Contents 0 Preliminaries 1. Some Abstract Nonsense . . 2. Representable Functors . . . 3. Tensor Products . . . . . . . 4. Background from Topology .

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3 3 8 10 15

1 Galois Theory of Fields 19 1. A Review of Classical Galois Theory . . . . . . . . . . . . . . 19 2. Infinite Galois Extensions . . . . . . . . . . . . . . . . . . . . 23 ´ 3. Finite Etale Algebras . . . . . . . . . . . . . . . . . . . . . . . 30 2 Fundamental Groups in Topology 1. Covers . . . . . . . . . . . . . . . . . 2. Group Actions and Galois Covers . . 3. The Main Theorems of Galois Theory 4. Construction of the Universal Cover .

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35 36 37 43 49

3 Locally Constant Sheaves 55 1. Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2. Locally Constant Sheaves and Their Classification . . . . . . . 58 3. Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4 Riemann Surfaces 1. Complex Manifolds . . . . . . . . . . 2. Branched covers of Riemann Surfaces 3. Relation with Field Theory . . . . . 4. Topology of Riemann Surfaces . . . .

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67 67 72 76 86

5 Enter Schemes 91 1. Prime Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2. Schemes – Mostly Affine . . . . . . . . . . . . . . . . . . . . . 93 1

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CONTENTS 3. 4. 5. 6.

First Examples of Schemes . . Quasi-coherent Sheaves . . . . Fibres of a Morphism . . . . . Special Properties of Schemes

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103 106 109 113

6 Dedekind Schemes 1. Integral Extensions . . . . . . . . . . . . 2. Dedekind Schemes . . . . . . . . . . . . 3. Modules and Sheaves of Differentials . . 4. Invertible Sheaves on Dedekind Schemes

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119 119 122 128 136

7 Finite Covers of Dedekind Schemes 1. Local Behaviour of Finite Morphisms . . . 2. Fundamental Groups of Dedekind Schemes 3. Galois Branched Covers and Henselisation 4. Henselian Discrete Valuation Rings . . . . 5. Dedekind’s Different Formula . . . . . . .

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145 145 152 158 162 168

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173 . 173 . 179 . 185 . 191 . 191

8 Finite Covers of Algebraic Curves 1. Smooth Proper Curves . . . . . . 2. Invertible Sheaves on Curves . . . 3. The Hurwitz Genus Formula . . . 4. Abelian Covers of Curves . . . . . 5. Fundamental Groups of Curves .

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9 The Algebraic Fundamental Group 195 1. The General Notion of the Fundamental Group . . . . . . . . 195 2. The Outer Galois Action . . . . . . . . . . . . . . . . . . . . . 200

Chapter 0 Preliminaries Here we have assembled some basic results which will be needed in the sequel. Most of them should be familiar so the reader is invited to consult this chapter only if she finds it necessary; one notable exception might be the section on representable functors which we recommend to assimilate as soon as possible.

1.

Some Abstract Nonsense

In this text, we shall frequently formulate our assertions using notions of category theory, the theory which has been affectionately termed “abstract nonsense” by Steenrod who was one of its founding fathers. At first some readers may find its use appalling but we hope that in the long run they will appreciate its elegance and efficacity. The present section assembles the definitions and theorems that will emerge and re-emerge from now on. Definition 1.1 A category consists of objects as well as morphisms between pairs of objects; given two objects A, B of a category C, the morphisms from A to B form a set, denoted by Hom(A, B). (Notice that in contrast to this we do not impose that the objects of the category form a set.) These are subject to the following constraints. 1. For any object A, the set Hom(A, A) should contain a distinguished element idA , the identity morphism of A. 2. Given two morphims φ ∈ Hom(B, C) and ψ ∈ Hom(A, B), there should exist a canonical morphism φ ◦ ψ ∈ Hom(A, C), the composition of φ and ψ. The composition of morphisms should satisfy two natural axioms: • Given φ ∈ Hom(A, B), one should have φ ◦ idA = idB ◦ φ = φ. 3

4

CHAPTER 0. PRELIMINARIES • (Associativity rule) For λ ∈ Hom(A, B), ψ ∈ Hom(B, C), φ ∈ Hom(C, D) one should have (φ ◦ ψ) ◦ λ = φ ◦ (ψ ◦ λ).

Some more definitions: a morphism φ ∈ Hom(A, B) is an isomorphism if there exists ψ ∈ Hom(B, A) with ψ ◦ φ = idA , φ ◦ ψ = idB ; we denote the set of isomorphisms between A and B by Isom(A, B). If the objects themselves form a set, one can associate an oriented graph to the category by taking objects as vertices and defining an oriented edge between two objects corresponding to each morphism. With this picture in mind, it is easy to conceive what the opposite category C op of a category C is: it is “the category with the same objects and arrows reversed”; i.e. for each pair of objects (A, B) of C, there is a canonical bijection between the sets Hom(A, B) of C and Hom(B, A) of C op preserving the identity morphisms and composition. A subcategory of a category C is just a category D consisting of some objects and some morphisms of C; it is a full subcategory if given two objects in D, HomD (A, B) = HomC (A, B), i.e. all C-morphisms between A and B are morphisms in D. Examples 1.2 Some categories we shall frequently encounter in the sequel will be the category Sets of sets (with morphisms the set-theoretic maps), the category Ab of abelian groups (with group homomorphisms) or the category Top of topological spaces (with continuous maps). Both Ab and Top are naturally subcategories of Sets but they are not full subcategories; on the other hand, Ab is a full subcategory of the category Groups of all groups. Here we really take “all” sets, (abelian) groups or topological spaces as objects of our category; for example, the one-element set consisting of the author and the one-element set consisting of you, dear reader represent different objects of Sets even if they are isomorphic. This poses serious set-theoretical (not to say philosophical) problems; to see how they are circumvented, one may consult texts on category theory such as MacLane [1]. Now comes the second basic definition of category theory. Definition 1.3 A (covariant) functor F between two categories C1 and C2 consists of a rule A 7→ F (A) on objects and a map on sets of morphisms Hom(A, B) → Hom(F (A), F (B)) which sends identity morphisms to identity morphisms and preserves composition. A contravariant functor from C1 to C2 is a functor from C1 to C2op . Examples 1.4 Here are some examples of functors. 1. The “most famous” functor is the identity functor idC of any category C which leaves all objects and morphisms fixed.

5

1.. SOME ABSTRACT NONSENSE

2. Other basic examples of functors are obtained by fixing an object A of a category C and considering the covariant functor Hom(A, ) (resp. the contravariant functor Hom( , A)) from C to the category Sets which associates to an object B the set Hom(A, B) (resp. Hom(B, A)) and to a morphism φ : B → C the set-theoretic map Hom(A, B) → Hom(A, C) (resp. Hom(C, A) → Hom(B, A)) induced by composing with φ. 3. An example of a functor whose definition involves “real mathematics” and not just abstract nonsense is given by the set-valued functor on the category Top which sends a space into its set of connected components. Here to see that this is really a functor one has to use the fact that a continuous map between topological spaces sends connected components to connected components. Definition 1.5 If F and G are two functors with same domain C1 and target C2 , a morphism of functors Φ between F and G is a collection of morphisms ΦA : F (A) → G(A) in C2 for each object A ∈ C1 such that for any morphism φ : A → B in C1 the diagram Φ

A F (A) −−− → G(A)

 

F (φ)y

Φ

  yG(φ)

B F (B) −−− → G(B)

commutes. The morphism Φ is an isomorphism if each ΦA is an isomorphism; in this case we shall write F ∼ = G. Remark 1.6 In the literature the terminology “natural transformation” is frequently used for what we call a morphism of functors. However, we prefer the latter name as it comes from the fact that given two categories C1 and C2 one can define a new category called the functor category of the pair (C1 , C2 ) whose objects are functors from C1 to C2 and whose morphisms are morphisms of functors. Here the composition rule for some Φ and Ψ is induced by the composition of the morphisms ΦA and ΨA for each object A in C1 . Disposing of the definition of an isomorphism between functors, we can now give one of the notions which will be ubiquitous in what follows. Definition 1.7 Two categories C1 and C2 are equivalent if there exist functors F : C1 → C2 and G : C2 → C1 such that F ◦ G ∼ = idC2 and G ◦ F ∼ = idC1 . op They are anti-equivalent if C1 is equivalent to C2 .

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CHAPTER 0. PRELIMINARIES

One sees that equivalence of categories has all properties that equivalence relations on sets have, i.e. it is reflexive, symmetric and transitive. Also, the seemingly asymmetric definition of anti-equivalence is readily seen to be symmetric. In practice, when one has to establish an equivalence of categories it often turns out that the construnction of one functor is easy but that of the one in the reverse direction is rather cumbersome. The following general lemma enables us to make do with the construction of only one functor in concrete situations. Before stating it, we introduce some terminology. Definition 1.8 A functor F : C1 → C2 is faithful if for any two objects A and B of C1 the map of sets FAB : Hom(A, B) → Hom(F (A), F (B)) induced by F is injective; it is fully faithful if all the maps FAB are bijective. The functor F is is essentially surjective if any object of C2 is isomorphic to some object of the form F (A). Lemma 1.9 Two categories C1 and C2 are equivalent if and only if there exists a functor F : C1 → C2 which is fully faithful and essentially surjective. There is an analogous characterisation of anti-equivalent categories with fully faithful and essentially surjective contravariant functors (defined in the obviuos way). Proof: For sufficiency, fix for all objects V of C2 an isomorphism iV : ∼ F (A) → V with some object A ∈ C1 . Such an A exists by the second condition; if actually V = F (A), choose iV as the identity. Then define −1 −1 G : C2 → C1 by setting G(V ) = A and G(φ) = FAB (iW ◦ φ ◦ iV ) for φ ∈ Hom(V, W ), where FAB is the bijection appearing in the definition of fully faithfulness. We have G ◦ F = idC1 by the particular choice of iF (A) . To ∼ construct an isomorphism F ◦ G → idC2 , put ΦV = iV for all objects V of C2 . The definition of G(V ) shows that this is indeed an isomorphism between (F ◦ G)(V ) and V and the definition of G(φ) for morphisms φ : V → W shows that the isomorphism is functorial. For necessity, assume there exist a functor G : C2 → C1 and isomorphisms ∼ ∼ of functors Φ : F ◦ G → idC2 and Ψ : G ◦ F → idC1 . The second property is immediate: given an object C of C2 , it is isomorphic to F (G(C)) via Φ. For fully faithfulness fix any two objects A, B of C1 and consider the sequence of maps Hom(A, B) → Hom(F (A), F (B)) → Hom(G(F (A)), G(F (B))) → Hom(A, B) induced respectively by FAB , GF (A),F (B) and Ψ. Their composite is the identity as Ψ is an isomorphism of functors. Thus given two morphisms

1.. SOME ABSTRACT NONSENSE

7

φ, ψ ∈ Hom(A, B) with F (φ) = F (ψ), following their images in the above sequence gives φ = ψ, i.e. F is a faithful functor. Since the situation is symmetric in F and G, we conclude that G is faithful as well. We still have to show that any λ ∈ Hom(F (A), F (B)) is of the form F (φ) for some φ ∈ Hom(A, B). For this define φ as the image of λ by the composition of the last two maps above. Since the last map is a bijection, chasing this φ through the above sequence gives G(λ) = G(F (φ)), whence λ = F (φ) by faithfulness of G. Note that in the above proof the construction of G depended on the axiom of choice: we had to pick for each V an element iV of the set of isomorphisms of V with F (A). Different choices define different G’s but the categories are equivalent with any choice. This suggests that the notion of equivalence of categories means that “up to isomorphism the categories have the same objects and morphisms” but, as the following example shows, this doesn’t mean at all that there are bijections between objects and morphisms. Example 1.10 (Linear algebra) Here is an example which despite its fastidious appearence contains something very familiar. Consider a field k and the category FinVectk of finite dimensional k-vector spaces (with linear maps as morphisms). We show that this category is equivalent to a category C which we define as follows. Objects of C are to be closed intervals of the form [1, n] in the ordered set of integers (that is, the objects are {1}, {1, 2}, {1, 2, 3} etc.) plus a distinguished object 0. For the morphisms, define Hom([1, n], [1, m]) to be the set of all n by m matrices; here the identity morphisms are given by the identity matrices and composition by multiplication of matrices. There is also a canonical morphism from each [1, n] to 0 (the “zero morphism”) and one in the reverse direction (the “inclusion of 0”); composition of zero morphisms with matrices gives zero morphisms and similarly for the inclusion of 0. (Notice that this is an example of a category where morphisms are not set-theoretic maps.) To show the asserted equivalence, we introduce an auxiliary subcategory 0 C and show it is equivalent to both categories. This category C 0 is to be the full subcategory of FinVectk spanned by the standard vector spaces k n . The equivalence of C 0 and FinVectk is immediate from the criterion of the previous lemma applied to the inclusion functor of C 0 to FinVectk : the first condition is tautological as we took C 0 to be a full subcategory and the second holds because any finite dimensional vector space is isomorphic to some k n . Let’s now show the equivalence of C 0 and C. Define a functor F : C 0 → C by associating to k n some fixed numbering of its standard basis vectors and by sending morphisms to their matrices with respect to the standard bases

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CHAPTER 0. PRELIMINARIES

in the source and target spaces; zero morphisms and inclusions of 0 are sent to the corresponding distinguished morphisms. The fact that F is indeed a functor hides a non-trivial result of linear algebra, namely that the matrix of the composition of two linear maps φ and ψ is the product of the matrix of φ with that of ψ. One constructs an inverse functor G by reversing this procedure: [1, n] is mapped to k n and a matrix to the linear map it defines with respect to the standard bases. It is immediate to check that in this case F ◦ G and G ◦ F are actually equal to the appropriate identity functors.

2.

Representable Functors

Another basic definition we shall constantly exploit is due to Grothendieck. Definition 2.1 Let C be a category. A functor F : C → Sets is representable by an object A of C if there is a canonical isomorphism of functors F ∼ = Hom(A, ). Similarly, a contravariant functor G is representable by A if G ∼ = Hom( , A). Examples 2.2 To get the flavour of this definition, let us look at some examples. 1. Consider the forgetful functor Ab → Sets which associates to an abelian group its underlying set. This functor is representable by Z since A ∼ = HomAb (Z, A) as a set and this isomorphism is functorial. 2. The set-valued functor on the category of (commutative) rings given by R → R × R is representable by the polynomial ring in two variables Z[T1 , T2 ]. Indeed, each element corresponds to a set-theoretic map from the two-element set T = {T1 , T2 } to R; each such map corresponds in turn to a ring homomorphism Z[T1 , T2 ] → R since Z[T1 , T2 ] is the free commutative ring generated by T . Again this bijection is manifestly functorial. (Notice that it was essential here to pass from T to Z[T1 , T2 ] since we had to represent the elemets of R×R as ring homomorphisms.) 3. Let D be a fixed domain. Consider the contravariant set-valued functor on the category of fields given by F 7→ Hom(D, F ), the homomorphisms being ring homomorphisms. This functor is represented by the fraction field of D; indeed, any element of Hom(D, F ) factors through the fraction field in a canonical way. 4. We can give a similar example from topology: let M be a fixed metric space and consider the contravariant set-valued functor on the category

9

2.. REPRESENTABLE FUNCTORS

of complete metric spaces which associates to such a space X the set of continuous maps from M to X. This functor is represented by the completion of M. The most important property of representable functors recieved its attribution in reminiscence of a coffee-house conversation between Yoneda and MacLane. Lemma 2.3 (Yoneda’s Lemma) Suppose F and G are representable functors from a category C to the category Sets, represented by objects A and B, respectively. Then any morphism of functors from F to G is induced by a unique morphism in Hom(B, A). Similarly, if F and G are contravariant functors representable by objects A and B, then any morphism between them is induced by a unique morphism in Hom(A, B). Here perhaps an explanation of the term “induced” is needed. In the covariant case, for instance, this means that given a morphism φ : B → A, composition by φ gives a morphism of sets Hom(A, C) → Hom(B, C) for any object C of C which is clearly compatible with morphisms C → D, i. e. it is a morphism of functors. Proof: We treat only the covariant case as the other is similar. Let Φ be a morphism of functors from F ∼ = Hom(A, ) to G ∼ = Hom(B, ). The identity morphism idA is canonically an element of F (A) ∼ = Hom(A, A); applying Φ we get a canonical element φ of G(A) = Hom(B, A). We claim that Φ is induced by φ in the sense explained above. This means that for a fixed object C of C, any element γ ∈ F (C) ∼ = Hom(A, C) is mapped to ∼ γ ◦ φ ∈ Hom(B, C) = G(C) by Φ. But γ is nothing but the image of idA by the morphism F (γ) : F (A) → F (C) and similarly γ ◦ φ is the image of φ by G(γ) : G(A) → G(C). Our claim now follows from the diagram F (A) −−−→ G(A)   y

  y

F (C) −−−→ G(C) which commutes since Φ is a morphism of functors. The last thing to be proved is the unicity of φ which follows from the fact that any φ inducing Φ should be the image of idA in Hom(B, A). Corollary 2.4 The object representing a representable functor is unique up to unique isomorphism.

10

CHAPTER 0. PRELIMINARIES

This somewhat succinct assertion means that if A and A0 are two objects representing the same functor F , then the set Isom(A, A0 ) consists of exactly one element. Innocent as it looks, this corollary will be invaluable in the sequel.

3.

Tensor Products

In this section we assemble some basic facts concerning tensor products of modules. This might be helpful for some readers as the construction, which is sometimes absent from the basic algebraic curriculum, will be of constant use in forthcoming chapters. We state only those basic properties that will be needed and restrict ourselves to the case of a commutative base ring A; for more information, see textbooks on algebra such as Lang [1]. Let R be a commutative ring and A, B, C three R-modules. Recall that a map f : A × B → C is called R-bilinear if it satisfies the relations f (a1 + a2 , b) = f (a1 , b) + f (a2 , b) f (a, b1 + b2 ) = f (a, b1 ) + f (a, b2 )

(1)

f (ra, b) = f (a, rb) = rf (a, b) for all elements r ∈ R, a, a1 , a2 ∈ A and b, b1 , b2 ∈ B. Now consider the set-valued functor BilinA×B on the category of R-modules defined by BilinA×B (C) = {R-bilinear maps A × B → C}. It is indeed a fuctor since any R-homomorphism φ : C → C 0 induces a map BilinA×B (C) → BilinA×B (C 0 ) by the rule f 7→ φ ◦ f . Theorem 3.1 The functor BilinA×B is representable. We denote by A ⊗R B the R-module representing the above functor and call it the tensor product of A and B over R. By the Corollary to the Yoneda Lemma we see that it is uniquely determined up to unique isomorphism as an object of the category of R-modules. We shall often drop the subscript R in the sequel if the base ring is clear from the context. Proof: We proceed in two steps. Step 1. We first show the representability of the functor MapsA×B defined by MapsA×B (C) = {set theoretic maps A × B → C}.

11

3.. TENSOR PRODUCTS

All we have to do here is to find some R-module R[A × B] with the property that any set-theoretic map A × B → C extends to a unique R-homomophism R[A × B] → C. But this is precisely the property defining the appropriate free R-module. So we may take R[A × B] as the free R-module whose free generating set consists of the elements of A × B (that is, ordered pairs of the form (a, b) with a ∈ A and b ∈ B). Step 2. Observe that for fixed C any element of BilinA×B (C) is also an element of MapsA×B (C) so it defines an R-homomorphism f : R[A×B] → C by Step 1. The extra condition required is that f should satisfy the relations (1). So if we define A ⊗R B as the quotient of R[A × B] by the R-submodule generated by elements of the form (a1 + a2 , b) − (a1 , b) − (a2 , b), (a, b1 + b2 ) − (a, b1 ) − (a, b2 ), (ra, b) − r(a, b), (a, rb) − r(a, b), the condition becomes that f should pass to the quotient and define an R-homomorphism A ⊗R B → C. But this is the property required of the representing object. Denote by a ⊗ b the image of the element (a, b) of R[A × B] in A ⊗R B. Remark 3.2 It is immediate from the construction that any element of P A ⊗R B can be written as a finite sum ai ⊗ bj but this representation is far from unique (see the example below). Similarly, if A (resp. B) is generated over R by a system of elements ai (i ∈ I) (resp. bj (j ∈ J)), then A ⊗R B is generated by the system of elements ai ⊗ bj with (i, j) ∈ I × J. Again there is no question of uniqueness in general; however, as we shall shortly see, in the case of free R-module uniqueness does hold. Example 3.3 Let m and n be coprime integers and consider Z/mZ and Z/nZ with their Z-module structure. Then Z/mZ ⊗Z Z/nZ = 0. Indeed, since m and n are coprime, for any element a ∈ Z/mZ there is some x with nx = a, and so for any b ∈ Z/nZ one has a ⊗ b = (nx) ⊗ b = x ⊗ (nb) = x ⊗ 0 = 0. More generally, the same argument shows that if A and B are abelian groups such that nA = A and nB = 0 (in this case, A is called n-divisible and B n-torsion), then A ⊗ B = 0.

12

CHAPTER 0. PRELIMINARIES

We now list some basic properties of the tensor product all of which follow from the defining property of the tensor product using the (corollary to the) Yoneda lemma. Readers may work out the proofs as an exercise in representable functors. Proposition 3.4 Let A, B, C and Ai (i ∈ I) be R-modules. Then there are canonical isomorphisms 1. A ⊗ B ∼ = B ⊗ A; 2. (A ⊗ B) ⊗ C ∼ = A ⊗ (B ⊗ C); 3. A ⊗R R ∼ = A. 4. (

L

i∈I

L Ai ) ⊗ B ∼ = (Ai ⊗ B). i∈I

The two last statements here imply: Corollary 3.5 If both A and B are free R-modules of rank m and n, respectively, then A ⊗ B is again a free R-module of rank mn. Next observe that given two R-module homomorphisms Φ : A1 → A2 and ψ : B1 → B2 , one gets a natural morphism of functors BilinA2 ×B2 → BilinA1 ×B1 by composing with (φ, ψ). By the Yoneda lemma this is induced by an R-homomorphism φ ⊗ ψ : A1 ⊗ B1 → B1 ⊗R B2 . By examining the proof of Theorem 3.1 one checks immediately that φ ⊗ ψ is the map which assosiates φ(a1 )⊗ψ(b1 ) ∈ A2 ⊗B2 to a1 ⊗b1 ∈ A1 ⊗B1 . In the particular case B1 = B2 = B and ψ = id one gets R-homomorphisms A1 ⊗ B → A2 ⊗R B. We shall refer to such maps as “tensoring with B”. Thus tensoring with B is a (covariant) functor on the category of R-modules. On the other hand, the set HomR (A, B) of R-homomorphims from A into B is naturally equipped with an R-module structure by putting (φ + ψ)(a) = φ(a) + ψ(a) and (rφ)(a) = rφ(a) for r ∈ R, a ∈ A, b ∈ B and we saw that A → HomR (A, B) is a contravariant functor. The above two functors are linked by the following relation, labelled as “la formule ch`ere `a Cartan” by Grothendieck. Proposition 3.6 There is a canonical isomorphism of R-modules HomR (A ⊗R B, C) ∼ = HomR (A, HomR (B, C)) which is covariantly functorial in C and contravariantly in A and B.

13

3.. TENSOR PRODUCTS

Proof: By the corollary to the Yoneda lemma it is enough to construct an isomorphism of functors Ψ between BilinA×B and HomR (A, HomR (B, ). To do this, observe first that for any f ∈ BiadA×B the map b 7→ f (a, b) is an Rhomomorphism by R-bilinearity of f . Then one constructs Ψ by calling Ψ(f ) the map a 7→ (b 7→ f (a, b)) which is again seen to be an R-homomorphism. To prove that Ψ is an isomorphism of functors, it is enough to construct a morphism of functors inverse to Ψ. One checks that it is the one defined by associating the R-bilinear map (a, b) 7→ φa (b) to a homomorphism a 7→ φa in HomR (A, HomR (B, C)). As a consequence we derive a fundamental property called the right exactness of the tensor product. Proposition 3.7 Given an exact sequence of R-modules 0 → A1 → A2 → A3 → 0,

(2)

tensoring with an R-module B induces an exact sequence A1 ⊗R B → A2 ⊗R B → A3 ⊗R B → 0.

(3)

For the proof we need the following lemma. Lemma 3.8 Let A1 , A2 , A3 be R-modules and i

p

A1 → A2 → A3 → 0

(4)

a sequence of R-homomorphisms. This is an exact sequence if and only if for any R-module C the sequence induced by composition of R-homomorphisms 0 → HomR (A3 , C) → HomR (A2 , C) → HomR (A1 , C)

(5)

is an exact sequence of R-modules. Proof: The proof that exactness of (4) implies that of (5) is easy and is left to the readers. The converse is not much harder: taking C = A3 /A2 shows that injectivity of the second map in (5) implies the surjectivity on the right in (4), and taking C = A2 /im(i) shows that if moreover (5) is exact in the middle, then the surjection A2 /im(i) → A3 has a section A3 → A2 /im(i) and thus im(i) = ker(p). Proof of Proposition 3.7. By the previous lemma it is enough to show that the sequence 0 → HomR (A3 ⊗R B, C) → HomR (A2 ⊗R B, C) → HomR (A1 ⊗R B, C)

14

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is exact for any R-module C. By Proposition 3.6, this sequence is isomorphic to the sequence (where we dropped the subscript R) 0 → Hom(A3 , Hom(B, C)) → Hom(A2 , Hom(B, C)) → Hom(A1 , Hom(B, C)), whose exactness follows from the other implication of the previous lemma together with the assumption. Remarks 3.9 Here some remarks are in order. 1. It is not true in general that sequence (3) is exact on the left. For instance, take R = Z, L = Z, M = Q, B = Z/mZ for some m > 1. Tensoring the inclusion Z → Q by Z/mZ we get a map Z ⊗ Z/mZ → Q ⊗ Z/mZ. But here the first group is isomorphic to Z/mZ by Proposition 3.4 (3) and the second is 0 by Example 3.3. 2. If B is such that all exact sequences of type (2) remain exact when tensored with B, then B is said to be flat over R. This notion will come up in Chapter 9. In the next chapter, we shall use the tensor product in the following situation. Suppose given a ring homomorphism φ : R → S. Via this map S becomes an R-module and thus we may consider the tensor product A ⊗R S with any R-module A. On A ⊗R S we can define a canonical S-module structure by putting s(a ⊗ s0 ) = a ⊗ (ss0 ) for any a ∈ A and s, s0 ∈ S. We shall speak of the S-module A ⊗R S as the S-module obtained by base change from A. Note that Proposition 3.4 (3), (4) imply that if A is a free R-module with free generators ai (i ∈ I), then A ⊗R S is also free with free generators ai ⊗ 1 (i ∈ I). Combining this with Proposition 3.7, we get: Corollary 3.10 If A is presented as a quotient of a free R-module F by a submodule L generated by “relations” fi (i ∈ I), then A ⊗R S is the quotient of the free module F ⊗ S by the submodule generated by fi ⊗ 1 (i ∈ I). Thus, loosely speaking, A ⊗R S “has the same generators and relations over S as A over R”. Remark 3.11 It is also worth noting that in the above situation given an R-module A as well as an R-module B which is also an S-module such that the actions of R and S on B are compatible, A ⊗R B inherits an S-module structure from that on B. Then the following refined version of Theorem 3.6 holds: for any S-module C there exists a natural isomorphism HomS (A ⊗R B, C) ∼ = HomR (A, HomS (B, C)) with the same functoriality properties as above.

4.. BACKGROUND FROM TOPOLOGY

15

Finally we shall examine the situation where both A and B have an R-algebra structure. Then the tensor product also inherits an R-algebra structure in a natural way, the multiplication being defined by (a⊗b)(a0 ⊗b0 ) = (aa0 ) ⊗ (bb0 ) and then extending by linearity. (To see that this indeed gives a well-defined map (A ⊗ B) × (A ⊗ B) → (A ⊗ B) observe that the map A × B × A × B → (A ⊗ B) given by (a, b, a0 , b0 ) 7→ aa0 ⊗ bb0 gives R-bilinear maps A × B → A ⊗ B when one fixes the first and second (resp. third and fourth) arguments; then apply Theorem 3.1.) We have natural analogues for R-algebras of most of the results proved above for R-modules. We only note the analogue of the preceding corollary: Corollary 3.12 If A is a quotient of a polynomial ring R[T1 , . . . , Tn ] by an R-subalgebra generated by polynomials fi , then A⊗R S can be presented as the quotient of the polynomial ring S[T1 , . . . , Tn ] by the S-subalgebra generated by the elements fi ⊗ 1. Proof: The ring R[T1 , . . . , Tn ] is a free R-module generated by the monomials in T1 , . . . , Tn , hence by the previous corollary S ⊗R R[T1 , . . . , Tn ] is a free S-module on the same generators. The definition of the product operation implies that this latter ring is indeed isomorphic to S[T1 , . . . , Tn ]. A similar argument for the relations using Proposition 3.7 finishes the proof.

4.

Background from Topology

We shall assume throughout that readers are familiar with the basic notions of general topology, including subspaces and quotient spaces, connected components and the theory of compact subsets – the latter will not be assumed to be necessarily Hausdorff spaces in this text. On all this (and much more) see basic textbooks such as Schubert [1]. Here we review some basic facts from the rudiments of homotopy theory; they will be needed in Chapter 2. Definition 4.1 Let X be a topological space. A (continuous) path in X is a continuous map f : [0, 1] → X, where [0, 1] is the closed unit interval. The endpoints of the path are the points f (0) and f (1); if they coincide then the path is called a closed path or a loop. Two paths f, g : [0, 1] → X are called homotopic if f (0) = g(0), f (1) = g(1) and there is a continuous map H : [0, 1]×[0, 1] → X with H(0, y) = f (y) and H(1, y) = g(y) for all y ∈ [0, 1]. Lemma 4.2 Homotopy of paths is an equivalence relation.

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Proof: For reflexivity take H as the composition of the second projection p2 : [0, 1] × [0, 1] → [0, 1] with f , for symmetry replace H by its composition with the map [0, 1]×[0, 1] → [0, 1]×[0, 1], (x, y) 7→ (1−x, y). For transitivity given a homotopy H1 between paths f and g and a homotopy H2 between g and h, construct first a continuous map H3 : [0, 2] × [0, 1] → X by piecing together H1 and the composition of H2 with the map [1, 2] × [0, 1] → [0, 1] × [0, 1], (x, y) 7→ (x − 1, y) and then compose H3 by the map [0, 1] × [0, 1] → [0, 2] × [0, 1], (x, y) 7→ (2x, y) to get a homotopy between f and h. Now given two paths f, g : [0, 1] → X with g(0) = f (1), define their composition f • g : [0, 1] → X by setting (f • g)(x) = f (2x) for 0 ≤ x ≤ 1/2 and (f • g)(x) = g(2x − 1) for 1/2 ≤ x ≤ 1. It is an easy exercise to verify that this composition passes to the quotient modulo homotopy equivelence, i.e. if f1 , f2 are homotopic paths with f1 (1) = f2 (1) = g(0) then so are f1 • g and f2 • g, and similarly if g1 is homotopic to some g2 with g1 (0) = g2 (0) = f (1), then f • g1 is homotopic to f • g2 . Construction 4.3 In view of the above remark, using the • operation we may define a group law on the set π1 (X, x) of homotopy classes of closed paths f : [0, 1] → X with f (0) and f (1) both equal to some fixed base point x by mapping the classes of such closed paths f1 , f2 : [0, 1] → X to the class of the product f1 • f2 . There is no difficulty in checking the group axioms; in particular, the unit element is the class of the constant path [0, 1] → {x} and the inverse of a class given by a path f : [0, 1] → X is the class of the path f −1 obtained by composing f with the map [0, 1] → [0, 1], x 7→ 1 − x. Definition 4.4 The group π1 (X, x) thus obtained is called the fundamental group of X with base point x. Our next goal is to show that for “nice spaces” the isomorphism class of the fundamental group does not depend on the choice of base point x. For this we need a definition: the space X is path-connected if for any two points x, y in X there is some path f : [0, 1] → X with f (0) = x and f (1) = y. Lemma 4.5 Path-connected spaces are connected.

4.. BACKGROUND FROM TOPOLOGY

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Proof: If we can write a space X as a disjoint union of non-empty open subsets U1 and U2 , then no couple of points xi in Ui (i = 1, 2) can be joined by a path f : [0, 1] → X for otherwise the inverse images of the open subsets Im (f ) ∩ Ui of Im (f ) would give a decomposition of [0, 1] into nonempty open subintervals which is impossible. Remark 4.6 Notice that by the same argument we may prove the stronger statement that a space X is connected if there is some pont x ∈ X that can be joined to any other point of X by a closed path. A maximal path-connected subset of a space X is called a path component of X; we can then write X as a disjoint union of its path components. We say that X is locally path-connected if each point of X has a basis of open neighbourhoods consisting of path-connected sets. Lemma 4.7 The path components of a locally path-connected space X are both open and closed subsets of X. Proof: It suffices to show that any path component is open since writing it as the complenent of the disjoint union of union of the other path components we get that it is also closed. For openness note that by assumption any point x ∈ X has a path-connected open neighbourhood U in X. But U must then be contained in the path component of x. Since in a connected space the only nonempty subset which is both open and closed is the space itself, we get the Corollary 4.8 Any connected and locally path-connected space is path-connected. Now we can prove: Lemma 4.9 In a path-connected space the fundamental groups π1 (X, x) and π1 (X, y) are isomorphic for any pair of base points x, y. Proof: It is immediate to check that given a path f : [0, 1] → X with f (0) = x and f (1) = y, the map π1 (X, x) → π1 (X, y), g 7→ f −1 • g • f induces such an isomorphism, its inverse being given by h 7→ f • h • f −1 . Note that the isomorphism in the lemma is not unique as it depends on the choice of the path f ; it is unique, however, up to an inner automorphism of π1 (X, y).

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Definition 4.10 We say that a path-connected space X is simply connected if π1 (X, x) = 1 for some base point x. Here by the previous lemma we may take any x ∈ X as base point. Also, note the easy fact that in a simply connected space X any two paths f, g : [0, 1] → X with f (0) = g(0) and f (1) = g(1) are homotopic (because by simply connectedness f • g −1 is homotopic to the trivial path). In Chapter 2 we shall use the following definition: Definition 4.11 A space X is locally simply connected if any point has a basis of open neighbourhoods consisting of simply connected open sets. Example 4.12 It is easy to give examples of locally simply connected spaces: since any open ball in Rn is simply connected (one may contract continuously any closed path to a point), any open subset of Rn is locally simply connected. More generally, this holds for spaces that can be covered by open subsets homeomorphic to open subsets of Rn , e.g. for (topological) manifolds (see Chapter 4). Finally, we note the important fact that the fundamental group defines a functor in a natural way. For this, define the category of pointed topological spaces to be the category whose objects are pairs (X, x) with X a topological space and x a point of X; a morphism (X, x) → (Y, y) in this category is a continuous map X → Y sending x to y. Then the rule (X, x) 7→ π1 (X, x) defines a functor from this category to the category of groups. Indeed, it is again a straightforward matter to check that any continuous map φ : X → Y sending x to y induces a group homomorphism φ∗ : π1 (X, x) → π1 (Y, y) by mapping a closed path f : [0, 1] → X with f (0) = f (1) = x to φ ◦ f .

Chapter 1 Galois Theory of Fields 1.

A Review of Classical Galois Theory

In this section we review the classical theory of Galois extensions of fields. We only aim at refreshing the readers’ memory and therefore skip the more involved proofs; for these we shall refer to standard textbooks on algebra. Let k be a field. Recall that an extension L|k is called algebraic if each element α of k is a root of some polynomial with coefficients in k (which we may as well assume to be monic). If this polynomial is irreducible over k, it is called the minimal polynomial of α. When L is generated as a k-algebra by the (algebraic) elements α1 , . . . , αk ∈ L, we write L = k(α1 , . . . , αk ). Of course, one may find many different sets of such αi ’s. A field is algebraically closed if it has no algebraic extensions other than itself. An algebraic closure of k is an algebraic extension k¯ which is algebraically closed; its existence is perhaps the least evident fact in the theory and can only be proved by means of Zorn’s lemma or some other equivalent form of the axiom of choice. We record it in the following proposition which also contains some important properties of the algebraic closure. Proposition 1.1 Let k be a field. 1. There exists an algebraic closure k¯ of k. It is unique up to (non-unique) isomorphism. 2. For any algebraic extension L of k there exists an embedding L → k¯ leaving k elementwise fixed. 3. In the previous situation take an algebraic closure L of L. Then the ¯ embedding L → k¯ can be extended to an isomorphism of L onto k. 19

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For the proof, see Lang [1], Chapter V, Corollary 2.6 and Theorem 2.8 or van der Waerden [1], §72. Thus henceforth when speaking of algebraic extensions of k we may and ¯ do assume that they are subfields of a fixed algebraic closure k. Any finite extension L of k is algebraic. Its degree over k, denoted by [L : K], is its dimension as a k-vector space. If L is generated over k by a single element with minimal polynomial f , then [L : K] is equal to the degree of f . If M|L|k is a tower of finite extensions, then one has the formula [M : k] = [M : L][L : k]. All this is proven by easy computation. An element of an extension L|k is called separable over k if its minimal polynomial has no multiple roots. A finite extension L|k is separable if all of its elements are separable over k. There is another characterisation of this property which will be particularly useful in Section 3. Lemma 1.2 Let L|k be a finite extension of degree n. Then L has at most n ¯ with equality if and only if L|k is separable. distinct k-homomorphisms to k, Proof: Assume first L is generated over k by a single element α with minimal polynomial f . Any k-homomorphism L → k¯ is then determined by the ¯ The number image of α which must be one of the roots of f contained in k. of distinct roots is at most n with equality if and only if L|k is separable. The general case reduces to the above one by using the multiplicativity of the degree in a tower of finite field extensions. Corollary 1.3 If M|L|k is a tower of finite extensions, then M|k is separable if and only if both M|L and L|k are. Proof: The “only if” part follows from the definition of separability. The “if” part is verified by the criterion of the proposition. The following important property of finite separable extensions is usually referred to as the theorem of the primitive element. Proposition 1.4 Any finite separable extension can be generated by a single element. For the proof, see Lang [1], Chapter V, Theorem 4.6 or van der Waerden [1], §46. Now recall that given two algebraic extensions L, M of k viewed as sub¯ their compositum LM is the smallest subfield of k¯ containing both fields of k, L and M.

1.. A REVIEW OF CLASSICAL GALOIS THEORY

21

Lemma 1.5 If L, M are finite separable extensions of k, their compositum is separable as well. Proof: By definition of LM there exist finitely many elements α1 , . . . , αk of L such that LM = M(α1 , . . . , αk ). Put M0 = M and Mi = Mi−1 (αi ) for i > 0. Then Mk = LM and each Mi is separable over Mi−1 for αi is separable over k and a fortiori over Mi−1 . One concludes by a repeated application of Proposition 1.2. In view of the above lemma forming the compositum of all finite separable subextensions of k¯ yields an extension k s of k with the property that each ¯ element of k s generates a finite separable subextension of k|k. Moreover, by ¯ definition each finite separable subextension of k|k is contained in k s which ¯ When in the sequel we thus merits to be called the separable closure of k in k. shall refer to “a separable closure of k” we shall mean its separable closure in some chosen algebraic closure. Also, henceforth any (possibly infinite) subextension of k s |k will be called separable. The field k is perfect if all finite extensions of k are separable. Examples of perfect fields are fields of characteristic 0, algebraically closed fields and finite fields (see van der Waerden, §45). A typical example of a non-perfect field is a rational function field F(T ) of one variable over a field F of characteristic p: here adjoining a p-th root ξ of the indeterminate T defines an inseparable extension in view of the decomposition X p − T = (X − ξ)p . For perfect fields it is immediate from the definition that their algebraic and separable closures coincide. Now we come to the fundamental definition in Galois theory. Given an extension L of k, denote by Aut(L|k) the group of field automorphisms of L fixing k elementwise. It acts on L and fixes some subextension of L|k elementwise. Definition 1.6 An algebraic extension L of k is called a Galois extension of k if it is separable and the elements of L which remain fixed under the action of Aut(L|k) are exactly those of k. In this case Aut(L|k) is denoted by Gal (L|k) and called the Galois group of L over k. Example 1.7 The absolute Galois group. The most natural example of a Galois extension of k is that of the separable closure k s (which coincides with the algebraic closure for perfect fields). It is certainly separable over k, so to check that it is Galois we only have to show that any element α of k s not contained in k gets moved by an appropriate automorphism of k s over k. Indeed, define an automorphism of the extension k(α) generated by α

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over k (itself a subfield of k s ) by sending α to another root of its minimal polynomial. An application of the third part of Proposition 1.1 then shows that this automorphism can be extended to an automorphism of the algebraic ¯ To conclude one only has to remark that any automorphism of k¯ closure k. maps k s onto itself since such an automorphism acts on an element β of k¯ by sending it to another root of its minimal polynomial; if β is separable, this polynomial has no multiple roots. We shall call the Galois group of k s over k the absolute Galois group of k and denote it by Gal (k). However, k s is an infinite extension of k whereas in the classical theory only finite Galois extensions of k are investigated. The historical reason for this, as all readers know, is that these are the extensions that arise naturally when one studies the problem of solving polynomial equations of one variable, as the next basic example indicates. Example 1.8 Splitting fields. Let k be a field and f a polynomial with coefficients in k and without multiple roots. Take an algebraic closure k¯ of ¯ generated by all roots of k and look at the finite subextension K|k of k|k the polynomial f . We contend that K|k is a Galois extension. To see this, note first that K is separable over k by virtue of Lemma 1.5; indeed, it is the compositum of the finitely many separable subextensions k(αi )|k, where ¯ Next, note that any k-automorphism of α1 , . . . , αk are the roots of f in k. k¯ maps K onto itself for it only permutes the roots αi ∈ K of f . Now to see that Aut(K|k) fixes only k, pick any element α ∈ K \ k. Mapping it to another root β ∈ k¯ of its minimal polynomial over k defines a k-isomorphism ¯ again by the third k(α) ∼ = k(β) which extends to an automorphism σ of k, part of Proposition 1.1. But then by the previous observation we must have σ(K) = K, so that β ∈ K and the restriction of σ to K is an element of Aut(K|k) moving α. The field K is usually called the splitting field of the polynomial f . Remark 1.9 It should be pointed out that from our point of view there is a more natural way of looking at finite Galois extensions: if a finite group G acts on a field K, then K is a Galois extension of the field of invariants K G . Of course here it is only separability that needs to be proven, which follows from the fact that any element of α ∈ K is a root of the polynomial Q f = (x − σ(α)), where σ runs over a system of (left) coset representatives of the stabiliser of α in G. This polynomial lies in K G [x] for it is invariant by the action of G and has no multiple roots by construction.

2.. INFINITE GALOIS EXTENSIONS

23

The link with the approach of the previous example is the following: taking α to be a primitive element of the extension K|K G , we can describe K as the splitting field of f over K G . Now the main theorem of Galois theory for finite Galois extensions reads as follows. Theorem 1.10 Consider a finite Galois extension L|k with Galois group G. Associating to a subfield M of L containing k the subgroup H of G consisting of those automorphisms which leave M elementwise fixed gives a bijection between subextensions of L|k and subgroups of G. The subfield M is Galois over k if and only if H is a normal subgroup of G; in this case, the Galois group of M over k is isomorphic to G/H. For the proof, see Lang [1], Chapter VI, Theorem 1.1 or van der Waerden [1], §58. The natural question that arises now is how to extend this theorem to infinite Galois extensions. The main difficulty is that for an infinite extension it will no longer be true that all subgroups of the Galois group arise as the subgroup fixing some subextension M|k. The first example of a subgroup which doesn’t correspond to some subextension has been found by Dedekind, who, according to Wolgang Krull, already had the feeling that “die Galoissche Gruppe gewissermaßen eine stetige Mannigfaltigkeit bilde”. It was Krull who then cleared up the question in a classic paper (Krull [1]); a modern version of it will be described in the next section.

2.

Infinite Galois Extensions

In the previous section, we defined the separable closure k s of a field k as ¯ the union of all finite separable subextensions of k|k. In fact, it is already the union of the finite Galois subextensions because of the following lemma which will be used several times in this section. Lemma 2.1 Any finite separable subextension of k s |k can be embedded into a Galois subextension. Proof: By the theorem of the primitive element (Proposition 1.4), each finite separable subextension is of the form k(α) with an appropriate element α and we may embed k(α) into the splitting field of the minimal polynomial of α which is already Galois over k by Example 1.8.

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Now given a tower of finite Galois extensions M|L|k, Theorem 1.10 provides us with a canonical surjection φM L : Gal (M|k) → Gal (L|k),

(1.1)

so one expects that if we somehow “pass to the limit in M”, then Gal (L|k) would actually become a quotient of the absolute Galois group Gal (k) itself and just as k s is determined by its finite Galois subextensions, Gal (k) would be determined by its finite quotients. The following definition gives us the key how to pass to the limit. Definition 2.2 A (filtered) inverse system of groups (Gi , φij ) consists of: • a partially ordered set (I, ≤) which is directed in the sense that for all (i, j) ∈ I there is some k ∈ I with i ≤ k, j ≤ k; • for each i ∈ I a group Gi ; • for each i ≤ j a homomorphism φij : Gj → Gi such that for i ≤ j ≤ k, φik = φij ◦ φjk . Q

The inverse limit of the system is the subgroup of the direct product i∈I Gi consisting of sequences (gi ) such that φij (gj ) = gi for all i ≤ j. It is denoted by lim Gi (we shall not specify the inverse system in the notation if it is clear ← from the context). Clearly, this notion is not specific to the category of groups and one can define the inverse limit of sets, rings, modules, even of topological spaces in an analogous way. Remark 2.3 Notice that the fact that the index set is directed was not used in the above definition. Thus we may speak more generally of inverse limits of inverse systems with respect to any partially ordered index sets; the word “filtered” in the above definition refers to the fact that the index set is directed. While we are at this matter, let us also define the dual notion of direct limits. Here we shall restrict to abelian groups in the main definition (but see the remark afterwards). Definition 2.4 A (filtered) direct system of abelian groups (Ai , φij ) consists of: • a partially ordered set (I, ≤) which is directed in the above sense;

25

2.. INFINITE GALOIS EXTENSIONS • for each i ∈ I an abelian group Ai ;

• for each i ≤ j a homomorphism φij : Ai → Aj such that for i ≤ j ≤ k, φik = φjk ◦ φij . L

The direct limit of the system is the quotient of the direct product i∈I Ai by the subgroup generated by elements of the form ai − φij (ai ). It is denoted by lim Ai . →

Remark 2.5 Note that this notion does not extend to arbitrary groups but it immediately does to abelian groups with additional structure (rings, modules etc.) Also, we can define the direct limit of a direct system of sets (Si , φij ) ` by taking the quotient of the disjoint union i∈I Si by the equivalence relation which is generated by pairs (si , sj ) ∈ Si × Sj with φik (si ) = φjk (sj ) for some k ≥ i, j. Notice that this generalises the notion of set-theoretic union: indeed, if (Si : i ∈ I) is a system of sets with the property that for any Si , Sj there is some k with Si ∪ Sj ⊂ Sk , we can make it into a direct system by putting i ≤ j if Si ⊂ Sj and define φij to be the inclusion map. Now it is S straightforward to check that the direct limit of this system is precisely Si . We shall need an analogue of the above: Proposition 2.6 Suppose that a ring R is a union of subrings Ri such that for any Ri , Rj there is some Rk with Rk ⊃ Ri ∪ Rj . Then partially ordering by inclusion turns the system (Ri ) into a direct system whose direct limit is R. L

Proof: Consider the natural homomorphism i∈I Ri → R given by the sum of components. Since the φij are now the inclusion maps, the kernel of this map is generated by the relations defining the direct limit; surjectivity is equally easy. Corollary 2.7 The separable closure k s of a field k is a direct limit of its finite Galois subextensions. Proof: It remains only to show that the system of finite Galois subextensions is directed which means that the compositum of any two finite Galois subextensions can be embedded into some finite Galois extension. Since the compositum is separable, this follows from Lemma 2.1. (In fact, the compositum is already Galois by Lang [1], Chapter VI, Theorem 1.14.) The next proposition can now be by no means surprising.

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Proposition 2.8 Let k be a field with fixed separable closure k s . Then the Galois groups of finite Galois subextensions of k s |k together with the homomorphisms φM L of (1.1) form an inverse system whose inverse limit is isomorphic to the absolute Galois group Gal (k). Proof: Only the last statement needs a proof. For this, define a group Q homomorphism φ : Gal (k) → Gal(L|k) (where the product is over all finite Galois subextensions L|k) by sending an automorphism σ of k s over k to the direct product of its restrictions to the various subfields L indexing the product. This map is injective since if an automorphism σ does not fix an element α of k s , then its restriction to any finite Galois subextension containing k(α) is nontrivial (such an extension exists by Lemma 2.1). On the other hand, Theorem 1.10 assures that the image of φ is contained in lim Gal (L|k). It is actually all of lim Gal (L|k) which one sees as follows: ← ← take an element (σL ) of lim Gal (L|K) and define an automorphism σ of ← k s by putting σ(α) = σL (α) with some finite Galois L containing k(α). The fact that σ is well defined follows from the fact that by hypothesis the σL form a compatible system of automorphisms; finally, σ maps to (σL ) ∈ lim Gal (L|K) by construction. ← Note that projection to the components of the inverse limit induces by virtue of the proposition natural surjections Gal (k) → Gal (L|k) for each finite Galois subextension L, just as we expected. Now we come to the Krull-Dedekind idea about the extension of Galois theory to the infinite case. This is to endow Gal (k) with a suitable topology for which the closed subgroups would turn out to be precisely those corresponding to some field extension. Of course for any reasonable topology on Gal(k) one would like the projections Gal (k) → Gal (L|k) to be continuous so the only judicious choice is to endow the groups Gal (L|k) with the disQ crete topology and put the product topology on Gal (L|k). Fortunately, we have: Lemma 2.9 Let (Gi , φij ) be an inverse system of finite groups endowed by the discrete topology. Then the inverse limit lim Gi is a closed topological ← Q subgroup of the product Gi . Q

Proof: Take an element g = (gi) ∈ Gi . If g ∈ / lim Gi , we have to show ← that it has an open neighborhood which does not meet lim Gi . By assumption ← Q for some i and j we must have φij (gj ) 6= gi . Now take the subset of Gi consisting of all elements with i-th component gi and j-th component gj . It

2.. INFINITE GALOIS EXTENSIONS

27

is a suitable choice, being open (by the discreteness of the Gi and by the definition of topological product) and containing g but avoiding lim Gi . ← Definition 2.10 A topological group which is the inverse limit of finite groups is called a profinite group. Examples 2.11 Here are some examples of profinite groups. 1. Given any group G, the set of its finite quotients can be turned into an inverse system: let I be the index set formed by the normal subgroups of finite index partially ordered by the following relation: Ui ≤ Uj iff Ui ⊃ Uj . Then if Ui ≤ Uj are such normal subgroups, we have a quotient map φij : G/Uj → G/Ui . The inverse limit of this system is ˆ There called the profinite completion of G, customarily denoted by G. ˆ is a canonical homomorphism G → G with dense image; in fact, as the next proposition will imply, this map is an inclusion. 2. Take G = Z in the previous example. Then I is just Z since each subgroup of finite index is generated by some m ∈ Z. The partial order is induced by the divisibility relation: m|n iff mZ ⊃ nZ. The completion ˆ is usually called zed hat, although some people call it the Pr¨ Z ufer ring ˆ is the free (it indeed has a natural ring structure). One can show that Z profinite group generated by one element in the following sense: it has a canonical topological generator 1 coming from the natural inclusion ˆ mapping it to any element in a profinite group G extends to a Z ⊂ Z; ˆ → G. continuous homomorphism Z 3. In the previous example, taking only powers of some prime p in place of m we get a subsystem of the inverse system considered there; in fact it is more convenient to index it by the exponent of p. With this convention the partial order becomes the usual (total) order of Z. The inverse limit is Zp , the (additive) group of p-adic integers; in fact, it is also a ring, just as in the previous example. One can show using the ˆ is the direct product of Chinese Remainder Theorem that the ring Z the rings Zp for all primes p. 4. By Proposition 2.8, the absolute Galois group of any field F furnishes an example of a profinite group. When F is finite, we get nothing ˆ Indeed, from the theory new: Gal (F ) is canonically isomorphic to Z. of finite fields (see e.g. Lang [1], Chapter V, §5) we know that for each positive integer n, F has a unique finite extension Fn of degree n, which is moreover Galois with group Gal (Fn |F ) ∼ = Z/nZ, so the

28

CHAPTER 1. GALOIS THEORY OF FIELDS inverse system one gets is exactly that of the second example above. As a contrast, determining Gal (Q) is one of the greatest mysteries in arithmetic.

