | e × J ≃ O_{J}, and hence we find a ψ : J → J such that

L̃ = (Id ×ψ)^{∗}(N).

Restriction to the diagonal yields

Δ^{∗}(L̃) = L^{⊗2} = L_{ψ}.

The shows that the homomorphism NS′(J × J) ≃ End(J) ⟶ NS(J) is surjective up to 2-torsion which means that the image contains 2NS(J).

Then it becomes clear that

NS(J) ⟶ End_{sym}(J) ⊂ End(J)

because End(J) is torsion free and 2NS(J) is mapped into the symmetric tensors. It is also clear that the kernel of this map is 2-torsion and hence it is injective because: The group NS(J) is torsion free. We only have to exclude 2-torsion. Let us consider a line bundle L on J for which the 2c(L) = 0, i.e.

L^{⊗2} = M ∈ Pic^{00}(J).

We can write M = M_{1}^{⊗2} with M_{1} ∈ Pic^{00}(J) (???) and hence we get

(L ⊗ M_{1}^{−1})^{⊗2} = O_{J}

in other words we may assume that c(L) ≠ 0 but L^{⊗2} = O_{J}.

Etale Cohomology

In the previous sections I gave some indications that the modules T_{ℓ}(J) should be considered a substitute for the first homology. If our ground field is ℂ, and if

J(ℂ) = ℂ^{g}/Γ,

then Γ = H_{1}(J(ℂ), ℤ) and

Γ ⊗ ℤ_{ℓ} ≃ T_{ℓ}(J).

If we are over an arbitrary ground field k, and if we pick a prime ℓ which is different from char(k) = p, then we have seen that

J[ℓ^{α}](\overline{k}) ≃ (ℤ/ℓ^{α}ℤ)^{2g},

and since the map ℓ^{α} : J → J is separable, we even see that all the ℓ^{α}-division points are actually defined over the separable closure k_{s} of k.

Hence we get an action of the Galois group Gal(k_{s}/k) = Gal(\overline{k}/k) on J[ℓ^{α}](\overline{k}) and passing to the limit we get a continous homomorphism

ρ : Gal(\overline{k}/k) → GL(T_{ℓ}(J)).

This means that T_{ℓ}(J) provides a much richer structure than just a free ℤ_{ℓ}-module.

The cyclotomic character.

There is a much simpler construction of such Galois modules. We consider the multiplicative group scheme G_{m}. This is by definition

G_{m} = Spec(k[T, T^{−1}]),

and we have a multiplication

m : G_{m} × G_{m} ⟶ G_{m}

which on the level of k-algebras is induced by the map

\begin{matrix}k[T, T^{−1}] & ⟶ & k[T, T^{−1}] ⊗ k[T, T^{−1}] \\ T & ⟶ & T ⊗ T. \end{matrix}

This induces a group structure on the rational points in any k-algebra B: We have G_{m}(B) = B^{∗}, and the group structure is multiplication

\begin{matrix}B^{∗} × B^{∗} & ⟶ & B^{∗} \\ (b_{1}, b_{2}) & ⟶ & b_{1}b_{2}. \end{matrix}

Again we can pick a prime ℓ ≠ p = char(k), and we know that

G_{m}[ℓ^{α}](\overline{k}) = \{ζ ∈ \overline{k}^{∗} | ζ^{ℓ^{α}} = 1\} ≃ ℤ/ℓ^{α}ℤ.

This gives us a representation of the Galoisgroup

α : Gal(\overline{k}/k) ⟶ GL(T_{ℓ}(G_{m})),

and since T_{ℓ}(G_{m}) = lim_{←} G_{m}[ℓ^{α}] ≃ ℤ_{ℓ}, we see that α yields a character

α : Gal(\overline{k}/k) ⟶ ℤ_{ℓ}^{∗}.

This character is called the Tate character or the cyclotinic character. We denote by ℤ_{ℓ}(1) the module ℤ_{ℓ} on which Gal(\overline{k}/k) acts by the character α.