Proposition 2.12 Let G = lim Gi be a profinite group. Then ←

1. G has a system Ui of open normal subgroups which forms a basis of neighborhoods of the unit element; 2. G is a Hausdorff space; 3. G is compact; 4. the open subgroups of G are precisely the closed subgroups of finite index. Proof: For the first assertion, one may take as the Ui the kernels of the continuous projections G → Gi ; these are open normal subgroups by construction and their intersection is the unit element, for any nontrivial element of G has a nontrivial image in some Gi . The second assertion follows from the first in view of the following well-known trick: take elements x, y ∈ G and let Ui be an open neighbourhood of the identity such that xy −1 ∈ / Ui ; then Ui x and Ui y are disjoint open neighbourhoods of x and y. For the third assertion, observe first that finite discrete groups are compact; by Tikhonov’s theorem (Chapter 0, Proposition ??) so is their product. To conclude the proof it remains to apply Lemma 2.9 and the fact that a closed subset of a compact topological space is itself compact (Chapter 0, Lemma ??). Finally, note that any open subgroup U is closed since its complement is a disjoint union of cosets gU which are themselves open (the map U 7→ gU being a homeomorphism in a topological group); by compactness of G, these must be finite in number. The converse follows by a similar but simpler argument. Remark 2.13 In fact, there is a nice topological characterisation of profinite groups as being those topological groups which are compact and totally disconnected (i.e. their only connected subsets are the one-element sets). See eg. Gruenberg [1] for a proof. Now let’s return to Galois theory. Observe first that if L is a subextension of k s |k, then k s is also the separable closure of L. Hence it is Galois over L and Gal (L) is naturally identified with a subgroup of Gal (k). Proposition 2.14 Let L be a subextension of k s |k. Then Gal (L) is a closed subgroup of Gal (k). It is open if and only if L is a finite extension of k.

2.. INFINITE GALOIS EXTENSIONS

29

Proof: Let’s begin with the second assertion. If L is a finite separable extension of k, embed it in a finite Galois extension M|k using Lemma 2.1. Then Gal (M|k) is one of the standard finite quotients of Gal (k) and it contains Gal (M|L) as a subgroup. Let UL be the inverse image of Gal (M|L) by the natural projection Gal (k) → Gal (M|k). Since the projection is continuous and Gal (M|k) has the discrete topology, UL is open. It thus suffices to show UL = Gal (L). We have UL ⊂ Gal (L) for any element of UL fixes L. On the other hand, any element of Gal (L) maps to an element in Gal (M|L) in Gal (M|k), whence the reverse inclusion. The converse will follow from Proposition 2.12 (4) once we have established the first assertion. So assume now L|k is an arbitary subextension of k s |k and write it as a union of finite subextensions Li |k. By what we have just proven, each Gal (Li ) is an open subgroup of Gal (k), hence it is also closed by Proposition 2.12 (4). Their intersection is precisely Gal (L) which is thus closed as well. We have finally arrived at the main theorem of this section. Theorem 2.15 (Main Theorem of Galois Theory – first version) Let k be a field, G its absolute Galois group. Then associating to each subextension L of k s |k its absolute Galois group Gal (L) gives a bijection between subextensions of k s |k and closed subgroups of G. Here finite extensions correspond to open subgroups. The subfield L is Galois over k if and only if H is a normal subgroup of G; in this case we have an isomorphism Gal (L|k) ∼ = G/H. Proof: Associating to a subextension L|k of k s |k the absolute Galois group Gal (L) gives a closed subgroup by the previous proposition. Its fixed field is exactly L for k s is Galois over L. Conversely, given a closed subgroup H ⊂ G, it fixes some extension L|k and is hence contained in Gal (L). To show equality, let σ be an element of Gal (L), and take any fundamental open neighbourhood UM of the identity in Gal (L), corresponding to a Galois extension M|L. Now H ⊂ Gal (L) surjects onto Gal (M|L) by the natural projection since by assumption any element of M \ L is moved by some element of H; in particular some element of H must map to the same element in Gal (M|L) as σ. Hence H contains an element of the coset σUM and, as UM was chosen arbitrarily, this implies that σ is in the closure of H in Gal (L). But H is closed by assumption, whence the claim. The assertion about finite extensions now follows from the preceding proposition and the last assertion is proven exactly as in the finite case (see references for Theorem 1.10). Remarks 2.16 We conclude with the following observations.

30

CHAPTER 1. GALOIS THEORY OF FIELDS 1. The theorem holds more generally for any infinite Galois extension. Statement and proof are literally the same. 2. To see that the Galois theory of infinite extensions is really different from the finite case we must exhibit non-closed subgroups in the Galois group. We have already seen an example in the second and third examples of 2.11: the absolute Galois group of a finite field is isomorphic ˆ which in turn contains Z as a non-trivial dense (hence non-closed) to Z ˆ The origsubgroup; there are in fact many copies of Z embeddded in Z. inal example of Dedekind was very similar though it did not involve an absolute Galois group but the Galois group of an infinite subextension of Q|Q: he took the field Q(µp∞ ) obtained by adjoining to Q all roots of unity of p-power order. One easily verifies Gal (Q(µp∞ )|Q) ∼ = Zp which also contains Z as a dense subgroup. 3. However, Dedekind did not determine the Galois group itself (profinite groups have not yet been discovered at the time); he just showed the existence of a non-closed subgroup. His proof was generalised in Krull[1] to establish the existence of non-closed subgroups in the Galois group of any infinite extension as follows. First one shows that given a non-trivial Galois extension K2 |K1 , any automorphism of K1 may be extended to an automorphism of K2 in at least two ways. From this one infers by taking an infinite chain of non-trivial Galois subextensions of an infinite Galois extension L|k that Gal (L|k) is uncountable. By the same argument (and the first remark above), any infinite closed subgroup of Gal (L|k) is uncountable, hence countably infinite subgroups are never closed.

3.

´ Finite Etale Algebras

In this section we give a second variant of the Main Theorem which is often referred to as “Grothendieck’s formulation of Galois theory”. We start again from a base field k of which we fix separable and algebraic ¯ Let L be a finite separable extension of k; here we do not closures k s ⊂ k. consider L as a subextension of k s . We know that L has only finitely many k-algebra homomorphisms into k¯ (the number of these is equal to [L : k] by Lemma 1.2); actually the images of these homomorphisms are contained in k s . So we may consider the finite set Homk (L, k s ) which is endowed by a natural left action of Gal (k) given by (g, φ) 7→ g ◦ φ for g ∈ Gal (k), φ ∈ Homk (L, k s ).

´ 3.. FINITE ETALE ALGEBRAS

31

Lemma 3.1 The above left action of Gal (k) on Homk (L, k s ) is continuous and transitive. In the case when L is a Galois extension of k, this left Gal (k)set is isomorphic to the (left) coset space Gal (k)/U, where U is the stabiliser of any element of Homk (L, k s ). Proof: The stabiliser U of an element φ consists of the elements of Gal (k) fixing φ(L). Hence by Proposition 2.14, U is open in Gal (k) which means precisely Gal (k) acts continuously on Homk (L, k s ). If L is generated by a primitive element α with minimal polynomial f , any φ ∈ Homk (L, k s ) is given by mapping α to a root of f in k s . Since Gal (k) permutes these roots transitively, the Gal (k)-action on Homk (L, k s ) is transitive. As the stabiliser of the element gφ is just gUg −1, we find that the stabilisers of the elements of Homk (L, k s ) are all conjugate open subgroups. Moreover, mapping gφ to gU induces an isomorphism of the Gal (k)-set Homk (L, k s ) with the left coset space of U. In the case when L is a Galois extension of k, we get from Theorem 2.15 that all stabilisers are equal to the same open normal subgroup U of Gal (k), whence the second assertion. If M is another finite separable extension of k, any k-homomorphism φ : L → M induces a map Homk (M, k s ) → Homk (L, k s ) by composition with φ. This map is clearly Gal (k)-equivariant so we have obtained a contravariant functor from the category of finite separable extensions of k to the category of finite sets with continuous transitive left Gal (k)-action. Theorem 3.2 Let k be a field with fixed separable closure k s . Then the contravariant functor which associates to a finite separable extension L|k the finite Gal (k)-set Homk (L, k s ) gives an anti-equivalence between the category of finite separable extensions of k and the category of finite sets with continuous and transitive left Gal (k)-action. Here Galois extensions give rise to Gal (k)-sets isomorphic to some finite quotient of Gal (k). Proof: We have already seen the last statement. To prove the rest, we check that Homk ( , k s ) satisfies the conditions of Chapter 0, Lemma 1.9. We begin by the second condition, i.e. that any continuous transitive left Gal (k)set S is isomorphic to some Homk (L, k s ). Indeed, pick a point s ∈ S. The stabiliser of s is an open subgroup Us of Gal (k) which fixes a finite separable extension L of k. Now define a map of Gal (k)-sets Homk (L, k s ) → S by the rule g ◦ i 7→ gs, where i is the natural inclusion L → k s and g is any element of G. This map is well-defined since the stabiliser of i is exactly Us and is readily seen to be an isomorphism; in fact, both Gal (k)-sets become isomorphic to the coset space G/Us by the maps sending i (resp. s) to Us .

32

CHAPTER 1. GALOIS THEORY OF FIELDS

For the first condition we have to show that given two finite separable extensions L, M of k, the set of k-homomorphisms L → M corresponds bijectively to the set of Gal (k)-maps Homk (M, k s ) → Homk (L, k s ). Since both Homk (M, k s ) and Homk (L, k s ) are transitive Gal (k)-sets, any Gal (k)map f between them is determined by the image of a fixed φ ∈ Homk (M, k s ). As f is Gal (k)-equivariant, any element of the stabiliser U of φ fixes f (φ) as well, whence an inclusion U ⊂ V , where V is the stabiliser of f (φ). By what we have just seen, taking the fixed subfields of U and V respectively, we get an inclusion of subfields of k s which is none but the extension f (φ)(L) ⊂ φ(M). Denoting by ψ : φ(M) → M the map inverse to φ we readily see that ψ◦f (φ) is the unique element of Homk (L, M) inducing f . If we wish to extend the previous anti-equivalence to Gal (k)-sets with not necessarily transitive action, the natural replacement for finite separable extensions of k is the following. Definition 3.3 A finite dimensional k-algebra A is ´etale (over k) if it is isomorphic to a finite direct sum of separable extensions of k. Theorem 3.4 (Main Theorem of Galois Theory – second version) Let k be a field. Then the functor which associates to a finite ´etale k-algebra A the finite set Homk (A, k s ) gives an anti-equivalence between the category of finite ´etale k-algebras and the category of finite sets with continuous left Gal (k)-action. Here separable field extensions give rise to sets with transitive Gal (k)-action and Galois extensions to Gal (k)-sets isomorphic to finite quotients of Gal (k). Proof: This follows from the previous theorem in view of the remark that L given a decomposition A = Li into a sum of fields and an element φ ∈ s Homk (A, k ), then φ induces the injection of exactly one Li to k s ; indeed, if φ(Li ) 6= 0, then being a field, Li injects to k s , and on the other hand, a sum Li ⊕ Lj cannot inject into k s since k s has no zero-divisors. Thus Homk (A, k s ) decomposes into the disjoint union of the Homk (Li , k s ); this is in fact its decomposition into Gal (k)-orbits. To conclude this section, we wish to give another characterisation of finite ´etale k-algebras which ties in with more classical treatments. The reader who needs some background on tensor products may look back to Chapter 0 first. Proposition 3.5 Let A be a finite dimensional k-algebra. Then the following are equivalent: 1. A is ´etale.

´ 3.. FINITE ETALE ALGEBRAS

33

¯ 2. A ⊗k k¯ is isomorphic to a finite sum of copies of k; 3. A ⊗k k¯ has no nilpotent elements. In the literature, finite dimensional k-algebras satisfying the third condition of the proposition are often called separable k-algebras. The proposition thus provides a structure theorem for these. For the proof we need the following lemma which is the commutative version of the Weddernburn-Artin theorem. Lemma 3.6 Let F be a field, A a finite-dimensional F -algebra. Then A is a direct product of finite field extensions of F if and only if A has no nonzero nilpotent elements. Proof: (after Fr¨ohlich-Taylor [1]) One implication is trivial. For the other, by decomposing A into a finite direct sum of indecomposable F -algebras, we may assume that A is indecomposable itself. Notice that under this restriction A can have no idempotent elements other than 0 and 1; indeed, if e 6= 0, 1 were an idempotent then A ∼ = Ae ⊕ A(1 − e) would be a nontrivial direct sum decomposition since e(1 − e) = e − e2 = 0 by assumption. The lemma will follow if we show that any nonzero element x ∈ A is invertible and thus A is a field. Since A is finite dimensional over F , the descending chain of ideals (x) ⊃ (x2 ) ⊃ . . . (xn ) ⊃ . . . must stabilise and thus for some m we must have xn = xn+1 y with an appropriate y. By iterating this formula we get xn = xn+i y i for all positive integers i, in particular xn = x2n y n . Thus xn y n = (xn y n )2 , i. e. xn y n is an idempotent. By what has been said above there are two cases. If xn y n = 0, then xn = (xn )(xn y n ) = 0 which is a contradiction since x 6= 0 by assumption and A has no nonzero nilpotents. Otherwise, xn y n = 1 and thus x is invertible. Remark 3.7 The lemma already implies that a finite-dimensional algebra over a perfect field is ´etale if and only if it contains no nilpotents. So the somewhat less manageable third condition of the proposition arises only over non-perfect fields. Proof of Proposition 3.5. Let’s begin by proving that the first condition implies the second. We may clearly restrict to finite separable extensions L of k. We then have L = k[X]/(f ) with some polynomial f which decomposes ¯ By Chapter 0, Corollary as a product of n distinct factors (X − αi ) in k. ∼ ¯ ), so we conclude by the chain of isomorphisms 3.12 L ⊗k k¯ = k[X]/(f ¯ ¯ k[X]/(f ) = k[X]/(X − α1 ) . . . (X − αn ) ∼ =

n M i=1

¯ k[X]/(X − αi ) ∼ =

n M i=1

¯ k,

34

CHAPTER 1. GALOIS THEORY OF FIELDS

the middle isomorphism holding by the Chinese Remainder Theorem (see e.g. Lang [1], Chapter II, Theorem 2.1). The derivation of the third condition from the second is immediate; actually, the lemma applied to A ⊗k k¯ shows that they are equivalent. Now to derive the first condition from the second, let A be the quotient of A by its k-subalgebra of nilpotent elements. The lemma implies that A is a sum of finite extension fields of k. Since k¯ contains no nilpotent elements, any k-algebra homomorphism A → k¯ factors through A and hence through one of its decomposition factors L. By lemma 1.2, the number of k-algebra homomorphisms L → k¯ can equal at most the degree of L over k, with equality ¯ has at most dim k (A) elif and only if L|k is separable, whence Homk (A, k) ements with equality iff A = A and A is ´etale. On the other hand, we have a canonical bijection of finite sets ¯ ∼ ¯ k). ¯ Homk (A, k) = Homk¯ (A ⊗k k, ¯ tensoring [To see this, observe that given a k-algebra homomorphism A → k, ¯ by k¯ and composing by the multiplication map gives a k-homomorphism ¯ ¯ ¯ ¯ A ⊗k k → k ⊗k k → k; on the other hand the natural inclusion k → k¯ induces a k-homomorphism A ∼ = A ⊗k k → A ⊗k k¯ which composed by ¯ ¯ homomorphisms A ⊗k k → k gives a map from the set on the right hand side to that on the left which is clearly inverse to the previous construction.] The ¯ assumption now implies that the set on the right hand side has dim k¯ (A ⊗k k) ¯ elements. But by Chapter 0, Proposition ??, dim k¯ (A ⊗k k) = dim k A so A is indeed separable.

Chapter 2 Fundamental Groups in Topology In the last section we saw that when studying extensions of some field it is plausible to conceive the base field as a point and a finite separable extension (or, more generally, a finite ´etale algebra) as a finite discrete set of points mapping to this base point. Galois theory then equips the situation with a continuous action of the absolute Galois group which leaves the base point fixed. It is natural to try to extend this situation by taking as a base not just a point but a more general topological space. The role of field extensions would then be played by certain continuous surjections, called covers, whose fibres are finite (or, even more generally, arbitrary discrete) spaces. We shall see in this chapter that under some restrictions on the base space one can develop a topological analogue of the Galois theory of fields, the part of the absolute Galois group being taken by the fundamental group of the base space. We shall also find an analogue of the separable closure, called the universal cover, whose group of relative automorphisms over the base space will be exactly the fundamental group. However, we point out a notable difference already at this stage: in contrast to the field-theoretic case the fundamental group will not be profinite in general and the orbits of points of the universal cover under its action will not necessarily be finite either. This indicates that when we shall be looking for an algebraic theory that subsumes both the Galois theory of the previous chapter and the topological constructions of the present one, we shall have to restrict to covers with finite fibres. 35

36

1.

CHAPTER 2. FUNDAMENTAL GROUPS IN TOPOLOGY

Covers

Let us begin with some seemingly uninteresting terminology. Definition 1.1 Let X be a topological space. A space over X is a topological space Y together with a continuous map p : Y → X. The category of spaces over X is the category whose objects are spaces over X and a morphism between to objects pi : Yi → X (i = 1, 2) is given by a continuous map f : Y1 → Y2 which makes the diagram f

Y1 −−−→ Y2 

p1  y

id

 p y 2

X −−−→ X

commute. The category of spaces over X, denoted by TopX is naturally a subcategory of the category Top of topological spaces. It is readily seen that a morphism f : Y1 → Y2 of spaces over X is an isomorphism in TopX if and only if f is an isomorphism in Top, i.e. it is a homeomorphism. Now the basic definition of this section is the following. Definition 1.2 A covering space of a topological space X, or a cover of X for short, is a space Y over X where the projection p : Y → X is subject to the following condition: each point of X has an open neighbourhood V for which p−1 (V ) decomposes as a disjoint union of open subsets Ui of Y such that the restriction of p to each Ui induces a homeomorphism of Ui with V . We define a morphism between two covers of a space X to be a morphism of spaces over X. Thus we get a natural full subcategory CovX of the category TopX . Example 1.3 Take a discrete topological space I and form the topological product X × I. The first projection X × I → X makes X × I into a space over X which is immediately seen to be a cover. It is called the trivial cover. Trivial covers may at first seem very special but as the next proposition shows, every cover is locally a trivial cover. Proposition 1.4 A space Y over X is a cover if and only if each point of X has an open neighbourhood V such that the restriction of the projection p : Y → X to p−1 (V ) is isomorphic in the category TopV to a trivial cover.

37

2.. GROUP ACTIONS AND GALOIS COVERS

Proof: One implication follows from the previous example and the other is ` easily seen as follows: given a decomposition p−1 (V ) ∼ Ui for some index = i∈I

set I as in the definition of covers, the map associating to ui ∈ Ui the couple ` (p(ui ), i) defines a homeomorphism of Ui onto V × I, where I is endowed i∈I

with the discrete topology. It is immediate from the construction that this is an isomorphism in TopV if we view V × I as a trivial cover of V .

Notice that with the notation of the previous proof, the set I is the fibre of p over the points of V . The proof shows that the points of X such that the fibre of p is I as a discrete set form an open subset of X; for varying I they give a decomposition of X into a disjoint union of open subsets. Thus: Corollary 1.5 If X is connected, the fibres of p are all equal to the same discrete set I. Notice that this does not mean at all that the cover is trivial. Indeed, let us give an example of a non-trivial cover with a connected base. Example 1.6 Consider a rectangle XY ZW and divide the sides XY and ZW into two equal segments by the points P and Q. Identifying the sides XY and ZW with opposite orientations we get a M¨obius strip on which the image of the segment P Q becomes a closed curve C homeomorphic to a circle. The natural projection of the boundary B of the M¨obius strip onto C coming from the perpendicular projection of the sides XW and Y Z of the rectangle onto the segment P Q makes B a space over C which is actually a cover since locally it is a product of a segment by a two-point space, i.e. a trivial cover of the segment. However, the cover itself is non-trivial since B is not homeomorphic to a disjoint union of two circles.

2.

Group Actions and Galois Covers

Perhaps the most basic statement of classical Galois theory is that given a finite Galois extension L|k, all intermediate extensions arise as invariants under some subgroup H of Gal (L|k). When one seeks an analogy in the context of covers the first difficulty one has to face that it is not completely evident to find a counterpart of an innocently looking fact hidden in the previous statement: that the set of invariants under H is a subfield of L. This will be our first goal. Henceforth we shall fix a base space X which will be assumed locally connected (i.e. each point has a basis of neighbourhoods consisting of connected

38

CHAPTER 2. FUNDAMENTAL GROUPS IN TOPOLOGY

open subsets). Given a cover p : Y → X, its automorphisms are to be automorphisms of Y in the category CovX , i.e. topological automorphisms compatible with the projection p. They clearly form a group that we shall denote by Aut(Y |X). By convention all automorphisms will be assumed to act from the left. Note that for each point x ∈ X, Aut(Y |X) maps the fibre p−1 (x) onto itself, so p−1 (x) is endowed by a natural action of Aut(X). Next we wish to find a necessary and sufficient condition for a topological automorphism of Y to be an element of Aut(Y |X). Lemma 2.1 Any automorphism φ of a connected cover p : Y → X having a fixed point must be trivial. Corollary 2.2 Any automorphism of a cover having a fixed point fixes the connected component of the point elementwise. Instead of proving the lemma we establish a more general statement which will also be needed later. The lemma follows from it by taking Z = Y , f = id and g = φ. Proposition 2.3 Let p : Z → X be a cover, Y a connected topological space, f, g : Y → Z two continuous maps satisfying p ◦ f = p ◦ g. If there is a point y ∈ Y with f (y) = g(y), then f = g. Proof: Suppose y ∈ Y is as above, z = f (y) = g(y). Take some connected open neighbourhood V of p(z) satisfying the condition in the definition of a cover (such a V exists since X is locally connected) and let Ui ∼ = V be the component of p−1 (V ) containing z. By continuity f and g must both map some open neighbourhood W of y into Ui . Since p ◦ f = p ◦ g and p maps Ui homeomorphically onto V , f and g must agree on W . The same type of reasoning shows that if f (y 0) 6= g(y 0) for some point y 0 ∈ Y , f and g must map a whole open neighbourhood of y 0 to different components of p−1 (V ). Thus the set {y ∈ Y : f (y) = g(y)} is nonempty, open and closed in Y so by connectedness it is the whole of Y . To proceed further, it is convenient to introduce some terminology (adopting the convention of Fulton [1] in contrast with the term “properly discontinuous” of classical parlance). Definition 2.4 Let G be a group acting from the left on a topological space Y . The action of G is even if each point y ∈ Y has some open neighborhood U such that the open sets gU are pairwise disjoint for all g ∈ G.

2.. GROUP ACTIONS AND GALOIS COVERS

39

This being done, we can state: Proposition 2.5 If p : Y → X is a connected cover, then the action of Aut(Y |X) on Y is even. Proof: Let y be a point of Y and let V be a connected open subset of X ` such that p−1 (V ) = Ui as in Definition 1.2 and such that a subset in this decomposition, say Ui , contains y. We contend that Ui satisfies the condition of the above definition. Indeed, since Y is connected, Lemma 2.1 applies and shows that any nontrivial φ ∈ Aut(Y |X) should map Ui isomorphically onto some Uj , with j ∈ I different from i. Moreover, if φ(Ui ) = ψ(Ui ) for some φ, ψ ∈ Aut(Y |X), then by composing with p we see that φ should coincide with ψ pointwise on Ui and thus we conclude by applying Lemma 2.1 to φ ◦ ψ −1 that φ = ψ. Now that we have found a necessary condition for an automorphism of a connected space Y to be an automorphism of some cover, we shall show that it is also sufficient, which will furnish the promised topological analogue of constructing field extensions by taking invariants under group actions. Recall that if a group G acts on a topological space Y , one may form the quotient space Y /G whose underlying set is by definition the set of orbits under the action of G and the topology is the finest one which makes the projection Y → Y /G continuous. Proposition 2.6 If G is a group acting evenly on a connected space Y , the projection pG : Y → Y /G turns Y into a cover of Y /G. The automorphism group of this cover is precisely G. Proof: For the first statement, note that pG is surjective and moreover each x ∈ Y /G has an open neighbourhood of the form V = pG (U) with a U as in Definition 2.4. This V is readily seen to satisfy the condition of Definition 1.2. For the second statement, notice first that we may naturally view G as a subgroup of Aut(Y |(Y /G)). Now given an element φ in the latter group, look at its action on an arbitrary point y ∈ Y . Since the fibres of pG are precisely the orbits of G we may find g ∈ G with φ(y) = gy. Applying Lemma 2.1 to the automorphism φ ◦ g −1 we get g = φ. Example 2.7 With this tool at hand, one can give lots of examples of covers. 1. Let Z act on R by translations (which means that the automorphism defined by n ∈ Z is the map x 7→ x + n). We obtain a cover R → R/Z, where R/Z is immediately seen to be homeomorphic to a circle.

40

CHAPTER 2. FUNDAMENTAL GROUPS IN TOPOLOGY 2. The previous example can be generalised to any dimension: take any basis {x1 , . . . , xn } of the vector space Rn and make Zn act on Rn so that the i-th direct factor of Zn acts by translation by xi . This action is clearly even and turns Rn into a cover of what is called a linear torus; for n = 2, this is the usual torus. The subgroup Λ of Rn generated by the xi is usually called a lattice; thus linear tori are quotients of Rn by lattices. 3. For an integer n > 1 denote by µn the group of n-th roots of unity. Multiplying by elements of µn defines an even action on C∗ := C \ {0}, whence a cover pn : C∗ → C∗ /µn . In fact, the map z 7→ z n defines a natural homeomorphism of C∗ /µn onto C∗ (even an isomorphism of topological groups) and via this homeomorphism pn becomes identified with the cover C∗ → C∗ given by z 7→ z n . Note that this map does not extend to a cover C → C; this phenomenon will be studied further in Chapter 4.

Remark 2.8 It is easy to show that quite generally if a finite group G acts without fixed points on a Hausdorff space Y , its action is even and thus it furnishes a cover Y → Y /G for connected Y . Now given a connected cover p : Y → X, we may form the quotient of Y by the action of Aut(Y |X). It is immediate from the definition of cover automorphisms that the projection p factors as a composite of continuous maps p

Y → Y /Aut(Y |X) → X where the first map is the natural projection. Definition 2.9 A cover p : Y → X is said to be Galois if Y is connected and the induced map p above is a homeomorphism. Remark 2.10 Note the similarity of the above definition with that of a Galois extension of fields. This analogy is further confirmed by remarking that the cover pG in Proposition 2.6 is Galois. Proposition 2.11 A connected cover p : Y → X is Galois if and only if Aut(Y |X) acts transitively on the fibres p−1 (x) of p.

2.. GROUP ACTIONS AND GALOIS COVERS

41

Proof: Indeed, the underlying set of Y /Aut(Y |X) is by definition the set of orbits of Y under the action of Aut(Y |X) and so the map p is one-toone precisely when each such orbit is equal to a whole fibre of p, i.e. when Aut(Y |X) acts transitively on the fibres. Remark 2.12 In fact, for a connected cover p : Y → X to be Galois it suffices for Aut(Y |X) to act transitively on one fibre. Indeed, in this case Y /Aut(Y |X) is a connected cover of X where one of the fibres consists of a single element; it is thus isomorphic to X by Remark 1.5. Example 2.13 We give an example of a Galois cover. Let X = Rn /Λ, Λ∼ = Zn be a linear torus and m > 1 be an integer. The multiplication-by-m map of Rn maps the lattice Λ into itself and hence induces a map X → X. It is easily seen to be a Galois cover of degree mn . Now we can state the topological analogue of Chapter 1, Theorem 1.10. Theorem 2.14 Let p : Y → X be a Galois cover, H a subgroup of G = Aut(Y |X). Then p induces a natural map pH : Y /H → X which turns Y /H into a cover of X. Conversely, if Z → X is a connected cover fitting into a commutative diagram f

Y −−−→ Z 

p y

id

 q y

X −−−→ X

then f : Y → Z is a Galois cover and actually Z ∼ = Y /H with some subgroup H of G. In this way we get a bijection between subgroups of G and intermediate covers Z as above. The cover q : Z → X is Galois if and only if H is a normal subgroup of G, in which case Aut(Z|X) ∼ = G/H. Before starting the proof we need an general lemma on covers. Lemma 2.15 Assume given a locally connected space X, a connected cover q : Z → X and a continuous map f : Y → Z. If the composite q ◦ f : Y → X is a cover then so is f : Y → Z.

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Proof: Let z be a point of Z, x = q(z) and V a connected open neighbourhood of X satisfying the property of Definition 1.2 for both p = q ◦ f and ` ` q: p−1 (V ) = Ui and q −1 (V ) = Vj . Here for each Ui its image f (Ui ) is a connected subset of Z mapping onto V by q, hence there is some j with f (Ui ) ⊂ Vj . But this is in fact a homeomorphism since both sides get mapped homeomorphically onto V by q. This implies in particular that the image of f is open. Now to prove the lemma we show first that f is surjective. It is enough to see by connectedness of Z that the complement of f (Y ) in Z is open, so assume that some point z of Z has no preimage by f . Then the whole component Vj of q −1 (V ) containing z must be disjoint from f (Y ) since otherwise by the previous argument the whole of Vj would be contained in f (Y ) which is a contradiction, whence the claim. To conclude the proof of the lemma it remains to notice that the preimage of the above Vj is a disjoint union of some Ui ’s. Proof of Theorem 2.14: We have already seen above that pH exists as a continuous map. By Proposition 1.4, over sufficiently small subsets V of X, p−1 (V ) is of the form V × F with F a discrete set (the fibre of p over each point of V ) endowed by an H-action. The open set p−1 H (V ) ⊂ Y will then be isomorphic to a product of V by the discrete set of H-orbits of F , so by applying Proposition 1.4 again we conclude that pH : Y /H → X is a cover. For the converse, apply the previous lemma to see that f : Y → Z is a cover. Then H = Aut(Y |Z) is a subgroup of G so to see that the cover is Galois it suffices by Proposition 2.11 to see that H acts transitively on each fibre of f . So take a point z ∈ Z and let y1 and y2 be two points of f −1 (z). They are both contained in the fibre p−1 (q(z)) so since p : Y → X is Galois, we have y1 = φ(y2 ) with some φ ∈ G. We are done if we show φ ∈ H, which is equivalent to saying that the subset S = {y ∈ Y : f (y) = f (φ(y))} is equal to the whole of Y . But this follows from Proposition 2.3, applied to our actual Y , Z and f as well as g = f ◦ φ. It is immediate that the two constructions above are inverse to each other so only the last statement remains. One implication is easy: if H is normal in G, then G/H acts naturally on Z = Y /H and this action preserves the projection q so we obtain a group homomorphism G/H → Aut(Z|X) which is readily seen to be injective. But Z/(G/H) ∼ = Y /G ∼ = X, so G/H ∼ = Aut(Z|X) and q : Z → X is Galois. For the reverse implication we employ an elegant argument found by G. Braun. Assume that Z → X is a Galois cover. We first show that under this assumption any element φ of G = Aut(Y |X) induces an automorphism of Z over X. In other words, we need an automorphism ψ : Z → Z which can be

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43

inserted into the commutative diagram: φ

Y −−−→ Y  

  yf

fy

Z

 q y

Z id

 q y

X −−−→ X

For this take a point y ∈ Y with image x = (q◦f )(y) in X. By commutativity of the diagram f (y) and f (φ(y)) are in the same fibre q −1 (x) of q. Since Aut(Z|X) acts transitively on the fibres of q, there is an automorphism ψ ∈ Aut(Z|X) with the property that ψ(f (y)) = f (φ(y)). In fact, ψ is the unique element of Aut(Z|X) with this property for if λ ∈ Aut(Z|X) is another one, Lemma 2.1 implies that ψ ◦ λ−1 is the identity. We contend that ψ is the map we are looking for, i.e. the maps ψ ◦ f and f ◦ φ are the same. Indeed, both maps are continuous maps from the connected space Y to Z which coincide in the point y and moreover their compositions with q are equal, so the assertion follows from Proposition 2.3. Now it is manifest that in this way we obtain a homomorphism G → Aut(Z|X). Its kernel is none but H = Aut(Y |Z) which is thus a normal subgroup in G.

3.

The Main Theorems of Galois Theory for Covers

The principal result of the last section was the topological analogue of the classical formulation of Galois theory. Now we turn to the analogues of the two versions of the Main Theorem of Galois Theory formulated in Chapter 1. This is only possible under some restrictions on the base space X. So assume henceforth that X is a connected and locally simply connected space and fix a base point x ∈ X. Consider the set-valued functor Fibx on the category CovX of covers of X which associates to a cover p : Y → X the fibre p−1 (x). This is indeed a functor since any morphism of covers respects by definition the fibre over x. Theorem 3.1 Let (X, x) be a pointed topological space as above. ˜ → X. 1. The functor Fibx is representable by a Galois cover π : X ˜ 2. The group Aut(X|X) is naturally isomorphic to the fundamental group π1 (X, x).

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˜ → X as in the theorem is called a universal Definition 3.2 A cover π : X cover of X. By definition, the universal cover has the property that cover maps from ˜ → X to a fixed cover p : Y → X correspond bijectively (and in a π : X ˜ functorial way) to points of the fibre p−1 (x) ⊂ Y . In particular, since X ∼ ˜ = itself is a cover of X via π, we have a canonical isomorphism Fibx (X) ˜ ˜ ˜ Hom(X, X); by this isomorphism the identity map of X corresponds to a canonical element x˜ of the fibre π −1 (x) called the universal element. Now for an arbitrary cover p : Y → X and element y ∈ π −1 (x) the cover map ˜ → Y corresponding to y via the isomorphism Fibx (Y ) ∼ ˜ Y) πy : X = Hom(X, maps x˜ to y by commutativity of the diagram = ˜ X) ˜ −−∼ ˜ Hom(X, −→ Fibx (X)

  y

∼

  y

˜ Y ) −−=−→ Fibx (Y ) Hom(X, where the vertical maps are induced by πy . ˜ is the representing object of a fuctor, it is unique up to unique Since X isomorphism according to the Yoneda Lemma; note, however, that it a priori depends on the base point x. In fact, Corollary 3.8 below will show that changing the base point induces a (generally non-unique) isomorphism between the two corresponding universal covers. We postpone the proof of the theorem to the next section. It is there that we shall also show the following a priori non-obvious statement: Lemma 3.3 Under the assumptions of the theorem, assume that p : Y → X is a cover of X and q : Z → Y is a cover of Y . Then p ◦ q : Z → X is a cover of X. In the rest of the section we take the statements of Theorem 3.1 and Lemma 3.3 for granted and discuss some important corollaries and complements. For these we need the following easy lemma whose proof is left to the readers. Lemma 3.4 Let X be a topological space. If X satisfies any of the properties: 1. X is Hausdorff; 2. X is locally connected; 3. X is locally path-connected;

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45

4. X is locally simply connected, then so does any cover Y → X of X. This being said, we see that if X satisfies the condition required by the theorem, then so does any connected cover p : Y → X. Proposition 3.5 Take (X, x) as in the previous theorem, and let p : Y → X ˜ of X is a universal cover be a connected cover. Then the universal cover X −1 of Y , i.e. it represents Fiby for some y ∈ p (x). ˜ represents Fibx , the point y corresponds to a canonical Proof: Since X ˜ → Y which turns X ˜ into a cover of Y by virtue ofLemma 2.15. map πy : X Our task is to show that this cover represents the functor Fiby . So take a cover q : Z → Y and pick a point z ∈ q −1 (y). Since by Lemma 3.3 the composition p ◦ q turns Z into a cover of X with z ∈ (p ◦ q)−1 (x), the point z ˜ → Z of covers of X mapping the universal corresponds to a morphism πz : X ˜ to z. It is now enough to see that πz is also a morphism of point x˜ of X covers of Y , i.e. πy = q ◦ πz . But p ◦ πy = p ◦ q ◦ πz by construction and moreover both πy and q ◦ πz map the universal point x˜ to y so the assertion follows from Lemma 2.3. ˜ is its own universal cover; hence it is simply Corollary 3.6 The space X connected. Proof: Only the simply connectedness needs a proof but it follows from ˜ ∼ ˜ X) ˜ ∼ the second statement of the theorem in view of π1 (X) = Aut(X| = 1. Combining the above results with Theorem 2.14, we get the following analogue of the first version of the main theorem of Galois theory discussed in Chapter 1. Theorem 3.7 Let X be a connected, locally connected and locally simply connected topological space and x ∈ X a fixed base point. Then for each connected cover p : Y → X and base point y ∈ p−1 (x) there is a natural injective homomorphism π1 (Y, y) → π1 (X, x) which endows the universal ˜ with an action of π1 (Y, y) so that we have X/π ˜ 1 (Y, y) ∼ cover X = Y . In this way we get a bijection between connected covers of Y and subgroups of π1 (X, x) where Galois covers correspond to normal subgroups. In the next section we shall show that the homomorphism π1 (Y, y) → π1 (X, x) whose existence is stated in the theorem is indeed the natural map induced by the projection p. See Remark 4.8 (1).

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The theorem allows one to give a simple criterion for a cover to be universal which also settles the question of independence of the choice of base point. ˜ → X is a Corollary 3.8 For X as in the theorem, a connected cover π : X universal cover of X (i.e. it represents Fibx for some x ∈ X) if and only if ˜ is simply connected. X Proof: We have already seen one implication in Corollary 3.6. The other implication follows from the Theorem since it implies the chain of isomor˜0 ∼ ˜ 0 /π1 (X, ˜ x˜) ∼ ˜ with some universal cover X ˜ 0 and point phisms X = X = X ˜ x˜ ∈ X. Of course, what is really hidden behind this corollary is the independence of the fundamental group of base points. This will become more transparent in the next section where we shall explicitly construct a universal cover. The corollary allows one to detect universal covers explicitly. Example 3.9 Since Rn is simply connected for any n, we see that the first two examples in 2.7 actually give universal covers of the circle and of linear tori, respectively. On the other hand, the third example there does not give a universal cover since C∗ is not simply connected. However, the complex plane C, being a 2-dimensional R-vector space, is simply connected and the exponential map C → C∗ , z 7→ exp(z) is readily seen to be a cover. Hence C is the universal cover of C∗ . Example 3.10 Let us investigate in more detail an example that will serve in Chapter 4. Let D˙ be the punctured complex disc {z ∈ C : z 6= 0, |z| ≤ 1}. As in the previous example, the exponential map z 7→ exp(z) restricted to the left half plane L = {z ∈ C : Re z ≤ 0} furnishes a universal cover ˙ This map is in fact a homomorphism from the additive group L to of D. ˙ its kernel is isomorphic to Z, being given by the multiplicative group D; integer multiples of 2πi. Thus if we let Z act on L by translation by 2πi, D˙ becomes the quotient of L by this action. But then by the theorem any connected cover of D˙ is isomorphic to a quotient of L by a subgroup of Z. These subgroups are Z itself and the subgroups kZ for integers k > 1; the corresponding covers of D˙ are L and D˙ itself via the map z 7→ z k . We next state the second form of the main theorem of Galois theory for covers. So that it can be completely analogous to the field case, it is convenient to make some convention about left and right actions which is enabled by the following definition.

3.. THE MAIN THEOREMS OF GALOIS THEORY FOR COVERS

47

Definition 3.11 Let G be any group. The opposite group of G, denoted by Gop , is the group having the same elements as G but which has the product rule (α, β) 7→ βα. The help of this definition is that any right action by a group on some set or space defines canonically a left action of the opposite group, simply by the rule (α, x) → xα. Now in our present situation, in which the fundamental group plays the role of the absolute Galois group, it is a right action of π1 (X, x) on the fibres which naturally arises, in contrast with the left action in the previous chapter. Indeed, observe that if p : Y → X is a cover, then the fibre p−1 (x) is equipped with a natural right action by π1 (X, x) as follows: a point y ∈ ˜ → Y ; on the p−1 (x) corresponds by Theorem 3.1 to a homomorphism πy : X ˜ via the isomorphism in other hand, the natural left action of π1 (X, x) on X Theorem 3.1 (2) induces a right action on cover maps to Y : here the action of α ∈ π1 (X, x) maps πy to πy ◦ α. This action is called the monodromy action on the fibre p−1 (x). (Again we shall see in the next section how to obtain this action in a more explicit way; see Remark 4.8 (2).) So from this canonical right action of π1 (X, x) on the fibres we obtain a canonical left action by π1 (X, x)op and we can state: Theorem 3.12 Let X and x be as in the previous theorem. Then the functor Fibx induces an equivalence of the category CovX with the category of left π1 (X, x)op -sets. Here connected covers correspond to π1 (X, x)op -sets with transitive action and Galois covers to coset spaces of normal subgroups. Proof: The proof is strictly parallel to that of Chapter 1, Theorem 3.4: we check that the functor satisfies the conditions of Chapter 0, Lemma 1.9. For full faithfulness we have to show that given two covers p : Y → X and q : Z → X, any map f : Fibx (Y ) → Fibx (Z) of π1 (X, x)op -sets comes from a unique map Y → Z of covers of X. For this we may assume Y , Z are ˜ via the map πy : X ˜ →Y connected and that actually Y is a quotient of X corresponding to some fixed element y ∈ Fibx (Y ). In fact, πy makes Y ˜ by the stabiliser Uφ of φ; let ψy : Y → X/U ˜ φ isomorphic to the quotient of X be the inverse map. Since Uφ injects into the stabiliser of f (φ) via f , the ˜ → Z corresponding to f (φ) induces a map X/U ˜ φ→Z natural map πz : X by passing to the quotient; composing it with ψy gives the required map Y → Z. For the second condition we have to show that any right π1 (X, x)-set S is isomorphic to the fibre of some cover of X. For S transitive we may take ˜ by the action of the stabiliser of some point; in the general the quotient of X

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case we decompose S into its π1 (X, x)-orbits and take the disjoint union of the corresponding covers. Remark 3.13 Notice the discrepancy with Chapter 1, Theorem 3.4: here we obtain an equivalence of categories whereas there we had an anti-equivalence. This problem will be resolved in Chapter 5 where we introduce the category of finite ´etale k-schemes which is the opposite category to that of finite ´etale k-algebras; for the k-schemes we will thus indeed get an equivalence of categories with the category of Gal (k)-sets. We now state a corollary of the above theorem that is even closer to Chapter 1, Theorem 3.4 in its formulation and which will be invoked in subsequent chapters. First a definition: call a cover Y → X finite if it has finite fibres. Corollary 3.14 Let X and x be as in the theorem. Then the functor Fibx induces an equivalence of the category of finite covers of X with the category dx)op -sets. Connected covers correspond to finite of finite continuous left π1 (X, dx)op -sets with transitive action and Galois covers to coset spaces of open π1 (X, normal subgroups. dx)op denotes the profinite completion of π (X, x)op (Chapter Here π1 (X, 1 1, Example 2.11 (1)).

Proof: The group π1 (X, x)op acts on fibres of a connected cover through a finite quotient if and only if the cover is finite. Whence an action by dx)op factoring through a finite quotient. To show it is continuous, it π1 (X, suffices to show that the stabiliser of any point, which is a subgroup of finite index, contains a normal subgroup of finite index, for then these are both dx)op by definition. But the latter property holds for open subgroups of π1 (X, any group (take the intersection of conjugates of a subgroup of finite index). dx)op on a finite set factors Conversely, any continuous action of π1 (X, through a finite quotient, which is also a quotient of π1 (X, x)op , whence the result. We conclude this section by proving a useful compatibility. Remark 3.15 Let p : Y → X be a cover. For a point x ∈ X, we have two canonical left group actions on the fibre p−1 (x): one is the action by Aut(Y |X) and the other is the monodromy action by π1 (X, x)op . We contend that these two actions are compatible, i.e. we that have (φ(y))α = φ(yα)

4.. CONSTRUCTION OF THE UNIVERSAL COVER

49

for y ∈ π −1 (x), φ ∈ Aut(Y |X) and α ∈ π1 (X, x). To check this, use again ˜ → Y of covers; φ(y) then Theorem 3.1 to identify y with a morphism πy : X becomes just φ ◦ πy and yα becomes πy ◦ α where α is again viewed as an ˜ Thus the desired compatibility is simply a consequence automorphism of X. of the associativity of the composition rule for continuous maps.

4.

Construction of the Universal Cover

The section is devoted to the proof of Theorem 3.1. As its second statement suggests, the universal cover must have something to do with path homotopies, so we shall make use of their theory and refer the readers in need to Chapter 0 for background. The crucial lemma to be used in construction is the following important result on “lifting paths and homotopies”. Lemma 4.1 Let p : Y → X be a cover, y a point of Y and x = p(y). 1. Given any path f : [0, 1] → X with f (0) = x, there is a unique path f˜ : [0, 1] → Y with f˜(0) = y and p ◦ f˜ = f . 2. Assume moreover given a second path g : [0, 1] → X homotopic to f . Then the unique g˜ : [0, 1] → Y with g˜(0) = y and p ◦ g˜ = g has the ˜ i.e. we have f˜(1) = g˜(1). same endpoint as f, Actually, the proof will show that in the second situation the liftings f˜ and g˜ are homotopic but this will not be needed later. Proof: For the first statement, note first that uniqueness follows from Proposition 2.3 applied with X, Y and Z replaced by our actual X, [0, 1] and Y . The existence is immediate in the case of a trivial cover. To reduce the general case to this, for each x ∈ f ([0, 1]) choose some open neighbourhood Vx satisfying the condition in the definition of a cover. The sets f −1 (Vx ) form an open covering of the interval [0, 1] from which we may extract a finite subcovering since the interval is compact. We may then choose a subdivision 0 = t0 ≤ t1 ≤ . . . ≤ tn = 1 of [0, 1] such that each closed interval [ti , ti+1 ] is contained in some f −1 (Vx ) (simply put a ti where two open intervals of the form f −1 (Vx ) overlap); hence the cover is trivial over each f ([ti , ti+1 ]). We can now construct f˜ inductively: given a lifting f˜i of the path f restricted to [t0 , ti ] (the case i = 0 being trivial), we may construct a lifting of the restriction of f to [ti , ti+1 ] starting from f˜i (ti ); piecing this together with fi gives fi+1 .

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For the second statement, notice that given a homotopy h : [0, 1] × [0, 1] → X with h(0, t) = f (t) and h(1, t) = g(t), one can construct a lifting ˜ : [0, 1] × [0, 1] → Y with p ◦ h ˜ = h, h(0, ˜ t) = f (t) and h(1, ˜ t) = g(t). The h construction is similar to that for f : first choose a sufficiently fine subdivision of [0, 1] × [0, 1] into small subsquares Sij so that over each h(Sij ) the cover is trivial. That this may be done is assured by a well-known fact from the topology of compact metric spaces called Lebesgue’s lemma (see e.g. Schubert [1], Theorem 7.4.4; we have used a trivial case of it above). Then proceed by piecing together liftings over each subsquare, moving “serpent˜ 0) = y) towards the point wise” from the point (0, 0) (for which we put h(0, (1, 1). Note that by uniqueness of path lifting it is sufficient to find a local lifting which coincides with the previous one at the left corner of the side where two squares meet; they will then coincide over the whole of the common side. Again by uniqueness of path lifting we get successively that the path ˜ t) is f˜ (since both are liftings of f starting from y), the path t 7→ h(0, ˜ 0) is the constant path [0, 1] → {y} and that t 7→ h(1, ˜ t) is none but s 7→ h(s, ˜ ˜ g˜. Finally, s 7→ h(s, 1) is a path joining f(1) and g˜(1) which lifts the constant path [0, 1] → {f (1)}; by uniqueness it must coincide with the constant path [0, 1] → {f˜(1)}, whence f˜(1) = g˜(1). Here is a first application. Proposition 4.2 Any cover of a simply connected and locally path-connected space is trivial. Proof: It is enough to show that given a connected cover p : Y → X of a space X satisfying the assumptions of the proposition, the map p is injective. For this, note first that Y is locally path-connected by Lemma 3.4 and so by Chapter 0, Corollary 4.8 it is actually path-connected. Now assume there exist points y0 6= y1 in Y with p(y0 ) = p(y1 ). By path-connectedness of y there is a path f˜ : [0, 1] → Y with f˜(0) = y0 and f˜(1) = y1 . The path f˜ is the unique lifting starting from y0 of the path f = p ◦ f˜ which is a closed path around x = p(y0) = p(y1 ). Since X is simply connected, this path is homotopic to the constant path [0, 1] → {x} of which the constant path [0, 1] → {y0 } provides the unique lifting to Y starting from y0 . But this is a contradiction with the second statement of the previous lemma according to which liftings of homotopic paths should have the same endpoints. It is now easy to give a proof of Lemma 3.3. We restate it as a corollary: Corollary 4.3 Let X be a locally simply connected space. Given two covers p : Y → X and q : Z → Y , their composite p ◦ q : Z → X is also a cover of X.

4.. CONSTRUCTION OF THE UNIVERSAL COVER

51

Proof: Given any x ∈ X, choose a simply connected neighbourhood U. According to the proposition, the restriction of p to p−1 (U) gives a trivial cover of U. Repeating this argument for q over each of the connected components of p−1 (U) (which are simply connected and locally path-connected themselves, being isomorphic to U) we get that the restriction of p ◦ q to (p ◦ q)−1 (U) is a trivial cover of U. Let us now proceed to the proof of Theorem 3.1. We construct the ˜ as follows. The points of X ˜ are to be homotopy classes of paths space X starting from x. The projection π is defined in the following straightforward ˜ is represented by a path f : [0, 1] → X with fashion: any point y˜ ∈ X f (0) = x and we put π(˜ x) = f (1) = y. Note that this gives a well-defined map by the second statement of Lemma 4.1. We define next the topology ˜ by taking as a basis of open neighbourhoods of a point y˜ the following on X sets U˜ : we start from a simply connected neighbourhood U of π(˜ y ) and if ˜ f : [0, 1] → X is a path representing y˜, we define Uy˜ to be the set of homotopy classes of paths obtained by composing the homotopy class of f with the homotopy class of some path g : [0, 1] → X with g(0) = y and g([0, 1]) ⊂ U. Notice that since U is assumed to be simply connected, two such g having the same endpoints have the same homotopy class. Thus in more picturesque terms, U˜ is obtained by “continuing homotopy classes of paths arriving at y to other points of U”. This gives indeed a basis of open neighbourhoods of y˜ ˜y˜ and V˜y˜, their intersection U˜y˜ ∩ V˜y˜ contains for given two neighbourhoods U ˜ y˜ with some simply connected neighbourhood W of y contained in U ∩ V ; W one also sees immediately that π is continuous with respect to this topology. It is equally immediate that we have obtained a cover of X as the inverse image by π of any simply connected neighbourhood of a point y will be the ˜y˜ for all inverse images y˜ of y. Finally note disjoint union of the open sets U that there is a “universal element” x˜ of the fibre π −1 (x) corresponding to the homotopy class of the constant path. ˜ is connected. Lemma 4.4 The space X ˜ is path-connected, for which we show Proof: It is enough to see that X ˜ that there is a path in X connecting the universal point x˜ to any other point x˜0 . Indeed, let f : [0, 1] → X be a path representing x˜0 . The multiplication map m : [0, 1] × [0, 1] → [0, 1], (s, t) 7→ st is continuous, hence so is f ◦ m and the restriction of f ◦m to any subset of the form {s}×[0, 1]; such a restriction defines a path fs from x˜ to f (s), with f0 the constant path [0, 1] → {˜ x} and ˜ f1 = f . The definition of the topology of X implies that the map associating to s ∈ [0, 1] the homotopy class of fs is continuous and thus defines the

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˜ beginning at x˜. path we need; in fact, it is the unique lifting of f to X Alternatively, one may start by taking this unique lifting and check by going through the construction that its endpoint is indeed x˜0 . ˜ → X defined above represents the functor Fibx . Lemma 4.5 The cover X Proof: We have to show that for any cover p : Y → X any point y of the fibre p−1 (x) corresponds in a canonical and functorial manner to a morphism ˜ → X to p : Y → X. Now if πy exists, the fact that it πy of covers from π : X ˜ its image should be a morphism of covers implies that for any point x˜0 ∈ X, πy (˜ x0 ) should map by p to the endpoint f (1) of any path f : [0, 1] → X representing x˜0 . So there is only one reasonable choice for πy : given a point ˜ ˜ represented by a path f : [0, 1] → X, we map it to f(1) x˜0 ∈ X where ˜ ˜ f : [0, 1] → Y is the unique path lifting f with f (0) = y whose existence is guaranteed by the first part of lemma 4.1. The second part of the lemma implies that this map is well defined; there is no difficulty in checking that this is indeed a map of covers. The map y 7→ πy is a bijection between ˜ → X to p : Y → X for p−1 (x) and the set of morphisms from π : X an inverse is given by mapping a morphism f to the image f (˜ x) of the distinguished element x˜. Finally, we have obtained an isomorphism between ˜ Y ). Indeed, for any morphism the functors Y → Fibx (Y ) and Y → Hom(X, 0 0 of p : Y → X to some cover p : Y → X mapping y to some y 0 ∈ Y 0 the ˜ Y ) → Hom(X, ˜ Y 0 ) maps πy to πy0 since these are the induced map Hom(X, maps mapping x˜ to y and y 0, respectively. ˜ → X is Galois. Lemma 4.6 The cover π : X ˜ Proof: By Remark 2.12 it is enough to show that Aut(X|X) acts transi−1 tively on the fibre π (x). By Lemma 4.5 for any point y˜ of the fibre π −1 (x) ˜ → X ˜ compatible with π and mapping we have a continuous map πy˜ : X the universal element x˜ to y˜. We show that πy˜ is an automorphism. Since ˜ is connected, by Lemma 2.15 πy˜ endows X ˜ with a structure of a cover X x). Since over itself – in particular it is surjective. Take an element z˜ ∈ πy−1 ˜ (˜ ˜ π ◦ πy˜ : X → X is also a cover of X according to Lemma 3.3 (proven in Corollary 4.3), we may apply Lemma 4.5 to this cover to get a continuous ˜ →X ˜ with πz˜(˜ and surjective map πz˜ : X x) = z˜ and π ◦ πy˜ ◦ πz˜ = π. But ˜ by Lemma 2.3. By πy˜ ◦ πz˜(˜ x) = x˜, hence πy˜ ◦ πz˜ is the identity map of X surjectivity of πz˜ this implies that πy˜ is injective and we are done. It remains to prove the second statement of Theorem 3.1. We recall it in a separate lemma:

4.. CONSTRUCTION OF THE UNIVERSAL COVER

53

∼ ˜ Lemma 4.7 There is a natural isomorphism Aut(X|X) = π1 (X, x). ˜ is endowed by a natural left action of π1 (X, x) Proof: First observe that X ˜ and an element α ∈ π1 (X, x) with defined as follows: given a point x˜0 ∈ X respective path representatives f and fα , we may take the composition fα • f and then take the homotopy class of the product. It is straightforward to ˜ →X ˜ thus obtained is continuous and compatible check that the map φα : X with π, i.e. it is a cover automorphism. Moreover, we have obtained a group ˜ homomorphism π1 (X, x) → Aut(X|X) which is injective since any nontrivial α moves the distinguished element x˜. It remains to prove the surjectivity of ˜ this homomorphism. For this purpose, take an arbitrary φ ∈ Aut(X|X) and 0 0 ˜ a point x˜ ∈ X represented by some path f : [0, 1] → X. The point φ(˜ x ) is −1 then represented by some g : [0, 1] → X. Now g • f is a closed path around x in X with (g • f −1 ) • f = g. Let α be the class of g • f −1 in π1 (X, x); we show that φα = φ. Indeed, the automorphism φ ◦ φ−1 ˜0 and thus it is α fixes x ˜ and Lemma 2.1. the identity by connectedness of X Remarks 4.8 We conclude this section by showing how the above construction makes some statements of the previous section more explicit. 1. First we clear up the issue invoked after the statement of Theorem 3.7. Namely, if p : Y → X is a connected cover, then by virtue of ˜ by some subgroup H of π1 (X, x). the theorem it is a quotient of X ˜ On the other hand, since X is also a universal cover of Y , we have H ∼ = π1 (Y, y) where y is some base point, whence an injection iy : π1 (Y, y) → π1 (X, x) which depends, of course, on the choice of y. Now if we choose y in the fibre over x, we claim that iy is nothing but the natural inclusion induced by mapping closed paths passing through y ˜ represented by the to paths through x. Indeed, take a point x˜0 ∈ X path f : [0, 1] → X. We know by construction that x˜0 as a point of the universal cover of Y is represented by the homotopy class of some path in Y starting from y. This path, however, must map to f by the projection p, hence it is the unique lifting f˜ of f to Y starting from y. Now we know by the proof of the previous lemma that the images of x˜0 under the action of π1 (Y, y) are given by composing f˜ with closed paths through y; the projection p maps the situation to composing f with paths through x which is precisely what we had to show. 2. Secondly we give a more concrete description of the monodromy action of π1 (X, x)op , i.e. the canonical right action of π1 (X, x) on the fibre p−1 (x) of some connected cover p : Y → X. By Lemma 4.5 any point

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CHAPTER 2. FUNDAMENTAL GROUPS IN TOPOLOGY ˜ → Y and y of the fibre corresponds to a morphism of covers πy : X ˜ represented by paths f to the proof shows that πy maps points of X points of the form f˜(1) where f˜ : [0, 1] → Y is the lifting of f with ˜ = y; in particular y, being the image of the class of the constant f(0) path c : [0, 1] → {x}, corresponds to the constant path [0, 1] → {y}. We invoke again the proof of the previous lemma to see that if an element of π1 (X, x) is represented by a path fα , it acts on c by mapping it to the class of fα . Hence the action of α on p−1 (x) maps y to f˜α (1), where f˜α : [0, 1] → Y is the canonical lifting of fα starting from y. This is the promised concrete description.

Chapter 3 Locally Constant Sheaves In this chapter we give a reinterpretation of the second form of the main theorem of Galois Theory for covers in terms of locally constant sheaves. Esoteric as these objects may seem to a novice, they stem from reformulating in a modern language very classical considerations from analysis, such as the study of local solutions of holomorphic differential equations. These theories predate the invention of covers themselves, so what we present here as a reformulation of the theory of the previous chapter in fact gives a natural framework for the study of older concepts, as the example of the last section will demonstrate.

1.

Sheaves

In this section we introduce the notion of a sheaf, to which the first step is the following definition. Definition 1.1 Let X be a topological space. A presheaf of sets F on X is a rule which associates to each nonempty open subset U ⊂ X a set F (U) and each inclusion of open sets V ⊂ U a map ρU V : F (U) → F (V ), the maps ρU U being identity maps and the identity ρU W = ρV W ◦ ρU V holding for a tower of inclusions W ⊂ V ⊂ U. Similarly, one defines a presheaf of groups (or abelian groups, or rings, etc.) by requiring that the F (U) be groups (abelian groups, rings...) and the ρU V homomorphisms. Elements of F (U) are called sections of the presheaf over U. Remark 1.2 Here is a more fancy formulation. Let’s associate a category XTop to our space X by taking as objects the nonempty open subsets U ⊂ X and by defining Hom(V, U) to be the one-element set consisting of the natural inclusion V → U whenever V ⊂ U and to be empty otherwise. 55

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Then a presheaf of sets is just a contravariant functor XTop → Sets and similarly a presheaf of abelian groups, for instance, is a contravariant functor XTop → Ab. With this interpretation, we see immeadiately that presheaves of sets (abelian groups, etc.) on a fixed topological space X form a category: for morphisms one takes morphisms of contravariant functors. By definition, this means that a morphism of presheaves Φ : F → G is a collection of maps (or homomorphisms) ΦU : F (U) → G(U) such that for each inclusion V ⊂ U the diagram ΦV F (V ) −−− → G(V )  

ρF UV y

Φ

  G yρU V

U F (U) −−− → G(U)

commutes. Example 1.3 The basic example to bear in mind is that of continuous realvalued functions defined locally on open subsets of X; in this case, the maps ρU V are given by restriction of functions to some open subset. Motivated by the example, given an inclusion V ⊂ U, we shall also use the more suggestive notation s|V instead of ρU V (s) for a section s ∈ F (U). The presheaf in the above example has a particular property, namely that continuous functions may be patched together over open sets. More precisely, given two open subsets U1 and U2 and continuous functions fi : Ui → R for i = 1, 2 with the property that f1 (x) = f2 (x) for all x ∈ U1 ∩ U2 , we may unambiguously define a function f : U1 ∪ U2 → R by setting f (x) = fi (x) if x ∈ Ui . In fact, most presheaves we shall encounter in the sequel will share this property so it is convenient to axiomatise it and state it as a definition. Definition 1.4 A presheaf F (of sets, abelian groups, etc.) is a sheaf if it satisfies the following two axioms: 1. Given any nonempty open set U and any covering {Ui : i ∈ I} of U by nonempty open sets, if two sections s, t ∈ F (U) satisfy s|Ui = t|Ui for all i ∈ I, then s = t. 2. For any open covering {Ui : i ∈ I} of U as above, given a system of sections {si ∈ F (Ui ) : i ∈ I} with the property that si |Ui ∩Uj = sj |Ui ∩Uj whenever Ui ∩ Uj 6= ∅, there exists a (by the previous property unique) section s ∈ F (U) such that s|Ui = si for all i ∈ I.

1.. SHEAVES

57

We define the category of sheaves (of sets, abelian groups, etc.) on a space X to be the full subcategory of the corresponding presheaf category, i.e. a morphism of sheaves is just a morphism of the underlying presheaves. We conclude this section by some more examples. Example 1.5 First, an easy example of a presheaf of sets which is not a sheaf. Take a set S containing at least two elements s1 6= s2 . Given a topological space X, define a presheaf FS by setting FS (U) = S for all nonempty open sets U ⊂ X and ρU V = idS for all open inclusions V ⊂ U. This is indeed a preshaf; in fact, it is the unique contravariant functor from XTop to the category constisting of the single object S and single morphism idS . Now if X is not an irreducible topological space, i.e. if there exist two nonempty disjoint open subsets U1 , U2 in U, then FS is not a sheaf since there is no section s ∈ FS (U1 ∪ U2 ) with s|U1 = s1 and s|U2 = s2 . Examples 1.6 Next we turn to examples of sheaves. 1. As we have seen above, for any topological space X, real or complex valued continuous functions defined locally on some open subset of X form a sheaf of abelian groups, and even a sheaf of rings with the usual ring operations for functions. More generally, if Y is any topological space (or abelian group, ring, etc.), continuous functions U → Y defined on some open U ⊂ X with the natural restriction operators define a sheaf of sets (resp. abelian groups, rings, etc.) on X. 2. If D is a connected open subset of C, we can define a the sheaf of holomorphic functions on D to be the sheaf of rings whose sections over some open subset U ⊂ D are complex functions holomorphic on U. This construction carries over to any complex manifold. Similarly one can define the sheaf of analitic functions on some real analytic manifold, or the sheaf of C ∞ functions on a C ∞ manifold, and so on. We shall encounter concrete examples of these in the next chapter on Riemann surfaces. 3. Constant sheaves. A special case of the first example is when we take Y to be a discrete topological space (or abelian group, etc.) In this case the corresponding sheaves are called constant sheaves: the name comes from the fact that over a connected open subset U the sections are just constant functions, i.e. the sheaf associates Y to U. Note that in contrast to example 1.5, here we indeed get a sheaf: the anomaly encountered there does not arise for the sheaf associates Y ×Y to a disjoint union of nonempty open subsets.

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CHAPTER 3. LOCALLY CONSTANT SHEAVES 4. Skyscraper sheaves. Here is a more eccentric example. Fix an abelian group A and a point x of a given topological space X. Define a presheaf of abelian groups Fx on X by the rule Fx (U) = A if x ∈ U and Fx (U) = 0 otherwise, the restriction morphisms being the obvious ones. This presheaf is easily seen to be a sheaf, called the skyscraper sheaf over x.

2.

Locally Constant Sheaves and Their Classification

Given an open subset U of a topological space X, there is an obvious notion of the restriction of a presheaf F from X to U: one simply considers the sections of F only over those open sets which are contained in U. This remark enables us to define the main objects of study in this chapter. Definition 2.1 A sheaf F on a topological space is locally constant if each point of X has an open neighbourhood U such that the restriction of F to U is isomorphic (in the category of sheaves on U) to a constant sheaf. In fact, as we shall instantly show, these are very familiar objects. Assume henceforth that all spaces are locally connected. First a definition which will ultimately explain the use of the terminology “section” for sheaves. Definition 2.2 Let p : Y → X be a space over X, U ⊂ X an open set. A (continuous) section of p over U is a continuous map s : U → Y such that p ◦ s = idU . Now let p : Y → X be a space over X. To this space we can associate a presheaf FY of sets on X as follows: for an open set U ⊂ X define FY (U) as the set of sections of p over U and given an inclusion V ⊂ U define the restriction map FY (U) → FY (V ) by restricting sections to V . Proposition 2.3 Let p : Y → X be a space over X. Then the presheaf FY is in fact a sheaf. If moreover p : Y → X is a cover, then FY is locally constant. Here FY is constant if and only if the cover is trivial. We shall call the sheaf FY in the proposition the sheaf of local sections of Y . (Note that FY depends on Y as a space over X so there is actually a slight abuse of notation here.)

2.. LOCALLY CONSTANT SHEAVES AND THEIR CLASSIFICATION59 Proof: The sheaf axioms follow from the fact that the sections over U are continuous functions U → Y and hence satisfy the patching properties. Now assume p : Y → X is a cover. Given a point x ∈ X, take a connected open neighbourhood V of x over which the cover is trivial, i.e. isomorphic to V × F where F is the fibre over x. The image of any section V → Y is a connected open subset mapped isomorphically onto V by p, hence it must be one of the connected components of p−1 (V ). Thus sections over V correspond bijectively to points of the fibre F and the restriction of FY to V is isomorphic to the constant sheaf defined by F . We get the constant sheaf if and only if we may take V to be connected component containing x in the above argument. Thus for instance Example 1.6 which shows a simple example of a nontrivial cover also gives, via the preceding proposition, a simple example of a locally constant but non-constant sheaf. Now we shall show that conversely, given any locally constant shaf F on a space X, one can cook up a cover pF : XF → X whose sheaf of local sections will be exactly F . However, just as before, we shall first present a construction working for any presheaf F and then show that in the case of locally constant sheaves we obtain covers. To begin with, notice that in the previous construction when we start from a cover p : Y → X, then for any point x ∈ X the set FY (U) is just the fibre p−1 (x) over x for all connected open neighbourhoods of x over which the cover is trivial; thus, roughly speaking, FY (U) is “equal to p−1 (x) for all sufficiently small U”. In fact, this statement can be made precise for more general spaces over X as follows. Define a partial order on the system of open neighbourhoods of x by setting U ≤ V whenever V ⊂ U. This partially ordered set is directed since for any U, V we have U, V ≤ U ∩ V . Now the sets FY (U) form a direct system indexed by this partially ordered set with respect to the restriction maps of the sheaf FY . Before stating the next lemma, recall that a continuous map p : Y → X is called a local homeomorphism if any y ∈ Y has an open neighbourhood mapped homeomorphically onto its image by p. This is the same as saying that for any y the image p(y) has an open neighbourhood over which p admits a continuous section with open image. Lemma 2.4 Assume that p : Y → X is a local homeomorphism. Then mapping a section s ∈ FY (U) to s(x) induces a bijection of sets lim F (U) ∼ = p−1 (x) →

where the direct limit is taken over all open neighbourhoods U of x.

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CHAPTER 3. LOCALLY CONSTANT SHEAVES Note that covers satisfy the assumption of the lemma.

Proof: Recall that by definition elements of the direct limit on the left are sections of F over some open neighbourhood of x modulo the equivalence relation that identifies two sections if they coincide over some smaller open neighbourhood. This shows that the map of the lemma is well defined and also that it is injective since if two sections si : Ui → Y (i = 1, 2) both map x to the same point y ∈ p−1 (x), then s1 (U1 ) ∩ s2 (U2 ) must be an open neighbourhood of y mapped homeomorphically onto U1 ∩ U2 by p and the si must coincide over U1 ∩ U2 . Finally surjectivity is immediate from the assumption that p is a local homeomorphism, for it implies that any element y ∈ p−1 (x) has an open neighnourhood over which p has a section mapping x to y. This prompts the following definition. Definition 2.5 Let F be a presheaf of sets (abelian groups, etc.) on a topological space X and let x be a point of X. The stalk Fx of F at x is the direct limit of the direct system of sets (abelian groups, etc.) formed by the F (U) for the open sets U containing x and by the natural restriction morphisms F (U) → F (V ) attached to the inclusions V ⊂ U. Notice that any morphism of presheaves F → G induces a natural morphism on the stalks Fx → Gx . Hence taking the stalk of a presheaf at some point defines a functor from the category of presheaves of sets (abelian groups, etc.) on X to the category of sets (abelian groups, etc.) Now we can finally begin the promised construction. Construction 2.6 The space associated to a presheaf of sets. Let F be a presheaf of sets on X. We define a space pF : XF → X over X as follows. As a set, Fx is to be the disjoint union of the stalks Fx for all x ∈ X. The natural projection pF is then induced by the constant maps Fx → {x}. To define the topology on XF , note first that given an open set U ∈ X, any section s ∈ F (U) defines a map is : U → XF which maps x ∈ U to the image of s in the stalk Fx . This being said, we may take on XF the topology for which the sets is (U) for all open U ⊂ X and all s ∈ F (U) form a basis of open sets. The construction shows that the projection pF is continuous for this topology; it is even a local homeomorphism as all the is are continuous sections of pF over the appropriate U. Remark 2.7 Note that pF may have continuous sections which are not of the form is : indeed, given an open cover {Uj : j ∈ I} of U and sections

2.. LOCALLY CONSTANT SHEAVES AND THEIR CLASSIFICATION61 sj ∈ F (Uj ) which agree over the intersections Uj ∩ Uk , the sj may not patch together to give a section of F over U if F is not a sheaf but the continuous maps isj do path together to give a continuous section of pF over U. We can infer something positive from the previous remark, namely that any section s ∈ F (U) is naturally a section of the sheaf of local sections of pF : XF → X over U. The latter sheaf is usually denoted F ] and is called the sheaf associated to the presheaf F . Moreover, the above argument gives a natural morphism of presheaves F → F ] . Proposition 2.8 If F is a sheaf, then the sheaf of local sections of the space pF : XF → X constructed above is isomorphic F : in fact, the natural map F → F ] is an isomorphism. If moreover F is locally constant, then pF endows XF with the structure of a cover over X. Proof: The first statement is immediate from the above considerations. To prove the second one, take a point x ∈ X and a connected open neighbourhood U of x over which F is constant. Then, as we have seen above, we have Fx ∼ = Fy for all y ∈ U, hence p−1 = F (U) ∼ F (U) is isomorphic to U × Fx . Remark 2.9 The proposition implies that any morphism of presheaves F → G with G a sheaf factors through the natural morphism F → F ] ; in other words, the sheaf F ] represents the functor on the category of sheaves that associates to each G the set of presheaf morphisms F → G. For any (locally connected) topological space denote by LocX the full subcategory of the category TopX of spaces over X whose objects are spaces p : Y → X for which p is a local homeomorphism. Note that the rule Y 7→ FY is a functor from the category LocX to the category of sheaves of sets: any morphism φ : Y → Z in LocX induces a morphism FY → FZ of sheaves by composing sections with the map φ (if p : Y → X and q : Z → X are the respective structure maps, then composing a section s of p with φ gives indeed a section of q, in view of the equalities q ◦ φ ◦ s = p ◦ s = id). Proposition 2.10 The functor Y → FY induces an equivalence of the category LocX with the category of sheaves of sets on X. In this equivalence covers of X correspond to locally constant sheaves of sets on X.

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Proof: It suffices to prove the first statement and again it is convenient to apply the criterion of chapter 0, Lemma 1.9. The second condition follows from the previous proposition. For fully faithfulness, assume given a morphism of sheaves of the form φF : FY → FZ , with p : Y → X and q : Z → X two objects of LocX . For any point x ∈ X, φ induces a map on the stalks φF ,x : FY,x → FZ,x , i. e. a map on the fibres p−1 (x) → q −1 (x) by Lemma 2.4. Define a map φ : Y → Z by setting φ(y) = φF ,x (y), where x = p(y). To see that φ is continuous, choose an open neighbourhood U of y such that q|U is a homeomorphism. Then φ−1 (U) = p−1 (q(U)) is an open set over which φ is equal to the composition of continuous maps q −1 ◦ p. Thus φ is a map of spaces over X and one checks by going through the constructions that it is indeed the unique map Y → Z inducing φF . In the rest of this section we consider only locally constant sheaves. Combining Proposition 2.10 with Theorem 2.14, we get: Theorem 2.11 Assume X is a connected and locally simply connected topological space and fix a point x ∈ X. Then the category of locally constant sheaves of sets on X is equivalent to the category of sets endowed with a left action of π1 (X, x)op . This equivalence is induced by the functor mapping a sheaf F to its stalk Fx at x. Now we consider sheaves with values in sets with additional structure. Recall that for a group G a left G-module is an abelian group A endowed by a left action of G satisfying σ(a + b) = σa + σb for all a, b ∈ A and σ ∈ G. Extending the action of G by linearity we see that this is the same as giving a left module over the (generally non-commutative) group ring Z[G]. Theorem 2.12 Let X and x be as above. Then the category of locally constant sheaves of abelian groups on X is equivalent to the category of left π1 (X, x)op -modules. This equivalence is induced by the functor mapping a sheaf F to its stalk Fx at x. More generally, for any commutative ring R the category of locally constant sheaves of R-modules on X is equivalent to the category of left modules over R[π1 (X, x)op ]. Proof: We only consider abelian groups, the case of modules being similar. First we have to show that for any point x ∈ X the stalk Fx of a sheaf of abelian groups is a left π1 (X, x)op -module, i.e. the action of π1 (X, x) is compatible with the addition law. For this let F × F be the sheaf defined by (F × F )(U) = F (U) × F (U) for all U; its stalk over a point x is just Fx × Fx . Then the addition law on F is a morphism of sheaves F × F → F given

3.. LOCAL SYSTEMS

63

over an open set U by the formula (s1 , s2 ) 7→ s1 + s2 ; one sees immediately that the morphism Fx × Fx → Fx obtained by applying the stalk functor is the addition law on Fx . But this latter map is a map of π1 (X, x)op -sets, which means precisely that σ(s1 + s2 ) = σs1 + σs2 for all s1 , s2 ∈ Fx and σ ∈ π1 (X, x)op . One still has to verify that a morphism of sheaves induces an additive map on the stalks, that any π1 (X, x)op -module comes from an sheaf of abelian groups, not just a sheaf of sets, and that any morphism of π1 (X, x)op -modules is induced by an additive morphism of sheaves. All these can be verified by applying the same trick to the construction proving the previous theorem; we leave the details to the reader.

3.

Local Systems

In this section we investigate a most interesting special case of the preceding construction, which is also the one that first arose historically. Definition 3.1 A complex local system on X is a locally constant sheaf of finite dimensional complex vector spaces. If X is connected, all stalks must have the same dimension, which is called the dimension of the local system. With this definition we can state the following corollary of Theorem 2.12: Corollary 3.2 Assume X is a connected and locally simply connected topological space and fix a point x ∈ X. Then the category of complex local systems on X is equivalent to the category of finite dimensional left representations of π1 (X, x)op . Thus to give a local system on X is the same as giving a homomorphism π1 (X, x)op → GL(n, C) for some n. This representation is called the monodromy representation of the local system. The following example shows where to find local systems “in nature”. It uses the straightforward notion of a subsheaf of a sheaf F : it is a sheaf whose sections over each open set U form a subset (subgroup, subspace etc.) of F (U). Example 3.3 Let D be a complex domain (i.e. a connected open subset of the complex plane). Consider over D a homogenous n-th order linear differential equation y (n) + an−1 y (n−1) + . . . + a1 y 0 + y = 0

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where the ai are holomorphic functions over D. We can employ a classical trick which consists of considering the y (i) as indeterminates and adding (n − 1) new equations of the form (y (i) )0 = y (i−1) for 2 ≤ i ≤ n. Thereby solving the above equation becomes equivalent to solving a system of n homogenous first order linear equations which may be written in the form x0 = Ax

(3.1)

where the variable x is a holomorphic function D → Cn and A is a nonsingular matrix of holomorphic functions. Thus, a solution of the equation (3.1) over an open subset U ⊂ D is a section of the sheaf On of Cn -valued holomorphic functions. This sheaf is the n-fold direct product of the sheaf O of Example 1.6 (2) with itself and is clearly a sheaf of complex vector spaces. Now over each U linear combinations of solutions of (3.1) are also solutions of (3.1) and given an open covering {Ui : i ∈ I} of U, any section x ∈ On (U) whose restrictions to each Ui satisfy (3.1) is a solution to (3.1) over U. Thus the local solutions of the equation (3.1) form a subsheaf of On . Moreover, by a classical theorem due to Cauchy (see eg. Forster [1], Theorem 11.2 or any basic text on differential equations) each point of D has an open neighbourhood U ⊂ D for which S(U) is an n-dimensional vector space over C or, in more down-to-earth terms, there exist n holomorphic solutions x1 , . . . , xn ∈ S(U) such that any solution of (3.1) over U can be written uniquely as a linear combination of the xi . Since the restrictions of the xi to smaller open sets obviously have the same property, we see that the restriction of S to U is a constant sheaf of complex vector spaces. Thus the sheaf S is a local system of dimension n. Remark 3.4 According to Corollary 3.2, the local system S of the above example is uniquely determined by an n-dimensional left representation ρ of π1 (X, x)op , where x is some fixed point of D (notice that D is locally simply connected). Let us first describe ρ explicitly. We know that it is given by the canonical right action of π1 (X, x) on the stalk Sx which is an n-dimensional vector space over C. Take a closed path f : [0, 1] → D representing an element α ∈ π1 (X, x) and take an element s ∈ Sx which is, in classical terminology, a germ of a (vector-valued) holomorphic function satisfying the equation (3.1). Now s is naturally a point of the fibre over x of the cover pS : DS → D associated to S by Proposition 2.8. The action of α on s is then described by Remark 4.8 (2) as follows: s is mapped to the element f˜(1) ˜ of the fibre p−1 S (x) = Sx , where f is the unique lifting of f to DS . By looking at the construction of the unique lifting in the proof of Lemma 4.1, one can make this even more explicit as follows: there exist open subsets U1 , . . . , Uk of D such that the f −1 (U1 ), . . . , f −1 (Uk ) give an open covering of [0, 1] “from

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left to right”(in particular, x ∈ U1 ∩ Uk ) and moreover S is constant over each Ui and there are sections si ∈ S(Ui ) such that the restrictions of si and si+1 to Ui ∩ Ui+1 coincide for all 1 ≤ i ≤ k − 1 and s1 (resp. sk ) maps to s (resp. sα) in Sx . Classically this is expressed by saying that sα is the analytic continuation of the holomorphic function germ s along the the path f representing α. (Notice that the existence and the uniqueness of sα are guaranteed by the fact that S is a locally constant sheaf and hence DS → D is a cover. Had we worked with the bigger sheaf On instead of S, the analytic continuation of an arbitrary germ may not have been possible.) Example 3.5 Let us illustrate the above theory by working out the simplest nontrivial case in detail. Take as D an open disc in the complex plane centered around 0, of radius 1 < R ≤ ∞, with the point 0 removed. We choose 1 as base point for the fundamental group of D. We study the local system of solutions of the first order differental equation x0 = f x

(3.2)

where f is a holomorphic function on D which will be assumed to extend meromorphically into 0. It is well known that the solutions to (3.2) in some neighbourhood of a point x ∈ D are constant multiples of functions of the form exp ◦ F where F is a primitive of f . Thus the solution sheaf is a locally constant sheaf of 1-dimensional complex vector spaces. The reason why it is locally constant but not constant is that, as we learn from complex analysis, the primitive F exists locally but not globally on the whole of D. For instance, a well defined primitive F1 of f exists, for instance, over U1 = D \ (0, −iR) and another primitive F2 over U2 = D \ (0, iR); we may assume that F1 and F2 coincide over U1 ∩ U2 , so in particular at -1 (they may only differ by a constant anyway). Thus the local system of solutions to (3.2) is isomorphic over each Ui (i = 1, 2) to the constant sheaf defined by the one-dimensional subspace of O(Ui ) generated by exp ◦ Fi . Now we compute the monodromy representation π1 (D, 1)op → GL(1, C) of this local system. It is well known (and follows easily from the third example in 3.9) that π1 (D, 1) ∼ = Z; a generator α is given by the class of the path g : [0, 1] → D, t 7→ e2πit which “goes counterclockwise around the unit circle”. In particular, π1 (D, 1) is commutative and hence the same as π1 (D, 1)op . Moreover, any one-dimensional representation of π1 (D, 1) is determined by the image m of α in GL(1, C) ∼ = C∗ . By the recipe of the previous remark, in our case m can be descibed as follows: given any holomorphic function germ φ defined in a neighbourhood of 1 and satisfying (3.2) with x = φ, the analytic continuation of φ along the path g representing α is precisely mφ. But we may take for φ the fuction exp ◦ F1 ; when continuing it analytically

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along g, we obtain exp ◦ F2 (since we have to switch from F1 to F2 when the path “goes out of U1 ”). Thus m = exp (F1 (1))(exp (F2 (1)))−1 = exp (F1 (1) − F1 (−1) + F2 (−1) − F2 (1)) Z

= exp (

0

1

f ◦ g) = exp (2πiRes (f ))

by the Residue Theorem (see e.g. Rudin [1], Theorem 10.42), where Res (f ) denotes the residue of f at 0. So we have obtained a formula for m in terms of the coefficient f occuring in the equation (3.2). Conversely, one sees that for any one-dimensional monodromy representation we may find a differential equation of type (3.2) whose local system has the given monodromy; if m is the image of α, one may take, for example, x0 (z) = (µ/z)x(z) with µ ∈ C satisfying exp (2πiµ) = m. (In fact, the argument works for arbitrary n-dimensional representations ρ : π1 (D, 1) → GL(n, C): if B is a constant matrix B with exp (2πiB) = ρ(α), the monodromy representation of sheaf of local solutions of the equation (3.1) with A(z) = (1/z)B is exactly ρ. See Forster [1], Thm. 11.10 for more details.) Remark 3.6 The above example is the solution of the so-called RiemannHilbert problem in the simplest case. This question, a special form of which was first raised by Riemann in 1857 and which later became the 21st of the famous Hilbert problems, asks the following: given a finite set of points x1 , . . . , xk ∈ C, does any n-dimensional right representation ρ of π1 (C \ {x1 , . . . , xk }, x) (with some x 6= xi ) arise as the monodromy representation of the local system of solutions to a system of n homogenous linear differential equations with coefficients having poles only at the xi ? Moreover, it is required that the system should be of Fuchsian type (which means roughly that the local behaviour of the solution sheaf in the neighbourhood of points contained in {x1 , . . . , xn } ∪ {∞} should be like in the example above; see Forster [1] for a precise formulation). Riemann and Hilbert themselves showed that the answer was positive in some particular cases. The affirmative answer in general came as early as 1908 by work of Plemelj [1]. However, the area has remained a very active field of research up to the present day, for far-reaching generalisations have been obtained since. As a milestone we may mention Deligne’s fundamental work [1] extending the setting to arbitrary dimension, to which Katz [1] gives a nice introduction.

Chapter 4 Riemann Surfaces This chapter forms a bridge between the preceding chapters and those to come. Indeed, Riemann surfaces furnish the best examples for the theory of covers which we have developed so far and at the same time motivate the algebraic theory of the forthcoming chapters. In particular, as we shall see, the theory of Riemann surfaces subsumes the theory of complex algebraic curves, so the study of the former yields sometimes motivation, sometimes ingredients for the study of the latter.

1.

Complex Manifolds

In short, Riemann surfaces are none but complex manifolds of dimension 1, so we begin with the definition of these. Let U be an open subset of Cn . Recall that a function U → C is called holomorphic if it is holomorphic in each variable; a map f : U → Cm is holomorphic if all of its coordinate functions fi : U → C are holomorphic functions for i = 1, . . . , m. Let now X be a connected Hausdorff space. A complex atlas of dimension n on X is an open covering U = {Ui : i ∈ I} of X together with maps fi : Ui → Cn mapping Ui homeomorphically onto an open subset of Cn such that for each pair (i, j) ∈ I 2 the map fj ◦ fi−1 : fi (Ui ∩ Uj ) → Cn is holomorphic. The maps fi are called coordinate charts. Two complex atlases U = {Ui : i ∈ I} and U 0 = {Ui0 : i ∈ I 0 } on X are equivalent if their union (defined by taking all the Ui and Ui0 as a covering of X together with all coordinate charts) is a also complex atlas. Note that the extra condition to be satisfied here is that the maps fj0 ◦ fi−1 should be holomorphic on fi (Ui ∩ Uj0 ) for all Ui ∈ U and Uj ∈ U 0 . Definition 1.1 An n-dimensional complex manifold is a connected Haus67

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dorff space X together with an equivalence class of complex atlases. A onedimensional complex manifold is called a Riemann surface. We shall often refer to the equivalence class of atlases occuring in the above definition as the complex structure on X. Examples 1.2 Let us proceed to some basic examples. 1. Any open subset U ⊂ Cn is endowed with a structure of a complex manifold by the trivial covering U = {U} and the inclusion i : U → Cn . In particular, any open domain D ⊂ C is a Riemann surface. 2. Complex tori. We endow the 2-dimensional real torus R2 /Z2 with the structure of a Riemann surface; one may similarly turn arbitrary even-dimensional linear tori into complex manifolds. Consider C as a 2-dimensional real vector space and let c1 , c2 ∈ C be a basis. The ci generate a discrete subgroup Λ of C isomorphic to Z × Z; let T be the topological quotient. Now cover C by sufficiently small open discs Di such that neither of them contains two points congruent modulo Λ. The image of each Di by the projection C → T is an open subset of T by definition of the quotient topology and the projection maps Di homeomorphically onto its image. The images of the Di thus form an open covering of T . (Since T is compact, being the continuous image of the fundamental parallelogram spanned by the ci , one can even extract a finite subcovering out of the above one.) It is straightforward to check that we have thus obtained a complex atlas. 3. Projective spaces. Consider Cn+1 \ {(0, . . . , 0)} as a topological space and make the multiplicative group C∗ of C act on it by the rule t(x0 , . . . , xn ) = (tx0 , . . . , txn ) for all t ∈ C∗ . The quotient space by this action is easily seen to be a connected Hausdorff space; it is called the n-dimensional complex projective space and denoted by Pn (C). By definition, any point of Pn (C) can be represented by an (n+1)-tuple (x0 , . . . , xn ), with (x0 , . . . xn ) and (y0 , . . . yn ) identified if (x0 , . . . xn ) = (ty0 , . . . tyn ) with some t ∈ C∗ ; we call the (n + 1)-tuple (x0 , . . . xn ) the homogenous coordinates of the point. Thus points of Pn (C) correspond bijectively to 1-dimensional subspaces of the complex vector space Cn . Now define a complex atlas on Pn (C) as follows. Let Ui be the open subset whose points have homogenous coordinates (x0 , . . . , xn ) with xi 6= 0. The map fi : (x0 , . . . , xi , . . . xn ) 7→ ((x0 /xi ), . . . , (xi−1 /xi ), (xi+1 /xi ), . . . , (xn /xi ))

1.. COMPLEX MANIFOLDS

69

defines a homeomorphism of Ui onto Cn ; indeed, an inverse is given by mapping the point (t1 , . . . , tn ) ∈ Cn to the point of Pn (C) having homogenous coordinates (t0 , . . . , ti−1 , 1, ti , . . . tn ). One checks immediately that the maps (fj ◦ fi−1 )|Cn \{tj =0} are holomorphic. Consider now the unit sphere S = {(x0 , . . . , xn ) ∈ Cn+1 : |x0 |2 + . . . |xn |2 = 1} ⊂ Cn+1 \ {0}. The restriction of the projection Cn+1 \{0} → Pn (C) to S is surjective; in fact, the preimage of each point x ∈ Pn (C) consists of the two antipodal points cut out from S by the line corresponding to x. Since S is known to be compact, Pn (C) as its continuous image by p is compact as well. The one-dimensional case is the projective line P1 (C): as a topological space, it is just C plus a point ∞ having homogenous coordinates (0, 1); the complex atlas consists of the two open sets U0 = C and U1 = P1 (C) \ {0}; the holomorphic map f1 ◦ f0−1 on C \ {0} is the function z 7→ 1/z. 4. Smooth plane curves. Let X be the closed subset of C2 defined as the locus of zeros of a polinomial f ∈ C[x, y]. Assume that there is no point of X where the partial derivatives ∂x f and ∂y f both vanish. We can then endow X with the structure of a Riemann surface as follows. In the neighbourhood of a point where ∂y f is nonzero, define a complex chart by mapping a point to its x-coordinate and similarly for points where ∂x f is nonzero we take the y-coordinate. By the inverse function theorem for holomorphic functions, in a sufficiently small neighbourhood the above mappings are indeed homeomorphisms. Secondly, the holomorphic version of the implicit function theorem implies that in the neighbourhood of points where both x and y define a complex chart, the transition function from x to y is holomorphic, i.e. when ∂y f does not vanish at some point, we may express y as a holomorphic function of x and vice versa. So we indeed have a complex atlas. If now F ∈ C[x, y, z] is a homogenous polynomial of degree d in 3 variables, i.e. F (tx, ty, tz) = td F (x, y, z) for all t 6= 0, then F induces a well defined function on P2 (C), we may thus regard its locus of zeroes Y in P2 (C) which is a closed, hence compact subset. As above, if there is no point in P2 (C) where all partial derivatives of F vanish, then Y can be endowed with the structure of a (compact) Riemann surface. In fact, since this is a local task, we may carry out the above construction

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CHAPTER 4. RIEMANN SURFACES on the three canonical copies of C2 covering P2 (C); on the intersections we of course get the same result. Of course, one can find lots of polynomials satisfying the condition on partial derivatives: the Fermat curves defined by xm + y m + z m are one family of examples.

Remark 1.3 The last example above may be generalised to a system of m (homogenous) polynomial equations in Cn (or Pn (C)): if their Jacobian matrix has rank n − m in each point, we get a complex manifold. Next we define mappings of complex manifolds, beginning with mappings to C. Let X be a complex manifold. A holomorphic function on an open subset U ⊂ X is a continuous map f : U → C such that for any open set Ui belonging to an atlas defining the complex structure on X the map (f ◦ fi−1 )|fi (U ∩Ui ) is holomorphic. By the definition of a complex atlas this notion is independent of the atlas chosen. The holomorphic functions on U form a ring O(U) with respect to the natural addition and multiplication of functions. Thus the rule U 7→ O(U) together with the natural restriction maps for functions defines a sheaf O on X, the sheaf of holomorphic functions. Now let X1 and X2 be two complex manifolds. A continuous map φ : X1 → X2 is holomorphic in the neighbourhood of a point x1 ∈ X1 if there exist open sets U1 ⊂ X1 and U2 ⊂ X2 containing respectively x1 and f (x1 ) and belonging to some atlas defining the complex structures on X1 (resp. X2 ) such that the induced map f2 ◦ φ ◦ (f1 )−1 |f1 (U1 ) is holomorphic. Again one checks that this definition is independent of the choice of the open sets Ui . The map φ is holomorphic if it is holomorphic in some neighbourhood of each point of X1 . Complex manifolds together with holomorphic maps thus form a category of which the category of Riemann surfaces is a full subcategory. We see that a holomorphic function on U ⊂ X1 is none but a holomorphic map U → C (with C endowed by its canonical complex structure as in the first example above). Now we give a second description of the category of complex manifolds using sheaves. For this we first need a definition from sheaf theory. Definition 1.4 Let F be a sheaf on a topological space X1 and φ : X1 → X2 a continuous map. Then we define the push-forward φ∗ F of F by setting φ∗ F (U) = F (φ−1 (U)) for all open U ⊂ X. One immediately checks that φ∗ F is indeed a sheaf. If we now assume X1 and X2 to be complex manifolds, on X1 the sheaf O1 is a subsheaf of the sheaf C1 of continuous functions and φ∗ O1 will then

1.. COMPLEX MANIFOLDS

71

be a sheaf on X2 which is a subsheaf of φ∗ C1 . Now if f is a holomorphic function on U ⊂ X2 , then f ◦ φ is at least a continuous function on φ−1 (U). Thus the rule f 7→ f ◦ φ defines a morphism of sheaves (of rings) O2 → φ∗ C1 . In fact, we have: Lemma 1.5 A continuous map φ : X1 → X2 of complex manifolds is holomorphic if and only if the induced morphism O2 → φ∗ C1 takes its values in the subsheaf φ∗ O1 of φ∗ C1 . Proof: For necessity we need to note merely that if φ is holomorphic, then f ◦ φ is holomorphic for each f ∈ O2 (U). Sufficiency follows from the fact that if a neighbourhood U2 ⊂ X2 of a point φ(x1 ) ∈ φ(X1 ) belongs to a complex atlas, the coordinate chart f2 : U2 → Cn is given by n holomorphic functions on U2 . The assumption means that these holomorphic functions give holomorphic functions on φ−1 (U2 ) when composed with φ. If now U1 ⊂ φ−1 (U2 ) belongs to a complex atlas on X1 , the composition of the above functions with f1−1 |f1 (U1 ) will still remain holomorphic, whence the claim. The lemma enables one to give a new description of complex manifolds. Observe that the complex structure one can define on a given connected Hausdorff space (if it is possible anyway) is by no means unique. For instance, the tori obtained by taking the quotient of C by some discrete subgroup Λ are all homeomorphic, but the complex structures we defined in the above example may differ for different Λ. Thus the statement of the following corollary is nonvacuous. Corollary 1.6 The complex structure on a given complex manifold X is uniquely determined by the underlying topological space and by the sheaf O of holomorphic functions. Proof: By unwinding the definitions, one sees that two complex atlases U and U 0 on the same topological space X are equivalent (and thus define the same complex structure) precisely if the identity map of X is a holomorphic map between the manifolds (X, U) and (X, U 0 ). By the previous lemma, this condition may be checked by looking at the sheaves of holomorphic functions with respect to the two complex structures. Remark 1.7 The above corollary motivates the general definition of complex analytic spaces as found in Gunning [1], for instance.

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Branched covers of Riemann Surfaces

In this section we study holomorphic maps between Riemann surfaces. Henceforth we shall always tacitly assume that the maps under consideration are non-constant, i.e. their image is not just a single point. We have seen among the examples of Chapter 2 that the map z 7→ z k defines a cover of C× by itself but its extension to C does not. But in both cases we get not only a continuous map but a holomorphic map of Riemann surfaces. The next proposition shows that locally any holomorphic map of Riemann surfaces looks like this, from which we shall be able to deduce the topological properties of holomorphic mappings. Proposition 2.1 Let φ : Y → X be a holomorphic map of Riemann surfaces. Denote by D the unit disc in C. Then for any point y ∈ Y with image x in X there exist complex charts fy : Uy → C and fx : Ux → C satisfying: • y ∈ Uy , x ∈ Ux ; • fy (Uy ) and fx (Ux ) are both contained in D; • fx (x) = fy (y) = 0; • there is a positive integer k for which the diagram φ

Uy −−−→ Ux  

fy y

z7→z k

  yfx

D −−−→ D

commutes. Proof: By performing affine linear transformations in C and by shrinking Ux and Uy if necessary, one may find charts fy and fx satisfying all conditions of the proposition except perhaps the last one. In particular, fx ◦ φ ◦ fy−1 is a holomorphic function in a neighbourhood of 0 which vanishes at 0. As such, it must be of the form z 7→ z k H(z) where H is a holomorphic function with H(0) 6= 0. It is a known fact from complex analysis that by shrinking Uy if necessary we may find a holomorphic function h on f (Uy ) with hk = H (indeed, we may choose h to be exp((1/k) log H)), where log is a branch of the logarithm in a neighbourhood of H(0) 6= 0). Thus replacing fy by its composition with the map z 7→ zh(z) satisfies the required properties. Corollary 2.2 Any holomorphic map between Riemann surfaces is open (i.e. maps open sets onto open sets).

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73

Proof: Indeed, the map z → z k is open. Definition 2.3 The integer k of the proposition is called the ramification index or branching order of φ at y and is denoted by ey . The points y with ey > 1 are called branch points. We denote the set of branch points of φ by Sφ . Corollary 2.4 The fibres of φ and the set Sφ are discrete subsets of Y . Proof: Indeed, the proposition implies that any point has a punctured neighbourhood in which φ is one-to-one and any branch point has an open neighbourhood in which it is the only branch point. Now we restrict our attention to compact Riemann surfaces. Proposition 2.5 Let φ : Y → X be a holomorphic map of Riemann surfaces. Then: 1. The fibres of φ and the set Sφ are finite. 2. The map φ is surjective with finite fibres. 3. The restriction of φ to Y \ φ−1 (φ(Sφ )) is a cover of X \ φ(Sφ ). 4. For any x ∈ X we have X

ey = n,

y∈φ−1 (x)

where n is the cardinality of the fibres of the previous cover. Proof: The first statement follows from the previous corollary since discrete subsets of a compact space are finite. The second statement holds because φ(X) is open in Y (by Corollary 2.2), but it is also closed (being the continuous image of X which is compact) and Y is connected. For (3) note that by Proposition 2.1 any of the finitely many preimages of x ∈ X \ φ(Sφ ) has an open neighbourhood mapping homeomorphically onto some open neighbourhood of x; the intersection of these is a distinguished open neighbourhood of x as in the definition of a cover. For (4), given x ∈ X, by the same argument we find an open neighbourhood Uy of any preimage y of x that surjects onto a suitably small fixed open neighbourhood U of x and such that φ behaves in Uy as in Proposition 2.1. Over x0 ∈ U \ {x} the map φ behaves as a cover and has n distinct preimages by statement (3). But for each preimage y of

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X, the open neighbourhood Uy contains exactly ey preimages of x0 , as the map z 7→ z ey has the same property in a sufficiently small open disc around the origin in C. The statement follows from this by summing over y. Continuous maps Y → X with the topological properties stated in the proposition are usually called finite branched covers or finite ramified covers of compact Riemann surfaces. We now would like to prove a converse, and show that if we throw away finitely many points (or none) from a compact Riemann surface X, any finite cover of the remaining space can be embedded into a compact Riemann surface Y mapping holomorphically onto X such that the map Y → X extends the structural map of the cover. The first step in this program is accomplished by the following lemma. Proposition 2.6 Let X be a Riemann surface, p : Y → X a connected cover of X as a topological space. Then Y can be endowed with a unique complex structure for which p becomes a holomorphic mapping. In fact, the proof will show that it is enough to require that p is a local homeomorphism. Proof: Any point y ∈ Y has a neighbourhood V that projects homeomorphically onto a neighbourhood U of p(y). Take a complex chart f : U 0 → C with U 0 ⊂ U; f ◦ p will the define a complex chart in a neighbourhood of y. It is immediate that we thus obtain a complex atlas and uniqueness follows from the fact that for any complex structure on Y the restriction of p to U must be an analytic isomorphism. Notice that the lemma solves our problem in the case where we throw away no points. Now comes the general case. Proposition 2.7 Assume given a compact Riemann surface X, a finite set x1 , . . . , xn of points of X and a finite connected cover φ0 : Y 0 → X 0 , where X 0 = X \ {x1 , . . . , xn }. Then there exists a compact Riemann surface Y containing Y 0 as an open subset and a holomorphic map φ : Y → X with φ|Y 0 = φ0 . Proof: We assume for simplicity that n = 1; for n > 1 one performs the same construction as below simultaneously for all points x1 , . . . , xn and gets a similar result. By performing an affine linear transform in C if necessary, we may find a complex chart mapping a connected open neighbourhood U of x = x1

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75

homeomorphically onto the open unit disc D ⊂ C. The restriction of φ0 to φ0−1 (U \ {x}) is a finite cover, hence φ0−1 (U \ {x}) decomposes as a finite disjoint union of connected components Vi each of which is a cover of U \{x}. Now each Vi is isomorphic to a finite connected cover of the punctured disc D˙ = D \ {0} via the complex chart and hence by Chapter 2, Example 3.10 it must be isomorphic to the cover D˙ → D˙ given by z 7→ z k for some k > 1 (in fact, we considered the closed disc there but the open case follows immediately from it). Now choose “abstract” points yi for each i and define Y as the disjoint union of Y 0 with the yi. Define an extension φ of φ0 to Y by mapping the yi to x and a topology on Y by taking as a basis of open neighbourhoods of each yi the inverse image by φ|Vi∪{yi } of a basis of open neighbourhoods of x contained in U. Finally, endow Y 0 with the complex structure defined in the previous lemma and define a complex chart containing each yi by extending the isomorphism of Vi with D˙ discussed above such that yi maps to 0. It follows from the proof of the previous lemma that these “new charts” are holomorphically compatible with those of the lemma, so that Y becomes a Riemann surface with this structure. Moreover, φ is a holomorphic map as it is holomorphic away from the yi by the lemma and in the neighbourhood of these looks like z 7→ z k . We also see that Y is compact, for it is mapped onto the compact space X by the continuous surjection φ which is an open mapping as well (according to Corollary 2.2). In fact, the Riemann surface Y is unique up to isomorphism, as the following result will imply. Proposition 2.8 In the situation of the previous proposition, assume given a holomorphic map of compact Riemann surfaces ψ : Z → X having branch points only over the xi , such that there is an isomorphism of covers λ : ∼ = Y0→ Z 0 , where Z 0 = Z \ ψ −1 ({x1 , . . . , xn }. Then λ extends uniquely to a holomorphic isomorphism from) Y to Z over X. Proof: Again we reason for n = 1 and employ the notation of the previous proof. Let Wi be a connected component of ψ −1 (U). Then we may find a connected component of φ−1 (U \ {x}), say Vi , such that λ(Vi ) ⊂ Wi . We contend that Wi \ λ(Vi ) consists of a single element zi ∈ ψ −1 (x). Indeed, if it contained another element zi0 , then the ramification index of ψ at both zi and zi0 would be equal to the cardinality of the fibres of the connected cover of U \ {x} obtained by restricting ψ to λ(Vi ). Choosing points in the fibre ψ −1 (x) lying in other components and applying the same argument, we see that this would contradict the formula in Proposition 2.5 (4). So our only choice is to extend λ by mapping yi to zi and similarly for

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the other components. It is immediate that in this way we obtain a map of spaces over X which is compatible with complex structures (for complex charts which map both Vi and λ(Vi ) to a punctured disc, the extension just maps 0 to 0). Corollary 2.9 Under the assumptions and notations of Proposition 2.7, 1. the Riemann surface Y is unique up to isomorphism (over X); 2. any automorphism of the cover Y 0 → X 0 extends uniquely to an automorphism of Y over X. Remark 2.10 By the second statement of the corollary, the automorphism group of Y as a space over X is the same as that of Aut(Y 0 |X 0 ). Therefore it makes sense to call Y a finite Galois branched cover of X if Y 0 is Galois over X 0 . Indeed, we see by using the continuity of automorphisms that Aut(Y |X) acts transitively on all fibres, including the ones containing the branch points.

3.

Relation with Field Theory

We begin with a basic definition. Definition 3.1 Let X be a Riemann surface. A meromorphic function on X is a holomorphic function f on an open subset X \ S, where S ⊂ X is a discrete set of points, such that for any complex chart φ : U → C containing some s ∈ S, the complex function f ◦ φ−1 : f (U) → C has a pole in f (s). Note that meromorphic functions on a Riemann surface X form a ring with respect to the usual addition and multiplication of functions; we denote this ring by M(X). In fact, it is none but the ring of global sections of the sheaf of meromorphic functions MX on X; this sheaf associates to an open subset U ⊂ X the direct sum of the rings MX (Ui ), where the Ui are the connected components of U. Lemma 3.2 The ring M(X) is a field. Proof: For any nonzero f ∈ M(X) the function 1/f will be seen to give an element of M(X) once we show that the zeros of f form a discrete subset. Indeed, if not, then it has a limit point x. Composing f with any complex chart containing x we get a holomorphic function on some complex domain whose set of zeros has a limit point. By the Identity Principle of complex

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77

analysis (Rudin [1], Theorem 10.18) this implies that the function is identically 0, so f is 0 in some neighbourhood of s. Now consider the set of those points y of X for which f vanishes identically in a neighbourhood of y. This set is open by definition but it is also closed for it contains any of its boundary points by the previous argument. Since X is connected, this implies f = 0, a contradiction. Nonconstant meromorphic functions play a crucial role in the theory of Riemann surfaces for the following reason. Lemma 3.3 Any nonconstant meromorphic function f on a Riemann surface X defines a holomorphic map X → P1 (C) by mapping x to f (x) if x is not a pole of f and to the point ∞ if x is a pole. Proof: Any x ∈ X has a neighbourhood U such that f is holomorphic on U \ {x}; by shrinking U if necessary we may assume f (U \ {x} ⊂ C \ {0}. Recall that on C \ {0} both standard complex charts of P1 (C) are defined; the first is given by the identity map and the second by z 7→ 1/z. There is an i ∈ {0, 1} for which (fi ◦ f )(U \ {x}) is a bounded open subset of C, thus fi ◦ f extends to a holomorphic function on U by Riemann’s Removable Singularity Theorem (see e.g. Rudin [1], Theorem 10.20). Thus if X is a compact Riemann surface, then by virtue of Proposition 2.5 any nonconstant function in M(X) defines a branched cover of P1 (C). For general X we cannot hope, of course, for this being a cover since P1 (C) is simply connected. The question thus arises whether any compact Riemann surface carries non-constant meromorphic functions, for this would imply that any compact Riemann surface is a branched cover of P1 (C). Surprisingly, this is a deep theorem of Riemann which we state without proof. Theorem 3.4 (Riemann’s Existence Theorem) Any compact Riemann surface X admits a nonconstant meromorphic function. Moreover, for any point x we may find a meromorphic function on X that is holomorphic on X \ {x}. The theorem can be deduced from the Riemann-Roch theorem for compact Riemann surfaces; see e.g. Forster [1], Theorem 14.12 where it is proven using a related cohomological result. We shall use the second statement through the following corollary: Corollary 3.5 Let x1 , . . . , xn be a finite set of points on a compact Riemann surface X. Then for any sequence a1 , . . . , an of complex numbers there exists f ∈ M(X) such that f is holomorphic at all the xi and f (xi ) = ai .