A. Grothendieck introduced the étale cohomology group for schemes. We want to give some very general ideas how this can be done.

The decisive step is to extend the notion of a topological space or the notion of a topology on a space.

Let us fix a field k and let us consider schemes X/ Spec(k) of finite type. We define some new “topology” on X by saying that the open sets are morphisms

Ũ ⟶ U ⊂ X

where U ⊂ X is Zariski open and Ũ → U is an étale covering. These new open sets form a category in an obvious sense. A morphism is a diagram

\begin{matrix}Ũ_{1} & ⟶ & Ũ_{2} \\ ↓ & & ↓ \\ U_{1} & ⟶ & U_{2} \end{matrix}

where U_{1} → U_{2} is an inclusion and the diagram is commutative (the vertical arrows being étale coverings).

This category has fibered products. For an ordinary topology this fibered product of two objects (open subsets in the space) is given by the intersection of the two open suebsets. In the general context this becomes more complicated. Another difference is this: For an ordinary topological space the set of morphisms between two subjects is either empty or consists of jest one element. This is no longer true.

The étale topology is already extremely interesting for X = Spec(k). There are not so many Zariski open subsets, but an étale open set is simply a finite separable extension Spec(k′) = X′ → X. The fibered product is

X′ ×_{X} X′′ = Spec(k′ ⊗_{k} k′′)

which then suddenly may consist of several points. If for instance k′/k is a finite normal extension, then

k′ ⊗_{k} k′ ≃ ⨁_{σ: \begin{aligned}k′ & → & k′ \\ & ↖ ↗ \\ & k \end{aligned}} k′.

We define Et_{X} to be the category of these morphisms

Ũ ⟶ U ⊂ X.

If we have a morphism X′ →^{f} X, and if U is open in X, then f^{−1}(U) is open in X′ and if Ũ → U is an étale covering, then Ũ′ = U′ ×_{U} Ũ is an ŕmetale covering of U′.

For an object Ũ → U is Et_{X} we can introduce the notion of a covering. This is a family of

Ũ_{i} ⟶ U_{i} ⊂ Ũ i ∈ I

such that ∪ U_{i} = Ũ.

Now we can define sheaves. This are contravariant functors

F : Et_{X} ⟶ C

where C is any reasonable category (for instance sets, rings, abelian groups and so on) which satisfy the two sheaf conditions. For any covering {Ũ_{i}} of U ∈ Et_{X} we have

F(U) ⟶ ∏_{i} F(Ũ_{i}) ⟶ ∏_{(i,j)} F(U_{i} ×_{U} U_{j})

is an exact sequence.

This notion is already non trivial for the case X = Spec(k). What does it mean that M is a sheaf on X, say with values in the category of abelian groups.

This means that we have a functor M which attaches to any finite separable extension k′/k, an abelian group M(k′) and for any morphism of k-algebras

k′  ⟶^{σ} k′′
↖ ↗
k

we have a group homomorphism

σ′ : M(k′) ⟶ M(k′′).

If our separable extension k′/k is not a field, then k′ = ⊕k′_{i} where these k_{i} are fields and M(k′) = ⊕M(k′_{i}).

If our separable extension k′/k is not a field, then k′ = ⊕k′_{i} where these k_{i} are fields and M(k′ = ⊕M(k′_{i}).

We have to formulate what the sheaf conditions mean. A covering of Spec(k′) → Spec(k) is a k-algebra homomorphism

\begin{aligned}k′ & ⟶ & L \\ & ↖ ↗ \\ & k \end{aligned}

i.e. Spec(L) → Spec(k′ which sends the identity of k′ to the identity of L (Note that k′ is not necessarily a field.).

Now it is clear that it suffices to know the value of M on finite separable field extensions L/k.