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Proof: It is enough to find fi with fi (xi ) = 1 and fi (xj ) = 0 for i 6= j as then we may define f as a linear combination of the fi . To find fi take first n − 1 meromorphic functions gj (1 ≤ j ≤ n, j 6= i) such that gj has a P pole only at xj ; for some constant c ∈ C the sum c + gj takes a non-zero P value at xi . Hence the function 1/(c + gj ) is 0 in xj for j 6= i and takes a non-zero value at xi ; dividing by a constant we thus get fi . Consider now a holomorphic map φ : Y → X of compact Riemann surfaces. The map φ induces a map of sheaves of meromorphic functions MX → φ∗ MY ; in particular, taking global sections we get an inclusion M(X) ⊂ M(Y ) of fields of global meromorphic functions. Proposition 3.6 If φ : Y → X is a holomorphic map of compact Riemann surfaces which has degree d as a branched cover, then the induced field extension M(Y )|M(X) has degree d. Proof: We first show that the extension has degree at most d. For this it suffices to see that any meromorphic function in M(Y ) has degree at most d over M(X). Indeed, we claim that this implies M(Y ) = M(X)(f ), with f a meromorphic function on Y of maximal degree d0 ≤ d over M(X). For if g is any other element of M(Y ), we have M(X)(f, g) = M(X)(h) with some function h ∈ M(Y ) according to the theorem of the primitive element. In particular we have M(X)(f ) ⊂ M(X)(h), but since h should also have degree at most d0 over M(X), this inclusion must be an equality, i. e. g ∈ M(X)(f ), what was to be seen. Now let’s prove that any meromorphic function f ∈ M(Y ) satisfies a (not necessarily irreducible) polynomial equation of degree d over M(X). Let S be the set of branch points of φ. Any x ∈ / φ(S) has some open neighbourhood U −1 such that φ (U) decomposes as a finite disjoint union of open sets V1 , . . . , Vd homeomorphic to U. Let si be the (holomorphic) section of φ mapping U onto Vi and put fi = f ◦ si . The function fi is then a meromorphic function on U. Put Y F (T ) = (T − fi ) = T n + an−1 T n−1 + . . . a0 . The coefficients aj , being the elementary symmetric polynomials of the fi , are meromorphic on U. Now if x1 ∈ / φ(S) is another point with distinguished open neighbourhood U1 , then on U ∩ U1 the coefficients of the polynomial F1 corresponding to the similar construction over U1 must coincide with those of F since the roots of the two polynomials are the same meromorphic functions over U ∩ U1 . Hence the aj extend to meromorphic functions on X \ φ(S). To see that they extend to meromorphic functions on the whole of X, one considers coordinate charts of the form fy : Uy → D in the neighbourhoods

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of branch points y as in Proposition 2.1. If f is holomorphic at y, f ◦ fy−1 is a bounded function on f (Uy \ {y}). Choosing similar charts for all preimages of x = φ(y), we infer from Proposition 2.1 that there is a complex chart fx : Ux → C such that all the aj ◦ fx−1 are bounded functions on fx (Ux \ {x}) and as such extend to holomorphic functions on the whole disc by Riemann’s removable singularity theorem. The case of f having a pole reduces to above, by arguing about some function z k f ◦ fy−1 which is already holomorphic on D. To see that the field extension is of exact degree d, take a point x ∈ X which is not the image of a branch point and let y1 , . . . , yd be its inverse images in Y . By Corollary 3.5 we may find f ∈ M(Y ) that takes distinct nonzero values at the yi. Now by what we have seen f satisfies a polynomial equation (φ∗ an )f n + . . . + (φ∗ a0 ) = 0, with ai ∈ M(Y ) and n ≤ d. Note that since f (yj ) 6= 0 for all j, the functions φ∗ ai can have no pole at the xi or, what amounts to the same, the ai can have no pole at x for otherwise the equation could not hold. Hence evaluating the ai at x we get a polynomial with coefficients in C that has d distinct roots (namely the f (yj )), whence n = d. Using Riemann’s existence theorem and the well-known fact that the field of meromorphic functions on P1 (C) is isomorphic to the rational function field C(T ), we get an important corollary: Corollary 3.7 The field of meromorphic functions on a compact Riemann surface is a finite extension of C(T ), that is, an algebraic function field of one variable over C. Another important consequence of the proposition is that associating to a compact Riemann surface Y the field M(Y ) gives a contravariant functor from the category of compact Riemann surfaces mapping holomorphically onto a fixed Riemann surface X to the category of finite field extensions of M(X). For example, for X = P1 (C) we get a contravariant functor defined on the whole category of compact Riemann surfaces, by preceding discussion. Our next goal is to prove that this functor induces an anti-equivalence of categories. We first prove that it is essentially surjective. Proposition 3.8 Let X be a compact Riemann surface and let L|M(X) be a finite extension of its field of meromorphic functions. Then there exists a compact Riemann surface Y mapping holomorphically onto X such that the induced field extension M(Y )|M(X) is isomorphic to L|M(X).

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Proof: Let n be the degree of the extension L|M(X), let α be a primitive element and let F be the minimal polynomial of α. The derived polynomial F 0 does not vanish identically, so since F is irreducible the ideal (F 0 , F ) of the polynomial ring M(X)[T ] (which is a principal ideal domain) must be the whole ring. Therefore there are functions a, b ∈ M(X) such that aF + bF 0 = 1. Denote by Fz the complex polynomial obtained from F by evaluating its coefficients at a point z ∈ X where all of them are holomorphic. From the above equation we infer that Fz and Fz0 may have a common root only at those points z ∈ X where one of the functions a, b has a pole. Therefore if we denote by T ⊂ X the discrete set consisting of poles of the coefficients of F as well as those of a and b, we get that on X 0 = X \ T all coefficients of F are holomorphic and Fz (x) = 0 implies Fz0 (x) 6= 0. Now denote by F the subsheaf of the sheaf of those holomorphic functions on X 0 consisting over U ⊂ X 0 of those φ ∈ OX 0 (U) for which F (φ) = 0. We contend that F is a locally constant sheaf of sets each fibre of which consists of n elements. Indeed, by the holomorphic version of the implicit function theorem (see e.g. Griffiths-Harris [1], p. 19) for each point x ∈ X with Fz (x) = 0 for some z the condition Fz0 (x) = 0 implies that there is a holomorphic function φx defined in a neighbourhood of z with F (φx ) = 0 and φx (z) = x. For each of the n distinct roots of Fz we thus find n different functions φx which are thus sections of F in some open neighbourhood of z. In a connected open neighbourhood V the sheaf F cannot have more sections since the product of the polynomials (T − φx ) already gives a factorisation of F in the polynomial ring M(V )[T ]. Now from Chapter 3, Proposition 2.8 we get a cover pF : XF0 → X 0 . Next we can apply Proposition 2.7 to each of the connected components Yi0 of X 0 F and get compact Riemann surfaces Yi mapping holomorphically onto X. We now show that i = 1, i.e. XF is connected. Indeed, define a function f on XF by putting f (φx ) = φx (pF (φx )). One sees by the method of proof of Proposition 3.6 that f extends to a meromorphic function on each Yi and by applying Proposition 3.6 we see that f as an element of M(Yi) has a minimal polynomial G over M(X) of degree at most di, where di is the cardinality of the fibres of the cover Yi0 → X 0 . But since manifestly F (f ) = 0, G must divide F in the polynomial ring M(X)[T ], whence F = G by irreducibility of F and finally d = n. This proves that XF is connected and we may denote the single Yi by Y . Finally, mapping f to α induces an inclusion of fields M(Y ) ⊂ L which must be an equality by counting degrees.

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Remark 3.9 One may replace the use of the implicit function theorem by using ideas from residue calculus in complex analysis as in Forster [1], Corollary 8.8. Theorem 3.10 Let X be a compact Riemann surface. Then the functor which associates to each pair (Y, φ) with Y a compact Riemann surface and φ : Y → X a (non-constant) holomorphic map the induced field extension M(Y )|M(X) induces an anti-equivalence of categories from the category of compact Riemann surfaces over X to that of finite field extensions of M(X). Moreover, here Galois branched covers correspond to Galois field extensions. Proof: Essential surjectivity was proven in the previous proposition. Full faithfulness follows from the following two facts. The first is that given a pair (Y, φ) as in the theorem and a generator f of the extension M(Y )|M(X) with minimal polynomial F , then the cover of X given by the restriction of φ to the complement of branch points and inverse images of poles of coefficients of F is canonically isomorphic to the cover XF defined in the previous proof. This isomorphism is best defined on the associated sheaf of local sections: just map a local section si to the holomorphic function f ◦ si . The isomorphism extends to an isomorphism of Y with the compactification of XF defined in the previous proof; in particular, morphisms Y → Z of compact Riemann surfaces over X corresponds bijectively to morphisms of such compactifications. The second fact is that given a tower of finite field extensions M|L|M(X), by the previous proof there is are canonical maps of Riemann surfaces YL → X and YM → YL corresponding to the extensions L|M(X) and M|L, respectively. But their composite is a holomorphic map YM → X and the map YM → YL is thus a canonical holomorphic map over X inducing the extension M|L. For the last statement we remark first that any automorphism of Y over X induces an automorphism of M(Y )|M(X) via the functor of the theorem. Using Proposition 3.6 we see that Y is Galois over X if and only if Aut(Y |X) is of order d = [M(Y ) : M(X)] which holds precisely if M(Y ) is Galois over M(X) by classical Galois theory. Remark 3.11 Let X be a compact Riemann surface such that M(X) can be written in the form C[x, y]/(f ) where the “homogenisation” F (z) = z d f (x/z, y/z) of the degree d polynomial f is such that its partial derivatives don’t vanish simultaneously at any point of the complex projective plane. It can be shown using the theorem that M(X) is isomorphic to the complex plane curve defined by the equation F = 0.

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Note that by virtue of Theorem 3.4 (and the lemma preceding it) any compact Riemann surface X admits a non-constant holomorphic map φ : X → P1 (C) and furthermore given any holomorphic map Y → X we may regard it as a map of spaces over P1 (C) by composing with φ. Hence the theorem immediately implies Corollary 3.12 Associating to a compact Riemann surface its field of meromorphic functions defines an anti-equivalence of the category of compact Riemann surfaces with non-constant holomorphic maps with that of finite field extensions of C(T ). Before stating the next corollary, define a non-connected compact Riemann surface to be a finite disjoint union of compact Riemann surfaces and extend the notion of a holomorphic map to this larger category in the obvious way. Then the theorem implies that the category of not necessarily connected compact Riemann surfaces equipped with a holomorphic map onto a fixed compact Riemann surface X is anti-equivalent to that of finite ´etale M(X)algebras. Hence combining with Chapter 1, Theorem 3.4 we get: Corollary 3.13 Let X be a compact Riemann surface. Then the category of not necessarily connected compact Riemann surfaces admitting a holomorphic map onto X is equivalent to that of finite continuous left Gal (M(X))-sets. This statement is rather similar to Chapter 2, Corollary 3.14 except that here we allow branch points as well. But the category of finite covers of X is a subcategory of that of finite branched covers, so by comparing the dx)op is a quotient of Gal (M(X )) in a two corollaries one expects that π1 (X, natural way. To confirm this intuition, we need some preliminaries. First a definition. Definition 3.14 Let p : Y → X and q : Z → X be Hausdorff spaces over the same topological (Hausdorff) space X. The fibre product of Y and Z over X is defined as the closed subset Y ×X Z = {(y, z) ∈ Y × Z : p(y) = q(z)} ⊂ Y × Z endowed with the subspace topology. Remarks 3.15 Here are some remarks. 1. We leave checking that Y ×X Z is indeed closed in Y × Z as an exercise to the readers.

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2. Note that the definition contains as a special case: • the direct product Y × Z (when X is a point); • the fibre of p over a point x ∈ X (when Y = {x} and q is the natural inclusion). 3. A basic property of Y ×X Z is that it represents the set-valued contravariant functor S 7→ {(φ, ψ) ∈ Hom(S, Y ) × Hom(S, Z) : p ◦ φ = q ◦ ψ} on the category of topological spaces. Indeed, given (S, φ, ψ) as above, the pair (φ, ψ) defines a canonical map S → Y ×X Z which gives back φ and ψ by composition with the natural projections of Y ×X Z onto Y and Z; moreover, the construction is functorial in S. Lemma 3.16 Let X be a connected Hausdorff space, p : Y → X be a continuous map and q : Z → X be a cover. Then the projection p1 : Y ×X Z → Y is a cover of Y . Hence if moreover Y → X is a cover as well and X is locally simply connected, the map p ◦ p1 : Y ×X Z → X is also a cover. The same conclusion holds if Y → X and Z → X are finite covers of X. Proof: For the first statement take y ∈ Y and let V be an open neighbourhood of p(y) such that q −1 (V ) decomposes as a disjoint union of open subsets Vi . Then the inverse image of U = q −1 (V ) by p1 decomposes as the disjoint union of the open subsets U ×V Vi . The second and third statements follow from the first by noting that under the assumptions any composition of covers is a cover. For X locally simply connected, this was shown in Chapter 2, Lemma 3.3; in the case of finite covers the claim follows by direct checking from the definition of a cover. Note that even if Y and Z are connected, their fibre product will not be connected in general. Now we can prove: Theorem 3.17 Let X be a compact Riemann surface and let X 0 be the complement of a finite (possibly empty) set of points in X. Let K be the composite in a fixed separable (=algebraic) closure M(X) of M(X) of all finite subextensions which arise from holomorphic maps of compact Riemann surfaces Y → X that restrict to a cover over X 0 . Then K is a Galois extension of M(X) with Galois group isomorphic to the profinite completion of π1 (X 0 , x)op (for some base point x ∈ X 0 ).

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Proof: We first show that any finite subsextension of K|M(X) is the field of meromorphic functions of some Y which restricts to a cover over X 0 . For this, we first remark that if K is contained in a finite extension L|M(X) which has this property, then so does K|M(X). Indeed, if the holomorphic map Y → X corresponding to K|M(X) had a branch point mapping to some x ∈ X 0 , then using the formula in Proposition 2.5 (4) one would get that x would have less than [K : M(X)] preimages in Y and thus less than [L : M(X)] preimages in Z, where Z → X is the map corresponding to L|M(X), so that Z → X would not restrict to a cover over X 0 . This being said, for our claim it suffices to show that given two subextensions Li |M(X) (i = 1, 2) coming from Riemann surfaces Yi → X which restrict to covers over X 0 , their composite in M(X) comes from some Y12 → X which is also a cover over X 0 . For this, take the fibre product Y1 ×X Y2 . Since it may not be connected, its ring of meromorphic functions is a finite ´etale M(X)-algebra. We contend that this algebra is none but L1 ⊗M(X) L2 . Indeed, the latter algebra represents the functor on the category of M(X)algebras associating to an algebra R the set of M(X)-bilinear homomorphisms L1 × L2 → R; the anti-equivalence of Theorem 3.10 (extended to ´etale M(X)-algebras) transforms this exactly to the defining property of fibre products as in Remark 3.15 (3). Now connected components of Y1 ×X Y2 correspond exactly to direct factors of L1 ⊗M(X) L2 , both corresponding to the factorisation of a minimal polynomial of a generator of L1 |M(X) into irreducible factors over L2 . But when we look at the fixed embeddings of the Li into M(X), the component coming from one of these factors becomes exactly the composite L1 L2 , and we are done. Next, K is Galois over M(X) for if a finite extension L|M(X) comes from a Riemann surface that restricts to a cover over X 0 , then so does the one inducing any conjugate of L, as one immediately sees by looking at the proof of Theorem 3.10. d0 , x) corresponds to a finite Galois cover of Now any finite quotient of π1 (X X 0 , which corresponds to a finite Galois branched cover of X by Proposition 2.7 and finally to a finite Galois subextension of K|M(X). By Theorem 3.10 and the first part of the proof we get in this way a bijection between d0 , x) and Gal (K|M(X)), respectively, and isomorphic finite quotients of π1 (X moreover this bijection is seen to be compatible with taking towers of covers on the one side and field extensions on the other. We get the statement of the proposition by passing to the inverse limit. Remark 3.18 The groups π1 (X 0 , x) are known from topology (see the next section). This makes it possible to determine the absolute Galois group of M(X) as follows. Any finite subextension of M(X) is contained in some ex-

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tension KX 0 corresponding to an X 0 ⊂ X by the proposition, hence the Galois group Gal (M(X)) is isomorphic to the inverse limit of the natural inverse system of Galois groups formed by the Gal (KX 0 |M(X)). (Note that this inverse limit is filtered, for any two groups Gal (KX 0 |M(X)), Gal (KX 00 |M(X)) are targets of a map from the group Gal (KX 0 ∩X 00 |M(X)).) But this result is interesting only in principle, for the groups in question are monstruosly big, as already the following considerations show. Consider the complement in P1 (C) of a finite set {x0 , . . . , xn } of points. It is known from topology (see the next section for a more general statement) that the fundamental group of this space has a presentation Gn =< γ0 , . . . , γn |γ0 . . . γn = 1 > where the generator γi is given by a closed path around the point xi . Now any finite group G is a quotient of some Gn : indeed, if G is generated by n elements g1 , . . . , gn , we may define a surjection Gn 7→ G by mapping γi to gi for i > 0 and mapping γ0 to (g1 . . . gn )−1 . Since the field M(P1 (C)) equals C(T ), we get the corollary: Corollary 3.19 Any finite group G is the Galois group of some finite extension L|C(T ). There is no currently known proof of this fact that uses purely algebraic methods. Remark 3.20 The question whether any finite group G is the Galois group of a finite regular Galois extension L|Q(T ) (regular meaning that Q has no algebraic extensions contained in L) is one of the most famous open problems in algebra. It would imply in particular that any finite group is a Galois group over Q: this follows from Hilbert’s irreducibility theorem which implies that for any irreducible polynomial F ∈ Q(T )[X] defining a Galois extension as above we may find infinitely many values t0 ∈ Q such that the polynomial obtained from F by substituting t0 defines a Galois extension of Q with the same group G (see e.g. Serre [3], Chapter 9). It can be shown by methods of algebraic geometry (see more on this in Chapter 9) that any finite Galois extension L|C(T ) comes by tensoring with C(T ) from a finite Galois extension L0 |Q(T ) with the same Galois group. To perform a “descent” of this extension from Q(T ) to Q(T ) is known to be possible in many concrete cases, including almost all finite simple groups. This is done by the (purely algebraic) rigidity method developed by Belyi, Fried, Matzat, Thompson and others to which a nice introduction can be found in Serre [4]; see also Malle/Matzat [1] and V¨olklein [1] for more extensive treatments.

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Remark 3.21 There is the following interesting analogue of the last corollary due to D. Harbater [1]: any finite group G arises as the Galois group of a finite Galois extension of Qp (T ), where Qp is the field of p-adic numbers (the fraction field of the ring Zp of p-adic integers encountered in Chapter 1, Example 2.11). See also Colliot-Th´el`ene [1] and Koll´ar [1] for other proofs and generalisations.

4.

Topology of Riemann Surfaces

In order to give a reasonable complete treatment of the theory of covers of Riemann surfaces, we have to mention several topological results which are proven by methods different from those encountered above. Since this material is well documented in several introductory textbooks in topology, we shall mostly state the results without proof, the book of Fulton [1] being our main reference. Our first topic is the topological classification of Riemann surfaces. This is a very classical theorem stemming from the early days of topology and is proven by a method commonly called as “cutting and pasting” (see Fulton [1], Theorem 17.4). Theorem 4.1 Any compact Riemann surface is homeomorphic to a torus with a finite number g of holes. The number g is called the (topological) genus of the Riemann surface. Here we have to explain what a torus with g holes means. There are several ways to conceive this. Perhaps the simplest one is to take a (usual 2dimensional) sphere and to attach g handles on it like on some mug. Another is to take what is called in topology a connected sum of g tori which is done as follows. Take first two copies of the usual torus (which are homeomorphic to the sphere with “one handle attached”), cut out a piece homeomorphic to a closed disc from each and glue the two pieces together by identifying the boundaries of the two holes just cut out. In this way we obtain a torus with two holes and the process may be continued g times. A third way generalises the fact that we may obtain the usual torus by identifying opposite sides of a square (with the same orientation, i.e. without such twists as in the construction of the Moebius band). To generalise this, −1 take a regular 4g-gon and label its sides clockwise by a1 , b1 , a−1 1 , b1 , . . . , ag , −1 −1 bg , ag , bg . Here the notation means that we consider the ai , bi with clock−1 wise orientation and the a−1 with counterclockwise orientation. Now i , bi −1 identify each ai with ai and bi with b−1 taking care of the chosen orientai tions (see the very suggestive drawings on pp. 240-241 of Fulton [1]). In this

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−1 way we get a sphere with g handles and the sides ai , bi , a−1 of our initial i , bi polygon get mapped to closed paths all of which go through a common point x.

Remark 4.2 The proof of the theorem uses several facts about the topology of the compact Riemann surfaces under consideration. The first is that they are topological manifolds of dimension 2, which means that the underlying continuous maps of the complex charts induce homeomorphisms of some neighbourhood of each point with some open subset of R2 . The second is that they are orientable manifolds, which can be expressed in this case by remarking that if fi : Ui → C (i = 1, 2) are some complex charts on our Riemann surface, then the R2 -valued differentiable real function coming from the real and complex parts of the map f1−1 ◦ f2 regarded as functions of two real variables has an everywhere positive Jacobian determinant; this is a consequence of the Cauchy-Riemann equations. Finally, one also has to use the fact that our compact Riemann surfaces can be triangulated; this we shall discuss below. The third representation of tori with g holes described above makes it possible to compute the fundamental group of a compact Riemann surface of genus g. Here it is convenient to take as a base point the point x where −1 all the closed paths coming from the ai , bi , a−1 meet; in fact, it can be i , bi shown that these paths generate the fundamental group. More precisely, one proves (cf. Fulton [1], Proposition 17.6): Theorem 4.3 The fundamental group of a compact Riemann surface X of genus g has a presentation of the form π1 (X, x) =< a1 , b1 , . . . , ag , bg |[a1 , b1 ] . . . [ag , bg ] = 1 >, −1 where the brackets [ai , bi ] denote the commutators a−1 i bi ai bi .

The proof uses the definition of the fundamental group in terms of closed paths and in particular the van Kampen theorem. Remark 4.4 In fact, by the same method that proves the theorem one can also determine the fundamental group of the complement of n + 1 points x0 , . . . , xn in a compact Riemann surface X of genus g. Here one has to add one generator γi for each xi given by a closed path going through x and turning around xi . We get a presentation of π1 (X \ {x1 , . . . , xn }, x) by < a1 , b1 , . . . , ag , bg , γ0 , . . . , γn |[a1 , b1 ] . . . [ag , bg ]γ0 . . . γn = 1 > . This gives an explicit presentation of the fundamental groups encountered in the previous section.

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Remark 4.5 Realising the groups described in the theorem as automorphism groups of the universal cover gives rise to a fascinating classical theory known as the theory of uniformisation; see Chapter IX of Shafarevich [1] for a nice introduction. The main result here, originating in work by Riemann and proven completely by Poincar´e and Koebe, is that any simply connected Riemann surface is isomorphic as a complex manifold to the projective line P1 (C), the complex plane C or the open unit disc D (see e.g. Forster [1], Theorem 27.6). Now one can produce compact Riemann surfaces as quotients of the above as follows. In the case of P1 (C) there is no quotient other than itself, for any automorphism of P1 (C) is known to have a fixed point. For C, one can prove that the only even action on it with compact quotient is the one by Z2 as in the second example of Chapter 2, Example 2.7, so the quotient is a torus C/Λ; this is in accordance with the case g = 1 of the theorem. All other compact Riemann surfaces are thus quotients of the open unit disc D by some even group action. Poincar´e studied such actions and showed that they come exactly from transformations mapping ai to a−1 and bi to b−1 in i i a 4g-gon with sides labelled as above; the only difference is that in this case the sides of the polygon are not usual segments but circular arcs inscribed into the unit disc, for he worked in the model of the hyperbolic plane named after him. We finally discuss triangulations of Riemann surfaces; we restrict ourselves to the compact case. So let X be a compact topological manifold of dimension 2. Then a triangulation consists of a finite set S0 of points of X (called the vertices of the triangulation), a finite set S1 of topological embeddings ι : [0, 1] → X (called the edges of the triangulation) and a finite set S2 of topological embeddings κ : ∆ → X, where ∆ is the unit triangle in R2 (called the faces of the triangulation) subject to the following conditions: • any ι ∈ S1 maps the endpoints of [0, 1] into S0 ; • any κ ∈ S2 maps the vertices of ∆ into S0 ; • the composition of each κ ∈ S2 by any of the three natural inclusions [0, 1] → ∆ giving the edges of the triangle is either an edge ι ∈ S1 or a “reversed edge” given by t 7→ ι(1 − t) for t ∈ [0, 1]; • the sets ι((0, 1)) are pairwise disjoint for all ι ∈ S1 and do not contain points from S0 ; • the sets κ(∆◦ ), where ∆◦ is the interior of ∆, are pairwise disjoint and do not meet the images of the edges in S1 ;

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• the union of the images of the sets κ(∆) for κ ∈ S2 is X. Example 4.6 The 2-dimensional sphere (which is the underlying topological space of P1C ) has several natural triangulations; one of them is cut out by the equator and two meridians. Proposition 4.7 Any compact Riemann surface has a triangulation. The proposition is an immediate consequence of the above example, of Riemann’s Existence Theorem and the following lemma. Lemma 4.8 Let φ : Y → X be a branched cover of compact Riemann surfaces (eg. a holomorphic map). Than any triangulation of X can be lifted canonically to a triangulation of Y . Before giving the proof note the obvious fact that given a triangulation of a compact topological surface X and a point x ∈ X \ S0 , the triangulation can be refined in a canonical way to a triangulation whose set of vertices is S0 ∪ {x}: take the face κ for which x ∈ κ(∆) (if x is lying on an edge, take both faces meeting at that edge), consider the natural subdivision of ∆ given by joining κ−1 (x) to the vertices and replace κ and the corresponding elements of S1 by the restrictions of κ to the edges and faces of the smaller triangles arising from the subdivision. Proof: By refining the triangulation as above if necessary, we may assume that in the given triangulation of X the set S0 contains all images of branch points. Hence the restriction of φ to X \ φ−1 (S0 ) is a cover. As the open interval (0, 1) (resp. the subset ∆0 ⊂ ∆ obtained by omitting the vertices) are simply connected, taking their image by an edge ι (resp. a face κ) we obtain subsets of X over which the cover is trivial, so the restriction of ι to (0, 1) (resp. that of κ to ∆0 ) can be canonically lifted to each sheet of the cover. Using Lemma 2.1 in the neighbourhood of each point of φ−1 (S0 ) one sees that adding these points as vertices of edges and triangles defines a triangulation of Y . Now denote by si the cardinality of the set Si . The integer χ = s0 −s1 +s2 is an important invariant of the triangulation called the Euler characteristic. One checks immediately that it does not change if we refine a triangulation by the process described above. Hence by choosing common refinements of two triangulations we see that χ does not depend on the triangulation but is an invariant of the surface itself.

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Proposition 4.9 Let φ : Y → X be a holomorphic map of compact Riemann surfaces inducing a field extension of degree d. Then the Euler characteristics χX and χY of X and Y are related by the formula χY = dχX −

X

(ey − 1)

y

where the sum is over the branch points of φ and ey is the ramification index at the branch point y ∈ Y . Proof: In the process of lifting a triangulation to a branched cover as in the lemma the number of edges of the lifted triangulation is ds1 and the number of its faces is ds2 , where d is the degree of the cover. Those points of S0 which are not in the image of the branch locus have d preimages as well but with each branch point the number of preimages diminishes by ey − 1. Now it is a known topological fact that the Euler characteristic of a torus with g holes is 2 − 2g (see Fulton [1], p. 244). Hence: Corollary 4.10 The formula of the proposition can be rewritten as 2gY − 2 = d(2gX − 2) +

X

(ey − 1)

y

where gX and gY are the genera of X and Y , respectively.

Chapter 5 Enter Schemes In this chapter and the next one we collect the technical toolkit needed to study the analogues of the theory developed in the previous chapters within the realm of arithmetic and algebraic geometry. The convenient category to work in is Grothendieck’s category of schemes, so we shall discuss as much of their theory as we shall need for subsequent applications. As always, the word “ring” in this chapter will mean commutative ring with unit. Also, when referring to compact topological spaces, we do not assume that they are Hausdorff spaces.

1.

Prime Spectra

Recall that a subset S of a ring A is called multiplicatively closed if 1 ∈ S, 0 ∈ / S and for any f, g ∈ S we have f g ∈ S. A prime ideal P is an ideal such that the set A \ P is multiplicatively closed. An equivalent formulation of this is that the quotient ring A/P should be a (nontrivial) domain, i.e. it should have no zero-divisors. From this formulation it follows easily that any maximal ideal of A (i.e. an ideal contained in no other proper ideal of A than itself) is always a prime ideal since in this case the quotient ring is a field. We now turn the set of prime ideals of an arbitrary ring A into a topological space. Definition 1.1 The prime spectrum Spec A of A is the topological space whose points are prime ideals of A and a basis of open sets is given by the sets D(f ) := {P : P is a prime ideal withf ∈ / P} for all f ∈ A. 91

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For this definition to be correct, we must verify that the system of the sets D(f ) is closed under finite intersections. But we have for all f, g ∈ A D(f ) ∩ D(g) = D(f g)

(5.1)

for by definition a prime ideal avoids f g if and only if it avoids f and g. It follows from the definition that a closed subset in the topology of Spec A can be described as the set of prime ideals containing some fixed ideal I (generated by a system of elements {fi : i ∈ J} of A). Thus one-point sets given by maximal ideals are closed; in fact, maximal ideals give the only closed points of the prime spectrum since for any prime ideal P a closed subset containing P contains the maximal ideals containing P as well. This shows that in general the prime spectrum does not satisfy even the weakest of the separation axioms in topology. However, it enjoys a nice topological property: Proposition 1.2 For any ring A the prime spectrum Spec A is compact. First a lemma we shall also use later. Lemma 1.3 A system of elements {fi : i ∈ I} generates A if and only if the sets D(fi ) give an open covering of Spec A. Proof: Indeed, if the fi generate A, there can be no prime ideal of A containing all of them, which is equivalent to the D(fi ) covering Spec A. If, however, they do not generate A, then they are all contained (by Zorn’s Lemma) in some maximal ideal M which thus gives an element of Spec A not contained in any of the D(fi ). Proof of Proposition 1.2: Let {Ui : i ∈ I} be an open covering of Spec A; we may assume that each Ui is in fact some basic open set D(fi ). By the above lemma the fi generate A. In particular, there is a relation of the form a1 f1 + a2 f2 + . . . + an fn = 1

(5.2)

with ai ∈ A and f1 , . . . , fn chosen among the fi above. This means, however, that already f1 , . . . , fn generate A, i.e. the sets D(f1 ), . . . , D(fn ) cover Spec A. Equations of type (5.2) are sometimes referred to as algebraic analogues of partitions of unity. Examples 1.4 We conclude this section by some easy examples.

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1. The prime spectrum of a field consists of a single point, corresponding to the ideal (0). 2. The prime spectrum of Z is the space consisting of a closed point for each prime p, and a non-closed point corresponding to (0), called the generic point, whose closure is the whole space. Other closed subsets are only finite sets of primes; indeed, any ideal in I ⊂ Z is of the form mZ for some positive integer m; the prime ideals containing I are generated by the prime divisors of m. 3. The prime spectrum of C[x] consists of a closed point for each a ∈ C, plus a non-closed generic point corresponding to (0). The closed subsets are again finite sets of closed points. Indeed, C[x] is a principal ideal ring with prime elements the polynomials x − a (a ∈ C), and we may argue as in the previous case. 4. If A is isomorphic to a finite direct sum

n L

i=1

Ai , then Spec A is a dis-

joint union of clopen sets each of which is homeomorphic to one of the Spec Ai . To see this, observe first that if one writes ei for the idempotent given by putting 1 at the i-th component and 0 elsewhere, any pairwise product ei ej is 0 and hence no prime ideal P of A can avoid both ei and ej . However, P cannot contain all of the ej since the sum of these is 1. Thus we conclude that P concludes all of the ej except one, say ei , which implies that P is of the form A1 ⊕ . . . Ai−1 ⊕ Pi ⊕ Ai+1 . . . ⊕ An with a prime ideal Pi of Ai . The required decomposition of Spec A is then induced by the map P 7→ Pi . In particular, the prime spectrum of a finite ´etale algebra over some field k is a finite discrete set of points. This confirms our intuition that a finite ´etale k-algebra is like a cover of the one-point space arising as the prime spectrum of k.

2.

Schemes – Mostly Affine

The prime spectrum of a ring is a rather coarse invariant: for instance, it cannot even distinguish between two fields. We shall remedy this by defining some additional structure on the prime spectrum. To motivate the construction to come, let us reconsider the third example from the last section. Example 2.1 The ring C[x] is nothing but the ring of holomorphic functions on C having at worst a pole at infinity. The prime spectrum of this ring can

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be identified to C with a generic point (0) added; closed sets are finite sets not containing (0). Obviously one cannot recover C[x] from these data; we cannot even distinguish between constant functions. Remember, however, that we have seen in the previous chapter that a Riemann surface is uniquely determined by the underlying topological space plus the sheaf of holomorphic functions on it. If we restrict to the sheaf of holomorphic functions on C having at worst a pole at infinity, we can easily describe its sections over a set of the form D(f ) with the generic point removed (this is an open set in the complex topology). For instance, over D(x) (which with the generic point thrown away identifies to C∗ ) the sections are the rational functions whose denominator is a power of x, for thesections are meromorphic functions on P1 (C) and hence elements of C(T ); moreover, any denominator other than the xm has a zero elsewhere. We find an analogous result for D(x − a) for (a ∈ C); all other D(f ) are finite intersections of these, so the sections of the sheaf over D(f ) are just the restrictions of the sections over the D(x − a) with (x − a) dividing f . If we wish to define something analogous to this for any ring A, we first have to extend the notion of a rational function, i.e. give a meaning to fractions of elements in an arbitrary ring A. So let S be a multiplicatively closed subset of A. We would like to define a ring AS which is to be the “ring of fractions with numerator in A and denominator in S”. Example 2.2 When A is a domain, this is fairly easy to do since in this case A admits a fraction field K. Elements of K can be represented by fractions f /g with f, g ∈ A, g 6= 0, where f /g = f1 /g1 whenever f g1 = f1 g. We may then take AS to be the subring of those elements which can be written as fractions with denominators in S; this is indeed a subring as S is multiplicatively closed. Now to treat the general case, observe first that just as the fraction field K can be defined as the object representing a certain functor (see Chapter 0, Section 2), the ring AS of the previous example is easily seen to represent the set-valued functor F given by F (R) = {φ ∈ Hom(A, R) : φ(s) is a unit in R for all s ∈ S} on the category of rings. When A has zero-divisors, A has no fraction field, but the above functor F still exists. Proposition 2.3 The functor F is representable by a ring AS for any ring A and multiplicatively closed subset S.

2.. SCHEMES – MOSTLY AFFINE

95

The ring AS is called the localisation of A with respect to S. By the Yoneda lemma, it is determined up to unique isomorphism. Moreover, it is equipped with a canonical homomorphism φS : A → AS sending elements of S to units which corresponds to the identity map AS → AS . Proof: Define AS as a set to be the quotient of A × S by the equivalence relation: (f, s) ∼ (f 0 , s0 ) iff there is a t ∈ S with (f s0 − f 0 s)t = 0. One sees that this is indeed an equivalence relation; for transitivity, note that the equations (f s0−f 0 s)t = 0 and (f 0 s00−f 00 s0 )u = 0 imply (f s00−f 00 s)s0 tu = 0 (multiply the first equation by s00 u and the second by st). Denote by f /s the image of (f, s) in AS and define the addition and multiplication laws as for fractions; one checks that this is independent of the representatives chosen. Now given a homomorphism φ : A → R sending elements of S to units, define a homomorphism AS → R by sending f /s to φ(f )φ(s)−1 (note that units are never zero-divisors, so φ(s)−1 is a well-defined element of R). This is a well-defined map, for if (f 0, s0 ) is another representative for f /s, we have 0 = φ((f s0 − f 0 s)t) = (φ(f )φ(s0) − φ(f 0)φ(s))φ(t), whence φ(f )φ(s0) = φ(f 0 )φ(s) as φ(t) is a unit. Conversely, as any element of S maps to a unit in AS by the map φS : A → AS sending s to s/1, homomorphisms AS → R induce elements of F (R) by composition with φS . Thus we have obtained a bijection between F (R) and Hom(AS , R) which is immediately seen to be functorial. We now wish to compare the prime spectra of A and AS . Lemma 2.4 The map P 7→ φS (P )AS defines a canonical bijection between prime ideals P of A avoiding S and prime ideals of AS . Proof: Let P be a prime ideal of A avoiding S. By this last condition, the ideal φS (P )AS generated by φS (P ) does not contain units and hence is different from AS . Moreover, it is a prime ideal, for if (f /s)(g/t) ∈ φS (P )AS , then uf g ∈ P for some u ∈ S, whence f or g is in P and thus (f /s) or (g/t) is in φS (P )AS . For surjectivity, note the easy fact that for any prime ideal Q of AS the ideal φ−1 S (Q) is a prime ideal of A avoiding S; the assertion then follows from the equality φS (φ−1 S (Q))AS = Q. Similarly, injectivity follows −1 from φS (φS (P )AS ) = P ; the verification of these relations is left to the reader.

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Examples 2.5 The two key examples of localisation to be used in the sequel are the following. 1. Let S be the set {1, f, f 2 , f 3, . . .} of all powers of f for some f ∈ A. In this case elements of AS are represented by fractions with numerator in A and denominator a power of f ; we shall use the notation Af for this particular AS . The previous lemma implies that Spec Af is naturally homeomorphic to the open set D(f ). 2. Let P be a prime ideal of A and take S to be the complement of P ; it is multiplicatively closed by primeness of P . Adopting a common abuse of notation from the literature, we shall denote the localisation of A with respect to S by AP instead of AA\P . The points of Spec AP correspond to prime ideals of A contained in P ; in particular, AP has a unique maximal ideal generated by the image of P . Rings having a unique maximal ideal are usually called local rings. This example contains the case of fraction fields: take P to be the ideal (0) in a domain. Now we may turn to defining a sheaf of rings OX on the prime spectrum X of any commutative ring A. In obvious analogy with the example of C[x] described above, we define OX (D(f )) = Af for all f ∈ A. To proceed further, we need an easy lemma. Lemma 2.6 If f, g ∈ A are such that D(f ) ⊂ D(g), then the image of g in Af is a unit. Proof: Indeed, if g did not give a unit in Af , it would be contained in a maximal ideal Q. By Lemma 2.4 there is a unique prime ideal P of A whose image in Af generates Q. This P contains g but not f , a contradiction. Combining the lemma with Proposition 2.3, we get for any inclusion D(f ) ⊂ D(g) of basic open sets a canonical restriction homomorphism Ag → Af . Clearly for a tower of inclusions D(f ) ⊂ D(g) ⊂ D(h) the map Ah → Af thus obtained is the composition of the intermediate maps Ah → Ag and Ag → Af . So putting OX (D(f )) = Af , we have obtained “something which behaves like a presheaf on basic open sets”. That this indeed extends to a presheaf on X follows from the first statement of the following formal lemma (of which we advise the readers to skip the proof in a first reading).

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Lemma 2.7 Let X be a topological space and V a basis of open sets on X. Assume given for each V ∈ V a set (resp. abelian group, ring, etc.) F (V ) and for each inclusion V 0 ⊂ V of elements of V a map (resp. homomorphism) ρV V 0 : F (V ) → F (V 0 ) satisfying ρV V = idF (V) and ρV V 00 = ρV 0 V 00 ◦ ρV V 0 for each tower V 00 ⊂ V 0 ⊂ V of elements of V. 1. There exists a presheaf of sets (resp. abelian groups, rings, etc.) F on X whose sections and restriction maps over elements of V can be canonically identified to those given above. 2. Assume moreover that the F (V ) above satisfy the sheaf axioms for all coverings of elements of V by elements of V. Then there is a unique sheaf F on X whose sections and restriction maps over elements of V are those given above. 3. Finally assume given two sheaves F , G on X and for each V ∈ V a map φV : F (V ) → G(V ) such that for each inclusion V 0 ⊂ V of elements of V the diagram φV F (V ) −−− → G(V )  

ρF V V 0y

φ

0

 ρG y VV0

V → G(V 0 ) F (V 0 ) −−−

commutes. Then there is a unique morphism of sheaves φ : F → G with the φV given as above. Proof: For the first statement, consider for a given open set U ⊂ X the set VU of elements of V contained in U; this set is partially ordered by inclusion. The restriction maps φV V 0 for V 0 ⊂ V ⊂ U turn the system of F (V ) with V ∈ VU into an inverse system. Note that this is a non-filtered inverse system in the sense of Chapter 1, Remark 2.3. (Note also that here we are working with the ordering opposite to that used in the definition of stalks in Chapter 2, Section 5). Define F (U) as the inverse limit of this system. By definition, F (U) consists of sequences (fV ) indexed by all V ∈ VU with fV ∈ F (V ) having the property that fV 0 = φV V 0 (fV ) whenever V 0 ⊂ V . If U 0 ⊂ U, define a restriction map ρU U 0 by mapping the sequence (fV ) above to the sequence of those fV for which V ⊂ U 0 . There is no difficulty in checking that we have thus defined a presheaf. Moreover, for W ∈ V, the sections of F over W can be canonically identified with the elements of the prescribed set F (W ) as in this case the sequences (fV ) defining the inverse limit are given by restrictions of elements of the prescribed F (W ) to all elements of V contained in W . Thus for any U containing W , the restriction map ρU W

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can be identified to the map projecting a sequence (fV ) to fW ; in particular, a section in F (U) is uniquely determined by its restrictions to each W ∈ VU . For the second statement, note first that unicity follows from the first sheaf axiom since each open U ⊂ X can be covered by elements of V. So it suffices to show that the presheaf F we have just defined satisfies the sheaf axioms. By construction of F , for the first sheaf axiom it is enough to see that for any open cover {Ui : i ∈ I} of U, two sections (fV ), (gV ) ∈ F (U) restricting to the same section over each Ui restrict to the same section over each W ∈ VU . Since V is a basis of open sets (hence in particular closed under finite intersections), we may write each Ui as a union of some Vij ∈ VU in such a way that W itself is a union of some of the Vij . Now as (fV ) and (gV ) restrict to the same section over each Vjk , they must restrict to the same section over W by the assumption. The verification of the second sheaf axiom is similar and is left to the readers. Finally, the last statement follows from the fact that the maps φV induce a morphism of the inverse systems defining F (U) and G(U) for a general U as above. The map φU : F (U) → G(U) is then obtained by passing to the limit: explicitly, it maps a sequence (fV ) to the sequence (φV (fV )). Now we are ready to prove: Theorem 2.8 For any ring A, there is a unique sheaf of rings OX on X = Spec A for which OX (D(f )) = Af for all f ∈ A and the restriction maps OX (D(g)) → OX (D(f )) for D(f ) ⊂ D(g) are the natural maps Ag → Af described above. Proof: We have to check the hypothesis of the previous proposition, i.e. the sheaf axioms over the basic open sets D(f ). Notice that since Spec Af is naturally homeomorphic to D(f ) and OX (D(f )) = Af , we may replace A by Af and assume f = 1. Then for the first sheaf axiom we are given a covering of X by basic open sets D(fi ); by compactness of X we may assume the covering is finite, say X = D(f1 ) ∪ . . . ∪ D(fn ). To give a section of OX (X) = A restricting to 0 over each D(fi ) is to give an element g ∈ A satisfying fiki g = 0 (5.3) for all 1 ≤ i ≤ n with appropriate positive integers ki . Now by the definition of prime ideals we have D(fiki ) = D(fi ) for all i, so the D(fiki ) cover X as well and hence by Lemma 1.3 there exist gi ∈ A with g1 f1k1 + . . . + gn fnkn = 1

(5.4)

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Thus if we multiply each equation in (5.3) by gi and take the sum we get g = 0, as desired. For the second sheaf axiom, assume again given a covering of X by basic open sets D(fi ) (1 ≤ i ≤ n) and elements ai /fiki ∈ Afi (viewed as sections of a would-be sheaf over D(fi )) whose restrictions to the pairwise intersections D(fi ) ∩ D(fj ) = D(fi fj ) coincide. This latter property can be written k explicitly as (ai fj j − aj fiki )(fi fj )kij = 0 with some positive integer kij . By changing the ai if necessary we may assume ki = k for all i and kij = m for all i, j, where m is large enough. Thus ai fjk (fi fj )m = aj fik (fi fj )m

(5.5)

for all 1 ≤ i, j ≤ n. Now apply (5.4) with ki = k + m for all i to get some P P gi with gi fik+m = 1 and define a = gi ai fim . Using equation (5.5) we get i

i

for all j a chain of equations fjk+ma

=

n X

gi ai fjk (fi fj )m

i=1

=

n X i=1

gi aj fik (fi fj )m = aj fjm

X

gifik+m = aj fjm

i

which means that the image of a in Afj coincides with aj /fjk , as required. Definition 2.9 An affine scheme is a pair (X, OX ) consisting of a topological space X and a sheaf of rings OX on X such that X = Spec A for some ring A and OX is the sheaf occuring in the above theorem. We call OX the structure sheaf of X. By abuse of notation, we shall frequently write X or Spec A instead of the pair (X, OX ). Next an important fact: Proposition 2.10 If X = Spec A is an affine scheme, then the stalk OX,P of OX at any point P of X is canonically isomorphic to the localisation AP ; in particular, it is a local ring. Proof: Recall that the stalk at P is defined as the direct limit of the filtered direct system of the rings OX (U), for U containing P . Since basic open sets D(f ) are cofinal in the index set of this direct system, we may restrict to the rings Af . Then the proposition follows from the fact that the direct limit of these rings is obtained by “dividing out by all f ∈ / P ”. More precisely, it follows from the construction of lim Af that any f ∈ / P is a unit in lim Af , → → hence there is a homomorphism AP → lim Af . If an element g ∈ AP maps → to zero here, it means f n g = 0 for some f ∈ / P and n ≥ 0 and thus g = 0 in AP ; surjectivity is equally obvious. Now some definitions.

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Definition 2.11 A ringed space is a pair (X, F ) where X is a topological space and F is a sheaf of rings on X. A morphism of ringed spaces (X, F ) → (Y, G) is a pair (φ, φ] ) consisting of a continuous map φ : X → Y and a morphism φ] : G → φ∗ F of sheaves of rings on Y . Ringed spaces thus form a category with the morphisms just defined. Affine schemes are naturally objects of this category enjoying the additional property that the stalks of the structure sheaf are all local rings. Next notice that given a morphism of ringed spaces (φ, φ]) as above, for any x ∈ X the morphism φ] induces a ring homomorphism Gφ(x) → Fx on the stalks, for by definition Gφ(x) is the (filtered) direct limit of G(U) for the open sets U containing φ(x), whereas φ∗ F (U) = F (φ−1 (U)) and there is a natural map from the direct limit of the latter sets to Fx , for Fx is defined as the direct limit of all open neighbourhoods containing x. Definition 2.12 A locally ringed space is a ringed space (X, F ) such that the stalk Fx is a local ring for all x ∈ X. A morphism of locally ringed spaces is to be a morphism of ringed spaces for which the induced maps Gφ(x) → Fx on stalks described above are local homomorphisms, which means that the preimage of the maximal ideal of Fx is the maximal ideal of Gφ(x) . Thus the category of locally ringed spaces is a subcategory of that of ringed spaces. A scheme is a locally ringed space (X, OX ) such that X admits an open covering {Ui : i ∈ I} such that for all i the locally ringed spaces (Ui , OX |Ui ) are isomorphic (in the category of locally ringed spaces) to affine schemes. The category of schemes is defined as the full subcategory of that of locally ringed spaces whose objects are schemes. Construction 2.13 As we have already remarked, for any commutative ring A the affine scheme X = Spec A is naturally an object of the category of schemes. We now show that the map A 7→ Spec A is in fact a contravariant functor from the category of rings to that of schemes. For this we have to assign to any ring homomorphism φ : A → B a morphism (Spec (φ), Spec (φ)] ) : Spec B → Spec A of schemes. The definition of Spec (φ) is obvious: it maps a prime ideal P ∈ Spec B to φ−1 (P ) which is immediately seen to be a prime ideal of A. The map is continuous since the inverse image of a basic open set D(f ) is just D(φ(f )). Now for defining Spec (φ)] note that by the third statement of Lemma 2.7 it suffices to consider sections over the basic open sets D(f ). By construction of the structure sheaves, over D(f ) our task is to define a ring homomorphism Af → Bφ(f ) . But there is a canonical such homomorphism according to Lemma 2.3: it corresponds to the composite A → B → Bφ(f ) .

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Now consider the natural question: given an affine scheme X = Spec A, how can we recover A from X? The answer is easy: we have A = OX (X). Moreover, the rule X 7→ OX (X) is also a contravariant functor: given a morphism φ : X → Y of affine schemes, we have in particular a morphism of sheaves φ] : OY → φ∗ OX , whence a homomorphism OY (Y ) → φ∗ OX (Y ) = OX (X). Theorem 2.14 The functors A 7→ Spec A and X → OX (X) are inverse to each other. Thus the category of affine schemes is isomorphic to the opposite category of the category of commutative rings with unit. Proof: If Y = Spec B and the scheme morphism X → Y comes from a homomorphism λ : A → B, by construction the map OY (Y ) → OX (X) is none but λ. We are left to prove that given a morphism (φ, φ] ) : Spec B → Spec A of schemes, if λ : A → B is the ring homomorphism induced by taking global sections, then (φ, φ]) = (Spec (λ), Spec (λ)] ). For this, we have to show first that for P ∈ Spec B we have φ(P ) = λ−1 (P ). Indeed, φ] induces a map on the stalks φ]P : Aφ(P ) → BP which by definition makes the diagram A   y

λ

−−−→ B φ]

  y

P Aφ(P ) −−− → BP

commute. But φ]P is a local homomorphism, i.e. φ(P )Aφ(P ) = (φ]P )−1 (P BP ); on the other hand, the vertical maps are localisation maps, whence the assertion. The equality φ] = Spec (λ)] follows from the analogous commutative diagram λ A −−−→ B   y

φ]D(f )

  y

Af −−−→ Bλ(f ) for sections over basic open sets. This result enables us to reformulate once more the main theorem of Galois theory. Given a field k, define a finite ´etale k-scheme to be the affine scheme Spec A for A a finite ´etale k-algebra. By the theorem, the category of finite ´etale k-schemes (viewed as a full subcategory of the category of affine schemes) is isomorphic to the opposite category of that of finite ´etale k-algebras. Thus Chapter 1, Theorem 3.4 yields:

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Theorem 2.15 For any field k the category of finite ´etale k-schemes is equivalent to the category of finite sets equipped with a continuous left Gal (k)action. We shall prove a broad generalisation of this result in Chapter 9. But while we are at finite dimensional k-algebras, we close this section by a purely algebraic result about them to be used in forthcoming chapters, which can be given an elegant proof via the theory of affine schemes. Proposition 2.16 Let k be a field. Then any finite dimensional k-algebra decomposes as a finite direct sum of local rings whose maximal ideals consist of nilpotent elements. For the proof we need a lemma. Lemma 2.17 For any ring A the map e 7→ D(e) gives a bijection between idempotents of A and clopen subsets of the underlying space of Spec A. Proof: As we have, the basic open set D(e) is closed for D(1−e) is its open complement. Conversely, given a decomposition of the underlying space of X into the disjoint union of two open subsets U and V , consider the sections 1 ∈ OX (U) and 0 ∈ OX (V ). Since U and V are disjoint, the sheaf axioms imply that these sections patch together to give an element e ∈ A = OX (X) which is an idempotent as its restrictions to U and V are. One checks easily that the two constructions are inverse to each other. Corollary 2.18 Let A be a ring and I an ideal consisting of nilpotent elements. Given an idempotent e¯ ∈ A/I, there is a unique idempotent e ∈ A mapping onto e¯ by the projection A → A/I. Proof: Since I consists of nilpotent elements, it is contained in all prime ideals of A. Hence the map Spec (A/I) → Spec A induced by the projection is the identity on the underlying topological spaces and as such induces a bijection of clopen subsets (and of functions which are 1 on the subset and 0 on the complement). Proof of Proposition 2.16: It is enough to show that if A is indecomposable and I is its ideal of nilpotent elements, then A/I is a field. If it were not, then by Chapter 1, Lemma 3.6 it would decompose as a direct sum of fields so in particular it would contain a nontrivial idempotent. But then by the previous corollary A would contain one as well, which is a contradiction with A being indecomposable.

3.. FIRST EXAMPLES OF SCHEMES

3.