If L_{1}/L is a finite separable and normal extension, then it is clear that the Galois group Gal(L_{1}/L) acts upon M(L_{1}). The sheaf condition then means: For any L_{1} > L > k as above we have

M(L) ↪ M(L_{1}),

and M(L) is exactly the module of fixed points under the action of Gal(L_{1}/L) on M(L_{1}).

We can reformulate this slightly. If we choose a separable closure k_{s}/k, then we can restrict the functor M to finite extensions

k ⊂ L ⊂ k_{s}.

If now k ⊂ L_{1} ⊂ L_{2} ⊂ k_{s}, then we have M(k) ⊂ M(L_{1}) ⊂ M(L_{2}) and we can form the limit

M = lim_{\overrightarrow{K/k}} M(L) = ∪ M(L).

This is now a continous module for the Galois group. The group Gal(k_{s}/k) = lim_{←} Gal(L/k) acts on M and for any m ∈ M the stabilizer of m is an open subgroup in the Galois group. If in turn M is a module on which we have a continous action of Gal(k_{s}/k) (i.e. for any m ∈ M we have a finite extension k ⊂ L ⊂ k_{s} such that Gal(k_{s}/L)m = m, then we can define M(L) = M^{Gal(k_{s}/L)} and L → M(L) defines an étale sheaf on Spec(k).)

We return to the more general situation. We have our X/ Spec(k). If we consider sheaves with values in AB (or modules over some ring R), then we have the notion of an exact sequence of sheaves on Et_{X}: This is a sequence

0 → F′ → F → F′′ → 0,

such that for any Ũ the sequence

0 → F′(U) → F(U) → F′′(U)

is exact and such that for any s′′ ∈ F′′(U) we can find a covering

{Ũ_{α}}_{α}  Ũ_{α} → U_{α} ⊂ U

such that for any α the restriction s′′_{α} of s_{α} to Ũ_{α} is in the image of F(U_{α}) → F′′(U_{α}).

Now I claim that we can define the derived functor to the functor

F ⟶ F(X) = H^{0}(X, F),

and we call the derived group Hét^{i}(X, F). A short exact sequence in cohomology

\begin{matrix}0 → & H^{0}/(X, F′) → H^{0}(X, F) → H^{0}(X, F′′) →^{δ} \\ Hét^{1}(X, F′) → Hét^{1}(X, F) → Hét^{1}(X, F′′) →^{δ} \end{matrix}

(The only thing we need to know is the existence of enough injective objects ( see ???).

If for instance X = Spec(k), then we have see that an étale sheaf with values in (Ab) is simply a Gal(k_{s}/k)-module M with continous action.

An exact sequence of such sheaves is nothing else than an exact sequence of such modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0.

But if we look at sections over X = Spec(k), then we get

0 ⟶ M′(k) ⟶ M(k) ⟶ M′′(k),

and the last arrow needs not to be surjective. For m′′ ∈ M(k) we can find an extension L/k (normal over k) such that m′′ lifts to an element m in M(L) and in the section (???) we have seen that the obstruction to lift m′′ to an element in M(k) lies in

H^{1}(Gal(L/k), M′(L)).

Hence it is more or less clear that

Hét^{1}(X, M′) = lim_{⟶} H^{1}(Gal(L/k), M(L))
= H^{1}(Gal(k_{s}/k), M).

In this case we are back in the situation of ordinary group cohomology. Then is a slight difference to the situation in (HA....), since our group is the projective limit of finite groups. This requires some harmless continuity considerations. The groups Hét^{i}(X, M) = H^{i}(Gal(k_{s}/k), M) are called the Galois cohomology groups.

The geometric étale cohomology groups.

Let us consider a scheme of finite type X/k. We choose a separable closure k_{s} and an algebraic closure \overline{k} > k_{s}.

We form the scheme \overline{X} = X ×_{Spec(k)} Spec(\overline{k}).