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First Examples of Schemes

It is now time for some examples. Let us first take a new look of those of the first section, but this time considering the structure sheaves as well. Examples 3.1 1. For k a field, the underlying topological space of Spec k is a single point. The stalk of the structure sheaf at this point is k. 2. The generic stalk of Spec Z, i.e. the stalk of OSpec Z at the generic point (0) is Q. The inclusion Z → Q corresponds to a morphism Spec Q → Spec Z, identifiable as the inclusion of the generic point into Spec Z. At a closed point corresponding to the prime ideal (p) the stalk OSpec Z,(p) is isomorphic to the subring of Q formed by fractions whose denominator is not divisible by p. The maximal ideal of this ring is generated by p; we have OSpec Z,(p) /pOSpec Z,(p) ∼ = Fp . The natural projection Z → Fp corresponds to a map Spec Fp → Spec Z, the inclusion of the closed point (p). 3. The generic stalk of Spec C[x] is the rational function field C(x). At the closed point (x − a) the stalk consists of those elements of C[x] whose denominator does not vanish at a; the maximal ideal of OSpec C[x],(x−a) is generated by functions vanishing at a. The quotient by this maximal ideal is isomorphic to C; the image of a function by the projection OSpec C[x],(x−a) → C is its value at a. Here again a map C[x] → C[x]/(x − a) ∼ = C corresponds to the inclusion of the point a ∈ C. Thus by comparing the two last examples, we may think of elements of Q as functions on the space Spec Z. If the denominator of f ∈ Q is not divisible by a prime p, then f is “defined” in a neighbourhood of (p); its image in Fp is its “value” at p. This is the coarsest analogy one may observe; it will be considerably refined later. Example 3.2 Affine spaces. For a field k we define affine n-space over k as the affine scheme Ank = Spec k[x1 , . . . , xn ] (with k[x1 , . . . , xn ] the polynomial ring in n variables over k). An explanation for this name is provided by a form of a classical theorem of Hilbert’s called the Nullstellensatz (see e.g. Lang [1], Chapter IX.1): this says that if k is algebraically closed, then any maximal ideal of k[x1 , . . . , xn ] is of the form (x1 − a1 , . . . , xn − an ) with some ai ∈ k. Thus in this case we may identify the set of closed points of Ank with elements of k n . Note that the above statement is false even for n = 1 if k is not algebraically closed: for instance, the polynomial x2 + 1 generates a maximal ideal of R[x] not of the above form.

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We next give the basic example of a non-affine scheme. Before discussing it, an easy definition. Definition 3.3 An open subscheme of a scheme X is the ringed space consisting of an open subset U and the restriction OX |U of the structure sheaf of X to U. Indeed, one checks that U admits an open covering by affine schemes (use basic open sets, for instance). Example 3.4 Projective spaces. For k an infinite field, we construct a scheme Pnk by a similar procedure to that in the complex analytic case. So denote by Gm the multiplicative group of k and make it act on k[x0 , x1 , . . . , xn ] so that t ∈ Gm sends xi to txi for all 0 ≤ i ≤ n. (This is in accordance with the complex case for in this way the preimage of an ideal of the form (x0 −a0 , . . . xn −an ) is (x0 −ta0 , . . . xn −tan )). By Theorem 2.14 this action corresponds to a canonical action of Gm on An+1 ; moreover, each t ∈ Gm fixes k the closed point O = (x0 . . . , xn ). So let U be the open subscheme of An+1 k obtained by removing the point O and define a ringed space Pnk as follows: the underlying topological space is the quotient of U by the action of Gm and the structure sheaf OPnk is given by V 7→ OU (p−1 (V ))Gm , with p : U → Pnk the natural projection and the superscript Gm denoting invariants by the action of Gm . It is immediate that OPnk is indeed a sheaf of rings, and that the ringed space thus defined is a locally ringed space. Finally we have to define an open covering whose elements are isomorphic to affine schemes. For this let D+ (xi ) be the image of the basic open set D(xi ) ⊂ U in Pnk for m each 0 ≤ i ≤ n. By definition, we have OPnk (D+ (xi )) = k[x0 , . . . , xn ]G xi . But since k is infinite, each element of this ring can be represented by a fraction f /xdi with f a homogenous polynomial of degree d (indeed, these elements are manifestly invariant under Gm ; otherwise decompose a polynomial f into homogenous components fk of degree k and look at what happens for fk /xdi for k 6= d). Thus the k-homomorphism k[t0 , . . . , ti−1 , ti+1 , . . . , tn ] → k[x0 , . . . , xn ]xGim ,

tj 7→ xj /xi

is an isomorphism. We leave it to the reader to check that this induces an isomorphism of the ringed space (D+ (xi ), OPnk |D+ (xi ) ) with Ank . Remark 3.5 In general, it is possible to define projective spaces over any commutative ring A (not to mention any scheme...) The preceding construction may break down, however, so one should use another construction. The

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most down-to-earth method among several available ones is to define PnA by “patching” together the affine schemes D+ (xi ) ∼ = Spec A[(x0 /xi ), . . . , (xi−1 /xi ), (xi+1 /xi ), . . . , (xn /xi )] over the isomorphic open subschemes D(xj /xi ) of D+ (xi ) and D(xi /xj ) of D+ (xj ). How this patching can be done precisely will be explained in the next section (see Construction 5.3). The next definition enables us to define the basic objects of study in algebraic geometry, namely loci of zeros of systems of polynomials in affine or projective space. Definition 3.6 A morphism φ : Y → X of schemes is a closed immersion if the underlying continuous map is the inclusion of a closed subset of X and moreover there is a covering of X by affine open subschemes Ui = Spec Ai such that for all i the open subscheme of Y defined by φ−1 (Ui ) is isomorphic to an affine scheme Spec Bi with the induced maps Ai → Bi surjections. We say that Y is a closed subscheme of X if there is a closed immersion of Y into X. Remark 3.7 It can be shown that any closed subscheme of an affine scheme X = Spec A is of the form Spec A/I with some ideal I. However, the reader should be warned that in general it is possible to give several different closed subscheme structures on a given closed subset of the underlying topological space of a scheme. Example 3.8 An (irreducible) affine hypersurface of dimension n − 1 over a field k is the closed subscheme of Ank given by the quotient of the polynomial ring k[x1 , . . . , xn ] by the principal ideal generated by an irreducible polynomial f (here the covering of Ank is just the one-element covering by the whole space). Affine hypersurfaces of dimension 1 are also called plane curves. For instance, the quotient of k[x1 , x2 ] modulo the principal ideal generated by the polynomial x1 x2 − 1 defines an affine plane curve: the conic of equation x1 x2 = 1. A projective hypersurface is a closed subscheme Y of some Pnk which restricts to an affine hypersurface on each canonical open subset D+ (xi ) via the isomorphisms D+ (xi ) ∼ = Ank . As above, in dimension 1 we get projective plane curves. For instance, the locus of zeros of the homogenous polynomial X0 X1 − X22 in P2k defines a projective plane curve given on D+ (X0 ) by the affine plane curve of equation x1 = x22 , on D+ (X1 ) that of equation x0 = x22 and on D+ (X2 ) that of equation x0 x1 = 1 (notice that different types of affine conics arise from the same projective conic).

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Remark 3.9 If k = C, a plane curve defined by a polynomial whose partial derivatives do not vanish simultaneously carries the structure of a scheme and the structure of a Riemann surface as well.

4.

Quasi-coherent Sheaves

In Section 2 we saw that any ring A defines an affine scheme X = Spec A. Here we study how to associate sheaves on X to modules over the ring A, a construction that will be very useful in the next chapter. As the construction of affine schemes makes one expect, sheaves associated to A-modules should be, in some sense, modules over the structure sheaf OX . The following definition makes this precise. Definition 4.1 Let X be any scheme. A sheaf of OX -modules or an OX module for short is a sheaf of abelian groups F on X such that for each open U ⊂ X the group F (U) is equipped with an OX (U)-module structure OX (U) × F (U) → F (U) making the diagram OX (U) × F (U) −−−→ F (U)   y

  y

OX (V ) × F (V ) −−−→ F (V ) commute for each inclusion of open sets V ⊂ U. In the special case when F (U) is an ideal in OX (U) for all U we speak of a sheaf of ideals on X. Examples 4.2 Here are two natural situations where OX -modules arise. 1. Let φ : X → Y be a morphism of schemes. We know that on the level of structure sheaves φ is given by a morphism φ] : OY → φ∗ OX , whence an OY -module structure on φ∗ OX . 2. In the previous situation the kernel I of the morphism φ] : OY → φ∗ OX (defined by I(U) = ker(OY (U) → φ∗ OX (U))) is a sheaf of ideals on Y . This is particularly interesting when X is a closed subscheme of Y and φ is the natural inclusion. In this case we call I the sheaf of ideals defining X. 3. More generally, given any OX -module F one can define its annihilator as the ideal sheaf whose sections over an open set U consist of those f ∈ OX (U) for which f s = 0 for all s ∈ F (U). For instance, the annihilator of the OX -module 0 is OX .

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We may now proceed to construct OX -modules over affine schemes from modules over rings. For this we first need an algebraic concept. Definition 4.3 Let A be a ring, S ⊂ A a multiplicatively closed subset and M an A-module. The localisation of M by S is the AS -module MS given by M ⊗A AS . As in the case of rings, given an element f ∈ A or a prime ideal P , we shall use the notations Mf for M ⊗A Af and MP for M ⊗A AP . Construction 4.4 Let A be a ring and M an A-module. For any multiplicatively closed S ⊂ A there is a natural map M → MS obtained by tensoring the natural map A → AS by M and similarly there is a natural map Mg → Mf for any inclusion D(f ) ⊂ D(g). The sheaf axioms for OX imply that the Mf satisfy the sheaf axioms over basic open sets, so that ˜ over X which is an OX -module Lemma 2.7 may be applied to get a sheaf M by construction. ˜ is naturally a functor from the category of A-modules The rule M → M to the category of OX -modules and it is easy to check that it is fully faithful. One cannot expect, however, that in this way an equivalence of categories arises, as the following simple counter-example shows. Example 4.5 Let A be the local ring of the affine line A1k in 0, i.e. the localisation of the polynomial ring k[x] by the ideal (x). Then X = Spec A consists only of two points: a closed point coming from (x) and a so-called generic point η coming from the ideal (0). The stalks of OX are A in the closed point and k(x) in the generic point. Now define an OX -module F on X = Spec A by putting F (X) = A and F (η) = 0, the restriction F (X) → F (η) being the zero map. As the only nonempty open subsets of X are η and X itself, these data indeed define an OX -module whose A-module of global sections is A. But this OX -module is not isomorphic to A˜ as the stalks at η are different. Definition 4.6 Let X be a scheme. A quasi-coherent sheaf on X is an OX module F for which there is an open affine cover {Ui : i ∈ I} of X such that the restriction of F to each Ui = Spec Ai is isomorphic to an OUi -module ˜ i with some Ai -module Mi . If moreover each Mi is finitely of the form M generated over Ai , then F is called a coherent sheaf. Remark 4.7 It can be shown that for an affine scheme X = Spec A the func˜ establishes an equivalence between the category of A-modules tor M → M and that of quasi-coherent sheaves; since we shall not need this, we omit the proof and refer the interested reader to Hartshorne [1], Corollary II.5.5.

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We now return to the first example in 4.2 and investigate the question of determining whether a morphism φ : X → Y yields a quasi-coherent sheaf φ∗ OX in Y . Unfortunately, this is not true in general but Section II.5 of Hartshorne [1] contains several sufficient conditions. For our purposes the following easy condition on φ will suffice. Definition 4.8 A morphism φ : X → Y of schemes is affine if Y has an covering by affine open subsets Ui = Spec Ai such that for each i the open subscheme φ−1 (Ui ) of X is affine as well. Any morphism of affine schemes is obviously affine. We shall see other examples of affine morphisms in the next chapter. Lemma 4.9 If φ : X → Y is an affine morphism, then φ∗ OX and the ideal sheaf defined by the kernel of φ] are quasi-coherent sheaves on Y . Proof: Assume first X = Spec B and Y = Spec A are affine schemes. Then ˜ with B regarded as an A-module via the map λ : A → B φ∗ OX is just B inducing φ. Indeed, it is enough to check this over basic open sets D(f ) for which we may argue in the same way as in the second half of the proof of Theorem 2.14. Moreover, a similar reasong shows that the ideal sheaf on Y defined by the kernel of φ] is just I˜ with I = ker(λ). Once we have these results at hand, the general case of the lemma follows from the definition of affine morphisms and quasi-coherent sheaves. The lemma applies in particular to a closed immersion i : X → Y of schemes which is affine by definition. Thus to any closed subscheme of Y we may associate a quasi-coherent sheaf of ideals. We conclude this section by proving the converse. Proposition 4.10 The above construction gives a bijection between closed subschemes X ⊂ Y and quasi-coherent sheaves of ideals on Y . Proof: Given a quasi-coherent sheaf of ideals I on Y , we may take a covering of Y by affine open subschemes Uj = Spec Aj as in the definition of quasi-coherence. Define for each j a closed immersion ij : Xj → Uj as the map induced by the projection Aj → Aj /Ij , where Ij is the ideal for which S I|Uj ∼ = I˜j . To see that X = Xj is closed in X, note first that X ∩ Uj = Xj for all j (look at the restriction of I to basic open sets contained in the intersections Ui ∩ Uj ). But then any point of Uj ∩ (Y \ X) has an open neighbourhood contained in Uj \ (X ∩ Uj ), whence the claim. Finally, the ij endow X with the structure of a closed subscheme. It is manifest that the two constructions are inverse to each other.

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5.

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Fibres of a Morphism

In our study of covers in topology, the examination of fibres played a preeminent role. Since our ultimate goal is to study covers in the context of schemes, an unavoidable task is to define quite generally the fibre of a morphism of schemes as a scheme and not just a point set. Motivated by the discussion in Chapter 4, Section 3, we introduce more generally the notion of a fibre product of schemes and get the definition of fibres as a special case. As seen in Chapter 4, Remark 3.15 (3), given topological spaces Y → X, Z → X over the same space X, their fibre product can be defined as the space representing the functor. S 7→ {(φ, ψ) ∈ Hom(S, Y ) × Hom(S, Z) : p ◦ φ = q ◦ ψ} on the category of topological spaces over X. We can adopt the same definition in the context of schemes if we show that the similarly defined functor on the category of schemes equipped with morphisms to a fixed base scheme X is representable. We first prove representability in the category of affine schemes. Proposition 5.1 Assume given affine schemes Y = SpecA and Z = SpecB equipped with morphisms p : Y → X, q : Z → X into an affine scheme X = Spec R. Then the contravariant functor S 7→ {(φ, ψ) ∈ Hom(S, Y ) × Hom(S, Z) : p ◦ φ = q ◦ ψ} on the category of affine schemes is representable by Y ×X Z := Spec (A⊗R B). Proof: Indeed, by Theorem 2.14 the statement of the proposition is equivalent to saying that given ring homomorphisms µ : R → A and ν : R → B, the ring A ⊗R B represents the functor C 7→ {(κ, λ) ∈ Hom(A, C) × Hom(B, C) : κ ◦ µ = λ ◦ ν} on the category of commutative rings with unit. But this is precisely the defining property of the tensor product (of R-algebras). Remark 5.2 It is not true that the underlying topological space of a fibre product of affine schemes is the topological fibre product of the underlying topological spaces of the schemes. As an easy example, take X = Spec k with some field k, Y = Spec L, with L|k a finite separable extension of k, ¯ Then the topological fibre product of Y and Z over X is a Z = Spec k. one-element set, whereas L ⊗k k¯ is a direct sum of [L : k] copies of k¯ and hence its prime spectrum consists of [L : k] points.

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To extend this construction to arbitrary schemes, the idea is of course to cover them with open affine subschemes and then to “patch” the fibre products of these affine schemes together. How this can be done precisely is explained next. Construction 5.3 Assume given a family of schemes {Xi : i ∈ I} and for each (i, j) ∈ I 2 an open subscheme Uij ⊂ Xi . Assume moreover we are given isomorphisms φij : Uij → Uji satisfying the compatibility conditions φij = φ−1 ji and φij (Uij ∩ Uik ) = Uji ∩ Ujk ,

φik = φjk ◦ φij on Uij ∩ Uik .

We now construct a scheme X having an open covering {Ui : i ∈ I} such that each Ui is isomorphic to Xi as a scheme and via these isomorphisms the Uij correspond to the intersections Ui ∩Uj . The above compatibility relations for the φij are thus necessary conditions for such a scheme X to exist. Define the underlying set of X to be the disjoint union of those of the Xi modulo the equivalence relation which identifies points of Uij with those of Uji via φij . The compatibility conditions for the φij ensure that this is indeed an equivalence relation; we endow X with the quotient topology. The com` posite maps pi : Xi → Xi → X map each Xi homeomorphically onto an open subset Ui ⊂ X. Now to define the structure sheaf OX of X it suffices by Lemma 2.7 to define its sections over a basis of open sets in X in a compatible fashion. The open sets U which are contained in one of the Ui clearly form a basis. For such a U one is tempted to define OX (U) as OXi (p−1 i (U)), but the problem is that U may be contained in the intersection of several Ui . However, the rings obtained for each choice of Ui are all isomorphic via the φij , Q so to remedy this one defines OX (U) to be the subring of OXi (p−1 i (U)) U ⊂Ui

consisting of sequences of sections mapped to each other by the φij . More precisely, we take those sequences (si ) with φ]ij (sj ) = si for all (i, j) (here si is viewed as a section in φij∗(OXi |Uij )(p−1 j (U))). One defines restriction maps for subsets V ⊂ U as induced by the product of the restriction maps of the OXi ; in fact, any element of OX (V ) is determined by its components indexed by the sets Ui containing U. It is now straightforward to check the sheaf axioms over U as well as the fact that the ringed space thus obtained is a scheme. Armed with this patching construction, we may now construct fibre products of arbitrary schemes. Just as in topology, let us refer to a morphism p : Y → X of schemes as a scheme over X.

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Proposition 5.4 Given two schemes p : Y → X, q : Z → X over the same scheme X, the contravariant functor S 7→ FY Z (S) := {(φ, ψ) ∈ Hom(S, Y ) × Hom(S, Z) : p ◦ φ = q ◦ ψ} the category of schemes is representable by a scheme Y ×X Z. The scheme Y ×X Z is called the fibre product of Y and Z over X. It is equipped with two canonical morphisms into Y and Z making the diagram π

2 Y ×X Z −−− → Z

 

π1 y

Y

p

 q y

−−−→ X

commute (they correspond to the identity morphism of Y ×X Z). Proof: We show first that if Y , Z and X are all affine, then the scheme Y ×X Z defined in Proposition 5.1 is indeed a fibre product in the category of schemes. For this we have to see that for an arbitrary scheme S any element of FY Z (S) factors as a composite (π1 , π2 ) ◦ φ with some morphism φ : S → Y ×S Z. Choosing an affine open cover {Si : i ∈ I} of S, for each i we dispose of a morphism φi : Si → Y ×X Z with the above property according to Proposition 5.1. Since by definition for any affine open subset U ⊂ Si ∩ Sj the elements of FY Z (U) are in bijection with Hom(U, Y ×X Z), we see that the restrictions of φi and φj to the open subschemes Si ∩ Sj are the same for all (i, j). Hence there is a unique morphism φ agreeing with φi over Si (the existence of the underlying continuous map is straightforward; for the existence of φ] use Lemma 2.7 (3)). Still assuming X affine, choose affine open coverings {Yi : i ∈ I} and {Zj : j ∈ J} of Y and Z, respectively. First fix some l ∈ J. We then dispose of affine shemes Yi ×X Yl for each i ∈ I. Now note that quite generally if Y ×X Z represents the functor FY Z and U ⊂ Y is an open subscheme, then the open subscheme π1−1 (U) ⊂ Y represents the functor FU Z and as such is unique up to unique isomorphism by the Yoneda lemma. Applying this remark with Zl in place of Z, Yi (resp. Yj ) in place of Y and Yi ∩ Yj in place of U we see that there exist unique isomorphisms φij : Uij → Uji , where Uij (resp. Uji ) is the inverse image of Yi ∩Yj by the projection Yi ×X Zl → Yi (resp. Yj ×X Zl → Yj ). The uniqueness of the φij implies that the compatibility conditions in Construction 5.3 are satisfied, so we may patch the Yi ×X Zl together along the Uij to obtain a scheme Y ×X Yl . The projections Y ×X Zl → Y and Y ×X Zl → Zl are defined by patching the projections from the

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elements of the open covering {Yi ×X Zl : i ∈ I} of Y ×X Zl as in the previous paragraph. To show that Y ×X Zl represents FY Zl one considers for a pair (φ, ψ) ∈ FY Zl (S) the restrictions (φ|φ−1 (Yi ) , ψ) ∈ FYi Zl and patches the corresponding morphisms S → Yi ×X Zl together, again arguing as in the previous paragraph. Now by exactly the same method one shows that the schemes Y ×X Zl patch together to give a scheme Y ×X Z representing FY Z . Finally one extends the construction to arbitary X by choosing a covering of X by affine open subschemes Xk and noting that the open subschemes Yk = p−1 (Xk ) form an open covering of Y such that the fibre products Yk ×Xk q −1 (Xk ) represent the functors FYk Z where the Yk are viewed as schemes over X (indeed, given (φ, ψ) ∈ FYk Z (S) we must have ψ(S) ⊂ q −1 (Xk )), so one may repeat the previous procedure to patch the schemes Yk ×X Z = Yk ×Xk q −1 (Xk ) together. Now if we imitate the situation in topology, to define the fibre of a morphism Y → X at some point P of X we first need to define the inclusion morphism {P } → X. This is achieved as follows. Take an affine open neighbourhood U = Spec A. Then P is identified with a prime ideal of A and we dispose of a morphism A → AP which we may compose with the natural projection AP → AP /P AP =: κ(P ). We get a morphism Spec κ(P ) → U, whence by composition with the inclusion map U → X a morphism iP : Spec κ(P ) → X. Lemma 5.5 The morphism iP : Spec κ(P ) → X just defined does not depend on the choice of U. Proof: If V = Spec B is another affine open neighbourhood of P , then there is some affine open subscheme W ⊂ V ∩ U. We may assume that W as a subscheme of U is of the form D(f ) with some f ∈ A \ P . But the localisation map A → AP factors through Af (since f is a unit in AP ), which means that the map Spec AP → U factors through W . By symmetry, we get the same conclusion for V . Definition 5.6 Given a morphism φ : Y → X and a point P of X, the fibre of φ at P is the scheme YP := Y ×X Spec κ(P ), the fibre product being taken with respect to the maps φ and iP . We saw in Remark 5.2 that the underlying topological space of a fibre product is not a topological fibre product in general. However, the good news is:

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Proposition 5.7 Given a morphism φ : Y → X and a point P of X, the underlying topological space of the fibre YP is homeomorphic to the subspace φ−1 (P ) of the underlying space of Y . Proof: We may assume we are dealing with affine schemes Y = Spec B and X = Spec A. We first show there is a bijection between φ−1 (P ) and Spec B⊗A κ(P ) as sets, the homomorphism λ : A → B defining the A-module structure of B being the one corresponding to φ by Theorem 2.14. Now the above λ induces a map λ : A/P → B/λ(P )B and a point Q ∈ Spec B is in −1 φ−1 (P ) if and only if its image Q in B/λ(P )B satisfies λ (Q) = (0). This is the same as saying that λ(A/P ) ∩ Q = {0}, or else, putting S = λ(A/P ) \ {0}, that Q defines a prime ideal of the localisation (B/λ(P )B)S . But the latter ring is none but B ⊗A κ(P ). To see this, note first the isomorphism B/λ(P )B ∼ = B ⊗A (A/P ) coming from the exact sequence B ⊗A P → B ⊗A A → B ⊗A (A/P ) → 0 coming from tensoring with B the short exact sequence 0 → P → A → A/P → 0 of A-modules. Here we have B ⊗A A ∼ = B coming from the multiplication map b⊗a 7→ ba and so the image of B ⊗A P in B is exactly λ(P )B (since B is an A-module via λ). Now the natural map A/P → B ⊗A (A/P ) = B/λ(P )B is given by a 7→ 1 ⊗ a and the localisation of B ⊗A (A/P ) by the subset {1 ⊗ a : a ∈ (A/P ) \ {0}} is exactly B ⊗A κ(P ). In the above procedure we identified YP = Spec B⊗A κ(P ) with a subset of Spec B; by looking at basic open sets D(f ) one sees easily that the topology of YP corresponds to the subspace topology.

6.

Special Properties of Schemes

In this section we have assembled some technical notions concerning schemes that are to be used in the sequel. The reader is advised to take a brief glance at it and to come back later if necessary. The first definition is: Definition 6.1 A scheme X is called integral if for all open subsets U ⊂ X the ring OX (U) is an integral domain. This algebraic notion has a strong consequence for the underlying topological space of the scheme. Namely, call a topological space X irreducible if

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it cannot be written as a union of two closed subsets properly contained in X, or, equivalently, if any two open subsets have a nonempty intersection. Now the basic fact is: Lemma 6.2 The underlying topological space of an integral scheme is irreducible. Proof: Indeed, if U1 and U2 are nonempty disjoint open subsets of a scheme X, then the sheaf axioms imply that OX (U1 ∪ U2 ) is isomorphic to the direct sum OX (U1 ) ⊕ OX (U2 ),which is not an integral domain. Remark 6.3 In fact, a scheme is integral if and only if its underlying space is irreducible and if the rings OX (U) contain no nilpotent elements. See Hartshorne [1], Proposition II.3.1. Proposition 6.4 Let X be an integral scheme. 1. There is a unique point η ∈ X whose closure is the underlying space of X. 2. The stalk OX,η is a field K which is naturally isomorphic to the fraction field of any local ring of X. Proof: We begin with the first statement. For uniqueness, assume η1 , η2 both have the required property. Then any affine open subset U = Spec A contains both η1 and η2 : they correspond to prime ideals P and Q of A with the property V (P ) = V (Q) = U. Since A is an integral domain, this is only possible for P = Q = (0). This argument also shows the existence of η: indeed, define it as the point corresponding to the ideal (0) of A. Its closure in X contains U, hence it must be the whole of X by the previous lemma. The second statement is obvious from this construction: OX,η is none but the fraction field of A which is the common fraction field of all local rings of U; for the points of X \ U we work with other affine open subsets which all have a non-empty intersection with U by irreducibility of X and hence have a local ring in common. Definition 6.5 The point η of the proposition is called the generic point of X and the field K the function field of X. For an integral scheme X we say that f ∈ K is regular at a point P if f ∈ OX,P ; it has a zero at P if it is contained in the maximal ideal of OX,P .

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Lemma 6.6 ?? Let X be an integral scheme The sets of points Df = {P ∈ X : f is regular at P } and Df0 = {P ∈ X : f is regular and does not have a zero at P } are open in X. Proof: Let f be regular at P and take an affine open neighbourhood U = Spec A of P . We may write f = x/y with x, y ∈ A; thus f is regular in the open neighbourhood D(y) of P , which proves openness of Df . As for Df0 , it is the intersection Df ∩ Df −1 . Definition 6.7 An integral closed subscheme of some affine (resp. projective) n-space over a field k is called an affine (resp. projective) variety over k. Remark 6.8 In the literature one often finds a stronger condition imposed on affine and projective varieties X, namely that they should also be geometrically integral, which means integrality of the scheme X ×Spec k Spec k¯ for an algebraic closure k¯ of k. But in some texts no integrality condition is required at all. Next we state a finiteness condition which is always satisfied for affine and projective varieties. Recall that a ring is noetherian if all of its ideals are finitely generated. Definition 6.9 A scheme X is noetherian if it is compact and has a covering by affine open subschemes of the form Spec A with A a noetherian ring. Remark 6.10 It can be shown that any affine open subset of a noetherian scheme is of the form form Spec A with A a noetherian ring. See Hartshorne [1], Proposition II.3.2. We next introduce the notion of dimension for schemes. Of course, we would like affine and projective n-space to be n-dimensional, a point to be 0dimensional, a plane curve 1-dimensional, a surface 2-dimensional, etc. One heuristic approach is the following inductive “argument”: a curve should be of dimension 1 because its irreducible proper closed subsets are only points, a surface should have dimension 2 as it contains only curves and points as proper closed subsets etc. This approach is summarised in the following definition.

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Definition 6.11 The dimension of a scheme X is the supremum of the integers n for which there exists a strictly increasing chain Z0 ⊂ Z1 ⊂ . . . ⊂ Zn of irreducible closed subsets properly contained in X. Remark 6.12 The dimension is either a positive integer or infinite. It is mainly interesting for noetherian schemes because noetherian rings have no infinite ascending chains of prime ideals. However, there exist noetherian rings whose associated affine scheme has infinite dimension; see AtiyahMacdonald [1], Exercise 11.4 for an example due to Nagata. In order to be able to give examples in the affine case, we first prove an easy lemma. Lemma 6.13 Let X = Spec A be an affine scheme. Then any irreducible closed subset of X is of the form V (P ), with P a prime ideal of A. Proof: Let Z = V (I) be a closed subset of X. We may and do assume that I is the intersection of the prime ideals corresponding to the points of Z. Assume f g ∈ I for some f, g ∈ A. Then any prime ideal containing I must contain f or g, hence the union of the closed subsets V (I + (f )) and V (I + (g)) is Z. Therefore Z is irreducible if and only if one of them, say V (I + (f )) equals Z. By our assumption on I this is equivalent to f ∈ I, whence the claim. By the lemma, the dimension of Spec A is the supremum of the lengths of chains of prime ideals in A. In ring theory this is called the Krull dimension of A and is usually denoted by dim A. Thus for instance, the Krull dimension of a field is 0, that of Z is 1. But in general with the above definition the dimension is hard to determine in practice. It is not even clear that affine or projective spaces have the dimension we expect. Fortunately, this can be remedied by means of a criterion for wich we need to recall a definition first. Definition 6.14 The transcendence degree of a field extension K|k is the maximal number of elements of K algebraically independent over k; the transcendence degree of an integral domain A containing k is defined as the transcendence degree of its fraction field over k and is denoted by tr.degk A. Now comes the criterion which we only quote from the literature. Proposition 6.15 Let k be a field and A an integral domain which is a finitely generated k-algebra. Then the Krull dimension of A is equal to its transcendence degree over k.

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For a proof, see e.g. Matsumura [1], Theorem 5.6. Example 6.16 As immediate applications of the proposition, we get that Ank and Pnk both have dimension n as expected, and that affine and projective plane curves have dimension 1. In general, affine or projective varieties of dimension 1 are called curves, those of dimension 1 surfaces.

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Chapter 6 Dedekind Schemes In this chapter we introduce the main protagonists of the following two chapters, namely Dedekind schemes. These will be schemes characterised by certain special properties that are common to smooth algebraic curves and spectra of rings of integers in number fields. Analogies between algebraic numbers and functions on algebraic curves have already been noticed in the 19th century; since then, several axiomatisations of the common features have been proposed of which the notion of a Dedekind scheme seems to be particularly satisfactory.

1.

Integral Extensions

In this section we review the basic theory of integral extensions of rings. As this topic is well treated in many texts (e.g. in the books of Lang [1], [2]), we include proofs only for the easiest facts. Recall that given an extension of rings A ⊂ B, an element b ∈ B is said to be integral over A if it is a root of a monic polynomial xn +an−1 xn−1 +. . .+a0 ∈ A[x]. There is the following well-known characterisation of integral elements: Lemma 1.1 Let A ⊂ B an extension of rings. Then the following are equivalent for an element x ∈ B: 1. The element x is integral over A. 2. The subring A[x] of B is finitely generated as an A-module. 3. There is a subring C of B containing x which is finitely generated as an A-module. 119

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Proof: For the implication 1) ⇒ 2) note that if x satisfies a monic polynomial of degree n, then 1, x, . . . , xn−1 is a basis of A[x] over A. The implication 2) ⇒ 3) being trivial, only 3) ⇒ 1) remains. For this consider the A-module endomorphism of C given by multiplication by x. Its characteristic polynomial f is monic; by the Cayley-Hamilton theorem, f (x) = 0. Corollary 1.2 Those elements of B which are integral over A form a subring in B. Proof: Indeed, given two elements x, y ∈ B integral over A, the elements x − y and xy are both contained in the subring A[x, y] of B which is a finitely generated A-module by assumption. If all elements of B are integral over A, we say that the extension A ⊂ B is integral. Corollary 1.3 Given a tower extensions A ⊂ B ⊂ C with A ⊂ B and B ⊂ C integral, the extension A ⊂ C is also integral. If A is a domain with fraction field K and L is an extension of K, the integral closure of A in L is the subring of L formed by elements integral over A. We say that A is integrally closed if its integral closure in the fraction field K is just A. By the corollary above, the integral closure of a domain A in some extension L of its fraction field is integrally closed. Example 1.4 Any unique factorisation domain A is integrally closed. Indeed, if an element a/b ∈ K (with a, b coprime) satisfies a monic polynomial equation of degree n, then by multiplying with bn we see that an should be divisible by b which is only possible when b is a unit. In particular, the ring Z is integrally closed. Recall that a number field K is by definition a finite extension of Q. We denote by OK the integral closure of Z in K and call it the ring of integers of K. Of course, OK is integrally closed. We next collect some easy results that will be needed in subsequent sections. Lemma 1.5 Let A ⊂ B be an integral extension of domains. 1. If I is a nonzero ideal of B, then I ∩ A is a nonzero ideal of A. 2. If A is a field, then B is a field as well.

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Proof: For the first statement note that if a nonzero u ∈ I satisfies an equation un + an−1 un−1 + . . . + a1 u + a0 = 0 with ai ∈ k, then a0 ∈ I ∩ A. Since B is a domain, we may assume a0 6= 0, whence the assertion. The second statement follows from this for if I is a nonzero ideal of B, we must have I ∩ A = A when A is a field, hence 1 ∈ I and I = B. Remark 1.6 In fact, the converse of the second statement is also true: if B is a field, then A must be a field as well, but we shall not need this. Lemma 1.7 A domain A is integrally closed if and only if for any prime idal P of A the localisation AP is integrally closed. Proof: Denote by K the fraction field of A. One implication is easy: if an element x ∈ K satisfies a monic equation over AP , then by multiplying with a suitable common multiple s of the denominators of the coefficients one gets that sx is integral over A, hence sx ∈ A. For the converse, take an element x = a/b ∈ K integral over A. If b is not a unit in A, there is some maximal ideal P containing it. But AP is integrally closed and a/b is integral over it, so a/b ∈ AP which is absurd. Thus b is a unit and a/b ∈ A. Finally, a similar argument to that in the first part of the previous proof shows: Lemma 1.8 Let A be a domain with fraction field K ans let S ⊂ A be a multiplicatively closed subset. Then for any field extension L|K, the integral closure of the localisation AS in L is BS , where B is the integral closure of A in L. The last topic to be treated in this section is the question whether the integral closure of an integral domain A in a finite extension of its fraction field is a finitely generated A-module. Unfortunately, this property does not hold for arbitrary domains, even under the assumption that A is noetherian. But there are two classical sufficient conditions which we now quote from the literature. Proposition 1.9 Let A be an integrally closed noetherian domain with fraction field K and let L be a finite separable extension of K. Then the integral closure B of A in any finite extension of L is a finitely generated A-module, and hence a Noetherian ring. For the proof, see Atiyah-Macdonald [1], Corollary 5.17 or Lang [2], Chapter I, Proposition 6. The other sufficient condition is the following.

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Proposition 1.10 Let k be a field and A an integral domain which is a finitely generated k-algebra. If L is any finite extension of the fraction field K of A, then the integral closure of A in L is a finitely generated A-module. Note that the proposition is implied by the previous one when k is of characteristic 0. In the case of positive characteristic, there is a simple proof for polynomial rings in Shafarevich [1], Appendix, Section 8. The general case reduces to this case by applying Noether’s Normalisation Lemma (Lang [1], Chapter VIII, Theorem 2.1).

2.

Dedekind Schemes

We begin the discussion of Dedekind schemes by studying their local rings which enjoy very similar properties to rings of germs of meromorphic functions in a neighbourhood of some point of a Riemann surface. The first of the several equivalent definitions we are to give is perhaps the simplest one. Definition 2.1 A ring A is a discrete valuation ring if A is a local principal ideal domain which is not a field. Before stating the first equivalent characterisations, observe that if A is a local ring with maximal ideal P , then the A-module P/P 2 is in fact a vector space over the field κ(P ) = A/P , simply because multiplication by P maps P into P 2 . Proposition 2.2 For a local domain A with maximal ideal P the following conditions are equivalent: 1. A is a discrete valuation ring. 2. A is noetherian of Krull dimension 1 and P/P 2 is of dimension 1 over κ(P ). 3. A is noetherian and P is generated by a single nonzero element. For the proof we need the following well-known lemma which will be extremely useful in other situations as well: Lemma 2.3 (Nakayama’s Lemma) Let A be a local ring with maximal ideal P and M a finitely generated A-module. If P M = M, then M = 0.

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Proof: Assume M 6= 0 and let m0 , . . . , mn be a minimal system of generators of M over A. By assumption m0 is contained in P M and hence we have a relation m0 = p0 m0 + . . . , pn mn with all the pi elements of P . But here 1 − p0 is a unit in A (as otherwise it would generate an ideal contained in P ) and hence by multiplying the equation by (1 − p0 )−1 we may write m0 as a linear combination of the other terms, which is in contradiction with the minimality of the system. Here is an immediate corollary of the lemma. Corollary 2.4 Let A be a Noetherian local ring with maximal ideal P . Then \

P i = (0).

i

Moreover, if P i 6= (0), then P i 6= P i+1 . Proof: Denote by Q the intersection of the P i . Since A is Noetherian, Q is finitely generated. Moreover, P Q = Q and the lemma applies. The second statement is proved in a similar way. Another corollary of the lemma is the following strengthened form. Corollary 2.5 Let A, P , M be as in the lemma and assume given elements t1 , . . . , tm ∈ M whose images in the A/P -vector space M/P M form a generating system. Then they generate M over A. Proof: Let T be the A-submodule generated by the ti ; we have M = T + P M by assumption. Hence M/T = P (M/T ) and the lemma gives M/T = 0. Proof of Proposition 2.2: Assume A is a discrete valuation ring and P is generated by t. Then by Corollary 2.4 any nonzero prime ideal Q ⊂ P must contain a power of t. But being a prime ideal, it must then contain t itself, so that Q = P and A is of Krull dimension one. Also, the image of t generates the vector space P/P 2, whence the second condition. The third condition follows from the second by applying Corollary 2.5 with M = P . Finally, to show that the third condition implies the first, assume the maximal ideal P of A is generated by some element t. We first show that any element a ∈ A can be written uniquely as a product a = utn , with u a unit in A. Indeed, by Corollary 2.4 there is a unique n ≥ 0 for which a ∈ P n \ P n+1 and thus a can be written in the required form. If a = utn = vtn , then u = v since A is a domain. Now take an ideal I of A. As A is Noetherian, I can be generated

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by a finite sequence of elements a1 , . . . , ak . Write ai = uitni according to the above representation and let j be an index for which ni ≥ nj for all i. Each ai is a multiple of tnj and hence I = (tnj ) is principal. In the course of the above proof we have also shown: Corollary 2.6 Any element x 6= 0 of the fraction field of a discrete valuation ring A can be written uniquely in the form x = utn with u a unit in A, t a generator of the maximal ideal and n a (possibly negative) integer. The second condition of the lemma may seem a bit technical, but it is very useful for it is a special case of a more general notion coming from algebraic geometry. Definition 2.7 A noetherian local ring A with maximal ideal P is regular if its Krull dimension equals the (finite) dimension of P/P 2 over κ(P ). Thus discrete valuation rings are regular local rings of dimension 1. We now explain the origin of the name “discrete valuation ring”. Definition 2.8 For any field K, a discrete valuation is a surjection v : K → Z ∪ {∞} with the properties v(xy) = v(x) + v(y), v(x + y) ≥ min{v(x), v(y)}, v(x) = ∞ if and only if x = 0. The elements x ∈ K with v(x) ≥ 0 form a subring A ⊂ K called the valuation ring of v. Proposition 2.9 A domain A is a discrete valuation ring if and only if it is the valuation ring of some discrete valuation v : K → Z ∪ {∞}, where K is the fraction field of A. Proof: Assume first A is a discrete valuation ring. Define a function v : K → Z ∪ {∞} by mapping 0 to ∞ and any x 6= 0 to the integer n given by the previous corollary. It is immediate to check that v is a discrete valuation with valuation ring A. Conversely, given a discrete valuation v on K, the maximal ideal of its valuation ring is generated by any t with v(t) = 1 and we may apply the previous proposition. We can now discuss the example mentioned at the beginning of this section.

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Example 2.10 Let M be the sheaf of meromorphic functions on some Riemann surface X and x a point of X. Define v(f ) = m if f is holomorphic at x and has a zero of order m and define v(f ) = −v(1/f ) otherwise. Then v is a discrete valuation on K = M(X) whose discrete valuation ring is the stalk of M at x. The last characterisation of discrete valuation rings we shall need is the following. Proposition 2.11 A domain A is a discrete valuation ring if and only if A is noetherian, integrally closed and has a unique nonzero prime ideal. For the proof, which we have taken from Serre [1], we need a technical lemma. Lemma 2.12 Let A be a domain having a unique nonzero prime ideal P . Then P −1 6= A. Here P −1 denotes the set of elements x of the fraction field K of A with xA ⊂ P . Proof: Observe first that for any f ∈ P the localisation Af is a field and hence equals K itself. Indeed, for a maximal ideal M of Af the prime ideal M ∩ A doesn’t contain f , hence it can be only (0). But for any nonzero element y/f n ∈ M we would have 0 6= y ∈ M ∩ A, whence M = 0, as desired. Take now another x ∈ P ; by the above we may write x−1 = a/f m with some m, so f m = ax, and thus f m ∈ (x). Letting f vary in a finite set of generators of P we conclude that a sufficiently high power of P is contained in (x). Let P N be the least such power; we may thus find y ∈ P N −1 with y∈ / (x). But yP ⊂ (x), so yx−1 ∈ P −1 . However, yx−1 ∈ / A as y ∈ / (x). Proof of Proposition 2.11: The necessity of the conditions is immediate (the second condition follows from Example 1.4 and the last from the fact that any nonzero ideal of A is of the form (tn ) with t a generator of the maximal ideal; such an ideal can be prime only for n = 1). For sufficiency denote by P the maximal ideal of A; we have to show that it is principal. Evidently A ⊂ P −1 , so P ⊂ P −1P , the latter being the ideal of A generated by elements of the form xy with x ∈ P −1 and y ∈ P . Since P is maximal, there are two cases: either P −1P = A or P −1P = P . We now show that if the first case holds, then P is principal. Indeed, in this case there is a relation of the form x1 y1 + . . . + xn yn = 1 with xi ∈ P −1 and yi ∈ P . Here there is at least one i for which xi yi ∈ / P , so there is some unit

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u with uxi yi = 1. We contend that P = (yi ). Indeed, if z ∈ P , we have z = uzxi yi, but uzxi ∈ A since xi ∈ P −1 . To finish the proof we show that the case P −1 P = P cannot occur. We do this by showing that this assumption implies P −1 = A, in contradiction with lemma 2.12. So take x ∈ P −1. By assumption xP ⊂ P ; iterating this we get xn P ⊂ P for all n, so xn ∈ P −1 for all n. In particular, for any f ∈ P all powers of x are contained in the A-submodule of K generated by f −1 , which is a finitely generated A-module. But A is noetherian and a submodule of a finitely generated module over a noetherian ring is always finitely generated, hence by Lemma 1.1 x is integral over A. As A is integrally closed, this implies x ∈ A, as desired. Remark 2.13 The affine scheme Spec A is particularly simple for a discrete valuation ring. It consists only two points, a closed point x (corresponding to the maximal ideal) and a non-closed generic point η (corresponding to the ideal (0)). The stalk of the structure sheaf at η is the fraction field K of A and the stalk at x is A itself. Now we can pass from local rings to schemes and give our main definition. Definition 2.14 A normal scheme is a scheme whose local rings are integrally closed domains. A Dedekind scheme is an integral noetherian normal scheme of dimension 1. A Dedekind ring is a domain A such that Spec A is a Dedekind scheme. Remarks 2.15 Here some remarks are in order. 1. There is another restriction that is convenient to impose on the schemes we shall be looking at, namely that the local rings of X should be distinct when viewed as subrings of the function field K; in this case one says X is separated (over Z). This condition plainly holds in the affine case where the local rings are all localisations at different prime ideals; it also holds for closed subschemes of projective space. However, there is a pathological example, the “affine line with the origin doubled” which satisfies the definition of a Dedekind scheme given above but which is not separated. This is constructed by taking two copies of the affine line A1k over some field k and patching them over the open subset D(x) using the isomorphism given by the identity map. In this way we get two closed points coming from the two origins whose local rings are the same. Henceforth we shall tacitly assume that all integral schemes under consideration are separated.

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2. Call a noetherian scheme regular if all of its local rings are regular. Then by Proposition 2.2 Dedekind schemes are precisely regular integral schemes of dimension 1. Now let us draw some immediate consequences from the definition of Dedekind schemes. Proposition 2.16 Let X be a Dedekind scheme with function field K. 1. The closed subsets of X are just X and finite sets of closed points. 2. The local rings of X at closed points are discrete valuation rings with fraction field K. 3. Any affine open subset of X is of the form Spec A, with A an integrally closed domain. Proof: Since X is noetherian, it is compact, so for the first statement we may assume X = Spec A with a Noetherian ring A. In Spec A any closed subset is a finite union of irreducible closed subsets. (To see this, decompose any reducible closed subset Z as a union of nonempty closed subsets Z1 ∪Z2 ; if these are not irreducible, decompose them again - the process must terminate in finitely many steps as otherwise we would get an infinite strictly increasing chain of ideals of A which is impossible in a noetherian ring.) But any irreducible closed subset of X is a closed point by Chapter 5, Lemma 6.13 and the fact that all prime ideals of A are maximal. This proves the first statement; the second follows from Proposition 2.11 in view of the easy fact that any localisation of a noetherian ring A is noetherian (as the ideals of the localisation are all generated by ideals of A according to Chapter 4, Lemma 2.4). The third is a consequence of Lemma 1.7. Examples 2.17 We now give the two main examples of Dedekind schemes. 1. Let OK be the ring of integers of some number field K. Then Spec OK is a Dedekind scheme. Indeed, A is a domain, so Spec A is integral. That it is noetherian follows from Proposition 1.9. To prove that it is of dimension 1, we show that any nonzero prime ideal P of OK is maximal. Indeed, by the first part of Lemma 1.5, the intersection P ∩ Z is nonzero, hence generated by some prime number p. Now the induced extension Z/pZ ⊂ OK /P is still integral, so we may apply the second statement of the same lemma to conclude that OK /P is a field. Finally the normality of Spec A follows from Lemma 1.7.

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2. The second basic example of a Dedekind scheme is given a by onedimensional normal integral closed subscheme of affine (resp. projective) n-space over some field k. These we shall call smooth affine (resp. projective) curves over k. For the moment, this definition is tautologous but one can still give some concrete examples. For instance, the affine line A1k is a smooth affine curve (being the spectrum of the onedimensional unique factorisation domain k[t]) and the projective line P1k is a smooth projective curve over k for it is integral and can be covered by two copies of A1k . However, for a general closed subscheme of affine or projective space this ring-theoretic definition in the second example is rather hard to check. What is much more preferable is the smoothness condition encountered in our discussion of Riemann surfaces; for a plane curve this said that the partial derivatives of its defining equation should not simultaneously vanish at some point. In the next section we introduce an algebraic formalisation of the notion of differentials and prove a broad generalisation of this criterion.

3.

Modules and Sheaves of Differentials

In differential geometry, the tangent space at a point P on some variety is defined to consist of so-called linear derivarions, i.e. linear maps that associate a scalar to each function germ at P and satisfy the Leibniz rule. We begin by an algebraic generalisation of this notion. Definition 3.1 Let B be a ring and M a B-module. A derivation of B into M is a map d : B → M subject to the two conditions: 1. (Additivity) d(x + y) = dx + dy; 2. (Leibniz rule) d(xy) = xdy + ydx. Here we have written dx for d(x) to emphasise the analogy with the classical derivation rules. If moreover B is an A-algebra for some ring A (for example A = Z), an A-linear derivation is called an A-derivation. The set of Aderivations of B to M is equipped with a natural B-module structure via the rules (d1 + d2 )x = d1 x + d2 x and bdx = dbx. This B-module is denoted by DerA (B, M). Note that applying the Leibniz rule to the equality 1·1 = 1 gives d(1) = 0 for all derivations; hence all A-derivations are trivial on the image of A in B.

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In the example one encounters in (say) real differential geometry we have A = M = R, and B is the ring of germs of differentiable functions at some point; R is a B-module via evaluation of functions. Now comes a purely algebraic example. Example 3.2 Assume given an A-algebra B which decomposes as an Amodule into a direct sum B ∼ = A ⊕ I, where I is an ideal of B with I 2 = 0. Then the natural projection d : B → I is an A-derivation of B into I. Indeed, A-linearity is immediate; for the Leibniz rule we take elements x1 , x2 ∈ B and write xi = ai + dxi with ai ∈ k for i = 1, 2. Now we have d(x1 x2 ) = d[(a1 + dx1 )(a2 + dx2 )] = d(a1 a2 + a2 dx1 + a1 dx2 ) = x2 dx1 + x1 dx2 where we used several times the facts that I 2 = 0 and d(A) = 0. In fact, given any ring A and A-module I, we can define an A-algebra B as above by defining a product structure on the A-module A ⊕ I by the rule (a1 , i1 )(a2 , i2 ) = (a1 a2 , a1 i2 + a2 i1 ). So the above method yields plenty of examples of derivations. Now notice that for fixed A and B the rule M → DerA (B, M) defines a functor on the category of B-modules; indeed, given a homomorphism φ : M1 → M2 of B-modules, we get a natural homomorphism DerA (B, M1 ) → DerA (B, M2 ) by composing derivations with φ. Proposition 3.3 The functor M → DerA (B, M) is representable by a Bmodule ΩB/A . Proof: The construction is done in a similar way to that of the tensor product of two modules. Define ΩB/A to be the quotient of the free Bmodule generated by symbols dx for each x ∈ B modulo the relations given by the additivity and Leibniz rules as in Definition 3.1 as well as the relations d(λ(a)) = 0 for all a ∈ A, where λ : A → B is the map defining the Amodule structure on B. The map x → dx is an A-derivation of B into ΩB/A . Moreover, given any B-module M and A-derivation δ ∈ DerA (B, M), the map dx → δ(x) induces a B-module homomorphism ΩB/A → M whose composition with d is just δ. This implies that ΩB/A represents the functor M → DerA (B, M); in particular, d is the universal derivation corresponding to the identity map of ΩB/A . We call ΩB/A the module of relative differentials of B with respect to A. We shall often refer to the elements of ΩB/A as differential forms. Next we describe how to compute relative differentials of a finitely presented A-algebra.

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Proposition 3.4 Let B be the quotient of the polynomial ring A[x1 , . . . , xn ] by an ideal generated by finitely many polynomials f1 , . . . , fm . Then ΩB/A is the quotient of the free B-module on generators dx1 , . . . , dxn modulo the P B-submodule generated by the elements j (∂j fi )dxj (i = 1, . . . , m), where ∂j fi denotes the j-th (formal) partial derivative of fi . Proof: First consider the case B = A[x1 , . . . , xn ]. As B is the free Aalgebra generated by the xi , one sees that for any B-module M there is a bijection between DerA (B, M) and maps of the set {x1 , . . . , xn } into B. This implies that ΩB/A is the free A-module generated by the dxi . The general case follows from this in view of the easy observation that given any M, composition by the projection A[x1 , . . . , xn ] → B induces an isomorphism of DerA (B, M) onto the submodule of DerA (A[x1 , . . . , xn ], M) consisting of derivations mapping the fi to 0. Next some basic properties of modules of differentials. Lemma 3.5 Let A be a ring and B an A-algebra. 1. (Direct sums) For any A-algebra B 0 Ω(B⊕B0 )/A ∼ = ΩB/A ⊕ ΩB0 /A . 2. (Exact sequence) Given a map of A-algebras φ : B → C, there is an exact sequence of C-modules ΩB/A ⊗B C → ΩC/A → ΩC/B → 0. In particular, if φ is surjective, we have a surjection ΩB/A ⊗B C → ΩC/A . 3. (Base change) Given a ring homomorphism A → A0 , denote by B 0 the A0 -algebra B ⊗A A0 . There is a natural isomorphism ΩB/A ⊗B B 0 ∼ = ΩB0 /A0 . 4. (Localisation) For any multiplicatively closed subset S ⊂ B there is a natural isomorphism ΩBS /A ∼ = ΩB/A ⊗B BS .