Let us assume that we have a sheaf F over X with values in (Ab) or modules over a ring R. Then it is clear that

H^{0}(\overline{X}, F) = F(\overline{X}) = F(X ×_{Spec(k)} Spec(k_{s})

is a module for the Galois group Gal(\overline{k}/k) = Gal(k_{s}/k), and hence we see that all the cohomology groups

H^{i}(\overline{X}, F)

are modules for Gal(\overline{k}/k). These are the geometric cohomology groups.

We want to “compute” these cohomology groups in a couple of simple cases. Here “compute” means that we assume that usual tools are available.

We consider a smooth, projective and absolutely irreducible curve C/k. On this curve we have the étale sheaf O_{C,ét}^{∗}: For any Ũ → U ⊂ C in Et_{C} we put

O_{C,ét}^{∗}(Ũ) = O(Ũ)^{∗}

i.e. we consider the invertible regular functions on Ũ.

If ℓ is a prime which is different from the characteristic of k, then we consider the map

O_{C,ét}^{∗} ⟶ O_{C,ét}^{∗},

and here you see the whole point of the story. This map is certainly not surjective for the sheaf O_{C,Zar}^{∗} – this is our sheaf restricted to the Zariski topology. But if we have an open set U ⊂ C and an étale cover Ũ → U and an element f ∈ O(Ũ)^{∗}, then we can think of Ũ as an affine scheme and

Ũ′ = Spec(O(Ũ) [ ^{ℓ^{α}} \sqrt{f}]) ⟶ Ũ = Spec(O(Ũ)) is an étale covering. Hence we see that

O_{C,ét}^{∗} ⟶^{ℓ^{α}} O_{C,ét}^{∗}

is surjective. The kernel is the sheaf μ_{ℓ^{α}} where

μ_{ℓ^{α}}(Ũ) = {f ∈ O_{C}(Ũ)^{∗} | f^{ℓ^{α}} = 1}.

This is the sheaf of ℓ^{α}-roots of unity. Hence we get an exact sequence

1 → μ_{ℓ^{α}} → O_{C,ét}^{∗} → O_{C,ét}^{∗} → 1

of sheaves for the étale topology.

We write down the long exact sequence in cohomology, but before we do that I want to make it plausible that

Hét^{1}(C, O_{C,ét}^{∗}) = H_{Zar}^{1}(C, O_{C}^{∗}) = Pic(C).

This follows from Hilbert’s theorem 90 which says:

For any field K and any normal separable extension L/K we have H^{1}(Gal(L/K), L^{∗}) = 0.

This can be exploited to prove that any class ξ ∈ Hét^{1}(C, O_{C,ét}^{∗}) becomes trivial on a suitable small Zariski open subset, and this implies the assertion above.

We write our exact cohomology sequence, but we restrict the cohomology to \overline{C} = C ×_{k} \overline{k}. We get

O_{\overline{C},ét}(\overline{C})^{∗} = \overline{k}^{∗}

and

\overline{k}^{∗} ⟶^{ℓ^{α}} \overline{k}^{∗}

is surjective. Hence the sequence starts in degree one and we find

0 → Hét^{1}(\overline{C}, μ_{ℓ^{α}}) → H_{Zar}^{1}(\overline{C}, O∗ ; \overline{C}) → H_{Zar}^{1}(\overline{C}, O∗ ; \overline{C})

→ Hét^{2}(\overline{C}, μ_{ℓ^{α}}) → 0.

(Is it entirely obvious why it stops in degree 2?)

Now we have by definition

H_{Zar}^{1}(\overline{C}, O∗ ; \overline{C}) = Pic(\overline{C}),

and this sits in the exact sequence

0 → J\overline{C}(\overline{k}) → Pic(\overline{C}) → ℤ → 0,

and this implies

\begin{aligned}H^{1}(\overline{C}, μ_{ℓ^{α}}) & ≃ & J\overline{C}(\overline{k})[ℓ^{α}] ≈ (ℤ/ℓ^{α}ℤ)^{2g} \\ H^{2}(\overline{C}, μ_{ℓ^{α}}) & ≃ & ℤ/ℓ^{α}ℤ \end{aligned}

and finally

H^{0}(\overline{C}, μ_{ℓ^{α}}) = μ_{ℓ^{α}}(\overline{k}).

Now I recall that these étale cohomology groups are not just abelian groups, but they come with an action of the Galois group Gal(\overline{k}/k), and of course it is as it must be: The exact sequence is an exact sequence of Galois modules.

Therefore we see that the isomorphisms above are isomorphisms of Galois modules.

It may look a little bit strange that we took the coefficient sheaf. Indeed we could also take the constant sheaf ℤ/ℓ^{α}ℤ on C, this is smiply

ℤ/ℓ^{α}ℤ(Ũ) = ⨁ ; components of Ũ ℤ/ℓ^{α}ℤ.

This sheaf is certainly not isomorphic to μ_{ℓ^{α}}, but if we restrict to \overline{C}, these two sheaves become isomorphic. Hence we can construct an isomorphism

Hét^{i}(\overline{C}, ℤ/ℓ^{α}ℤ) ≃ Hét^{i}(\overline{C}, μ_{ℓ^{α}}).

But this is not necessarily an isomorphism of Galois modules, this is already clear in degree zero. Hét^{0}(\overline{C}, ℤ/ℓ^{α}ℤ) = ℤ/ℓ^{α}ℤ is the trivial Galois module

Hét^{0}(\overline{C}, ℤ/ℓ^{α}ℤ) = μ_{ℓ^{α}}(\overline{k})

is non trivial, it is the Galois module of ℓ^{α}-th roots of unity.

This Galoismodule is also denoted by ℤ/ℓ^{α}ℤ(1), we can define

ℤ/ℓ^{α}ℤ(−1) = Hom(ℤ/ℓ^{α}ℤ(1), ℤ/ℓ^{α}ℤ),

and ℤ/ℓ^{α}ℤ(n) for any integer n.

It is at least very plausible thte we have an isomorphism of Galois modules

H^{i}(\overline{C}, ℤ/ℓ^{α}ℤ) ≃ H^{i}(\overline{C}, μ_{ℓ^{α}}) ⊗ ℤ/ℓ^{α}ℤ(−1)

and especially

Hét^{1}(\overline{C}, ℤ/ℓ^{α}ℤ) ≃ J\overline{C}(\overline{k})[ℓ^{α}] ⊗ ℤ/ℓ^{α}ℤ(−1).

Now we may vary the α, we get a projective system, and we put

Hét^{i}(\overline{C}, ℤ_{ℓ}) =: lim_{⟵} Hét^{i}(\overline{C}, ℤ/ℓ^{α}ℤ),

and we showed

\begin{aligned}Hét^{0}(\overline{C}, ℤ_{ℓ}) & = & ℤ_{ℓ} \\ Hét^{1}(\overline{C}, ℤ_{ℓ}) & ≃ & T_{ℓ}(J) ⊗ ℤ_{ℓ}(−1) \\ Hét(\overline{C}, ℤ_{ℓ}) & ≃ & ℤ_{ℓ}(−1) \end{aligned}

as modules under the Galois group. (It turns out that the reasonable approach is: We start from torsion sheaves und define cohomology with coefficients in ℤ_{ℓ} by taking projective limits.)

The above definition of ℓ-adic cohomology groups works in full generality. We can take an arbitrary scheme X/k of finite type and we define

Hét^{i}(\overline{X}, ℤ_{ℓ}) = lim_{⟵} Hét^{i}(\overline{X}, ℤℓ^{α}ℤ),

and these modules are modules for the Galois group Gal(\overline{k}/k). (We may of course also consider cohomology groups

Hét^{i}(X, ℤ_{ℓ}) = lim_{⟵} Hét^{i}(X .ℤ/ℓ^{α}ℤ)

but they are still much more complicated.)

Now one has to prove many theorems to develop this theory and in principle I want to stop here.