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Proof: The first property is easy and left to the readers. For the second, note that for any C-module M we have a natural exact sequence 0 → DerB (C, M) → DerA (C, M) → DerA (B, M) of C-modules isomorphic to 0 → HomC (ΩC/B , M) → HomC (ΩC/A , M) → HomB (ΩB/A , M). The claim follows from this in view of the formal Lemma 3.8 of Chapter 0 and the isomorphism HomB (ΩB/A , M) ∼ = HomC (ΩB/A ⊗B C, M). This isomorphism is obtained by mapping a homomorphism ΩB/A → M to the composite ΩB/A ⊗B C → M ⊗B C → M where the second map is multiplication; an inverse is given by composition with the natural map ΩB/A → ΩB/A ⊗B C. If the map B → C is onto, then any B-derivation is a C-derivation as well, so ΩB/C = 0 and the first map in the exact sequence is onto. For base change, note first that the universal derivation d : B → ΩB/A is an A-module homomorphism and so tensoring it by A0 we get a map d0 : B 0 → ΩB/A ⊗A A0 ∼ = ΩB/A ⊗B B 0 = ΩB/A ⊗B B ⊗A A0 ∼ which is easily seen to be an A0 -derivation. Now any A0 -derivation δ 0 : B 0 → M 0 induces an A-derivation δ : B → M 0 by composition with the natural map B → B 0 . But δ factors as δ = φ ◦ d, with a B-module homomorphism φ : ΩB/A → M 0 , whence a map φ0 : ΩB/A ⊗B B 0 → M 0 constructed as above. Now one checks that δ 0 = φ0 ◦ d0 which means that ΩB/A ⊗B B 0 represents the functor M 0 7→ DerA0 (B 0 , M 0 ). For the localisation property, given an A-derivation δ : B → M, we may extend it uniquely to an A-derivation δS : BS → M ⊗B BS by setting δS (b/s) = (δ(b)s − bδ(s)) ⊗ (1/s2 ). (We leave it to the reader to check that for b0 /s0 = b/s we get the same result – this is much simpler in the case when there are no zero-divisors in S which is the only case we shall need.) This applies in particular to the universal derivation d : B → ΩB/A , and one argues as in the previous case to show that any A-derivation BS → MS factors uniquely through dS . As a first application of the theory of differentials we prove a characterisation of finite ´etale algebras over a field, to be used in forthcoming chapters. Proposition 3.6 Let k be a field and A a finite dimensional k-algebra. Then A is ´etale if and only if ΩA/k = 0.

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Proof: For necessity we may assume by compatibility of ΩA/k with direct sums that A is a finite separable field extension L of k. Then by the theorem of the primitive element A ∼ = k[x]/(f ) with some polynomial f ∈ k[x] and so by Proposition 3.4 the L-module ΩL/k can be presented with a single generator dx and relation f 0 dx = 0. But since the extension is separable, the polynomials f and f 0 are relatively prime and hence the image of f 0 in L∼ = k[x]/(f ) is not 0. Whence dx = 0 in ΩL/k and so ΩL/k = 0. For sufficiency, it is enough to show by virtue of Chapter 1, Proposition 1.2 that A ⊗k k¯ is ´etale over k¯ with k¯ an algebraic closure of k. So using the base change property of differentials we may assume k is algebraicaly closed. Moreover, using the direct sum property as above we may even assume that A is indecomposable. Denoting by I its ideal of nilpotent elements, Chapter 5, Proposition 2.16 gives that A/I is a field. Since k is algebraically closed, we cannot but have A/I ∼ = k and so we have a decomposition A ∼ = k⊕I of A as a k-module. Now to finish the proof we show that assuming I 6= 0 implies ΩA/k 6= 0. For this it is enough to show by the surjectivity property in Lemma 3.5 (2) that Ω(A/I 2 )/k 6= 0, so we may as well assume I 2 = 0. But then we are (up to change of notation) in the situation of Example 3.2 which shows that for I 6= 0 the projection d : A → I is a nontrivial k-derivation, which implies ΩA/k 6= 0. As a second application of differentials we give a criterion for a onedimensional closed subscheme of affine or projective space to be a smooth curve. For this it is enough to check that all local rings at closed points are discrete valuation rings. Since the proof works more generally for regular local rings, we state the result in this context. Proposition 3.7 Let k be a perfect field and let A be a localisation of a finitely generated n-dimensional k-algebra at some closed point P . Then A is a regular local ring if and only if ΩA/k is a free A-module of rank n. In particular, if n = 1, A is a discrete valuation ring if and only if ΩA/k is free of rank 1. Remark 3.8 Explicitly, if A is a localisation of the k-algebra B = k[x1 , . . . , xd ]/(f1 , . . . , fm ), then Proposition 3.4 and the localisation property of differentials imply that P the proposition amounts to saying that among the relations j (∂j fi )dxj = 0 there should be exactly d − n linearly independent ones, which in turn is equivalent by linear algebra to the fact that the k × m “Jacobian” matrix J = [∂j fi ] should have rank d − n. In fact, for k = C reducing the entries of

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J modulo the maximal ideal of A gives just the classical Jacobian matrix of the closed subscheme of Cd defined by the equations fi = 0 at the point P corresponding to A and the condition says that some open neighbourhood of P should be a complex manifold of dimension n. For the proof of the proposition we need two lemmas from algebra. The first of these is a form of Hilbert’s Nullstellensatz (which implies the one used in the previous chapter). Lemma 3.9 Let k be a field and let P be a maximal ideal in a finitely generated k-algebra A. Then the field A/P is a finite extension of k. For a proof, see Lang [1], Chapter IX, Corollary 1.2. See also AtiyahMacdonald [1] for four different proofs. The other lemma is from field theory. Lemma 3.10 Let k be a perfect field and let K|k be a finitely generated field extension of transcendence degree n. Then there exist algebraically independent elements x1 , . . . , xn ∈ K such that the finite extension K|k(x1 , . . . , xn ) is separable. For a proof, see Lang [1], Chapter VIII, Corollary 4.4. Corollary 3.11 In the situation of the lemma, the K-vector space ΩK/k is of dimension n, a basis being given by the dxi . Proof: We may write the field K as the fraction field of the quotient A of the polynomial ring k[x1 , . . . , xn , x] by a single polynomial relation f . Here f is the minimal polynomial of a generator of the extension K|k(x1 , . . . , xn ) multiplied with a common denominator of its coefficients. Now according to Proposition 3.4 the A-module ΩA/k has a presentation with generators dx1 , . . . , dxn , dx and a relation in which dx has a nontrivial coefficient because f 0 6= 0 by the lemma. The corollary now follows using Lemma 3.5 (4). Proof of Proposition 3.7: We give the proof under the additional assumption that there exists a subfield k ⊂ k 0 ⊂ A that maps isomorphically onto the residue field κ(P ) = A/P by the projection A → A/P . (Lemma 3.9 implies that this condition is trivially satisfied if k is algebraically closed.) In the remark below we shall explain how one can reduce the general case to this one. Notice that since k is perfect and k 0 |k is a finite extension by Lemma 3.9, we have Ωk0 |k = 0 by Proposition 3.6 (or the previous corollary). Hence by

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applying Lemma 3.5 (2) (with our k in place of A, k 0 in place of B and A in place of C) we get ΩA/k ∼ = ΩA/k0 , so we may as well assume k = k 0 ∼ = κ(P ). In this case the k-module P/P 2 is canonically isomorphic to ΩA/k /P ΩA/k . Indeed, the latter k-vector space is immediately seen to represent the functor M → Derk (A, M) for any k-vector space M viewed as an A-module via the quotient map A → A/P ∼ = k. On the other hand, the above functor is also 2 represented by P/P . To see this, note first that the Leibniz rule implies that any k-derivation δ : A → M is trivial on P 2, hence we may as well assume P 2 = 0. But then we are in the situation of Example 3.2 and we may observe that δ factors uniquely as δ = φ ◦ d, with d as in the quoted example and φ ∈ Homk (P, N). Now if ΩA/k is free of rank n, then ΩA/k /P ΩA/k ∼ = P/P 2 has dimension n. For the converse, observe first that the previous isomorphism and the corollary to Nakayama’s lemma (Corollary 2.5) gives that ΩA/k can be generated as an A-module by n elements dt1 , . . . , dtn . Were there a nontrivial relation P fi dti = 0 in ΩA/k , by the localisation property of differentials this relation would survive in ΩK/k , contradicting Corollary 3.11. This implies that ΩA/k is free. Remark 3.12 To reduce the general case of the proposition to the one discussed above it is convenient to use the completion Aˆ of A. This is the inverse limit of of the natural inverse system formed by the quotients A/P n of A. There is a natural map A → Aˆ which is injective for A noetherian by Corollary 2.4. The image of P gives a maximal ideal Pˆ of Aˆ with Pˆ i/Pˆ i+1 ∼ = P i/P i+1 for all i > 0. If A is of dimension 1, the case i = 1 of this isomorphism together with Corollary 2.5 implies that A is a discrete valuation ring if and only if Aˆ is. In general, we get that Aˆ is regular if and only if A is regular, for one can prove (see Atiyah-Macdonald [1], Corollary ˆ Also, the base 11.19) that the Krull dimension of A is the same as that of A. change property of differentials implies that ΩA/k is free of rank n if and only ˆ if ΩA/k is. Therefore it remains to see that Aˆ satisfies the condition at the beginning of the above proof. For this, let f ∈ k[x] be the minimal polynomial of a (separable) generator α of the extension κ(P )|k; it is enough to lift α to a ˆ This can be done by means of Hensel’s lemma (see Chapter root of f in A. 7, Section 4). In the remaining of this section we discuss quasi-coherent sheaves associated to modules of differentials. Namely, we shall define sheaves of relative differentials ΩY /X for certain classes of morphisms of schemes Y → X. In fact, one may define these for any morphism Y → X but since we did not

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develop the necessary background we refer the interested readers to the excellent treatment in Mumford’s notes [1] or to Section II.8 of Hartshorne [1]. What we propose instead is a more down-to-earth discussion of the special cases we shall need. Construction 3.13 First, if Y = Spec B and X = Spec A are both affine, ˜ B/A . Notice that according to we define ΩY /X as the quasi-coherent sheaf Ω the localisation property of differentials, over a basic open set D(g) = Spec Bg of X the sheaf ΩY /X is given by the Bg -module ΩBg /A . Construction 3.14 Next assume we have a morphism X → Spec k with an arbitrary scheme X; we shall use the abusive notation ΩX/k for the corresponding sheaf of differentials which we now construct. For any affine open covering of X by subsets Ui = Spec Ai the rings Ai are all k-algebras and the ˜ A /k is defined on Ui . Moreover, any basic open subset consheaf ΩUi /k = Ω i tained in Ui ∩ Uj is canonically isomorphic to both (Ai )fi and (Aj )fj , whence an isomorphism Ω(Ai )f /k ∼ = Ω(Aj )f /k. These isomorphisms are compatible i j for inclusions of basic open sets, so the third statement of Chapter 5, Lemma 2.7 applies to give an isomorphism (ΩUi /k )|Ui ∩Uj ∼ = (ΩUj /k )|Ui ∩Uj . These latter isomorphisms in turn are compatible over triple intersections Ui ∩ Uj ∩ Uk so we may patch the ΩUi /k together by the method of Chapter 5, Construction 5.3 (which adapts to the construction of quasi-coherent sheaves) to get ΩX/k . Finally one checks that if we use a different open covering we get an OX -module isomorphic to ΩX/k . Remark 3.15 Let X be an affine or a projective variety of dimension n. Then Proposition 3.7 may be rephrased by saying that X is a regular scheme if and only if the stalk of the sheaf ΩX/k at each point is free of rank n (for the generic point this follows by localisation). From the next section on, we shall call such sheaves locally free (see Lemma 4.3 below). Also, those X for which ΩX/k is locally free are usually called smooth (over k). In particular, an affine or projective variety of dimension 1 is a Dedekind scheme if and only if it is a smooth curve. Construction 3.16 Finally, the other case where we shall use relative differentials is that of an affine morphism φ : Y → X. In this case X is covered by affine open subsets Ui = Spec Ai whose inverse images Vi = Spec Bi form an open covering of Y and the Bi are Ai -modules via the maps λi : Ai → Bi arising from φ. Take fi ∈ Ai and put gi = λi (fi ). Then the inverse image of the basic open set D(fi ) = Spec (Ai )fi is none but D(gi ) which in turn is isomorphic to Spec (Bi ⊗Ai (Ai )fi ); indeed, one checks easily that (Bi ⊗Ai (Ai )fi )

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represents the functor defining the localisation (Bi )gi . Hence by the base change property of differentials we have canonical isomorphisms ΩVi /Ui (D(gi )) = ΩBi /Ai ⊗Bi (Bi )gi ∼ = Ω(Bi )gi /(Ai )fi = ΩD(gi )/D(fi ) , so we may patch the sheaves ΩVi /Ui together over inverse images of basic affine open subsets contained in Ui ∩ Uj by the same method as in the previous case.

4.

Invertible Sheaves on Dedekind Schemes

In this section we shall study some special coherent sheaves of fundamental importance for both the arithmetic and the geometry of Dedekind schemes. Here is the basic definition. Definition 4.1 A locally free sheaf on a scheme X is an OX -module F for which there exists an open covering U = {Ui : i ∈ I} of X such that the restriction of F to each Ui is isomorphic to OUnii for some positive integer ni . A trivialisation of F is a covering U as above and a system of isomorphisms OUnii ∼ = F |Ui . If X is connected, then the ni are all equal to the same number n called the rank of F . A locally free sheaf of rank 1 is called an invertible sheaf or a line bundle. Remark 4.2 For any locally free sheaf F and point P ∈ X with residue field κ(P ) the group FP ⊗ κ(P ) is a finite dimensional κ(P )-vector space. So we may think of a locally free sheaf as a family of κ(P )-vector spaces which is “locally trivial”. In fact, locally free sheaves correspond to vector bundles in the algebro-geometric context, whence the name line bundle in rank 1. Lemma 4.3 Any locally free sheaf is coherent. Moreover, if X is noetherian and connected, a coherent sheaf F on X is locally free of rank n if and only if it stalk FP at each point P is a free OX,P -module of rank n. Proof: For the first statement, take any affine open subset V = Spec A contained in one of the Ui as in the definition. Then by the assumption the restriction of F to V is isomorphic to the coherent sheaf defined by the free A-module A ⊕ . . . ⊕ A (with A repeated ni times). In the second statement necessity follows from the definitions by taking the direct limit. For sufficiency, assume FP is freely generated over OX,P by some generators t1 , . . . , tn . We may view the ti as sections generating F (U)

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for some suficiently small open neighbourhood U of P . By shrinking U if ˜ for some A-module M necessary we may assume U = Spec A and F = M generated by the ti . Since X is noetherian, M is the quotient of the free A-module of rank n by a submodule generated by finitely many relations among the ti . By assumption, any of the finitely many coefficients occuring in these relations vanishes when restricted to some open neighbourhood of P contained in U. Denoting by V the intersection of these neighbourhoods, the elements ti |V generate F |V freely over OV . Remark 4.4 A similar (but easier) argument as in the second part of the above proof shows that if F is a coherent sheaf on any scheme X and P is a point for which FP = 0 then there is some open neighbourhood V of P with F |V = 0. The crucial importance of invertible sheaves for the study of Dedekid schemes is shown by the following proposition. Proposition 4.5 Any nonzero coherent sheaf of ideals I on a Dedekind scheme X is invertible. Proof: By the criterion of Lemma 4.3, it is enough to check that IP is a free OX,P -module of rank 1 for each P ∈ X. If P is the generic point, this is obviously true since any nonzero ideal in a ring generates the unit ideal in its fraction field. If P is a closed point, we are done by the fact that OX,P is a principal ideal ring. Up to now, the notion of an invertible sheaf may well have seemed to be rather abstract, but now we explain a method for constructing invertible sheaves on Dedekind schemes. First a definition. Definition 4.6 A (Weil) divisor on a Dedekind scheme X is an element of the free abelian group Div(X) generated by the closed points of X. P

Thus a divisor is just a formal linear combination D = mi Pi of finitely many closed points of X. If P is a closed point, we define vP (D) to be equal to mi if P = Pi for some i and 0 otherwise. On the other hand, let K be the function field of X; it is the common fraction field of all local rings of X. Since the local ring OX,P is a discrete valuation ring, there is an associated discrete valuation vP on K which takes finite values on nonzero elements of K. By analogy with the case of meromorphic functions on a Riemann surface, for a nonzero element f 6= 0 of K we say that f has a zero of order m at P if m = vP (f ) > 0 and that f has a pole of order m at P

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if m = vP (f ) < 0. The following lemma shows that elements of K behave in a similar way to meromorphic functions on a compact Riemann surface (compare with the proof of Chapter 4, Lemma 3.2): Lemma 4.7 If X is a Dedekind scheme with function field K, any nonzero function f ∈ K has only finitely many zeros and poles. Proof: This follows from Chapter 5, Lemma ?? and the first statement of Proposition 2.16. Thanks to the lemma, we may define the divisor of a nonzero function f ∈ K as the divisor D with vP (D) = vP (f ) for all closed points P . In this way, we obtain a homomorphism div : K × → Div(X) where K × is the multiplicative group of K. Elements of the image of div are traditionally called principal divisors. Now denote by K the constant abelian sheaf on X defined by the additive group of K. It has an OX -module structure coming from the natural embedding of OX into K but is not a quasi-coherent sheaf. However, given any divisor D ∈ Div(X) we may define a subsheaf of K which is not only quasi-coherent but, as we shall see shortly, even invertible. Namely, define for any open subset U ⊂ X L(D)(U) := {f ∈ K(U) : vP (f ) + vP (D) ≥ 0 for all closed points P ∈ U}. One sees immediately that together with the restriction maps induced by those of K we get a subsheaf L(D) of K. One thinks of the sections of L(D) over U as functions with local behaviour determined by the “restriction of D to U”: they should be regular except perhaps at the points P ∈ U with vP (D) > 0 where a pole of order at most vP (D) is allowed; furthermore, at points with vP (D) < 0 they should have a zero of order at least −vP (D). Thus the sheaf L(0) is none but the image of OX in K via the natural embedding. Moreover, each L(D) becomes an OX -submodule of K with its natural OX -module structure. Proposition 4.8 Each L(D) is an invertible sheaf on the Dedekind scheme X. Moreover the rule D 7→ L(D) induces a bijection between divisors and invertible subsheaves of K. Here the term “invertible subsheaf” means that we consider invertible sheaves which are OX -submodules of K with its canonical OX -module structure.

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Proof: For the first statement, let P be a closed point of X and f ∈ K × a function with vP (f ) = vP (D) (for example, a power of a generator of the maximal ideal of OX,P ). Denote by S the set of closed points Q of X with vQ (D) 6= 0 or vQ (f ) 6= 0. According to the previous lemma, S is a finite set. Now let U be the open set (X \ S) ∪ {P }. Over U, the sections of L(D) are functions g which are regular outside P and vP (g) + vP (f ) ≥ 0. Hence the map of OU -modules induced by 1 7→ f −1 gives an isomorphism of OU onto L(D)|U . Since P was arbitrary, we get a covering of X by open subsets over each of which L(D) is isomorphic to the structure sheaf. Conversely, given an invertible subsheaf L of K there is a trivialisation of L over some open covering U which we may choose finite by compactness of X. For each Ui ∈ U denote by fi the image of 1 ∈ OX (Ui ) under the isomorphism OX (Ui ) ∼ = L(Ui ) arising from the trivialisation. Now define a divisor D by setting vP (D) = −vP (fi ) where i is an integer for which P ∈ Ui . Since the finitely many fi have finitely many zeros and poles according to the previous lemma, D is indeed a divisor. We still have to check that the definition of D does not depend on the choices made. First, if P ∈ Ui ∩ Uj , viewing fi and fj as elements of OX,P we see that there should exist functions u, v ∈ OX,P with fi = ufj and fj = vfi , whence both u and v are units in OX,P and vP (fi ) = vP (fj ). Secondly, the definition of D does not depend on the choice of the trivialisation for passing to another one induces an automorphism of the stalk LP viewed as a free OX,P -module of rank 1, and such an automorphism is given by multiplication with a unit of OX,P . Finally, one checks by going through the above construction that L = L(D). The proof shows that in the case when vP (D) ≥ 0 for all P the sheaf L(−D) is a subsheaf not only of K but also of OX and hence is an ideal sheaf. Now assume X = Spec A is affine and take a nonzero ideal I of A. According to Proposition 4.5, the coherent ideal sheaf I˜ is an invertible subsheaf of K, so the proposition applies and shows that I˜ can be identified with an invertible P sheaf of the form L(−D), where D = mi Pi is a divisor with all mi > 0. By definition, a global section of the latter sheaf is an element of A contained in the intersection ∩Pimi , where the Pi are viewed as prime ideals of A. But as the ideals Pimi are pairwise coprime, their intersection is the same as their product, so by taking global sections of I˜ = L(−D) we get: Corollary 4.9 In a Dedekind ring any ideal decomposes uniquely as a product of prime ideals. This applies in particular to rings of integers in number fields. For the ring Z it is none but the Fundamental Theorem of Arithmetic.

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Remark 4.10 Of course, we could have obtained this result without making all this d´etour amidst schemes and invertible sheaves. But having done so, we get as a bonus a geometric interpretation of the situation, namely that ideals in a Dedekind ring can be regarded as compatible systems of local solutions for a problem of finding functions with restricted behaviour at a finite set of points. This problem can be regarded as an analogue of the classical Mittag-Leffler and Weierstrass problems in complex analysis. Remark 4.11 The preceding arguments also enable one to prove directly that on X = Spec A all quasi-coherent ideal sheaves (in fect, all of them ˜ Indeed, are also coherent for A is noetherian) are isomorphic to some I. we already know that nonzero ideal sheaves are of the form L(−D), with vP (D) ≥ 0 for all P ; put I = L(−D)(X). By Chapter 4, Lemma 2.7 (3) and the sheaf axioms it is enough to show the existence of compatible isomorphisms I(U) ∼ = L(−D)(U) over basic open sets U = D(f ). But we ˜ have I(U) = I ⊗A Af and by the same argument as for U = X we get L(D)(U) = IAf ; the isomorphism is then given by the natural multiplication map I ⊗A Af → IAf . Returning to Proposition 4.8, one might be under the impression that the invertible sheaves that are subsheaves of K (and hence arise from divisors) are rather special. This is a misbelief, for we have quite generally: Proposition 4.12 Let X be an integral scheme whose underlying topological space is compact. Then any invertible sheaf L on X is isomorphic to a subsheaf of the constant sheaf K associated to the function field of X. Proof: By compactness we may choose a trivialisation of L over a finite covering of X by open subsets U1 , . . . , Un ; by irreducibility of X the intersection U0 = ∩Ui is a nonempty open subset in X. For each integer 0 ≤ i ≤ n let si be the image of 1 by the isomorphism OX (Ui ) ∼ = L(Ui ) coming from the chosen trivialisation. For each i > 0 there exists a section fi ∈ OX (U0 ) with fi s0 = si |U0 . Now we define an injective morphism L → K. According to the third part of Chapter 4, Lemma 2.7 for this it is enough to define a compatible system of embeddings L(U) → K(U) over those open subsets U with U ⊂ Ui for some i. These we define as the OX (U)-module homomorphisms induced by the maps si |U → fi |U , viewing fi as a global section of K. The definition does not depend on the choice of i, for if U ⊂ Ui ∩ Uj and s ∈ L(U) arises both as s = ai si |U and s = aj sj |U , we have s|U0 = ai fi s0 = aj fj s0 , whence the function ai fi − aj fj vanishes over the dense open subset U, so ai fi = aj fj in K = K(X) by Chapter 5, Lemma ??. Injectivity of the morphisms L(U) → K(U) is obvious.

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Remark 4.13 In fact, the compactness assumption in the proposition is superfluous. See Hartshorne [1], Proposition II.6.15. Next, note that Proposition 4.8 implies that over a Dedekind scheme it is possible to introduce an abelian group law on the set of invertible subsheaves of K, by the rule L(D1 ) × L(D2 ) → L(D1 + D2 ) for any two divisors D1 , D2 . In this way the map D 7→ L(D) becomes a group homomorphism. But in view of the above proposition, it is interesting to know that one may define a natural abelian group law on the set of isomorphism classes of invertible sheaves on any scheme X. For this we first need the notion of the tensor product F ⊗ G of two OX -modules F and G: this is none but the sheaf associated to the presheaf U → F (U) ⊗OX (U ) G(U). Remark 4.14 One checks easily that F ⊗G represents the set-valued functor on the category of OX -modules which maps an OX -module M to the set of OX -bilinear maps F × G → M. (Use the representability for each U of the functor associating OX (U)-bilinear maps F (U) × G(U) → M(U) to M(U) and conclude by Chapter 3, Remark 2.9.) Proposition 4.15 For any scheme X, tensor product of OX -modules induces an abelian group structure on the set of isomorphism classes of invertible sheaves on X. The unit element of this group is OX and the inverse of a class represented by an invertible sheaf L is the class of the sheaf L∨ given by U → HomOU (L|U , OU ). For the proof we need two general lemmata. Lemma 4.16 If F and G are two OX -modules on a scheme X and P is a point of X, we have a natural isomorphism on the stalks (F ⊗ G)P ∼ = FP ⊗OX,P GP . Proof: This follows by nasty checking from the definitions. Note that since the stalks of the sheaf associated to a presheaf are the same as that of the presheaf it is enough to work with the presheaf tensor product U → F (U) ⊗OX (U ) G(U). Lemma 4.17 A morphism φ : F → G of abelian sheaves on a topological space is an isomorphism if and only if for each point P the induced group homomorphisms on the stalks φP : FP → GP are.

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Proof: One implication is easy. For the other, assume that the φP are all isomorphisms. We have to show that the maps φU : F (U) → G(U) are all bijective. For injectivity, assume s ∈ F (U) maps to 0 by φU . Then by injectivity of the maps φP for P ∈ U we get that the image of s in the stalks FP is 0 for all P , whence a covering of U by open subsets over each of which s restricts to 0. Hence s = 0 by the first sheaf axiom. The proof of surjectivity is similar, using the second sheaf axiom (and the injectivity just proven). Proof of Proposition 4.15: The group law is well defined since the tensor product of modules respects isomorphisms. The abelian group axioms concerning commutativity, associativity and the unit element follow from the corresponding properties of the tensor product (of modules, which are clearly inherited by OX -modules). So only the axiom concerning the inverse remains. For each open set U define a morphism L(U) × HomOX (U ) (L(U), OX (U)) → OX (U) by the natural evaluation map (s, φ) 7→ φ(s). This is clearly compatible with restriction maps for open inclusions V ⊂ U and moreover it is OX (U)-bilinear, so it induces a morphism of OX -modules L ⊗ L∨ → OX . To show that it is an isomorphism, it is enough to look at the induced maps LP ⊗OX,P L∨P → OX,P on stalks by the previous two lemmata. But since LP is a free OX,P -module of rank 1, we have isomorphisms LP ⊗OX,P HomOX,P (LP , OX,P ) ∼ = OX,P ⊗OX,P HomOX,P (OX,P , OX,P ) ∼ = OX,P and we are done. Definition 4.18 The group of isomorphism classes of invertible sheaves on a scheme X is called the Picard group of X and is denoted by P ic(X). We finally establish the link between the group Div(X) of divisors on a Dedekind scheme and the group P ic(X). For this denote by [L] the class of an invertible sheaf L in the Picard group. By what we have seen so far, we dispose of a map Div(X) → P ic(X) given by D 7→ [L(D)]. Lemma 4.19 The above map D 7→ [L(D)] is a group homomorphism. Proof: We have already seen that the map D 7→ L(D) is a homomorphism. Recall that here the multiplication map L(D1 ) × L(D2 ) → L(D1 + D2 ) for two divisors D1 , D2 is given by multiplication of functions. This map is OX bilinear and as such induces a morphism of OX -modules L(D1 ) ⊗ L(D2 ) → L(D1 + D2 ). It is then enough to see that this latter map is an isomorphism, which can be checked on the stalks. The stalk of L(Di) (i = 1, 2) at a point P is a free OX,P -module generated by some fi ; that of L(D1 ) ⊗ L(D2 ) is

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generated by f1 ⊗ f2 and that of L(D1 + D2 ) by f1 f2 . Hence mapping f1 ⊗ f2 to f1 f2 indeed induces an isomorphism. Proposition 4.20 The sequence div

0 → OX (X)× → K × −→ Div(X)

D7→[L(D)]

−→

P ic(X) → 0

is exact, where OX (X)× is the multiplicative group of units of OX (X). Proof: Exactness at the first two terms on the left is immediate from the definitions and surjectivity on the right follows from Propositions 4.8 and 4.12. For exactness at the third term note first that for any divisor of the form D = div(f ) the invertible sheaf L(D) maps to the unit element of the Picard group as multiplying sections of L(D) over an open set U by f |U induces an isomorphism L(D) ∼ = OX . Conversely, if such an isomorphism is known, let f be the image of 1 ∈ OX (X) by this isomorphism. Then one checks easily that D = div(f −1 ). Remarks 4.21 1. Traditionally when X = Spec A, the cokernel of the map div is called the (ideal) class group of the Dedekind ring A. When A is the ring of integers in a number field, a classical theorem of the arithmetic of number fields asserts that this is a finite. See e.g. Lang [2], p. 100 or Neukirch [1], Chapter I, Theorem 6.3 for a proof of this fundamental fact. 2. The results of this section generalise to integral schemes that are locally factorial, i.e. their local rings are unique factorisation domains. In the general context closed points have to be replaced by closed subschemes of codimension one. See Hartshorne [1], Section II.6 for details.

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Chapter 7 Finite Covers of Dedekind Schemes In this chapter we study finite (branched) covers of Dedekind schemes, which will turn out to behave in a strongly analogous way to finite branched covers of compact Riemann surfaces. On the way, we also prove classical numbertheoretic results in a geometric manner which emphasises their analogy with the theory of branched covers.

1.

Local Behaviour of Finite Morphisms

We begin with some examples. Example 1.1 Consider the ring Z[i] of Gaussian integers; this is the ring of integers of the algebraic number field Q(i). The natural inclusion corresponds to a morphism of Dedekind schemes Spec Z[i] → Spec Z: we now describe its fibres. The fibre over the generic point (0) is Spec (Z[i] ⊗Z Q) ∼ = Q(i). = Spec ((Z[x]/(x2 + 1)) ⊗Z Q) ∼ = Spec Q[x]/(x2 + 1) ∼ Similarly, the fibre over a closed point (p) of Spec Z is Spec (Z[i] ⊗Z Fp ) = Spec (Fp [x]/(x2 + 1)). Now there are three cases: • If p ≡ 1 mod 4, then the polynomial x2 + 1 factors as the product of two distinct linear terms over Fp , hence (by the Chinese Remainder Theorem) the fibre is isomorphic to Spec (Fp ⊕ Fp ). • If p ≡ −1 mod 4, then the polynomial x2 + 1 is irreducible over Fp and hence the fibre is isomorphic to Spec Fp2 . 145

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• For p = 2 we have x2 + 1 = (x + 1)2 over Fp and hence the fibre is Spec (Fp [x]/(x + 1)2 ). The underlying topological space of this scheme is a single point (the maximal ideal (x + 1)) and the local ring at this point contains nilpotent elements (for example x + 1). One is thus tempted to regard the point in the fibre over (2) ∈ Spec Z as a kind of a branch point for this is the only point contained in a fibre which is degenerate in the sense that there are nilpotent functions on it. (We shall see shortly that though the fibres of the second type consist only of a single point, it is not reasonable to consider them as degenerate.) The following example confirms this intuition. Example 1.2 For Riemann surfaces, the basic example of a branched cover was the cover C → C, z 7→ z n (indeed, we saw that any branched cover is analytically isomorphic to this one in the neighbourhood of a branch point). Algebraically, this corresponds to the morphism Spec C[z] → Spec C[z] coming from the C-algebra homomorphism C[z] → C[z] induced by z 7→ z n . Introduce the variable y = z n in the second ring: we thus have an isomorphism C[z] ∼ = C[z, y]/(y − z n ) and the above homomorphism corresponds to mapping z to y. Now first look at the fibre over the generic point (0) ∈ Spec C[z]: it is Spec (C[z, y]/(y − z n ) ⊗C[z] C(z)) ∼ = Spec (C(y)[z]/(y − z n )) (don’t forget that the C[z]-module structure on C[z, y]/(y − z n ) occuring in the tensor product is given by z 7→ y). Hence the fibre is the spectrum of a degree n Galois field extension of the rational function field C(y). Now a closed point of the Dedekind scheme Spec C[z] is given by a maximal ideal (z − a) with some a ∈ C; the residue field of the local ring at this point is C[z]/(z − a) ∼ = C. Hence the fibre over this point is Spec (C[z, y]/(y − z n ) ⊗C[z] C) ∼ = Spec (C[z]/(a − z n )) since the C[z]-modules on the two terms of the tensor product are given respectively by z 7→ y and z 7→ a. Now there are two cases: • For a 6= 0 the polynomial z n − a splits into a product of n distinct linear terms over C and thus the fibre is Spec (C ⊕ . . . ⊕ C) (n copies). • For a = 0 the fibre is Spec C[z]/(z n ), a one-point scheme with nilpotent elements in its unique local ring.

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Our intuition is thus confirmed and it can be made more precise: in the first two cases of the first example and in the first case of the second, the fibre is a finite ´etale k-scheme, whereas in the remaining cases of the two examples (the “branch points”) it is not. This also explains why there were only two cases in the second example but three in the first: over C a finite ´etale algebra can only be a finite direct sum of copies of C, whereas over a non-separably closed field we may have non-trivial finite separable field extensions as well. The essence of the phenomenon encountered above is distilled in the definitions we are to give. First a natural restriction on morphisms which ensures in particular that they have finite fibres. Definition 1.3 A morphism of schemes φ : Y → X is called finite if X has a covering by open affine subsets Ui = Spec Bi such that for each i the open subscheme Vi = φ−1 (Ui ) of Y is an affine scheme Vi = Spec Ai and the ring homomorphism λi : Ai → Bi corresponding to φ|Vi turns Bi into a finitely generated Ai -module. In particular, any finite morphism is affine. This implies that for any P ∈ X the fibre YP is affine as well; the additional property of finite morphisms assures that YP is the spectrum of a finite dimensional κ(P )-algebra. Remark 1.4 When we shall be dealing with a finite morphism φ : Y → X, with X a Dedekind scheme, we shall always assume that the induced map OX,η → (φ∗ OY )η at the generic point η of X is nonzero (and hence is an injection, for OX,η is the function field of X). This seemingly innocent assumption has an important consequence (valid for any noetherian integral scheme in place of X): that the continuous map underlying φ is surjective. Indeed, if it were not, there would be a point P ∈ X over which the fibre is vacuous, i.e. is the spectrum of the zero ring. But then by Nakayama’s lemma (Chapter 6, Lemma 2.3), the stalk of φ∗ OY at P would be zero as well, so since φ∗ OY is a coherent sheaf, it would be 0 in an neighbourhood of P (by Chapter 6, Remark 4.4). But each such neighbourhood contains η, a contradiction. Examples 1.5 1. If K ⊂ L is an inclusion of number fields, then the induced morphism Spec OL → Spec OK is finite as OL is a finitely generated OK -module according to Chapter 6, Proposition 1.9.

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2. We shall prove in the next chapter that any nonconstant morphism of smooth projective curves is finite. For affine plane curves, this is not always true: consider, for instance, the curve Spec C[x, y]/(xy − 1) ∼ = Spec C[x, x−1 ]; this is the complex affine line with the point 0 removed. The natural inclusion map Spec C[x, x−1 ] → Spec C[x] is not finite; it is not even surjective. 3. One may nevertheless construct many examples of finite maps of affine curves: for instance, Spec C[x, y]/(y n − f ) → Spec C[x] is such for any f ∈ C[x]. Remark 1.6 It can be shown that if φ : Y → X is a finite morphism then any affine open cover of X satisfies the property of the definition. For basic open sets this is easy to check: if D(fi ) = Spec (Ai )fi ⊂ Ai is such, then the fact that Bi is a finitely generated Ai -module via λi immediately implies that (Bi )λi (fi ) is a finitely generated (Ai )fi -module. We can now begin the analysis of fibres of finite morphisms. Given a finite morphism φ : Y → X, the fibre YP over any point P of X is the spectrum of a finite dimensional κ(P )-algebra, so by Chapter 5, Proposition 2.16 decomposes as a finite disjoint union of schemes each of which has a single point Q of X as its underlying space (a point in the topological fibre) such that all elements of the maximal ideal of the local ring at Q (i.e. the germs of functions vanishing at Q) are nilpotent. Definition 1.7 Let X be a Dedekind scheme and φ : Y → X a finite morphism. We say that φ is ´etale at a point Q ∈ Y if the component of the fibre YP corresponding to Q is ´etale over Spec κ(P ), i.e. if it is the spectrum of a finite separable field extension of κ(P ). The morphism φ is a finite ´etale morphism if it is ´etale at all points of Y . In particular, all fibres of a finite ´etale morphism are finite ´etale schemes over residue fields of points of X. Remark 1.8 Though not used explicitly in the above definition of finite ´etale morphisms, the assumption that X is a Dedekind scheme is important here. When defining finite ´etale morphisms of general schemes, one needs an additional assumption which is automatically satisfied here (see Chapter 9). We can now state the main result of this section. Theorem 1.9 Let φ : Y → X be a finite morphism of Dedekind schemes. Then φ is ´etale at a point Q of Y if and only if the stalk of the sheaf of relative differentials ΩY /X at Q is 0.

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Proof: Take an affine open neighbourhood U = Spec A of P whose inverse image in Y is of the form V = Spec B and identify P and Q with the corresponding prime ideals of B and A as usual. Then by the localisation property of differentials the stalk of ΩY /X at Q is ΩBQ /A , with BQ regarded as an A-algebra via the composite map A → B → BQ . Next observe that the local component of the fibre YP corresponding to Q is precisely Spec (BQ ⊗A κ(P )). (Indeed, by definition the local component ¯ is the image of Q is the spectrum of the localisation (B ⊗A κ(P ))Q¯ , where Q in B ⊗A κ(P ). This localisation is obtained by localising first B/P B by the ¯ which is the same as image of (A/P ) \ {0} and then by the complement of Q localising B by Q first, then passing to the quotient by the image of P and finally localising by the image of (A/P ) \ {0}.) By the base change property of differentials we have an isomorphism Ω(BQ ⊗A κ(P ))/κ(P ) ∼ = ΩBQ /A ⊗BQ (BQ ⊗A κ(P )). Now assume ΩBQ /A = 0. Then the left hand side vanishes and so by Chapter 6, Proposition 3.6 BQ ⊗A κ(P ) is ´etale over κ(P ), i.e. it is equal to κ(Q) which is finite and separable over κ(P ). Conversely, if this is the case, then by applying the argument backwards we get that ΩBQ /A ⊗BQ κ(Q) ∼ = 0. But this latter ring is isomorphic to ΩBQ /A /QΩBQ /A and the assertion follows from Nakayama’s lemma (Chapter 6, Lemma 2.3). Taking Chapter 6, Remark 4.4 into account, we get as a first corollary: Corollary 1.10 Let φ : Y → X be a finite morphism of Dedekind schemes. Then the points of Y at which φ is ´etale form an open subset of Y . In particular, if the fibre of φ at the generic point is ´etale, then φ is ´etale at all but finitely many closed points of Y . Example 1.11 It may very well happen that the generic fibre is not ´etale and hence φ is nowhere ´etale. An example is given by the map Spec Fp [t] → Spec Fp [t] induced by the Fp -algebra homomorphism t 7→ tp (an analogue of Example 1.2 in characteristic p > 0). As already noted in Chapter 1, on the generic stalks this induces an inseparable field extension. Remark 1.12 The theorem shows that when X and Y are smooth complex curves, then φ is ´etale precisely over those closed points P ∈ X for which the associated holomorphic map gives a cover in some complex open neighbourhood of P . We illustrate this by the example when Y is an affine plane curve of equation f (x, y) = 0 and φ is the map which projects Y onto the x-axis;

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the general case is based on the same principle. In our case ΩY /X is the coherent OX -module associated to the C[x, y]/(f )-module with single generator dy and relation ∂y f . Hence its stalk is 0 precisely at those points (x, y) of X where ∂y f (x, y) 6= 0. But by the implicit function theorem, these are the points where the holomorphic map associated to φ is a local isomorphism; the points with ∂y f (x, y) = 0 are the branch points. We now introduce a useful concept originating in work of Dedekind. Definition 1.13 Let φ : Y → X be a finite morphism of Dedekind schemes which is ´etale at the generic point. Then the we define the different DY /X as the nonzero ideal sheaf on Y which is the annihilator of ΩY /X . Putting Theorem 1.9 together with the theory of the previous section we get: Corollary 1.14 For φ : Y → X as in the above definition, the different P DY /X is an invertible sheaf of the form L(D), where D = i mi Qi is a divisor supported exactly at those points Qi at which φ is not ´etale. Definition 1.15 The points Qi arising in the above corollary are called the branch points of φ or those points at which φ is ramified. Remark 1.16 In the case when Y = Spec OL and X = Spec OK are spectra of rings of integers in number fields we get using Chapter 6, Remark 4.11 ˜ with I an ideal of OL . Thus I is a product that the different is of the form I, of powers of those prime ideals of OL at which the map is not ´etale. This gives the link to the classical concept of the different in algebraic number theory. In the remaining of this section we investigate fibres of finite morphisms more closely, especially those containing branch points. For this the following observation is crucial. Proposition 1.17 Let φ : Y → X be a finite morphism of Dedekind schemes, inducing a field extension L|K on the generic stalks. Then φ∗ OY is a locally free OX -module of rank |L : K|.

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Proof: Let P be a point of X. As usual, we consider an affine open neighbourhood U = Spec A of P over which φ comes from a ring homomorphism λ : A → B. Here B is the integral closure of A in L, being integrally closed and finite over A. Now the stalk of φ∗ OY at P is the spectrum of the AP algebra BP = B ⊗A AP . This algebra can also be seen as the localisation of B by the multiplicatively closed subset λ(A \ P ). Indeed, given f ∈ A \ P , we have already seen in Chapter 6 during the construction of the sheaf of relative differentials that Bλ(f ) is canonically isomorphic to B ⊗A Af ; the statement follows from this by passing to the direct limit (a union in this case). Taking Chapter 6, Lemma 1.8 into account, this shows that BP is the integral closure of AP in L. In particular, since any element of L can be multiplied with an appropriate element of K to become integral over AP , any generating system of the finitely generated AP -module BP generates the K-vector space L. According to (the corollary to) Nakayama’s lemma we get such a generating system by choosing elements t1 , . . . , tn ∈ BP whose images modulo P BP form a basis of the κ(P )-vector space BP /P BP (the spectrum of this κ(P )-algebra is none but the fibre over P ). It remains to be seen that the ti are linearly independent over K, for this implies n = |L : K| as P well. So assume there is a nontrivial relation ai ti = 0 with ai ∈ K. By multiplying with a suitable power of a generator of P (viewed as the maximal ideal of AP ) we may assume that all ai lie in AP and not all of them are in P . But then reducing modulo P we obtain a nontrivial relation among the ti in BP /P BP , a contradiction. To derive the next result we need to introduce some notation and terminology. Recall that the fibre YP of φ over a point P ∈ Y decomposes as a finite disjoint union of spectra of local κ(P )-algebras each of which corresponds to a point Q in the topological fibre. We have already seen during the proof of Theorem 1.9 that if V = Spec B is an affine open set containing YP , then the local component corresponding to Q is the spectrum of BQ ⊗A κ(P ) ∼ = BQ /P BQ . Here BQ is a discrete valuation ring whose max¯ of BQ /P BQ. By Chapter 5, imal ideal QBQ induces the maximal ideal Q ¯ consists of nilpotent elements, so being finitely generated, Proposition 2.16 Q it is actually a nilpotent ideal (i.e. some power of it is 0). Definition 1.18 With the above notations, the smallest nonnegative integer ¯ n = 0 is denoted by e(Q|P ) and is called the ramification index n for which Q of φ at Q. The degree f (Q|P ) of the residue field extension κ(Q)|κ(P ) is called its residue class degree. In particular, φ is ´etale at Q if and only if e(Q|P ) = 1 and κ(Q) is separable over κ(P ). In the case where X = Spec B and Y = Spec A are

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affine there is a more classical definition of the ramification index: it is the multiplicity of Q in the product decomposition of the ideal P B in B (cf. Chapter 6, Corollary 4.9). Remark 1.19 Applying the above definition to Example 1.2 corroborates that the above definition of the ramification index is compatible with the one used for Riemann surfaces. The extension of non-trivial residue class degrees is, however, a phenomenon which does not arise over the complex numbers. Now we can state a fundamental equality of the arithmetic of Dedekind schemes which is the analogue of Chapter 4, Proposition 2.5 (4). Proposition 1.20 In the situation of the previous proposition, let P be a point of Y . Then we have the equality X

e(Q|P )f (Q|P ) = |L : K|

Q

where Q runs over the points of the topological fibre over P . Proof: During the previous proof we have seen that the dimension of the κ(P )-space BP /P BP is |L : K|. Since BP /P BP decomposes as a direct sum of its components BQ /P BQ it will be enough to show that the dimension of such a component over κ(P ) is precisely e(Q|P )f (Q|P ). For this, notice that since BQ is a discrete valuation ring, multiplication by the k-th power of a generator of QBQ induces an isomorphism BQ /QBQ ∼ = (QBQ )k /(QBQ )k+1 for any positive integer k. Hence (with notation as in the definition above) in the filtration ¯⊃Q ¯2 ⊃ . . . ⊃ Q ¯ e(Q|P ) = 0 BQ /P BQ ⊃ Q each successive quotient is isomorphic to BQ /QBQ ∼ = κ(Q) and so is an f (Q|P )-dimensional κ(P )-vector space. This proves our claim.

2.

Fundamental Groups of Dedekind Schemes

In this section we shall construct a profinite group which classifies finite ´etale covers of a fixed Dedekind scheme just as the absolute Galois group classifies finite ´etale algebras over a field or as the profinite completion of the topological fundamental group classifies finite covers of a compact Riemann surface. The method we shall follow will be the exact analogue of the procedure for Riemann surfaces in Chapter 4, Section 3; the technical details will be different, though.

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So to start the procedure, we have to study function fields of Dedekind schemes. Consider the functor which associates to a Dedekind scheme X its function field K. This is indeed a functor, for given another Dedekind scheme Y with function field L and a morphism φ : Y → X, we have an induced morphism K → L on generic stalks of structure sheaves. If the morphism φ is finite, we get in this way a finite field extension L|K. Call the morphism φ separable if the extension L|K is separable. Proposition 2.1 Let X be a Dedekind scheme with function field K. Then for any finite separable extension L|K there is a Dedekind scheme Y with function field L and equipped with a finite separable morphism Y → X. Furthermore, the scheme Y is unique up to isomorphism (over X). The Dedekind scheme Y is called the normalisation of X in L. Proof: First we prove uniqueness. Assume φ : Y → X and φ0 : Y 0 → X are two normalisations of X in L. Choose some affine open subset Spec A ⊂ X. By Remark 1.6, over any basic open set D(f ) = Spec Af ⊂ Spec A both φ and φ0 satisfy the condition for a finite map. Let D(g) (resp. D(g 0)) be the basic open set in Y (resp. Y 0 ) which is the preimage of D(f ). Then both OY (D(g)) and OY 0 (D(g 0)) are finitely generated Af -modules. Moreover, they are integrally closed with fraction field L (since their localisations at closed points are), so they are both isomorphic to the integral closure of Af in L via their embeddings in L. This yields an isomorphism D(g) ∼ = D(g 0). Using Chapter 6, Lemma 1.8 we see that these isomorphisms are compatible over intersections of basic open sets, therefore they define an isomorphism of Y with Y 0 over X. Now assume X = Spec A is affine with fraction field K and let B be the integral closure of A in L. To prove that Spec B is a Dedekind scheme one employs exactly the same argument as in the special case A = Z treated before in Chapter 6, Example 2.17 (note that it is here that we use separability of L|K). By Chapter 6, Proposition 1.9, B is a finitely generated A-module, so the morphism φA : Spec B → Spec A is finite. Before going over to the general case, notice that if U = Spec Af is a basic open subscheme of X, then Chapter 6, Lemma 1.8 implies that φ−1 A (U) is the normalisation of U, for it is the spectrum of the localisation of B by g = φ]A (f ). Now cover X with affine open subsets Ui . By the affine case, each Ui has a normalisation Vi equipped with a finite morphism Vi → Ui . It will suffice −1 to show that there exist isomorphisms φij : φ−1 i (Ui ∩ Uj ) → φj (Ui ∩ Uj ) compatible over triple intersections, for then the Vi may be patched together using the construction of Chapter 5, Construction 5.3. To do this, cover

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Ui ∩ Uj by basic affine open sets Wk ; their inverse images by φi and φj (which −1 cover φ−1 i (Ui ∩Uj ) and φj (Ui ∩Uj ), respectively) both give a normalisation of Wk in L by the remark in the previous paragraph and hence are canonically isomorphic by the uniqueness statement. Since moreover these isomorphisms are readily seen to be compatible over the intersections of the Wk (which are themselves basic open sets), we can conclude that they can be patched together to define the required φij : for the underlying continuous maps this is immediate and for the maps on the structure sheaves one uses the third part of Chapter 5, Lemma 2.7. For a fixed Dedekind scheme X define the category DedsX of finite separable Dedekind schemes over X as the category whose objects are Dedekind schemes Y equipped with a finite separable morphism Y → X and whose morphisms are finite morphisms compatible with the projections onto X. Corollary 2.2 The functor mapping an object Y → X to the induced extension of function fields induces an anti-equivalence between the category DedsX and the category of finite separable extensions of the function field K of X (with morphisms the inclusion maps). Proof: The proposition shows that the functor is essentially surjective; it remains to check that it is fully faithful. For this, note first that given a finite morphism Y → X inducing a finite separable extension L|K, there is an isomorphism of Y with the normalisation of X in L constructed in the above proof. Fixing such an isomorphism for each object of DedsX we get a canonical bijection between morphisms Y → Z in DedsX and morphisms between normalisations of X in finite separable extensions of K. But these in turn correspond bijectively to towers of extensions K ⊂ L ⊂ M for the normalisation in M of the normalisation of X in L is none but the normalisation of X in M. Now let X be a Dedekind scheme. A finite ´etale X-scheme is a scheme Y equipped with a finite ´etale morphism Y → X. The following result shows that such a Y can only be of a very special type. Proposition 2.3 Any finite ´etale X-scheme Y is a finite disjoint union of Dedekind schemes. Proof: Let Q be a closed point of Y and let BQ be the local ring of Y at Q whose maximal ideal we also denote by Q. Denote by P the image of Q in X and by AP its local ring, which is a discrete valuation ring with maximal ideal P generated by a single nonzero element t. The spectrum

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of BQ /P BQ = BQ /tBQ is the local component of Q in the fibre over P , so it is a finite field extension of A/P by assumption. Hence t generates a maximal ideal in BQ which is only possible for Q = (t). Furthermore, BQ is a noetherian local ring, being a localisation of a finitely generated AP -module. We now show that BQ is a discrete valuation ring. Combining what we know so far with Chapter 6, Lemma 2.2, it is enough to see that B is an integral domain. For this, let M ∈ Spec BQ be an inverse image of the generic point (0) of Spec AP . Since M as an ideal of BQ is properly contained in Q, we have t ∈ / M and m = bt for any m ∈ M with some b ∈ BQ . As M is a prime ideal, this forces b ∈ M, so that QM = M and finally M = 0 by Nakayama’s lemma. Now since X is noetherian and Y is finite over X, we conclude from the definitions that Y is noetherian as well. Furthermore, now that we know that the local rings of Y at closed points are discrete valuation rings, we may conclude that Y is normal and of dimension 1 (since all other local rings are localisations of those at closed points). So it remains to be seen that Y is a finite disjoint union of integral schemes. For this, let η1 , . . . , ηn be the finitely many points in the generic fibre Yη and let Yi be the closure of ηi in Y . We have Yi 6= Yj for i 6= j as the generic point of an irreducible closed subset is unique. But then the Yi are pairwise disjoint, for if say Y1 and Y2 had a a closed point Q in common, then, since we may assume Y1 not contained in Y2 , for an affine open subset Spec A ⊂ Y1 containing Q the irreducible closed subset Y2 ∩ Spec A ⊂ Spec A would define a nonzero prime ideal of A properly contained in Q, which is impossible as the local ring at Q is a domain of dimension 1. Hence the Yi are the finitely many connected components of Y ; endowing them with their open subscheme structure we get a decomposition as required. Now if we wish to continue our program parallel to Chapter 4, Section 3, we need the following lemma. Lemma 2.4 Let p : Y → X be a finite morphism of Dedekind schemes. 1. q : Z → Y be a second finite morphism of Dedekind schemes. Then p ◦ q is ´etale if and only if p and q are ´etale. 2. Let now q : Z → X be a Dedekind scheme finite and ´etale over X. Then the morphism Y ×X Z → Y is finite and ´etale and hence so is the composite Y ×X Z → X. Note that by our convention (Remark 1.4) all finite morphisms under consideration are surjective.

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Proof: For the first statement, we may assume (using Remark 1.6 if necessary) that X = Spec A, Y = Spec B, Z = Spec C are all affine. For a point P ∈ Spec A we have C ⊗A κ(P ) ∼ = C ⊗B B ⊗A κ(P ).