I want to formulate some of the important theorems:

1) If we have a scheme X/k of finite type, then the cohomology groups

H^{i}(\overline{X}, ℤ_{ℓ})

are finitely generated, and if X is irreducible, then H^{i}(\overline{X}, ℤ_{ℓ}) = 0 for i > 2 dim X.

2) If X/k is an irreducible affine scheme, then we even have

H^{i}(\overline{X}, ℤ_{ℓ}) = 0 for i > dim X

and

H^{i}(𝔸n ; \overline{k}, ℤ_{ℓ}) = \{ \begin{matrix}0 & i > 0 \\ ℤ_{ℓ} & for i = 0. \end{matrix}

3) If the field k = ℂ, then we can consider the étale topology on X but we also have the analytic topology on X(ℂ).

A little bit of thinking yields that we can have a continous map

Xét ⟵ X_{an}(ℂ)

which induces a map in cohomology

H^{i}(Xét, ℤ/ℓ^{α}ℤ) ⟶ H^{1}(X_{an}(ℂ), ℤ/ℓ^{α}ℤ),

and the comparison theorem asserts that this map is an isomorphism.

4) If X/k is projective non singular and absolutely irreducible, then we have in addition

H^{2d}(\overline{X}, ℤ_{ℓ}) = ℤ_{ℓ}(−d),

and the cup product (which of course has to be defined) induces a non degenerate pairing

H^{i}(\overline{X}, ℤ_{ℓ}) ⊗ ℚ_{ℓ} × H^{2d−i}(\overline{X}, ℤ_{ℓ}) ⊗ ℚ_{ℓ} ˜ ⟶H^{2d}(\overline{X}, ℤ_{ℓ}) ⊗ ℚ_{ℓ} ≃ ℚ_{ℓ}(−d).

5) We have a Künneth formula for products

H^{m}(\overline{X} × \overline{Y} , ℚ_{ℓ}) ≃ ⨁ H^{i}(\overline{X}, ℚ_{ℓ}) ⊗ H^{m−1}(\overline{X}, ℚ_{ℓ}).

6) If we have a subscheme Z ⊂ X, let us assume that it is absolutely irreducible of codimension r, then we can attach a cycle class

c(Y) ∈ H^{2r}(\overline{X}, ℚ_{ℓ})

to it. The subspace ℚ_{ℓ} ⊂ (Y) is isomorphic to ℚ_{ℓ}(−r) as a module under the Galois group Gal(\overline{k}/k).

We can extend this cycle map to arbitrary cycles in codimension r. First of all, if Y ⊂ X is not irreducible, then Y ×_{k} \overline{k} = ∑ Y_{i} where the Y_{i} are permuted by the Galois group. Then

c(Y) = c(Y ×_{k} \overline{k}) = ∑ c(Y_{i}),

and ℚ_{ℓ}(Y ⊂ (Y_{i}) = ℚ_{ℓ}(−r) under the Galois group Gal(\overline{k}/L_{i}) if Gal(\overline{ℚ}/L_{i}) is the stabilizer of Y_{i}.

Finally we extend to arbitrary cycles by linearity.

Of course we want: If we have two cycles Y_{1}, Y_{2} in codimension r_{1}, r_{2} respectively, and if they are intersecting in a reasonable manner, then

c(Y_{1} ∘ Y_{2}) = c(Y_{1}) ∪ c(Y_{2}).

Especially if D is a divisor, then we want

D^{d} = D · − · D = c_{1}(D)^{d}.

7) These formal properties imply a Lefschetz fixed point formula. If we have a morphism f : X → X of a smooth projective, absolutely irreducible variety into itself, then we may consider the two cycles

Γ_{f} ⊂ X × X , Δ ⊂ X × X,

where Δ is the diagonal. Let us assume (which is not really necessary) that Γ_{f} and Δ intersect transversally, this means that at any geometric point p ∈ \overline{Γ}_{f} ∩ \overline{Δ}, the tangent spaces span the tangent space of \overline{X} × \overline{X}. Then the intersection number

\overline{Γ}_{f} · \overline{Δ} = #(\overline{Γ}_{f} ∩ \overline{Δ}) = #fixed points of f.