(7.1)

Now if B ⊗A κ(P ) is a direct sum of finite separable extensions κ(Q)|κ(P ) and so is C ⊗B κ(Q) for each Q, we get that C ⊗A κ(P ) is a direct sum of separable extensions. For the converse we argue as in the beginning of the proof of Chapter 4, Theorem 3.17: if K ⊂ L ⊂ M are the respective function fields of X, Y and Z, it follows from Proposition 1.20 (in fact, already from Proposition ??) that C ⊗A κ(P ) is ´etale if and only if all of its residue fields are separable over κ(P ) and the sum of residue class degrees is [M : K]. Now if one of the residue fields of B ⊗A κ(P ) is inseparable over κ(P ) or if the sum of its residue class degrees is less than [L : K], then a counting shows that C ⊗A κ(P ) cannot be ´etale. This not being the case by assumption, Y is ´etale over X, and since in this case by formula (7.1) C ⊗A κ(P ) is just the direct sum of the C ⊗B κ(Q) for Q running over the points in the fibre YP , we see that Z → Y must be ´etale as well. For the second statement, finiteness of the morphism Y ×X Z → Y follows from the definitions. For ´etaleness again we may assume X = Spec A, Y = Spec B, Z = Spec C are all affine. Then for Q ∈ Spec B, we have (B ⊗A C) ⊗B κ(Q) ∼ = C ⊗A κ(Q). But the homomorphism A → κ(Q) factors as A → κ(P ) → κ(Q), where P is the image of Q in X. Now since by assumption C ⊗A κ(P ) is ´etale over κ(P ), the algebra C ⊗A κ(Q) ∼ = C ⊗A κ(P ) ⊗κ(P ) κ(Q) is ´etale over κ(Q) (one may use Chapter 6, Proposition 3.6 and Lemma 3.5 (3)). Remark 2.5 It can be shown that if p : Y → X is a finite ´etale morphism of Dedekind schemes and q : Z → Y any morphism of Dedekind schemes such that the composite p ◦ q is finite and ´etale, then q is also finite and ´etale. This follows immediately from the first statement of the lemma and the definition of finite morphisms once we know that q is surjective. The proof of this innocent-looking fact, however, requires more technique than we have seen so far. We shall return to this point in Chapter 9. Now let X be a Dedekind scheme with function field K and φ : Y → X a Dedekind scheme finite and ´etale over X. By Proposition 2.3, the fibre of φ over the generic point of X is the spectrum of a finite ´etale K-algebra which defines via the functor of Chapter 1, Theorem 3.4 a finite continuous

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Gal (K)-set SY . Moreover, the rule Y 7→ SY defines a functor from the category of schemes finite and ´etale over X to the category of finite continuous Gal (K)-sets. (The perceptive reader will have noticed that in order to make everything fit here we should consider only those morphisms between schemes over X which are finite and separable but if we admit the above remark this condition is satisfied by all morphisms of finite ´etale X-schemes.) Theorem 2.6 Let X be a Dedekind scheme with function field K. Fix a separable closure K s of K and let K et be the composite in K s of all finite subextensions L|K which correspond by Corollary 2.2 to Dedekind schemes ´etale over X. Then for each finite ´etale X-scheme φ : Y → X the action of Gal (K) on the set SY defined above factors through the quotient Gal (K et |K) and in this way we obtain an equivalence between the category of schemes finite and ´etale over X and the category of finite continuous left Gal (K et |K)sets. To be consistent with the notations in previous chapters, we call the opposite group of Gal (K et |K) the algebraic fundamental group of the Dedekind scheme X. Proof: The first statement of the theorem is immediate from the construction and implies via Chapter 1, Theorem 3.4 that the functor we are investigating is fully faithful. To see that any finite continuous Gal (K et |K)-set S is isomorphic to some SY , one first produces from Chapter 1, Theorem 3.4 a finite direct sum of finite subextensions of K et |K giving rise to S. Then it remains to see that any finite subextension of K et |K is the function field of some Dedekind scheme finite and ´etale over X. But this can be proven by exactly the same argument as in the proof of Chapter 4, Theorem 3.17, using Lemma 2.4. Remark 2.7 We shall study of fundamental groups of smooth curves in the next section; here we briefly discuss spectra of rings of integers in number fields. We know several classical facts concerning these from algebraic number theory. Firstly, a theorem of Minkowski states that π1 (Spec Z) is trivial. Secondly, a theorem of Hermite and Minkowski states that for any number field K the group π1 (Spec OK ) has only finitely many finite quotients of given order. Thirdly, one of the main results of class field theory (usually attributed to Hilbert) states that the maximal abelian quotient of π1 (Spec OK ) is finite for all K and isomorphic (up to a finite group of exponent 2 coming from socalled real places of K) to P ic(Spec OK ) (see the next chapter for a geometric

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analogue). Proofs of these classical theorems can be found in the books of Lang [2] and Neukirch [1], for example. But as for the groups π1 (Spec OK ) themselves and not just their finite quotients, our current knowledge is far from being ample. For several structural results, including the fact that they are topologically finitely generated (in contrast to the groups Gal (K)), see chapter X of the monograph of Neukirch-Schmidt-Wingberg [1], where the results are stated more generally for open subschemes of Spec OK .

3.

Galois Branched Covers and Henselisation

The perceptive reader has noted that though the main theorem of the last section was completely analogous to the topological situation, there was one point missing from the presentation, namely the analogue of Galois branched covers. In this section we first repair this sin of omission. We define (rather tautologically) a finite morphism φ : Y → X of Dedekind schemes to be Galois if the induced inclusion K ⊂ L of function fields is a finite Galois extension. Now there is a natural action of the (finite) Galois group G on X defined as follows. Take an affine open covering of X by Ui = Spec Ai whose inverse image consists of affine open sets Vi = Spec Bi . Then B is the integral closure of Ai in L, so that σ(Bi ) = Bi for all i and σ ∈ G and if Q is a prime ideal in Bi , then σ(Q) is also a prime ideal of Bi . From this we deduce an action of G on X as a topological space but we have actually more: from the automorphism σ|Bi : Bi → Bi we deduce an automorphism of Vi by Chapter 5, Theorem 2.14. These automorphisms are easily seen to be compatible over the intersections Vi ∩ Vj , so we get an action of G on X as a scheme. The next proposition shows that topologically this action by G has the property of a Galois cover. Proposition 3.1 The group G acts transitively on the fibres of φ. Proof: We may assume X = Spec A and Y = Spec B are affine. Let P be a prime ideal of A and let Q1 be a prime ideal of B with Q1 ∩ A = P . Let Q2 , . . . , Qr be the other prime ideals of B in the G-orbit of Q1 . Assume there is some prime ideal Q lying over P which is not among the Qi . Since Q and the Qi are all maximal ideals, we may apply the Chinese Remainder Theorem (Lang [1], Chapter II, Theorem 2.1) which gives an isomorphism B/(QQ1 . . . Qr ) ∼ = B/Q ⊕ B/Q1 ⊕ . . . ⊕ B/Qr , hence we may find x ∈ Q not contained in any of the Qi (we may even find an x mapping to 1 modulo

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each Qi and to 0 modulo Q). Now for any σ ∈ G the element σ(x) is still Q outside each Qi , hence the same holds for their product N(x) = σ∈G σ(x). But N(x) ∈ A, which means that N(x) ∈ / P . But since x ∈ Q, we have N(x) ∈ Q ∩ A = P , a contradiction. Remark 3.2 The proposition implies that X as a scheme is the quotient of Y by the action of G defined above. To make this precise, note that we may take the quotient of Y by the G-action in the category of locally ringed spaces just as in the construction of projective spaces: we take the quotient of the underlying space by the action of G, whence a canonical topological projection p : Y → Y /G; then for each open set U ⊂ Y /G we define OY /G (U) as the ring of G-invariant elements of OY (p−1 (U)). One checks that (Y /G, OY /G ) is indeed a locally ringed space and there is a canonical map ψ : Y /G → X with φ = ψ ◦ p. By the proposition ψ is an isomorphism on the underlying topological spaces. Now observe that if we choose an affine open covering of X by subsets Spec Ai whose inverse images in Y are of the form Spec Bi , then by construction the G-orbit of any point of Y is entirely contained in one of the Spec Bi . This implies that we may define the structure of a scheme on Y /G by choosing as an affine open covering the schemes Spec BiG , where BiG is the ring of G-invariant elements of Bi . By construction we have BiG ∼ = Ai ∼ and hence Y /G = X as schemes. Now let us draw some immediate consequences from the proposition concerning the local behaviour of a morphism φ : Y → X near branch points. Consider a point P of X and a point Q in the fibre over P . According to Proposition 3.1, any other point of the fibre YP over P is of the form σ(Q) with some σ ∈ G and by construction e(σ(Q)|P ) = e(Q|P ) and f (σ(Q)|P ) = f (Q|P ). Hence we may denote simply be e end f the common ramification index and residue class degree of the points in YP . Now let DQ be the stabiliser of Q with respect to the action of G on Y . Again by Proposition 3.1, the cosets of G mod DQ are in bijection with the points in YP . Hence Proposition 1.20 implies that the order of DQ is exatly ef . Now let LQ be the fixed field of DQ and X 0 the normalisation of X in LQ . Denote by Q0 the image of Q by the map φ0 : Y → X 0 . Lemma 3.3 The topological fibre of φ0 over Q0 consists only of Q. We have the equalities e(Q|Q0 ) = e,

f (Q|Q0 ) = f

and

e(Q0 |P ) = f (Q0 |P ) = 1.

In particular, the finite morphism X 0 → X is ´etale at Q0 .

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Proof: According to Proposition 3.1, the group DQ acts transitively on the fibre of φ0 over Q0 . On the other hand, it fixes Q, whence the first statement. The second statement follows from the already proven fact that the order of DQ is ef in view of the equalities e(Q|Q0 )e(Q0 |P ) = e and f (Q|Q0 )f (Q0 |P ) = f which hold quite generally and follow from the definitions. Remark 3.4 In arithmetic terminology, the group DQ is called the decomposition group of Q and LQ is called its decomposition field. Thus the local ring OX 0 ,Q of X 0 at Q has the property that its integral closure in L is a discrete valuation ring (the local ring of Q) and hence the ramification index and the residue class degree are easy to compute; in particular, the formula of Proposition 1.20 reduces to ef = [L : LQ ]. This property is not shared by the local ring OX,P of X at P : its integral closure in L has several maximal ideals corresponding to the points in the fibre YQ . We may of course localise at one of these to get an extension of discrete valuation rings but then we lose the finiteness property of the corresponding morphism of affine schemes. So we see that in order to study the local behaviour of φ at Q it is much better to work with OX 0 ,Q than with OX,P ; moreover, by the corollary we get the same ramification index and residue class degree. This procedure is a priori only available in the Galois case but if φ : Y → X is only seperable with function field extension L|K, we may embed L into a Galois extension M of K and study the corresponding morphism of Dedekind schemes. This prompts the idea that if we choose M to be the biggest Galois extension available, namely the separable closure, then by the generalising the above procedure we may reduce the study of the local behaviour of finite morphisms near branch points to a problem about finite extensions of discrete valuation rings. We now make this idea precise. First an easy lemma which could have figured earlier. Lemma 3.5 Let A be a discrete valuation ring with maximal ideal P and B an integral extension of A. Then there is a prime ideal Q of B with Q ∩ A = P. Note that here the morphism Spec B → Spec A is not assumed to be finite.

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Proof: It is enough to show that P B 6= B, for then P B is contained in some maximal ideal of B whose intersection with B is nonzero according to Chapter 6, Lemma 1.5 (1) and hence cannot be but P . So assume P B = B. P Then there are b1 , . . . , br ∈ B and p1 , . . . , pr ∈ P with i pi bi = 1. Hence the subring B 0 = A[b1 , . . . , br ] also satisfies P B 0 = B 0 . But B 0 is integral over A and hence a finitely generated A-module, so Nakayama’s lemma implies B 0 = 0 which is absurd. Now we can generalise the situation of Corollary 3.3 above. Construction 3.6 Let A be a discrete valuation ring with maximal ideal P and fraction field K. Fix a separable closure K s of K. Denoting by B the integral closure of A in K s , apply the lemma to find some Q ∈ Spec B lying above P . Let DQ be the stabiliser of Q with respect to the natural action of Gal (K) on Spec B (defined in the same way as for finite Galois extensions). Let K 0 be the fixed field of DQ and put B 0 = B ∩ K 0 , Q0 = Q ∩ B 0 . The localisation of B 0 at Q0 is called the henselisation of A and is denoted by Ah . Proposition 3.7 Let A, P , Ah , Q be as above. 1. The ring Ah is a discrete valuation ring with the same residue field as A. Its maximal ideal is generated by any generator of P . 2. The isomorphism class of Ah does not depend on the choice of the prime ideal Q. For the proof we need the following generalisation to Proposition 3.1 to infinite Galois extensions. Lemma 3.8 With notations as in the above construction, the group Gal (K) acts transitively on the maximal ideals of B lying over P . Proof: Take two such maximal ideals Q1 6= Q2 and for each finite Galois subextension L|K denote by XL the set of those elements of G = Gal (K) which when restricted to L map Q1 ∩ L onto Q2 ∩ L. Since the latter are prime ideals of the integral closure of A in L, Proposition 3.1 implies that XL 6= ∅ for any L. Moreover, each XL is a closed subset of G, for if some σ∈ / XL , then the whole left coset σGal (L) of the open subgroup Gal (L) is T contained in G \ XL . But G is compact, so we have X = XL 6= ∅. Any element of X maps Q1 onto Q2 .

L

Proof of Proposition 3.7: For the first statement, note that by Corollary 3.3 for any finite separable extension L|K the ring Ah ∩ L is a discrete valuation ring whose spectrum is ´etale over Spec A. Hence Ah is the union of

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an increasing chain of such discrete valuation rings, which shows that it is local (its maximal ideal being the union of that of the L ∩ Ah , and the same holding for its units) and its maximal ideal is generated by any generator of P . The same type of argument shows that the residue field of Ah is the same of that of A. The second statement follows from the lemma which implies that performing the above construction with a maximal ideal above P other than Q yields the ring σ(Ah ) for some σ ∈ Gal (K). We conclude this section by the following proposition which shows that henselisations do serve for the purpose they were constructed for. Proposition 3.9 Let Y → X be a finite separable morphism of Dedekind schemes. Let L (resp. K) be the function field of Y (resp. of X), and embed L|K into a separable closure K s |K. Fix a closed point Q of Y mapping to a point P of X. Let AP = OX,P be the local ring of X at P and BQ = OY,Q that of Y at Q. Finally, fix a henselisation AhP of AP , with fraction field K h ⊂ K s. 1. The integral closure of AhP in the composite field LK h is isomorphic to h the henselisation BQ of BQ . h 2. The finite map Spec BQ → Spec AhP thus obtained has ramification index e(Q|P ) and residue class degree f (Q|P ).

3. The stalk of the different DSpec BQh /Spec AhP of the above map at the closed h h point of Spec BQ is the ideal of BQ generated by the stalk of the different h DY /X at Q. (Here BQ is viewed as a subring of BQ .) h Proof: For the first statement, note that by construction BQ is a dish h crete valuation ring with fraction field LK integral over AP . The second statement follows from Corollary 3.3 by taking the intersection of LK h with each finite Galois extension of K containing L. For the proof of the third statement, take an affine open neighbourhood Spec A of P with inverse image Spec B in Y . Then the stalk of DY /X at Q is the annihilator of ΩBQ /A ∼ = ΩB/A ⊗B BQ by the localisation property of differentials, whereas DSpec BQh /Spec AhP is the annihilator of ΩBQh /AhP ∼ = ΩB/A ⊗A AhP by the first statement and the localisation property of differentials.

4.

Henselian Discrete Valuation Rings

By the results of the previous section, the study of the local behaviour of finite morphisms of Dedekind schemes can be reduced to the study of the induced

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morphisms on the henselisations. In this section we study the henselisations in general and determine their fundamental groups. First a general definition. Definition 4.1 A discrete valuation ring is called henselian if its integral closure in any finite extension of its fraction field is a discrete valuation ring. Remarks 4.2 1. This definition is in accordance with the classical definition of henselian valuations as in Neukirch [1]. For its relation to the more general concept of henselian local rings, see Remark 4.8 below. 2. An immediate consequence of the definition is that the integral closure of a henselian discrete valuation ring in any finite extension of its fraction field is again henselian. The following proposition shows that the above definition is not out of place here. Proposition 4.3 The henselisation Ah of any discrete valuation ring A is henselian. Before the proof we recall a well-known algebraic lemma. Lemma 4.4 Let L|K be a finite extension of fields of characteristic p > 0 and let K ⊂ L0 ⊂ L be the maximal separable subextension (i.e. the compositum of all separable extensions of K contained in L). Then there exists m a positive integer m such that xp ∈ L0 for all x ∈ L. For a proof, see Lang [1], Chapter V, Section 6. The extension L|L0 is called purely inseparable. We now have the following general lemma. Lemma 4.5 Let A be a discrete valuation ring with fraction field K of characteristic p > 0 and let L be a purely inseparable finite extension of K. Then the integral closure B of A in L is a discrete valuation ring. m

Proof: Let m be a positive integer for which xp ∈ K for all x ∈ L. Then m if v denotes the discrete valuation associated to A, the map x 7→ v(xp ) is a homomorphism from the multiplicative group of L to Z. Moreover, denoting by b a positive generator of its image in Z and setting w(0) = ∞, the formula m w(x) = (1/b)v(xp ) defines a discrete valuation w : L → Z ∪ {∞}. The valuation ring of w is precisely B, for an element x ∈ L is integral over A if m m and only if xp ∈ A. (Indeed, xp is always an element of K and is integral over A if and only if x is; now use the fact that A is integrally closed.)

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Remark 4.6 Quite generally, the integral closure B of any integrally closed local domain A in a finite purely inseparable extension L|K of its fraction field is always local. To see this, one shows that the elements x ∈ L such m that xp lies in the maximal ideal of A form a unique maximal ideal in B. Proof of Proposition 4.3: Let L|K h be a finite extension. The construction of Ah (and the arguments preceding it) imply that the integral closure A0 of Ah in the maximal separable subextension L0 ⊂ L is a discrete valuation ring. The proposition then follows by applying the above lemma with L0 in place of K and A0 in place of A. Now we have the following important characterisation of henselian discrete valuation rings. Proposition 4.7 Let A be a discrete valuation ring with maximal ideal P . Then the following are equivalent: 1. A is henselian. 2. Any integral domain B ⊃ A finitely generated as an A-module is a local ring. 3. Given a monic polynomial f ∈ A[x] whose reduction f¯ modulo P factors as f¯ = f¯1 f¯2 with f¯1 and f¯2 relatively prime monic polynomials in κ(P )[x], there exists a factorisation f = f1 f2 of f into the product of two relatively prime monic polynomials in A[x] such that f¯i = fi modulo P for i = 1, 2. 4. If f ∈ A[x] is a monic polynomial such that its reduction f¯ modulo P has a simple root α ¯ in κ(P ), then there is α ∈ A with f (α) = 0 and α ¯ = α modulo P . Proof: To show that (1) implies (2), assume there is an integral domain B ⊃ A with fraction field L that is finitely generated over A and has at least two maximal ideals P1 6= P2 . But then the common integral closure C of A and B in L has two different prime ideals lying above the Pi in contradiction with (1). This follows by applying Lemma 3.5 to the localisations of B (resp. C) at the Pi in place of the A (resp. B) that figures in the lemma. Now suppose that A satisfies (2) but some monic polynomial f ∈ A[x] provides a counterexample to the property (3). We may assume f is irreducible in A[x] for otherwise an irreducible factor would still give a counterexample. This implies that (f ) is a prime ideal of A, for A is a unique factorisation domain (in fact, to see this it would suffice to use that A is interally closed),

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and so B = A[x]/(f ) is an integral domain integral over A. But then by assumption B/P B ∼ = B ⊗A κ(P ) ∼ = κ(P )[x]/(f¯1 ) + κ(P )[x]/(f¯2 ), so B is not local. Since (4) is just a special case of (3), it remains to prove that (4) implies (1) for which we employ an argument from Nagata [1]. Assume that the integral closure B of A in some finite extension L|K has at least two distinct maximal ideals Q1 and Q2 . We then concoct a polynomial f ∈ A[x] which is a counterexample to (4). Thanks to Lemma 4.5 there is no harm in supposing L|K separable and even Galois. Denote by H1 the stabiliser of Q1 in G = Gal (L|K), by L1 its fixed field and by B 0 the integral closure of A in L1 . Let 1 = σ1 , . . . , σn be a system of two-sided representatives of G modulo H1 . We know from Proposition 3.1 that there are exactly n maximal ideals of B, namely Qi = σi (Q1 ). Denote by Q0i the image of each Qi in Spec B 0 ; by Lemma 3.3 we have Q0i 6= Q01 for any i 6= 1. Using the Chinese Remainder Theorem as in the proof of Proposition 3.1 we may thus find an element α lying in the intersection of the Q0i for i > 1 but not in Q01 (and hence not in K). Since α ∈ L1 , the σi (α) for 1 ≤ i ≤ n are exactly the distinct conjugates of α in L. Again using Proposition 3.1 and the fact that the σi form a twosided system of representatives we may find for each i 6= 1 some Qj 6= Q1 with σi (Qj ) = Q1 . Thus σi (α) ∈ Q1 if and only if i 6= 1. Now look at the minimal polynomial f = xn + an−1 xn−1 + . . . a0 ∈ A[x] of α over K. Here an−1 is up to sign the sum of the σi (α), so it does not lie in Q1 ∩ A = P . But for s < n − 1 the coefficient as is (still up to sign) a higher order symmetric polynomial of the σi (α) and hence already lies in P . Thus by reducing −an−1 modulo P we get a simple root of f¯ = xn + a ¯n−1 xn−1 . This contradicts (4) as f is irreducible over K. Remark 4.8 An analysis of the above proof shows that in proving the equivalence of statements (2)–(4) we did not use the assumption that A was noetherian of dimension 1, hence these are equivalent conditions for any integrally closed local domain. The construction of the henselisation also works in this generality. It gives an integrally closed local domain Ah with the same residue field as A and equipped with a local homomorphism A → Ah . Moreover, Ah is seen to satisfy condition (2) above (and a fortiori (3), (4)). In general one calls any local ring satisfying condition (3) a henselian local ring. It can then be shown (see Nagata [1], Theorem 4.11.7) that the henselisation Ah of an integrally closed local domain A represents the contravariant functor on the category of henselian local rings which associates to an object B the set of local homomorphisms A → B. Thus Ah is the “smallest”

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henselian local ring equipped with a local homomorphism A → Ah . In fact, if we take the above representability as the definition of the henselisation, then Ah can be shown to exist for any local ring A but then one has to use a different construction. See Milne [1], Section I.4 for more details concerning this point. Example 4.9 Besides the henselisation, there is a classical example for a henselian discrete valuation ring, namely that of a complete discrete valuation ring which we now explain. Quite generally, one calls a local ring A complete if the natural map A → Aˆ into its completion (see Chapter 6, Remark ??) is an isomorphism. Obviously Aˆ is complete for any local ring A, so as concrete examples we may mention the ring Zp of p-adic integers encountered in Chapter 1 (which is the completion of the localisation of Z at (p)) and the ring of formal power series k[[t]] over a field k (which is the completion of any discrete valuation ring with residue field k that contains a subring isomorphic to k). Now we show that any complete local ring A satisfies condition (4) of Proposition 4.7; this assertion is classically known as Hensel’s lemma, whence the term “henselian”. So let P be the maximal ideal of A and assume the reduction f¯ of f ∈ A[x] modulo P has a root a1 ∈ A/P which is simple, i.e. f¯0 (a1 ) 6= 0. We construct a lifting a ∈ A of a1 with f (a) = 0 by Newton’s method of successive approximation. Represent a by a coherent sequence (ai ), with ai ∈ A/P i and assume ai is already determined (this being the case for i = 1). Lift ai arbitrarily to an element bi ∈ A/P i+1 ; the ai+1 we are looking for must then be of the form ai+1 = bi + p, with p ∈ P i /P i+1. Keeping the notation f for the image of f in (A/P i+1 )[x], the element f 0 (bi ) is a unit in the local ring A/P i+1 for its image f 0 (a1 ) modulo P is nonzero. By the Taylor Formula of order 2 (which is quite formal for polynomials), we have f (bi + p) = f (bi ) + f 0 (bi )p + cp2 with some c ∈ A/P i+1 , but anyway we have p2 = 0, so since we are aiming at f (bi + p) = 0, we only have to choose p = −f (bi )/f 0 (bi ). The next proposition shows that finite ´etale covers of spectra of henselian discrete valuation rings have a simple description. Proposition 4.10 Let X = Spec A, where A is a henselian discrete valuation ring with maximal ideal P . 1. Let f be a polynomial whose reduction f¯ modulo P is irreducible and defines a finite separable extension of κ(P ). Then the ring B = A[x]/(f ) is a discrete valuation ring and the canonical morphism Spec B → X is finite and ´etale.

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2. Any finite ´etale X-scheme is a finite disjoint union of affine schemes Spec B, with B of the above type. 3. The functor Spec B 7→ Spec (B ⊗A κ(P )) defines an equivalence between the category of finite ´etale X-schemes and that of finite ´etale κ(P )schemes. Proof: For the first statement, note that B is finitely generated as an Amodule, and hence is noetherian. Since f is irreducible it is also an integral domain (see the proof of Proposition 4.7), and hence it is a local ring by Proposition 4.7 (2). By Chapter 6, Proposition 3.4, ΩB/A = 0 and so by base change ΩB⊗A κ(P )/κ(P ) = 0, hence Spec B is ´etale over X and P generates the maximal ideal of B which is thus principal, so B is a discrete valuation ring. For the second statement, note first that since the only affine covering of X is by X itself, any finite X-scheme is necessarily affine. If it is moreover ´etale, decomposing it into a finite disjoint union of components, we may assume it is integral and hence the spectrum of some integral domain B. By exactly the same argument as above, we see that B is a discrete valuation ring. Now to prove that B is of the required form, let f¯ be the minimal polynomial of a generator α ¯ of the separable field extension B ⊗A κ(P )|κ(P ) ¯ and lift f to a polynomial f ∈ A[x]. By proposition 4.7 (4), α ¯ lifts to a root of f in A, whence an injective morphism A[x]/(f ) → B. Here both rings are discrete valuation rings with the same residue field and their spectra are finite and ´etale over X, so by Proposition 1.20 their fraction field K must be the same. Thus both rings are equal to the integral closure of A in K. In the last statement essential surjectivity follows if we show that any finite separable extension L|κ(P ) is the residue field of some extension A[x]/(f ) as in (1). For this we only have to take as f some lifting in A[x] of the minimal polynomial of a generator of L|κ(P ). For fully faithfulness, assume B = A[x]/(f ), C = A[x]/(g) are such that Spec B, Spec C are ´etale over X and assume given a morphism B ⊗A κ(P ) → C ⊗A κ(P ). It is given by mapping a generator of the field extension B ⊗A κ(P )|κ(P ) to a root α ¯ of f¯ in C ⊗A κ(P ). Lifting α ¯ to a root α of f in C gives a homomorphism B → C inducing the above one by tensoring with κ(P ). To see that this morphism is unique, it is enough to see that α is the unique root of f in C lifting α. ¯ For this, by enlarging C if necessary we may assume that C ⊗A κ(P ) is Galois over κ(P ) (embed C ⊗A κ(P )|κ(P ) in a finite Galois extension and lift its defining polynomial to A[x]). Then f¯ decomposes as a product of distinct linear factors in C ⊗A κ(P ) and each of its roots lifts to a different root of f in C.

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Corollary 4.11 For X as in the proposition there is a canonical isomorphism π1 (X)op ∼ = Gal (κ(P )). Proof: The group π1 (X)op is the Galois group of the Galois extension K et |K defined in Theorem 2.6. Let Aet be the integral closure of A in K et ; as A is henselian, it is a local ring with maximal ideal P et . Any element of Gal (K et |K) maps Aet and P et onto themselves and hence defines a κ(P )automorphism of the field Aet /P et which is none but a separable closure of κ(P ) by the third statement of the proposition. Whence a homomorphism π1 (X)op → Gal (κ(P )) of which the proposition implies the bijectivity. Remark 4.12 Let X = Spec A be as above, and denote by K the fraction field of A. In this case the closed normal subgroup of Gal (K) which is the kernel of the canonical map Gal (K) → Gal (K et |K) (and hence of the map Gal (K) → Gal (κ(P )) defined by composing with the above isomorphism) is usually called the inertia subgroup.

5.

Dedekind’s Different Formula

In this section we harvest the fruits of our efforts in the two previous ones and complete our study of finite morphisms of Dedekind schemes; in particular we prove a classical formula of Dedekind computing the different. First an easy application of the ideas we have just seen. Proposition 5.1 Let A be a henselian discrete valuation ring with fraction field K and maximal ideal P , let B be its integral closure in a finite separable extension L|K and let Q be the maximal ideal of B. Assume further that the residual extension κ(Q)|κ(P ) is separable. Then there is a unique discrete valuation ring A ⊂ C ⊂ B with residue field κ(Q) and such that the map Spec C → Spec A is ´etale. Proof: Embed L in a separable closure K s and put M = L ∩ K et , with K et as in Theorem 2.6. Let C be the integral closure of A in M. The affine scheme Spec C is integral, finite and ´etale over Spec A by construction and hence C is a discrete valuation ring by Proposition 4.10 (2). The residue field of C is κ(Q), for otherwise, B being henselian by Remark 4.2 (2), arguing as in the proof of Proposition 4.10 (2) we would get a subring C 0 ⊂ B properly containing C with residue field κ(Q) and Spec C 0 → Spec A ´etale, contradicting the construction of C.

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Using Proposition 1.20 we get that in the situation of the proposition the finite morphism Spec B → Spec C has ramification index e(Q|P ) at Q and residue class degree 1. It is convenient to separate this property in a definition. Definition 5.2 A finite morphism φ : Y → X is called totally ramified over a closed point P of X if the underlying space of the fibre YP consists of a single point Q with f (Q|P ) = 1. In the case when Y is the spectrum of a discrete valuation ring, we simply say that φ is totally ramified if it is totally ramified over the closed point of X. Totally ramified extensions of discrete valuation rings have the following characterisation, somewhat analogous to Proposition 4.10 (1), (2) but valid for not necessarily henselian rings as well. Proposition 5.3 Let A ⊂ B be an extension of discrete valuation rings with maximal ideals P ⊂ Q, and fraction fields K ⊂ L, respectively. If the induced map φ : Spec B → Spec A is finite and totally ramified, then B = A[t] with a generator t of Q and the minimal polynomial of t over K is of the form f = xe + ae−1 xe−1 + . . . + a0 , where e = e(Q|P ), ai ∈ P for all i but a0 ∈ / P 2. Conversely, if A is a discrete valuation ring and B = A[t] an extension of the above type, then B is a discrete valuation ring and the map Spec B → Spec A is totally ramified. A polynomial f as in the statement of the proposition is called an Eisenstein polynomial. Proof: Let v be the discrete valuation associated to B. For the first part, note that the elements 1, t, . . . , te−1 are linearly independent over K. Indeed, assume given a linear combination ae−1 te−1 + . . .+ a1 t+ a0 with ai ∈ A. Since the ramification index is e, for all a ∈ A the valuation v(a) is divisible by e, and hence the integers v(ai ti ) are all distinct modulo e. From this we see P that v( ai ti ) = min v(ai ti ) and hence the sum cannot be 0. As [L : K] = e by Proposition 1.20 and t is integral over A, we indeed have L = K(t) and B = A[t] = A[x]/(f ) with some monic polynomial f . To see that f is of the above type, remark that by the above argument, f (t) = te +ae−1 te−1 +. . .+a0 can only be 0 if two of the terms with the smallest valuation have equal valuation. But for 0 < i < e the v(ai ti ) are distinct and nonzero modulo e; on the other hand v(te ) = e. Since v(a0 ) is divisible by e the only possibility that remains is v(a0 ) = e, v(ai ) ≥ e for 0 < i < e. Conversely, if B = A[t] = A[x]/(f ) is of the above type, f is irreducible in A[x] (same proof as over Z), so B is a domain that is finitely generated as

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an A-module and hence noetherian. The fibre over P is Spec κ(P )[x]/(xe ), from which we see that the last statement holds, but also that B is a local ring whose maximal ideal is generated by t. We can now prove the promised formula of Dedekind. To state it we need one more piece of terminology. Definition 5.4 Let φ : Y → X be a finite morphism of Dedekind schemes, and let Q be a branch point of φ mapping to a closed point P of X. We say that φ is tamely ramified at Q if the ramification index e(Q|P ) is not divisible by the characteristic of κ(P ). Otherwise φ is wildly ramified at Q. Proposition 5.5 (Dedekind’s Different Formula) Let φ : Y → X be a finite morphism of Dedekind schemes such that the corresponding extension L|K of function fields is separable. Let Q1 , . . . , Qn be the branch points of φ with (not necessarily distinct) images P1 , . . . , Pn in Y and let DY /X = P L( i mi Qi ) be the different of φ. Then we have mi = e(Qi |Pi ) − 1 if φ is tamely ramified at Qi , and mi ≥ e(Qi |Pi)

if φ is wildly ramified at Qi .

Proof: Using Proposition 3.9 we may localise and henselise at Qi and Pi , reducing thereby to the case X = Spec A and Y = Spec B with A ⊂ B henselian discrete valuation rings with maximal ideals P ⊂ Q. Consider the maximal ´etale subextension A ⊂ C ⊂ B. Since Spec C → Spec A is ´etale, we get from the exact sequence in Chapter 6, Lemma 3.5 (2) an isomorphism ΩB/A ∼ = ΩB/C . Thus we may assume A = C and so by Proposition 5.3 we have B = A[t] ∼ = A[x]/(f ) with f ∈ A[x] an Eisenstein polynomial. In this case ΩB/A is generated by dt and has f 0 = 0 as its single relation, so the different is the ideal generated by f 0 = ete−1 + (e − 1)ae−1 te−2 + . . . + a1 . If v is the valuation associated to B, here we have v(ai ) ≥ e for all i, hence all terms have valuation at least e except for the first in the case e ∈ / P (i.e. that of tame ramification), when v(ete−1 ) = e − 1. It is time for a concrete example. Example 5.6 Let ` be a prime number, ζ a primitive `-th root of unity, K = Q(ζ) and OK its ring of integers. We analyse the local behaviour of the map φ : Spec OK → Spec Z. For a prime number p (viewed as a closed point of Spec Z), denote by Zhp the henselisation of Z at p and Qhp its fraction field. Let Bp be the integral closure of Zhp in Qhp (ζ).

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• For ` 6= p, let C be the maximal ´etale subextension of Zhp ⊂ Bp given by Proposition 5.1. Being henselian, it must contain all roots of the equation x` −1 as its residue field does. In particular, it must contain ζ, so it must have the same fraction field as Bp . Since C is also a discrete valuation ring finitely generated as a ZhP -module, it must equal Bp , so Spec Bp is ´etale over Spec Zhp and a fortiori φ is ´etale at all points of OK not lying above `. • For ` = p, the maximal ´etale subextension is just Zhp so the map Spec Bp → Spec Zhp is totally ramified. We now show that the degree of the field extension Qhp (ζ)|Q is exactly p − 1, which, together with Proposition 1.20, will imply that the map Spec OK → Spec Z is totally ramified over (`). Indeed, the extension Qhp (ζ)|Q is of degree at most p−1. But it contains the element 1 − ζ which is a root of the polynomial F obtained by substituting 1 + y in place of x in xp−1 + xp−2 + . . . + 1. One sees immediately that F is an Eisenstein polynomial, so it is irreducible in Qhp [y]. • By Proposition 5.5 we get that the different DSpec OK /Spec Z is the ideal sheaf associated to the divisor (` − 2)S, where S is the unique point lying above (`). We could have obtained this result immediately if we knew that OK = Z[ζ] but this fact is not obvious; to prove it one commonly uses nearly all the information obtained above. Remark 5.7 As an amusing application of the previous example we show that any finite abelian group occurs as the Galois group of a finite Galois extension K|Q. Indeed, let A be a finite abelian group of order m and decomposing as a direct sum A ∼ = A1 ⊕ . . . ⊕ An with each Ai cyclic. By Dirichlet’s theorem of prime numbers in an arithmetic progression we may find n different prime numbers `1 , . . . , `n each congruent to 1 modulo m. Choose a primitive `i -th root of unity ζi for each i. The Galois group Gi of the Galois extension Q(ζi)|Q is cyclic of order `i − 1, hence divisible by m and as such has a quotient isomorphic to Ai . Denote by Ki the corresponding Galois extension of Q. For each i the map Spec OKi → Spec Z is ´etale at the points not mapping to `i but totally ramified at the unique point lying over `i ; a degree count shows that this implies that we must have Ki ∩ Kj = Q for i 6= j. Hence the composite K of the Ki is Galois over Q with group A.

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Chapter 8 Finite Covers of Algebraic Curves 1.

Smooth Proper Curves

In Chapter 6 we have already defined smooth affine (resp. projective) curves as Dedekind schemes equipped with a closed immersion into some affine (resp. projective) space over a field. Here we shall study them in a slightly (in fact, seemingly) more general setting, forgetting about the closed immersion but preserving some properties that are implied by are existence. Definition 1.1 Let k be a field. A scheme X is of of finite type over k if it admits an affine open covering by affine schemes associated to finitely generated k-algebras. Note that this implies the existence of a canonical morphism X → Spec k. A smooth curve over k is a Dedekind scheme of finite type over k. A morphism of smooth k-curves is a morphism compatible with the projections onto Spec k. We see that smooth affine and projective curves defined previously are smmoth curves in the above sense. Let now X be a smooth curve and let K be the function field of X. We view k as a subfield of K and identify the local rings at closed points of X with subrings of K (assumed to be distinct as in previous chapters). Definition 1.2 A smooth curve X is proper over k if any discrete valuation ring containing k whose fraction field is K is in fact a local ring at some closed point of K. 173

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Example 1.3 Clearly, affine curves are not proper, for the “points at infinity” are missing. For instance, if we view the affine line A1k = Spec k[t] as the projective line with a point ∞ deleted, then the local ring of ∞ is not a local ring of the affine line but it is a discrete valuation ring containing k whose fraction field is k(t). However, we have: Example 1.4 Any smooth projective k-curve is proper over k. To see this, we may assume that X is embedded as a closed subscheme into some Pnk by a map i : X → Pnk does not map X into any hyperplane of Pnk . Then from the defintion of closed subschemes we infer that the homogeneous coordinate functions x0 , . . . , xn on Pnk induce nonzero functions x¯0 , . . . , x¯n on X. Let now K be the function field of X and A a discrete valuation ring containing k with fraction field K. Denoting by v the discrete valuation associated with A, choose a pair (i, j) for which the valuation v(¯ xi /¯ xj ) is maximal. For any k 6= j we have v(¯ xk /¯ xj ) = v(¯ xi /¯ xj ) − v(¯ xi /¯ xk ) ≥ 0, hence all x¯k /¯ xj are contained in A. But X ∩ D+ (xj ) is an affine scheme which is precisely the spectrum of the k-algebra R generated by the x¯k /¯ xj for k 6= j, 0 so R ⊂ A. If P is the maximal ideal of A, then P = P ∩ R cannot be 0, for otherwise A would contain the fraction field of B which is K. Hence P 0 is maximal and the localisation RP 0 gives a local ring of X contained in A. Now we may conclude the argument by the following easy lemma. Lemma 1.5 Let A ⊂ B be two discrete valuation rings with the same fraction field K. Then A = B. Proof: The proof is based on the basic property of discrete valuation rings that any element of their fraction fields is either contained by them, or is the inverse of an element of their maximal ideal. So let t be a generator of the maximal ideal of B. Then t ∈ A, for otherwise we would have t−1 ∈ A ⊂ B, which is impossible for t is not a unit in B. Similarly, any unit u of B is contained in A, for otherwise u−1 would be in the maximal ideal of A, hence in that of B. We next prove a statement that can be regarded as the algebraic analogue of Riemann’s existence theorem. Recall that the main statement of the latter is the existence of a non-constant meromorphic function on a compact Riemann surface, which then endows it with the structure of a branched cover of the projective line. The existence of non-constant elements in the function field of a proper smooth curve is evident, so what remains to be proven is the following statement.

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Proposition 1.6 Let X be a smooth proper k-curve with function field K and f ∈ K a transcendental element over k. Then f induces a finite morphism φf : X → P1k . Before the proof, recall that given a finite separable extension L of the function field K of X, we constructed in Chapter 7, Proposition 2.1 the normalisation of X in L, i.e. a Dedekind scheme with function field L equipped with a finite morphism onto X. Now for smooth curves exactly the same argument shows that the normalisation exists in any finite extension L|K, except that instead of Chapter 6, Proposition 1.9 we have to use Proposition ?? from there. Moreover, we have: Lemma 1.7 The normalisation Y of a smooth proper curve X in a finite extension L|K of its function field is also a smooth proper curve. Proof: That Y is of finite type over the field k over which X is defined follows from the fact that it maps onto X by a finite morphism. To see that Y is proper, let A be a discrete valuation ring with fraction field L and maximal ideal P . Then A0 = A ∩ L is a discrete valuation ring of fraction field K, hence a local ring of X. The integral closure B 0 of A0 in L has finitely many maximal ideals one of which is P 0 = P ∩ B 0 . The localisation of BP0 0 is then a local ring of X contained in A; according to Lemma 1.5, we have A = BP0 0 . Proof of Proposition 1.6: Regard P1k as Spec k[f ] and Spec k[f −1 ] being patched together over Spec k[f, f −1 ]. Let U ⊂ X be the open complement of the set of poles of f and V ⊂ X that of its zeros. Take an affine open covering of U by subsets Ui = Spec Ai of U (in fact, we shall see later that U itself is affine). Since for all i we have f ∈ Ai , we have natural maps Ui → Spec k[f ] which are easily seen to agree when restricted to basic open sets contained in the intersections Ui ∩ Uj . Whence a morphism φUf : U → Spec k[f ]; in a similar way, one constructs φVf : V → k[f −1 ]. Again it can be readily checked that φUf and φVf agree over U ∩ V , whence the required map φf . It remains to be seen that φf is finite. For this, notice first that the normalisation Y of P1k in the field extension K|k(f ) is obtained by patching the normalisations of Spec k[f ] and Spec k[f −1 ] together, so the above construcλ tion shows that φf factors as a composite X → Y → P1k , where the second map is the finite map coming from the normalisation. So it is enough to show that λ is an isomorphism. But for all closed points P ∈ X we have an inclusion OX,P ⊃ OY,λ(P ) (for the latter ring is a localisation of the integral closure of k[f ] or k[f −1 ] in L which must be contained in the integrally closed ring OX,P ). By Lemma 1.5 we have equality here, whence the claim.

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Next we may draw the analogy between compact Riemann surfaces and proper smooth curves even closer. Recall that when studying branched covers of compact Riemann surfaces we always took non-constant holomorphic maps as morphisms in our category whereas in the analogous theory for Dedekind schemes we regarded only finite morphisms. Now we can show (as already promised in the last chapter) that for proper smooth curves non-constant and finite morphisms are the same. To be precise, call a morphism Y → X of schemes constant if it factors as Y → Spec κ(P ) → X for some closed point P of X. Obviously, such a morphism cannot be finite when X is of dimension at least 1. Proposition 1.8 Let φ : Y → X be a morphism of proper smooth curves over a field k. If φ is not constant, then φ is finite. Recall from the last chapter that the statement is false without assuming X proper. For the proof, we need the following easy but crucial lemma. Lemma 1.9 Any non-constant morphism φ : Y → X of proper smooth curves over a field k is surjective. Proof: Let K (resp. L) be the function field of X (resp. of Y ). Since φ is not constant, the point of Y underlying the image of the composite map Spec L → Y → X must be the generic point of Y . Whence a map K → L which cannot be 0 since φ is a morphism over k. Now let OX,P be the local ring of a closed point P of X. Its integral closure in L has finitely many maximal ideals; localising by one of them we get a discrete valuation ring A with fraction field L. Since Y is proper, we must have A = OY,Q for some closed point Q; moreover, Q must map to P for if we view OY,Q as a subring of L, the morphism of local rings induced by φ must be the inclusion of OY,Q ∩ K = OX,P in OY,Q . Proof of Proposition 1.8: Again let K (resp. L) be the function field of X (resp. of Y ). As in the lemma, we may view K as a subfield of L; in particular any transcendental element f ∈ K may be viewed as an element of L. By the previous proposition, we have a finite morphism φf : X → P1k attached to f ; its composition with φ is the finite morphism Y → P1k corresponding to f viewed as an element of L. Since moreover φ is surjective by the lemma, the definition of finite morphisms implies that φ must be finite. The proposition thus implies that we may restate the results of Chapter 7, Section 2 in the case of proper smooth curves by considering non-constant morphisms and arbitrary finite field extensions in our categories. We only note here the corollary:

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Corollary 1.10 Let k be a field. Then associating to a proper smooth curve over k its function field induces an anti-equivalence between the category of proper smooth curves over k with nonconstant morphisms and the category of finitely generated extensions of k of transcendence degree 1, with the inclusion maps as morphisms. Comparing this result for k = C with those of Chapter 4 we get an equivalence of categories between smooth proper complex curves and compact Riemann surfaces. We conclude this section by making this equivalence explicit and show that over C the set of closed points of any smooth proper curve (in fact, any smooth curve) can be endowed with the structure of a Riemann surface. If we knew that any smooth proper curve admits a closed immersion into some projective space (which we shall not prove), this would follow easily for then we could simply take the complex manifold structure coming from the embedding. We take another approach here which has the advantage of describing explicitly a complex atlas. It is based on the fact that “locally any smooth curve is ismorphic to a plane curve”. More precisely, let k be any algebraically closed field, X a smooth curve over k with function field K and P a closed point of X. Let x be a generator of the maximal ideal P of OX,P . Then we may write K = k(x, u) with some u integral over the polynomial ring k[x]. If f is the minimal polynomial of u over k(x), then K is the fraction field of the ring A = k[x, y]/(f ) which corresponds to an affine curve of equation f = 0 in the (x, y)-plane. As u is integral over k[x], we have A ⊂ OX,P . Therefore P ∩ A is a nonzero, hence maximal ideal of A containing x and as such must be of the form (x, u − a), where a is the image of u in A/P ∩ A ∼ = k. Thus P induces the point (0, a) on the curve. By Lemma 1.5, the local ring AP ∩A at this point equals OX,P if and only if it is a discrete valuation ring. According to the criterion of Chapter 6, Proposition 3.7, this is the case if and only if ΩA∩P/k /P ΩA∩P/k has dimension 1 over k. But ΩA∩P/k is generated by dx and dy with a single relation ∂1 f dx + ∂2 f dy; since ∂1 f gives an element of P ∩ A, we see that OX,P = AP ∩A if and only if ∂2 f (0, a) 6= 0. Lemma 1.11 In the above situation we may always choose u such that ∂2 f (0, a) 6= 0 and hence AP ∩A = OX,P . The proof will implicitly use the fact that K is separable over k(x). The reader may prove this as an exercise or assume k is of characteristic 0 (since we are mostly interested in the case k = C anyway).

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Proof: (after Lang [3]) If u does not satisfy already the condition of the lemma, we modify it as follows. Let u = a0 + a1 x + a2 x + ... be the power series expansion of u at P . (Recall that this is obtained as follows: reduce u modulo P to get a0 , so that u − a0 = b1 x; now reduce b1 modulo P to get a1 and so on.) For a fixed positive integer r write Pr for the polynomial a0 + . . . + ar xr . Let L ⊃ K be a finite Galois extension of k(x) and let u = u1 , u2 , . . . , un be the roots of f (viewed as a polynomial with coefficients in k(x)) in L. Let B be the integral closure of A in L and let Q be a point of Spec B above P . The localisation BQ is a discrete valuation ring with residue field k, hence we may speak of zeros and poles of elements of L at Q. Now write vi = (1/xr )(ui − Pr ) for all i. Then taking r large enough we may assume that vi has a pole at Q for i > 1 and that v1 has neither a pole nor a zero. The vi are permuted by the elements of Gal (L|k(x)) and hence are the roots of an irreducible polynomial F with coefficients in k(x); by clearing denominators we can actually assume F = φn y n + . . . + φ0 with φj ∈ k[x] and having no common divisor. This F will play the role of f and v1 the role of u before. Now write F = φn v2 . . . vn (y − v1 )(y/v2 − 1) . . . (y/vn − 1). Here w = φn v2 . . . vn lies in A for it equals v1−1 φ0 . Hence by reducing modulo n−1 Q we get F (0, y) = w(−1) ¯ (y − v¯1 ), where w ¯ (resp. v¯1 ) is the image of w (resp. v1 ) modulo Q. Note that F (0, y) 6= 0, for otherwise the φj would all be divisible by x. This implies w¯ 6= 0, whence 0 6= (−1)n−1 w¯ = ∂2 F (0, v¯1 ), which was to be seen. Using the lemma we may construct the Riemann surface structure on smooth complex curves as follows. Construction 1.12 In the situation above assume k = C. The plane curve Spec A is not necessarily smooth but both differentials of f can vanish at only finitely many points (as their loci of zeros define proper closed subsets). Away from these singular points the local rings of Spec A may be identified with local rings of X by the above discussion. Also, by the condition ∂2 f (0, a) 6= 0 furnished by the lemma the construction of the Riemann surface structure for smooth plane curves in Chapter 4 may be applied to the complement of the singular points and we get that x defines a complex chart centered around (0, a) in a sufficiently small open neighbourhood U not containing singular points. Identifying U with a subset of closed points of X we may declare U open in the complex topology of X and get a bijection of U onto some open subset in C via x; we declare this map to be a complex chart and in particular a homeomorphism.

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Now assume x0 is another generator of the maximal ideal of OX,P and perform the above construction for x0 in place of x. We have to show that in this way we get compatible complex charts; this will imply in particular that x and x0 define the same topology in a sufficiently small neighbourhood of P . To see this, observe that (again by the condition ∂2 f (0, a) 6= 0) we may apply the holomorphic implicit function theorem to the polynomial f around (0, a) and express u as a holomorphic function (i.e. a convergent power series) in x. As any element of OX,P = AP ∩A is a rational function in x and u it must be holomorphic as well. This applies in particular to x0 and shows that the two charts are compatible. Carrying out the above construction around each closed point of X thus gives a complex topology and a complex atlas on X, as required. We have just seen that local rings of X consist of holomorphic functions in the complex structure, from which it follows that any element f of the function field K is meromorphic and also (via Chapter 4, Lemma 1.5) that any morphism of smooth curves induces a holomorphic map of Riemann surfaces. To complete our discussion, we still need to show: Corollary 1.13 If X is a proper smooth curve, then the associated Riemann surface is compact. Proof: By what we have just observed, the finite, hence surjective morphism φf : X → P1C associated to a nonconstant function f ∈ K induces a holomorphic map. But any holomorphic map is open (Chapter 4, Corollary 2.2), so the compactness of X follows from that of P1 (C).

2.

Invertible Sheaves on Curves

Invertible sheaves have played an important role in our study of Dedekind schemes. For the special class of proper smooth curves they play an even more important part, as the presence of a base field enables one to establish several very useful results about them which are not available in the general case. Throughout this section (and the next), X will be a smooth proper curve over a perfect field k. We shall also make the additional assumption that k is algebraically closed in the function field K of X, i.e. any element of K \ k is transcendental over k; we shall call such elements non-constant. Remark 2.1 Curves (or more generally, varieties) satisfying the above additional assumption are usually called geometrically integral, for one may prove

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that this property is equivalent to the scheme X ×Spec k Spec k¯ being integral, where k¯ is an algebraic closure of k. Now let L be an incbvertible sheaf of X. Recall from Chapter 5, Section 4 that any such L is isomorphic to some sheaf the form L(D), with D a divisor on X. The presence of the base field k enables us to define a very important invariant of D. Definition 2.2 The degree of a divisor D = P deg(D) = i mi [κ(Pi ) : k].

P

i

mi Pi on X is the integer

In this way we obtain a homomorphism deg : Div(X) → Z. Now recall that in Chapter 5, loc. cit. we defined the divisor Div (f ) of a function f ∈ K × as the divisor D with vP (D) = vP (f ) for all closed points P of X. We may write Div (f ) as a difference Div 0 (f ) − Div ∞ (f ), where Div 0 (f ) is the divisor for which vP (Div 0 (f )) = vP (f ) if vP (f ) > 0 and 0 otherwise and similarly Div ∞ (f ) is the divisor with vP (Div ∞ (f )) = −vP (f ) if vP (f ) < 0 and 0 otherwise. These are called the divisor of zeros and the divisor of poles of f , respectively Lemma 2.3 Let f be a nonconstant function on X. Then deg(Div 0 (f )) = deg(Div ∞ (f )) = [K : k(f )] and hence deg(Div (f )) = 0. Proof: Consider the map φf : X → P1k induced by f . By construction, the underlying space of the fibre of φf over the point 0 of P1k is the set of zeros of f and that of the fibre over ∞ is the set of poles. Since here f is viewed as a coordinate function on P1k , it generates the maximal ideal of the local ring at 0 and its inverse generates that at ∞. This implies that for P a zero or a pole of f , vP (f ) is none but the ramification index of φf at P ; moreover, [κ(P ) : k] is the corresponding residue class degree. Hence the assertion follows from Chapter 7, Proposition 1.20. Corollary 2.4 There are no nonconstant regular functions on X, i.e. OX (X) = k. Proof: Indeed, a nonconstant function must have at least one pole by the proposition. Corollary 2.5 If D1 , D2 ∈ Div(X) are two divisors with L(D1 ) ∼ = L(D2 ), then deg D1 = deg D2 .

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Proof: This follows from the lemma and Chapter 5, Proposition 4.20. By the last corollary, we may define the degree of an invertible sheaf L on X as deg D for any D ∈ Div(X) with L(D) ∼ = L (such a D exists by Chapter 5, Propositions 4.8 and 4.12). Moreover, the corollary implies that the homomorphism deg : Div(X) → Z factors through the quotient Pic (X). The kernel of the map deg : Pic (X) → Z is denoted by Pic 0 (X). We next investigate global sections of invertible sheaves on X. If L is such a sheaf, then L(X) is naturally a vector space over k. Now we have the following basic finiteness result. Lemma 2.6 For any invertible sheaf L on X, the k-vector space L(X) is finite dimensional. Before giving the proof it is convenient to introduce a partial order on the group Div(X): we shall write D1 ≥ D2 if vP (D1 ) ≥ vP (D2 ) for all closed points P and D1 > D2 if D1 ≥ D2 and there is some P with vP (D1 ) > vP (D2 ). Proof: The results from Chapter 5, Section 4 enable us to choose some D with L(D) ∼ = L. Since for D1 ≥ D2 we obviously have L(D1 )(X) ⊃ L(D2 )(X), it is enough to prove the lemma for D ≥ 0. We use induction on deg(D). For D = 0 we have dim k L(0)(X) = 1 by Corollary 2.4. For the inductive step we show that if D ≥ 0 and P is a closed point of X, then the quotient space L(D + P )(X)/L(D)(X) has dimension at most [κ(P ) : k]. Indeed, define a homomorphism of k-vector spaces L(D + P )(X) → κ(P ) by mapping a function f ∈ L(D)(X) to (tm+1 f )(P ), where m = vP (D) and t is a generator of the maximal ideal of OX,P . (By assumption, tm+1 f ∈ OX,P ; its evaluation at P means its reduction modulo the maximal ideal.) The kernel of this homomorphism is exactly L(D)(X), whence the claim. Remark 2.7 The lemma is a special case of much more general finiteness results about coherent sheaves. See Hartshorne [1], Section 8.8 and the more general theorem of Grothendieck cited there. The dimension whose finiteness is asserted by the lemma is a very important invariant of the invertible sheaf L; roughly speaking, it measures the number of independent solutions to the Mittag-Leffler type problem of finding functions on X with zero and pole behaviour restricted by any D for which L = L(D). Also, the lemma applies in particular to ΩX/k which is an invertible sheaf on X since k is perfect, so we may define a very important invariant of X.

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Definition 2.8 The dimension g of the finite dimensional k-vector space ΩX/k (X) is called the genus of the curve X. See the next section for the relation over C with the topological genus defined in Chapter 4. Now we can state a famous theorem computing the dimension of L(X) over k. Note that the proof of the above lemma implies an estimate dim k L(D) ≤ deg D + 1 for D ≥ 0, but this is insufficient for most applications; the following theorem is much more precise. Theorem 2.9 (Riemann-Roch Formula) For any invertible sheaf L on X we have the formula dim k L(X) − dim k (ΩX/k ⊗OX L∨ )(X) = deg L − g + 1, where g is the genus of X. We shall not prove the theorem here for developing the necessary techniques would lead us to far astray. For modern proofs using the cohomology of coherent sheaves, see Hartshorne [1], Section IV.1 or the first chapter of Serre [2] (these authors assume k algebraically closed but the proof works with minor modifications in general). For a more traditional proof due to Weil, see e.g. Lang [3], Stichtenoth [1] or van der Waerden [1]. We content ourselves here by deriving some consequences, of which here are the first ones. Corollary 2.10 1. deg ΩX/k = 2g − 2. 2. If deg L > 2g − 2, then dim k L(X) = deg L − g + 1. Proof: The first assertion follows by substituting L = ΩX/k into the theorem. For the second, assume deg L > 2g − 2. We show that this implies dim k (ΩX/k ⊗OX L∨ )(X) = 0, for which by the first part it is enough to see that for any invertible sheaf L with deg L < 0 we must have dim k L(X) = 0. So write L = L(D) and assume dim k L(D)(X) > 0. This means there is a function f ∈ K with vP (f ) + vP (D) ≥ 0 for all closed points P . Multiplying by [κ(P ) : k] and taking the sum over P we get deg D = deg L ≥ 0 using Lemma 2.3, a contradiction. Another, perhaps unexcepted, corollary is the following. Corollary 2.11 Let U ⊂ X be an open subset properly contained in X. Then U is affine.

2.. INVERTIBLE SHEAVES ON CURVES

183

Proof: Let S = {P1 , . . . , Pn } be the complement of U. For a fixed Pi the second part of the previous corollary implies that for a sufficiently large positive integer m we have dim k L((m + 1)Pi )(X) − dim k L(mPi )(X) > 0 for all 1 ≤ i ≤ n, so for each i there exists a function fi ∈ K having a pole at Pi and regular elsewhere. Hence the set of poles of f = f1 + . . . + fn is exactly S. Now consider the morphism φf : X → P1k defined by f . The inverse image of the complement of ∞ (i.e. the normalisation of Spec k[f ] in the extension K|k(f )) is precisely U, and we are done. The Riemann-Roch formula is an invaluable tool in the classification of curves. We illustrate this by taking a look at the cases of two lowest genera. Example 2.12 Let X be of genus 0 and assume it has a k-rational point, i.e. a closed point P with κ(P ) = k. Corollary 2.10 (2) applies to the sheaf L = L(P ) and computes the dimension of L(X) to be 2, hence this space must contain besides the constant functions some function f with a simple pole at P and regular elsewhere. The morphism φf : X → P1k associated to f induces an isomorphism on function fields, for P is the only point above ∞ and has residue class degree 1; now apply Chapter 7, Proposition 1.20. Hence by Corollary 1.10 φf is an isomorphism and we get that smooth proper curves of genus 0 with a k-rational point are isomorphic to P1k . Remark 2.13 The assumption on the existence of a k-rational point is important if we wish to get an isomorphism defined over k (i.e. compatible with the projections to Spec k). Indeed, the projective plane curve of equation x20 + x21 + x22 = 0 can be shown to be smooth of genus 0 over Q but is not isomorphic over Q (or even R) to P1 . Example 2.14 Let now X be a smooth proper curve of genus 1 with a k-rational point P ; such a curve is called an elliptic curve over k. Here Corollary 2.10 (2) applies to the space L(2P )(X) and shows it has dimension 2; pick a basis consisting of the constant function 1 and some function x with a single pole of order 2 at P (it cannot have a simple pole for then X would be isomorphic to P1k by the previous example, which is impossible for P1k has genus 0 and X has genus 1). Next, the space L(3P )(X) has dimension 3; add a third basis element y having a pole of order 3 at P . Note in passim that the elements x and y generate K over k. Indeed, using Chapter 7, Proposition 1.20 as in the previous example we get that [K : k(x)] = 2 and [K : k(y)] = 3; since [K : k(x, y)] must divide both degrees here, it must be 1.

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Now apply the corollary to the space L(6P )(X) to see that that it is 6-dimensional. Since it contains the seven elements 1, x, x2 , x3 , xy, y, y 2, we must have a linear relation of the form a6 y 2 + a5 y + a4 xy = a3 x3 + a2 x2 + a1 x + a0 with ai ∈ k. Regard this formula as the equation of an affine curve in the (x, y)-plane. We contend that this curve must be smooth. Indeed, assume it has a point where both partial derivatives vanish; by performing a linear change of coordinates we may assume this is the point (0, 0). Then the equation must have the form a6 y 2 + a4 xy = a3 x3 + a2 x2 Dividing this equation by x2 shows that x is contained in the field k(y/x), whence k(y/x) = k(y/x, x) = k(x, y) = K. But using Corollary 1.10 as in the previous example, this implies that the morphism X → P1k associated to y/x is an isomorphism, which is again impossible. So we have shown that the affine scheme V = Spec k[s, t]/(a6 t2 + a5 t + a4 st − a3 s3 − a2 s2 − a1 s − a0 ) is a smooth plane curve. Now let U be the complement of P in X; by Corollary 2.11 U is the form Spec A, with A containing x and y, so sending s to x and t to y induces a morphism of smooth affine curves U → V ; arguing in a similar way as in the proof of Proposition 1.6, we see that this is actually an isomorphism. Finally, consider the projective plane curve Y with equation a6 x0 x22 + a5 x20 x2 + a4 x0 x1 x2 = a3 x31 + a2 x0 x21 + a1 x20 x1 + a0 x30 . One checks easily that there is a single point ∞ on this curve for which x0 = 0, that the complement of ∞ is isomorphic to V above and that the curve is smooth at ∞ as well. Now define a map X → Y by sending U to V by the above isomorphism and mapping P to ∞. The local rings OX,P and OY,P are then the same as subrings of K (being “the missing local rings from X and Y ”) and we get an isomorphism X ∼ =Y. We have thus proven that any elliptic curve is isomorphic to a cubic projective plane curve with an equation as above. Remark 2.15 In the above two examples we have constructed closed immersions of our curve X into some projective space. Quite generally, it is possible to find invertible sheaves on X using the Riemann-Roch formula from which one can construct closed embeddings by a similar method. See Hartshorne [1], Section IV.3 for details (over an algebraically closed field). In particular, any proper smooth curve is isomorphic to a projective curve; combining this with Corollary 2.11 we get that any smooth curve is either affine or projective.

3.. THE HURWITZ GENUS FORMULA

3.

185

The Hurwitz Genus Formula

In this section we prove a classical formula due to Hurwitz relating the different of a finite separable morphism φ : Y → X of smooth proper curves to the genera of the curves X and Y . Since both the different and the genus are defined using differentials, the existence of such a relation is by no means surprising. In order to derive the precise formula, we need some technical preliminaries. The first of these is the concept of the pullback of a quasi-coherent sheaf by a morphism, which we did not need up till now. As in the construction of sheaves of differentials in Chapter 6, we only define this concept for an affine morphism and refer the readers to Hartshorne [1], Section II.5 for the general case. ˜ a quasiConstruction 3.1 Let X = Spec A be an affine scheme and F = M coherent sheaf associated to an A-module M. Given a morphism φ : Y → X with Y = Spec B an affine scheme, we define the sheaf φ∗ F on Y as the quasi-coherent-sheaf associated to the B-module M ⊗A B. In general, given an affine morphism of arbitrary schemes φ : Y → X and a quasi-coherent sheaf F on X, we may find an affine open covering of X by subsets Ui = Spec Ai whose inverse images are of the form Spec Bi . By refining the covering of X if necessary, we may assume that over each Ui ˜ i with some Ai -module Mi and the definition the sheaf F is of the form M of quasi-coherent sheaves shows that over basic open subsets Uij contained ˜ i |U ∼ in the intersections Ui ∩ Uj there exist canoinical isomorphisms M ij = ∗ ˜ j |U . This shows that we may define a quasi-coherent sheaf φ F on Y by M ij performing the above construction over each Ui . One also checks by choosing a common refinement of two coverings of X that this definition does not depend on the covering chosen. Proposition 3.2 Let φ : Y → X be a finite separable morphism of smooth proper curves over a field k. Then we have an exact sequence λ

0 → φ∗ ΩX/k → ΩY /k → ΩY /X → 0 of invertible sheaves on Y . Proof: Choosing suitable affine open coverings of X and Y , it follows from Chapter 6, Lemma 3.5 (2) and the above construction applied to F = ΩX/k that we have an exact sequence as above except for the injectivity of λ. For this by the injectivity part of Chapter 6, Lemma 4. it is enough to check

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that the induced map λP is injective on all stalks at points P of Y . If P is the generic point, then by separability of φ we have that the stalk of ΩY /X at P is 0, so by the localisation property of differentials we have that that λP is a surjective morphism of 1-dimensional L-vector spaces (where L is the function field of Y ) and hence is an isomorphism. Assume now that λP has nontrivial kernel for some closed point P of X. As the OY,P -modules (φ∗ ΩX/k )P and (ΩY /k )P are free of rank 1 and λP is an OY,P -module homomorphism, this can only happen if λP =0 (look at the image of a generator), but then, again because the sheaves are invertible, we must have λ = 0 over an open subset U of Y . But since we have just seen that φ is ´etale over the generic point of X, by Chapter 7, Corollary ?? ΩY /X = 0 over a nonempty open subset V of Y and hence λ is surjective on V . Since Y is irreducible, we have U ∩ V 6= ∅ which is absurd. The next statement is essentially a reformulation of the above using the different. Proposition 3.3 In the situation of the previos proposition we have an isomorphism φ∗ ΩX/k ∼ = ΩY /k ⊗OY DY /X of invertible sheaves on Y . For the proof of the proposition we need an easy general statement about sheaves of modules. Lemma 3.4 Let Y be a scheme, L an invertible sheaf on Y and F an OY module whose stalk at each point is 0 except for finitely many closed points. Then the sheaf F ⊗OY L is isomorphic to F . Proof: Let P1 , . . . , Pn be the closed points where F has nontrivial stalk and for each Pi choose an open neighbourhood Ui over which L is trivial. Consider those open subsets V ⊂ Y for which either (i) V ⊂ Ui for some i or (ii) V does not contain any of the Pi . These subsets form a basis of open sets of Y , so by Chapter 5, Lemma 2.7 if we want to define a morphism φ : L ⊗OY F → F it is enough to define maps φV in a compatible way for V of the above type. Over a V of type (i) we take φV to be the composite of the isomorphisms L(V ) ⊗ F (V ) ∼ = OY (V ) ⊗ F (V ) ∼ = F (V ) coming from the trivialisation and over a V of type (ii) we define φV to be 0. It is straightforward to check that the φV so defined are compatible and thus define a φ we are looking for; by looking at the stalks we see moreover that φ is an isomorphism.

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Proof of Proposition 3.3: Tensoring the exact sequence of Proposition 3.2 by the dual Ω∨Y /k of ΩY /k and using the previous lemma we get a sequence Ω∨Y /k ⊗ φ∗ ΩX/k → OY → Ω∨Y /k ⊗ ΩY /X → 0. which is exact as one sees over a suitable open covering using the corresponding fact about tensor products of modules from Chapter 0. Furthermore, the sequence is also exact on the left, for as above we need only to check this on stalks where we are tensoring with a free module of rank one. So applying lemma 3.4 to the last term of the sequence we get an exact sequence 0 → Ω∨Y /k ⊗ φ∗ ΩX/k → OY → ΩY /X → 0. By definition the kernel of the surjection OY → ΩY /X is DY /X , whence the proposition by tensoring with ΩY /k . As a fairly easy consequence we get: Theorem 3.5 (Hurwitz Genus Formula) Let φ : Y → X be a finite separable morphism of smooth proper geometrically integral curves over a field k, of genus gY and gX , respectively. Let L|K be the corresponding extension of function fields. Then 2gY − 2 = [L : K](2gX − 2) − deg DY /X . If moreover φ is tamely ramified at all branch points Q (with image P in X), we have 2gY − 2 = [L : K](2gX − 2) +

X

(e(Q|P ) − 1)[κ(Q) : k].

Q

For the proof we need the following lemma. Lemma 3.6 In the situation of the theorem, for any invertible sheaf L on X we have deg φ∗ L = [L : K] deg L. Proof: The lemma follows using Chapter 7, Proposition 1.20 once we show P that writing L ∼ = L(D) with some divisor D = i mi Pi in Div(X), we have φ∗ L ∼ = L(

X i

mi

X

e(Q|Pi )Q).

Q∈YPi

To see this, observe that in Chapter 6, Section 4 we obtained the integer mi by embedding L into the constant sheaf K associated to the function field

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K and writing mi = vPi (f −1 ) with a generator f of the stalk LPi viewed as an element of K, where vPi is the valuation associated to the ring OX,Pi . Locally above Pi , taking the pullback of L by φ amounts to considering f as a function on Y via the embedding K ⊂ L. But for any point Q lying over Pi we have vQ (f −1 ) = e(Q|Pi )vPi (f −1 ). Proof of Theorem 3.5: The first formula follows by taking the degrees of the invertible sheaves that figure in the previous proposition and using the lemma. The second statement follows from the first by Dedekind’s Different Formula (Chapter 7, Theorem 5.5). The Hurwitz formula is very useful for computing genera of curves. A typical application is the following. Example 3.7 Let k be an algebraically closed field of characteristic 6= 2. Consider a smooth affine plane curve over k that can be defined by an equation of the form y 2 = (x − a1 ) . . . (x − a2m+1 ) with some integer m > 1 and all ai distinct. Making the usual substitution x = x0 /x2 , y = x1 /x2 we get a smooth projective plane curve Y which has a single point (0, 1, 0) at infinity. Such curves are called hyperelliptic. Now the projection to the x0 -axis is a finite morphism Y → P1k inducing a degree 2 extension of function fields. The branch points are the point at infinity and the points above the ai in the (x, y)-plane. Writing the Hurwitz formula gives 2gY − 2 = −4 + 2m + 2, i.e. gY = m. We now turn to applications to the theory of covers. It is well known from topology that the projective line and the affine line over C are simply connected. Using the Hurwitz formula, we can prove the following generalisation. Corollary 3.8 1. Any finite ´etale morphism φ : Y → P1k with Y a smooth proper geometrically integral curve is an isomorphism. 2. Let φ : Y → P1k be a finite morphism with Y a smooth proper geometrically integral curve. Assume that all of its branch points lie over the point at infinity and that φ is tamely ramified at these points. Then φ is an isomorphism. The tameness assumption in the second statement is essential: there are many non-trivial covers of P1k in positive characteristic that are ´etale over A1k and are wildly ramified at infinity.

3.. THE HURWITZ GENUS FORMULA

189

Proof: For the first statement, writing the Hurwitz formula for X = P1k and Y ´etale over X we get 2gY − 2 = −2[L : K] < 0. This can only happen for gY = 0 and L = K. For the second statement, we see by comparing Chapter 7, Proposition 1.20 with the Hurwitz formula in the tamely ramified case that the maximal contribution of the different can be [L : K] − 1 (in the case of a single branch point that is rational over k). Hence the right hand side of the formula can be at most −[L : K] − 1 < 0, which again only allows gY = 0 and L = K. Remark 3.9 The first part of the corollary implies that for k a perfect field the group π1 (P1k ) is isomorphic to Gal (k). This follows from results of the next section but the reader may check it as an exercise. A similar result holds for the so-called tame fundamental group of the affine line by the second statement. Another corollary is: Corollary 3.10 Any finite morphism φ : P1k → X with X a smooth proper geometrically integral curve over a field k is an isomorphism. Proof: Assume first that φ is separable. Then the Hurwitz formula implies that we must have −2 = [L : K](2gX − 2) − deg DP1 /X which is only possible for gX = 0 and L = K as deg DP1 /X is always non-positive. In general let K ⊂ L0 ⊂ L be the maximal separable subextension and write L (which is now the function field of P1k ) as k(x) with some function x. Then L = L0 (x) and we know from Chapter 7, Lemma ?? that the minimal n polynomial of x over L0 is of the form xp −a with some a ∈ L0 . In particular, n n L0 contains the field k(xp ) and since [k(x) : k(xp )] = [k(x) : L0 ] = pn we n actually have L0 = k(xp ). Thus L0 is a rational function field, so that the normalisation of X in L0 is isomorphic to P1k and we may conclude by the separable case. Remark 3.11 We have the following algebraic reformulation of the above corollary: any transcendental subextension of a purely transcendental extension k(x)|k is itself purely transcendental. This result is known as L¨ uroth’s theorem. For degree 2 transcendental extensions of C the analogous statement remains true but in degree 3 counterexamples have been found independently by Artin/Mumford, Clemens/Griffiths and Iskovskih/Manin in the beginning of the 70’s. We finish this section by considering the case k = C and relating the (second form) of the Hurwitz formula proven above to the topological formula

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(Chapter 4, Corollary 4.10) for branched covers of Riemann surfaces. As both formulae have the same shape (up to sign), we only have to prove: Proposition 3.12 Let φ : Y → X be a finite morphism of smooth proper curves over C. 1. If Q is a closed point of Y mapping to a closed point P of X, the ramification index e(Q|P ) is the same as the topological ramification index at Q defined in Chapter 4. 2. The genus gX of X (resp. gY of Y ) is equal to the topological genus of X (resp. Y) defined in Chapter 4. Proof: For the first statement, let tP be a generator of the maximal ideal of OX,P . Then φ induces a natural inclusion OX,P ⊂ OY,Q and by definition of the ramification index tP generates the e(Q|P )-th power of the maximal ideal of OY,Q , or in other words φ] tP has a zero of order e(Q|P ) at Q. But tP (resp. a generator tQ of the maximal ideal of OY,Q ) induces a complex chart of X around P (resp. Y around Q), which we denote by the same symbol. Then the above means that the holomorphic function tP ◦ φ ◦ t−1 Q has a zero of order e(Q|P ) at 0, which is the topological definition of the ramification index. For the second statement, note that because any complex curve admits a finite morphism onto P1C and because of the same shape of the formulae of Theorem 3.5 and Chapter IV, Corollary 4.10 it is enough to prove that P1C has genus 0 in both senses. This is tautological in the topological case and we have seen the algebro-geometric case in the previous section. Remark 3.13 One can give a more direct proof of the second statement above by using some difficult theorems about complex varieties. Indeed, one knows that for X a compact Riemann surface of genus g (in the topological sense), the singular cohomology group H 1 (X, C) is of dimension 2g over C. (One may prove this directly or deduce it from the structure of the fundamental group of X discussed in Chapter 4 via the Hurewicz isomorphism.) Then the Hodge decomposition theorem (Griffiths-Harris [1], p. 116) and Serre duality (ibid., p. 102 or Hartshorne [1], Corollary 7.13) imply that H 1 (X, C) decomposes as a direct sum of ΩX/C (X) and of its dual vector space, whence the result. This proof is not only more direct but is also more conceptual for it reduces the equality of the two notions of genera to fundamental relations between the topology and the differentiable structure of X.

4.. ABELIAN COVERS OF CURVES

4.

Abelian Covers of Curves

5.

Fundamental Groups of Curves

191

In this section we give an overview of some of the major results currently known about the structure of fundamental groups of smooth curves over algebraically closed fields. We begin by discussing smooth proper curves. To state the fundamental theorem, we first need to introduce some terminology: given a prime p, the maximal prime-to-p quotient (resp. maximal ˆ is defined as the inverse limit of all pro-p quotient) of a profinite group G finite quotients of π1 (X) of order prime to p (resp. of order a power of p). In fact, this construction can be done for any group G: the inverse limit of the natural inverse system of its finite quotients of order prime to p (resp. a power of p) is the profinite p0 -completion (resp. profinite p-completion) of G. We extend these definitions to p = 0 by declaring all integers prime to 0 (which is strange but useful). Theorem 5.1 (Grothendieck [1]) Let k be an algebraically closed field of characteristic p ≥ 0 and let X be a smooth proper connected curve of genus 0 g over k. Then π1 (X)(p ) is isomorphic to the profinite p0 -completion of the group Πg given by 2g generators a1 , b1 , . . . , ag , bg subject to the single relation [a1 , b1 ] . . . [ag , bg ] = 1. We only indicate the main steps of the proof of this difficult result. Step 1. First one proves the theorem for k = C using the topological result of Chapter 4, Theorem 4.3. To be able to apply this result, one uses the fact that any topological cover of X is in fact a smooth proper curve having the structure of an ´etale cover of X; this follows from Riemann’s Existence Theorem. Thus the algebraic fundamental group is the profinite completion of the topological one. Step 2. From this transcendental result one deduces the theorem for any algebraically closed field of characteristic 0 by applying a general theorem (to be stated precisely in the next section) according to which in characteristic 0 the algebraic fundamental group does not change by extensions of algebraically closed fields. This immediately applies to curves defined over algebraically closed subfields of C. But in fact any curve in characteristic zero can be defined over such a subfield: since its definition involves only finitely many coefficients of finitely many polynomial equations, the algebraic closure of the field generated over Q by these coefficients can be embedded in C as the latter has infinite transcendence degree over Q.

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Thus in characteristic 0 the theorem reduces to the topological result of Chapter 4. But to treat positive characteristic (which is the main contribution of Grothendieck) one needs further arguments. Step 3. One begins by showing that it is possible to “lift curves from positive characteristic to characteristic 0”. More precisely, one first shows that given any algebraically closed k of characteristic p > 0 there exists a discrete valuation ring A of with fraction field K and maximal ideal P such that K is of characteristic 0 and A/P ∼ = k. This is a classical algebraic construction (valid for any perfect k) and can be realised by using so called Witt vectors (see eg. Serre [1] or Chapter 20 of Mumford [2], due to G. Bergman). One then proves that there exists a scheme X → Spec A smooth and proper of relative dimension 1 over Spec A with closed fibre X (this amounts basically to the fact that the fibre over the closed point of Spec A is isomorphic to X and that over the generic point is a smooth proper curve). Intuitively, the generic fibre XK “reduces modulo P to X”. The existence of X is by no means easy to prove. Grothendieck originally used techniques of formal geometry to construct a so-called formal scheme of which he then deduced that it comes from an actual relative curve X . Loosely speaking, he first had a grasp on the completions of the local rings of X and only afterwards on the local rings themselves. Nowadays one would prefer to construct X by applying a form of Hensel’s Lemma to a suitable moduli stack parametrising smooth proper curves. But one then has to give a rigorous meaning to the preceding sentence which requires some hard work. In the notes of Popp [1] one finds an elementary version of this argument using only the classification of plane curves (which is elementary) but the price to be paid is to allow singular curves to enter the discussion. Step 4. The conclusion of the proof is then to prove the following so-called specialisation theorem: denoting by XK¯ the base-change of XK to an algebraic closure of K there is a natural homomorphism π1 (XK¯ ) → π1 (X) inducing an isomorphism on maximal prime-to-p quotients. The theorem then follows by applying the characteristic 0 result to XK¯ . The proof of the specialisation theorem is of more elementary nature than that of the previous step (it basically reposes on the so-called Zariski-Nagata purity theorem and Abhyankar’s Lemma) but still it requires some delicate arguments. We refer the reader to the original account in Grothendieck [1] or the excellent rendition of Orgogozo-Vidal in Bost/Loeser/Raynaud [1]. Now that we know the maximal prime-to-p quotient of π1 (X), we may ask the similar question about its maximal pro-p quotient; of course this question only arises in characteristic p > 0.

5.. FUNDAMENTAL GROUPS OF CURVES

193

Before stating the corresponding result, we need a definition: the p-rank of an abelian variety over an algebraically closed field is the dimension of the Fp -vector space given by the kernel of the multiplication-by-p map on A. Theorem 5.2 (Shafarevich) Let X be a smooth proper curve over an algebraically closed field of characteristic p > 0. Then π1 (X)(p) is a free profinite group whose rank equals the p-rank of the jacobian variety of X. One may infer that the previous two theorems elucidate the structure of fundamental groups of smooth proper curves over algebraically closed fields. This is indeed the case in characteristic 0 but in positive characteristic even if we know the maximal pro-p and prime-to-p quotients, the structure of the group itself remains a mystery. The theorems give, however, a good description of its maximal abelian quotient: this group is the direct sum of its maximal prime-to-p and pro-p quotients and hence the previous two theorems together suffice to describe it. To emphasize further the difference between the characteristic 0 and characteristic p > 0 case, let us mention a striking recent result of Tamagawa’s (building upon earlier work by Raynaud, Pop and Sa¨ıdi). Observe that in characteristic 0 the fundamental group is completely determined by the topology of the underlying complex curve and hence by the genus. Thus there are many curves having the same fundamental group. However, the positive characteristic case turns out to be completely different: Theorem 5.3 (Tamagawa) Let k be an algebraically closed field of characteristic p > 0 and let G be a profinite group. Then there are only finitely many smooth proper curves of genus g ≥ 2 over k whose profinite group is isomorphic to G. Of course, the theorem is interesting only for those G which actually arise as the fundamental group of some curve as above. We leave it to the readers as an easy exercise to treat the cases g = 0, 1. Now we turn to affine curves. It turns out that Grothendieck’s method for proving Theorem 5.1 can be refined to handle this case as well. Thereofre one has: Theorem 5.4 (Grothendieck [1]) Let k be an algebraically closed field of characteristic p ≥ 0 and let X be a smooth affine connected curve over k arising from a smooth proper curve X 0 ⊃ X of genus g by deleting n closed points.

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Then π1 (X)(p ) is isomorphic to the profinite p0 -completion of the group Πg given by 2g + n generators a1 , b1 , . . . , ag , bg , γ1, . . . , γn subject to the single relation [a1 , b1 ] . . . [ag , bg ]γ1 . . . γn = 1. Again the natural question about the maximal pro-p-quotient arises. About this no such definitive result is known as in the proper case, but the following important theorem, previously known as Abhyankar’s Conjecture, shows that any finite p-group arises as its quotient. Before stating it, let us intoduce some notation: for a finite group G denote by p(G) the (normal) subgroup generated by its p-Sylow subgroups. Hence G/p(G) is the maximal prime-to-p quotient of G. Theorem 5.5 (Raynaud [1], Harbater [2]) Let k be algebraically closed of characteristic p > 0 and let X be a smooth affine connected curve over k arising from a smooth proper curve X 0 ⊃ X of genus g by deleting n points. Then any group G for which G/p(G) can be generated by 2g + n − 1 elements arises as a quotient of π1 (X). Raynaud proved the crucial case X = A1k by explicitly constructing Galois covers using two radically different methods, based respectively on techniques from rigid analytic geometry and the theory of semi-stable curves. The proof also uses some previous results of Serre [5]. Harbater then observed how to reduce the general case to the case of the affine line; this reduction was later simplified by Pop [1]. See also the chapters by Chambert-Loir and Sa¨ıdi in Bost/Loeser/Raynaud [1].

Chapter 9 The Algebraic Fundamental Group 1.

The General Notion of the Fundamental Group

First we give the general definition of finite ´etale morphisms. Definition 1.1 A finite morphism φ : X → S of schemes is ´etale if each fibre XP of φ is the spectrum of a finite ´etale κ(P )-algebra and if the direct image sheaf φ∗ OX is locally free (of finite rank). We shall speak of finite ´etale covers if in addition φ is surjective. In the case when Y is a Dedekind scheme, we have seen in Setion 8 of Chapter 5 that the second condition is automatically satisfied, so this definition is a generalisation of the previous one. As immediate consequences of this definition, we get that any composition of finite ´etale morphisms is ´etale and that any base change of a finite ´etale morphism is again finite ´etale. Less immediate consequences are: Lemma 1.2 Let φ : X → S be a finite ´etale morphism. 1. The image of φ is clopen. 2. The image of any section s of φ (i.e. of a morphism s : S → X with φ ◦ s the identity of S) is clopen. 3. The image of the diagonal morphism X → X ×S X (corresponding to the identity maps of X the two components) is clopen. 195

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We omit the proof. Corollary 1.3 If Z → S is a connected S-scheme and f, g : Z → X are two S-morphisms into a finite ´etale S-scheme which coincide at some point of Z, then φ1 = φ2 . Here the term “coincide” should be understood in the scheme-theoretic sense, i.e. we require not only that there should be a point P ∈ Z such that f (P ) = g(P ) = Q ∈ X, but also that the maps κ(Q) → κ(P ) induced by f and g should be the same. Proof: By taking the fibre product of Z and Y over S and using the base change property of ´etale morphisms we may assume S = Z. Then we have to prove that if two sections of a finite ´etale cover X of a connected scheme S coincide at a point, then they are equal. This follows from the second statement of the lemma, for each such section being injective, it must be an isomorphism of S onto a connected component of X and hence it is determined by the image of a point. As in topology, this implies: Corollary 1.4 Any S-automorphism of a connected finite ´etale cover X → S acts without fixed points. Hence the automorphism group of a connected finite ´etale cover is finite. We have the following statement essentially proved in Chapter 5, Section 7: Lemma 1.5 If S is a locally noetherian scheme, φ : X → S is a finite ´etale cover and G is a finite group of S-automorphisms of X, then the quotient X/G is an S-scheme finite and ´etale over S. After these preparations, we may state the main theorem of this section. First some notations and terminology. A geometric point of a scheme S is a morphism s¯ : Spec Ω → S, where Ω is some algebraically closed field. The topological image of s¯ is a point s of S such that Ω is an algebraically closed extension of κ(s). Now consider the category Fet|S of schemes finite and ´etale over S, with morphisms the S-morphisms as usual, and fix a geometric point s¯ : Spec Ω → S. For any object X → S of Fet|S we may define its geometric fibre Fibs¯(X) over s¯ as the fibre product X ×Spec Ω S induced by s¯. Regarding F ibs¯(X) only as a set, we get a set-valued functor Fibs¯ on Fet|S called the fibre functor.

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Theorem 1.6 Assume further S is connected and locally noetherian. Then there is a profinite group π1 (S, s¯) such that for each X finite and ´etale over S the set Fibs¯(X) is canonically equipped with a continuous left action of π1 (S, s¯). This action is transitive if and only if X is connected. Moreover, the functor Fibs¯ induces an equivalence of Fet|S with the category of finite continuous left π1 (S, s¯)-sets. Definition 1.7 The group π1 (S, s¯) in the theorem is called the (algebraic) fundamental group of the pair (S, s¯). Remarks 1.8 1. There is an obvious analogy with the theory of covers of a connected and locally simply connected topological space. If we restrict to covers with finite fibres only, then the analogy is even more explicit: the category of these is easily seen to be equivalent to the category of finite sets equipped with a continuous action of the profinite completion of the fundamental group. (Note, however, an analogue of the above theorem can be proved by the method we shall employ for finite covers of any topological space.) 2. The theorem contains as a special case the case S = Spec k, for k a field. Here any finite ´etale S-scheme is the spectrum of a finite ´etale k-algebra. For a geometric point s¯ the fibre functor is given by Hom(Spec k s , X), where k s is the separable closure of k in Ω via the embedding given by s¯ and π1 (S, s¯) = Gal (k s |k). Note that this does not mean that the fibre functor is representable, for k s is not a finite ´etale k-algebra. However, its spectrum is the inverse limit of finite (Galois) subextensions which already are. This motivates the following definition. Definition 1.9 Let C be a category and F a set-valued functor on C. We say that F is strictly pro-representable if there exist • an inverse system of objects Pi indexed by a directed index set I; • elements pi ∈ F (Pi ) for each i ∈ I such that for each i ≤ j the element pj is mapped to pi by the induced map F (Pj ) → F (Pi ); • a functorial isomorphism lim Hom(Pi , X) ∼ = F (X) for each object X → induced by mapping a morphism φ : Pi → X to the image of pi by F (φ). (Notice that the inverse limit of the Pi may not exist but the direct limit of their Hom’s does in the category of sets.)

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Thus in the last example the fibre functor is strictly prorepresentable by the inverse system of spectra of finite (Galois) extensions conatined in the fixed separable closure. We shall prove that quite generally the fibre functor is strictly pro-representable, from which the theorem will follow by formal arguments. As in topology, we define a connected finite ´etale cover X → S to be Galois if its S-automorphism group acts transitively on the fibres. Now the key lemma is the generalisation of the well-known fact that any finite separable field extension can be embedded in a finite Galois extension and there is a smallest such extension, namely the Galois closure. Lemma 1.10 Let φ : X → S be a connected finite ´etale cover. Consider the full subcategory Gal|S of Fet|S spanned by Galois covers. Then the setvalued functor on Gal|S which maps a Galois cover Y → S to HomS (Y, X) is representable. In other words, there is a Galois cover P → S eqipped with an Smorphism π : P → X such that any S-morphism from a Galois cover to X factors through p. Proof: Let Fibs¯(X) be the finite set F = {¯ x1 , . . . , x¯n }. The x¯i induce a canonical geometric point x¯ of the n-fold fibre product X n = X ×S . . . ×S X giving x¯i in the i-th component. Let P be the connected component of X n containing the image of x¯ and let π : P → X be the map induced by the first projection of X n to X. Using the composition and base change properties of X we see that P becomes an object of Fet|S via φ ◦ π. Notice that P has the property that any point in Fibs¯(P ) can be represented by an n-tuple (¯ xσ(1) , . . . , x¯σ(n) ) for some permutation σ ∈ Sn . Indeed, n any point of Fibs¯(X )) corresponds to an element of F n so we only have to show that those points which are concentrated on P have distinct coordinates. But by the third statement of Lemma 1.2, if ∆ denotes the diagonal image of X in X ×S X, its inverse image πij−1 (∆) by any of the projections πij mapping X n to two of its components is a clopen subset. Since P is connected, πij−1 (∆) ∩ P 6= ∅ would imply πij−1 (∆) ⊃ P , which is impossible for x¯ hits X n away from any of the πij−1 (∆). Now to show that P is Galois over S, simply remark that any permutation σ ∈ Sn induces an S-automorphism φσ of X n by permuting the components. If φσ ◦ x¯ ∈ Fibs¯(P ), then φσ (P )∩P 6= ∅ and so φσ ∈ Aut(P/S) by connectedness of P . So Aut(P/S) acts transitively on one geometric fibre, from which we conclude as in topology (using Lemma 1.5) that P is Galois.

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Now if q : Q → X is an S-morphism with Q Galois, choose a preimage y¯ of x¯1 . By composing with appropriate elements of Aut(Q/S) we get n maps q = q1 , . . . , qn : Q → X such that qi ◦ y¯ = x¯i . Whence an S-morphism Q → X n which factors through P for it maps y¯ to x¯ and Q is connected. This concludes the proof. Proposition 1.11 Under the assumptions of the theorem, the fibre functor Fibs¯ is strictly prorepresentable. Proof: Take the index set I to be the set of all finite ´etale Galois covers Pi → S and define Pi ≤ Pj if there is a morphism Pj → Pi . This partially ordered set is directed (apply the previous lemma to any connected component of a fibre product Pi ×S Pj ). For each Pi ∈ I fix an arbitrary element pi ∈ FibS¯ (Pi ). Then since the objects are Galois, if there exists an S-map Pj → Pi then there is also an S-map mapping pj to pi ; such a map is even unique by Corollary 1.3. So we take the inverse system formed by the Pi of the above particular maps. To conclude the proof we only have to construct a functorial map Fibs¯(X) → lim Hom(Pi , X) inverse to the natural map. For → this, we may assume X is connected (otherwise take disjoint unions). Given an element x¯ ∈ Fibs¯(X), consider the Galois closure π : P → X given by the previous lemma. Since P is Galois, we may assume that F (π) maps the distinguished element p ∈ F (P ) to x¯. Now mapping x¯ to the class of p in lim Hom(Pi , X) is easily seen to be a good choice. →

Now to prove the theorem, define π1 (S, s¯) to be the opposite of the automorphism group of the inverse system constructed in the above proof. By definition, this automorphism group is the inverse limit of the Aut(Pi ) by the maps induced by those in the inverse system; it is profinite by Corollary 1.4. Finally, the opposite group simply means that we change the multiplication law from (x, y) 7→ xy to (x, y) 7→ yx; this ensures that if we agree that automorphisms of finite ´etale covers act from the left, then π1 (S, s¯) also acts from the left on the geometric fibres. The statements in the theorem are then proved as in the special case of fields, if we replace the arguments which involve taking the fixed field of an open subgroup of the Galois group by taking first an open normal subgroup V contained in a given open subgroup U of π1 (S, s¯) and then taking the appropriate quotient of the Galois cover corresponding to V . Remark 1.12 It can be checked by looking at the construction that, just as in the topological case, changing the base point induces an isomorphism of the corresponding fundamental groups. This isomorphism is unique up to an inner automorphism of one of the groups in question.

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Just as the topological fundamental group, the algebraic fundamental group is a functor on the category of pointed schemes. Proposition 1.13 Let φ : (S 0 , s¯0 ) → (S, s¯) be a map of pointed schemes, i.e. a morphism of schemes with φ ◦ s¯0 = s¯. 1. The map φ induces a canonical functor Fet|S → Fet|S 0 by mapping an object X → S to the fibre product X ×S S 0 → S 0 and a canonical morphism φ∗ : π1 (S 0 , s¯0 ) → π1 (S, s) of profinite groups. 2. If any finite ´etale cover X 0 → S is of the form X ×S S 0 , with X → S a finite ´etale cover, then φ∗ is injective. 3. The map φ∗ is surjective if and only if for any connected finite ´etale cover X → S the cover X ×S S 0 is connected as well. Proof: To show the existence of φ∗ , it will suffice to construct a compatible system of homomorphisms π1 (S/, s¯0 ) → AutS (Pi )op for each Pi ∈ i, where the Pi prorepresent the fibre functor associated to (S, s¯). Now to give an S-automorphism of Pi is the same according to the theorem as giving an automorphism of the π1 (S, s¯)-set Fibs¯(Pi ). But by definition of the fibre product we have a bijection of sets Fibs¯(Pi ) ∼ = Fibs¯0 (Pi ×S S 0 ), so via its action on the right hand side each element of π1 (S 0 s¯0 ) induces an automorphism of the left hand side set. To see that it commutes with the action of elements of π1 (S, s¯) notice first that each such element comes from an Sautomorphism of Pi . Whence an S 0 -automorphism of Pi ×S S 0 , inducing via the theorem an automorphism of Fibs¯0 (Pi ×S S 0 ) commuting with the action of π1 (S 0 , s¯0 ). This is what we wanted to see. The second statement follows from the observation, implied by the theorem, that an element of π1 (S, s¯) is the identity if and only if it acts trivially on the geometric fibres of each finite ´etale cover of S. For the third, use the fact that connected covers map via the fibre functor to sets with transitive π1 (S, s¯)-action. This immediately gives one implication. For the converse, notice that if connected covers pull back to connected covers, then Galois covers pull back to Galois covers, so that each automorphism of a Galois cover of S comes from a corresponding one over S 0 .

2.

The Outer Galois Action

If k is a field and X is a (locally noetherian connected) scheme equipped with a morphism X → Spec k, the functoriality of the fundamental group

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induces a map π1 (X, x¯) → Gal (k) once a geometric point x¯ is chosen. For ¯ = an algebraic closure k¯ of k we also dispose of a geometric point of X ¯ x X ×Spec k Spec k¯ induced by x¯, whence another map π1 (X, ¯) → π1 (X, x¯). These two maps are related by the following important proposition. Proposition 2.1 Assume X is a noetherian and geometrically connected ¯ is conscheme over a perfect field k. (The latter condition means that X nected.) Then once a geometric point x¯ is chosen, we have an exact sequence of profinite groups ¯ x 1 → π1 (X, ¯) → π1 (X, x¯) → Gal (k) → 1. The proposition also holds over non-perfect fields. For this one shows that the groups under consideration are not affected by purely inseparable extensions of the base field. For the proof we need a lemma. ¯ arises by base change to k¯ from Lemma 2.2 Any finite ´etale cover Y → X a cover YL → XL , where L is a suitable finite extension of k. The same holds for automorphisms of covers. Proof: We prove the first statement, the proof of the second being sim¯ is compact by assumption, it has a finite covering by open ilar. Since X subschemes Ui = Spec Ai such that the restrictions Y Ui of the cover Y to each Ui are defined by finitely many equations with coefficients in Ai ; these generate subalgebras A0i ⊂ Ai which in turn arise as quotients of some poly¯ 1 , . . . , xm ] by an ideal generated by finitely many polynomials. nomial ring k[x The coefficients of all these polynomials are contained in some finite Galois extension L|k. Similarly, one sees that the isomorphisms showing the compatibility of the Y Ui over the overlaps Ui ∩ Uj can be defined by equations involving only finitely many coefficients so we conclude by taking a suitably large L. Proof of the proposition: Surjectivity of the map π1 (X, x¯) → Gal (k) follows immediately from the criterion of the previous proposition. Indeed, any connected finite ´etale cover of k is the spectrum of a finite separable extension L|k, so its pullback is XL = X ×Spec k Spec L which is connected ¯ by the assumption, being the continuous image of X. ¯ Next observe that π1 (X, x ¯) is isomorphic to the projective limit of the natural projective system formed by the groups π1 (XL , x¯) with L running over the set of finite Galois extensions of k. Indeed, there are natural compatible ¯ x¯) → π1 (XL , x maps π1 (X, ¯) induced by functoriality, whence a map in the ¯ x¯) can limit. To show its injectivity, notice that a nontrival element of π1 (X,

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¯ be represented by a nontrivial automorphism of some Galois cover P → X and by the lemma this automorphism must come from an automorphism of some cover PL → XL for L large enough. For surjectivity, one represents an element of the inverse limit by a compatible system of automorphisms of Galois covers Pi,L → XL . Keeping those terms which correspond to covers ¯ we get a compatible system of remaining connected after base change to k, ¯ by base change, whence an element automorphisms of Galois covers P → X ¯ x¯). of π1 (X, On the other hand, for each finite Galois extension L|k there is an exact sequence of topological groups 1 → π1 (XL , x¯) → π1 (X, x¯) → Aut(XL /X)op → 1. Indeed, XL → X is a Galois cover, so the opposite of its automorphism group is naturally a quotient of π1 (X, x¯); moreover, one sees using Lemma 1.10 that the open kernel of the corresponding projection is exactly π1 (XL , x¯). Note that here the map π1 (X, x¯) → Aut(XL /X)op factors as a composite π1 (X, x¯) → Gal (k) → Gal (L|k) → Aut(XL /X)op where the first two maps are the natural projections and the third one maps an element of Gal (L|k) to the automorphism of XL it induces via its action on L; this is an injective map. Now passing to the projective limit and using the observation of the previous paragraph we get an exact sequence ¯ x¯) → π1 (X, x¯) → lim Aut(XL /X)op , 1 → π1 (X, ← which immediately yields that the sequence of the proposition is exact on the left. Exactness in the middle follows from the fact that the last map in the above exact sequence factors through an injective map Gal (k) → lim Aut(XL /X)op by the above remark. ←

The group π1 (X, x¯) is particularly interesting in the case when k ⊂ C for then the following results apply. Theorem 2.3 Let K|k be an extension of algebraically closed fields, X a noetherian connected k-scheme. Put XK = X ×Spec k Spec K as above and choose a geometric point x¯ of X. Then the natural map π1 (XK , x¯) → π1 (X, x¯) is an isomorphism in each of the following cases: 1. (Serre-Lang) X is proper over k (eg. X is projective); 2. X is a smooth curve and k is of characteristic 0.

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In fact, the second case can be deduced from the first using Abhyankar’s lemma. Theorem 2.4 The algebraic fundamental group of X a scheme of finite type over C is isomorphic to the profinite completion of the associated complex analytic space. We shall not define the general notion of a complex analytic case. In the case of a smooth proper curve, it is the associated compact Riemann surface. It is also a Riemann surface in the case of an affine curve: choose a smooth compactification and delete the extra points from the associated Riemann surface. Thus we see that in these cases the theorem follows if one knows that any finite (branched) topological cover of a compact Riemann surface is in fact algebraic; this can be deduced from Riemann’s existence theorem. All in all, we conclude from the above: Corollary 2.5 In the case of a smooth curve over a subfield of C, the algebraic fundamental group is an extension of the absolute Galois group of the base field by the profinite completion of the topological fundamental group. Remark 2.6 The latter group is known from topology: if the associated Riemann surface is topologically a torus with g holes and there are n deleted points, then the topological fundamental group has the presentation < a1 , b1 , . . . , ag , bg , γ1 , . . . , γn |[a1 , b1 ] . . . [ag , bg ]γ1 . . . γn = 1 >, the [, ] meaning commutators. For g = 0 and n ≥ 3 this is the free group on n−1 generators. For g ≥ 2 it has a quotient isomorphic to the free group on 2 generators. Since by the classification of finite simple groups any such group can be generated by 2 elements, they are all quotients of the topological, and hence the algebraic, fundamental group. So in these cases we get a group that is monstruously big. Now returning to the situation of Proposition 2.1, observe that the group ¯ x π1 (X, x¯) acts on its closed normal subgroup π1 (X, ¯) by conjugation, whence ¯ x¯)). The restriction of this map to a representation π1 (X, x¯) → Aut(π1 (X, ¯ x¯) is a homomorphism π1 (X, ¯ x¯) → Inn (π1 (X, ¯ x¯)), where the latter the π1 (X, group denotes the group of inner automorphisms. Thus by passing to the ¯ x¯)), where the quotient we get a representation ρ : Gal (k) → Out (π1 (X, ¯ x¯) defined simply latter group is the group of outer automorphisms of π1 (X, ¯ x¯)) by its subgroup Inn (π1 (X, ¯ x¯)). If k ⊂ C, as the quotient of Aut(π1 (X, this representation thus maps Gal (k) into the outer automorphism group of a “transcendental object”. So the following fact may be surprising.

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Theorem 2.7 (Belyi, 1979) Consider the projective line P1Q over the field of rational numbers equipped with three marked Q-rational points 0, 1 and ∞. Then for any geometric point x¯ away from the marked ones the representation ρ : Gal (Q) → Out (π1 (P1 \ {0, 1, ∞}, x ¯)) introduced above is faithful (i.e. has trivial kernel). Why the particular case of the projective line minus three points is so interesting is explained by the following proposition, also due to Belyi (which in fact will be the main ingredient of the proof). ¯ Proposition 2.8 Let X be a smooth proper algebraic curve defined over Q. Then there is a morphism X → P1 ´etale over P1 \ {0, 1, ∞}. Thus the fundamental group of the projective line minus three points classifies all smooth proper curves defineable over an algebraic number field (but the isomorphism class of a curve may be counted several times). This fact is so surprising that Grothendieck didn’t even venture to conjecture it. Note also that by virtue of Theorem 2.3 the converse is also true in the following form: any ramified cover of the complex projective line that is ´etale ¯ outside 0, 1, ∞ can be defined over Q. ¯ and Proof: In any case there is is a morphism p : X → P1 defined over Q ´etale above the complement of a finite set S of closed points (any rational function induces such a map, as we well know). In a first step, we reduce to the case where S consists of points defined over Q. For this, let P be a point for which the degree n = [κ(P ) : Q] is maximal among the points in S. Choose a minimal polynomial f for a generator of the extension κ(P )|Q and consider the map φf : P1 → P1 attached to f . Since f has coefficients in Q, this map is defined over Q. Now the composite φf ◦ p is ´etale outside the set S 0 = φf (S) ∪ {∞} ∪ φf (Sf ) where Sf is the branch locus of φf , i.e. the support of Div 0 (f 0 ). Now ∞ has degree 1 over Q; the points of Sf , and hence of φf (Sf ) have degree at most n − 1; finally, those in φf (S) has degree at most n. But since φf (P ) = 0, there are strictly less points of degree exactly n in S 0 than in S. Replacing p by φf ◦ p and S by S 0 we may continue this procedure until we arrive at n = 1. So assume all points in S are defined over Q. If S consists of at most three points, we are done by composing with an automorphism of P1 ; otherwise we may assume S contains 0, 1, ∞ and at least one more rational point α. The idea again is to compose p by a map φf : P1 → P1 associated to a

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well-chosen polynomial f . This time we seek f in the form z A (z − 1)B . Then φf will map both 0 and 1 to 0 and fix ∞. However, φf ◦ p will be also ramified over the images of those points where the derivative f 0 vanishes. These are given by the equation Az A−1 (z − 1)B + Bz A (z − 1)B−1 = 0, or else A(z − 1) + Bz = 0 which gives z = A/(A + B). So if we choose A and B such that α = A/(A + B), then φf ◦ p will be ramified over at least one (Q-rational) point less than p and we may continue the procedure. Proof of Theorem 2.8: We use the shorthand X for P1 \{0, 1, ∞}. Assume that ρ has a nontrivial kernel, fixing a (possibly infinite) extension L|k. Then the representation ρL : Gal (L) → Out (π1 (XL , x ¯))) is trivial which means ¯ x that any automorphism of π1 (X, ¯) induced by conjugating with an element ¯ x¯). This implies x ∈ π1 (XL , x¯) is in fact conjugation by an element y ∈ π1 (X, ¯ x¯) in π1 (XL , x¯), so that the latter that y −1 x is in the centraliser C of π1 (X, ¯ x¯). But as we have remarked above, in group is generated by C and π1 (X, ¯ this case π1 (X, x ¯) is isomorphic to the profinite completion of the free group on two generators which is known to have a trivial center. Hence C and ¯ x π1 (X, ¯) have trivial intersection, which implies that π1 (XL , x ¯) is actually ¯ x their direct product. Whence a quotient map π1 (XL , x¯) → π1 (X, ¯) which gives the identity by composing with the natural inclusion in the reverse ¯ x¯)-sets we get that any direction. Applying this to finite continuous π1 (X, ¯ comes by base change from a cover of XL . Combining finite ´etale cover of X this with the above proposition implies that any smooth proper curve defined ¯ can in fact be defined over L. But there are counter-examples to this over Q latter assertion even for curves of genus 1 (take an elliptic curve with jinvariant an algebraic number outside L).