Now we look at the cycle classes and apply the Künneth formal, then we get

c(\overline{Γ}_{f}) ∪ c(\overline{Δ}) = ∑_{i=0}^{2d}(−1)^{i} tr(f^{x} | H^{i}(\overline{X}, ℚ_{ℓ})),

and we get (under our assumtion #) fixed points of f = ∑_{i=0}^{2d}(−1)^{i} tr(f^{∗} | H^{i}(\overline{X}, ℚ_{ℓ})).

If the intersection is not proper, then we have to be more careful in counting the fixed points.

We apply these theorems to the special case of an absolutely irreducible projective, smooth variety over a finite field 𝔽_{q}, let us write X/𝔽_{q}

In this case we have the Frobenius morphism Φ : x → x^{q}, and the intersection of Γ_{Φ} and the diagonal is transversal. this gives us

#fixed points ofΓ_{Φ} = #X(𝔽_{q}) = ∑(−1)^{i} tr(Φ^{∗} | H^{i}(\overline{X}, ℚ_{ℓ})).

Of course we can apply this to all powers Φ_{q}^{n}, and then

\begin{aligned}t · \frac{Z′_{X}(t)}{Z_{X}(t)} = & ∑_{n=0}^{∞} #X(𝔽_{q^{n}})t^{n} = \\ & ∑_{n=0}^{∞}(∑_{i=0}^{2g}(−1)^{i} tr(Φ_{q}^{n} | H^{i}(\overline{X}, ℚ_{ℓ})))t^{n} \end{aligned}

and this yields the formula

Z_{X}(t) = ∏_{i=0}^{2g} det(Id −tΦ_{q} | H^{i}(\overline{X}, ℚ_{ℓ})^{(−1)^{i+1}}

for the Z-function of X. This is Grothendiecks theorem (???).

At this point we have

Z_{X}(t) = \frac{P_{1} ⋯ P_{2g−1}}{P_{0} ⋯ P_{2d}}

where

Z_{X}(t) ∈ ℤ[[t]]

but a priori we have

P_{i}(t) = det(Id −tΦ_{q} | H^{i}(\overline{X}, ℚ_{ℓ})) ∈ ℤ_{ℓ}[t].

But then in 1973 P. Deligne proved the following theorem which was anticipated or conjectured by A. Weil in (???).

Theorem (Weil conjectures): 1) The polynomials

P_{i}(t) = det(Id −tΦ_{q} | H^{i}(\overline{X}, ℚ_{ℓ})) ∈ ℤ[t]

and they are independent of ℓ.

2) The reciprocical roots

P_{i}(t) = ∏(1 − ω_{iν}t)

are algebraic integers and

|ω_{iν}| = q^{i/2} for allν.

In principle we have proved this theorem for the case of curves, except that we replaced H^{i}(\overline{C}, ℚ_{ℓ}) by the ad hoc defined group ℚ_{ℓ}(0), T_{ℓ}(J) ⊗ ℚ_{ℓ}(−1), ℚ_{ℓ}(−1).

Categories I.1

Commutative diagrams I.1

Functors I.2.

covariant, contravariant, representable I,2 additive, exact, II. 2. left and right exact, derived , right and left derived II. 3. II.2

universal property I.3 Yoneda Lemma I.3.1. directed (ordered) sets I.3.5. Complexes II 1.1.

homological, cohomological, exact II.1.1.

cohomology groups II. 1.1. cocycle II 1.1. coboundary II. 1.1. exact sequence II. 1.1 split exact sequence II. 3. acyclic modules ( for a given functor ) II 3.1. acyclic resolution II 3.1. module

injective, projective II.3.1., flat II.4.3.

resolution

injective, acyclic II. 3.1.

homotopic to zero II. 3.4.