Notes on Galois Theory, Riemann Surfaces and Schemes (T. Szamuely)

Contents

0 Preliminaries 31. Some Abstract Nonsense . . . . . . . . . . . . . . . . . . . . . 32. Representable Functors . . . . . . . . . . . . . . . . . . . . . . 83. Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 104. Background from Topology . . . . . . . . . . . . . . . . . . . . 15

1 Galois Theory of Fields 191. A Review of Classical Galois Theory . . . . . . . . . . . . . . 192. Infinite Galois Extensions . . . . . . . . . . . . . . . . . . . . 233. Finite Etale Algebras . . . . . . . . . . . . . . . . . . . . . . . 30

2 Fundamental Groups in Topology 351. Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362. Group Actions and Galois Covers . . . . . . . . . . . . . . . . 373. The Main Theorems of Galois Theory for Covers . . . . . . . . 434. Construction of the Universal Cover . . . . . . . . . . . . . . . 49

3 Locally Constant Sheaves 551. Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552. Locally Constant Sheaves and Their Classification . . . . . . . 583. Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Riemann Surfaces 671. Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 672. Branched covers of Riemann Surfaces . . . . . . . . . . . . . . 723. Relation with Field Theory . . . . . . . . . . . . . . . . . . . 764. Topology of Riemann Surfaces . . . . . . . . . . . . . . . . . . 86

5 Enter Schemes 911. Prime Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 912. Schemes – Mostly Affine . . . . . . . . . . . . . . . . . . . . . 93

1

2 CONTENTS

3. First Examples of Schemes . . . . . . . . . . . . . . . . . . . . 1034. Quasi-coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . 1065. Fibres of a Morphism . . . . . . . . . . . . . . . . . . . . . . . 1096. Special Properties of Schemes . . . . . . . . . . . . . . . . . . 113

6 Dedekind Schemes 1191. Integral Extensions . . . . . . . . . . . . . . . . . . . . . . . . 1192. Dedekind Schemes . . . . . . . . . . . . . . . . . . . . . . . . 1223. Modules and Sheaves of Differentials . . . . . . . . . . . . . . 1284. Invertible Sheaves on Dedekind Schemes . . . . . . . . . . . . 136

7 Finite Covers of Dedekind Schemes 1451. Local Behaviour of Finite Morphisms . . . . . . . . . . . . . . 1452. Fundamental Groups of Dedekind Schemes . . . . . . . . . . . 1523. Galois Branched Covers and Henselisation . . . . . . . . . . . 1584. Henselian Discrete Valuation Rings . . . . . . . . . . . . . . . 1625. Dedekind’s Different Formula . . . . . . . . . . . . . . . . . . 168

8 Finite Covers of Algebraic Curves 1731. Smooth Proper Curves . . . . . . . . . . . . . . . . . . . . . . 1732. Invertible Sheaves on Curves . . . . . . . . . . . . . . . . . . . 1793. The Hurwitz Genus Formula . . . . . . . . . . . . . . . . . . . 1854. Abelian Covers of Curves . . . . . . . . . . . . . . . . . . . . . 1915. Fundamental Groups of Curves . . . . . . . . . . . . . . . . . 191

9 The Algebraic Fundamental Group 1951. The General Notion of the Fundamental Group . . . . . . . . 1952. 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 chapteronly if she finds it necessary; one notable exception might be the section onrepresentable 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 ofcategory theory, the theory which has been affectionately termed “abstractnonsense” by Steenrod who was one of its founding fathers. At first somereaders may find its use appalling but we hope that in the long run theywill appreciate its elegance and efficacity. The present section assembles thedefinitions and theorems that will emerge and re-emerge from now on.

Definition 1.1 A category consists of objects as well as morphisms betweenpairs of objects; given two objects A, B of a category C, the morphisms fromA to B form a set, denoted by Hom(A,B). (Notice that in contrast to thiswe do not impose that the objects of the category form a set.) These aresubject to the following constraints.

1. For any object A, the set Hom(A,A) should contain a distinguishedelement idA, the identity morphism of A.

2. Given two morphims φ ∈ Hom(B,C) and ψ ∈ Hom(A,B), there shouldexist a canonical morphism φ ψ ∈ Hom(A,C), the composition ofφ and ψ. The composition of morphisms should satisfy two naturalaxioms:

• 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 isomorphismif there exists ψ ∈ Hom(B,A) with ψ φ = idA, φ ψ = idB; we denotethe set of isomorphisms between A and B by Isom(A,B). If the objectsthemselves form a set, one can associate an oriented graph to the category bytaking objects as vertices and defining an oriented edge between two objectscorresponding to each morphism. With this picture in mind, it is easy toconceive what the opposite category Cop of a category C is: it is “the categorywith 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 andHom(B,A) of Cop preserving the identity morphisms and composition.

A subcategory of a category C is just a category D consisting of someobjects and some morphisms of C; it is a full subcategory if given two objectsin D, HomD(A,B) = HomC(A,B), i.e. all C-morphisms between A and Bare morphisms in D.

Examples 1.2 Some categories we shall frequently encounter in the sequelwill be the category Sets of sets (with morphisms the set-theoretic maps), thecategory Ab of abelian groups (with group homomorphisms) or the categoryTop of topological spaces (with continuous maps). Both Ab and Top arenaturally subcategories of Sets but they are not full subcategories; on theother hand, Ab is a full subcategory of the category Groups of all groups.

Here we really take “all” sets, (abelian) groups or topological spaces asobjects of our category; for example, the one-element set consisting of the au-thor and the one-element set consisting of you, dear reader represent differentobjects of Sets even if they are isomorphic. This poses serious set-theoretical(not to say philosophical) problems; to see how they are circumvented, onemay 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 morphismsHom(A,B) → Hom(F (A), F (B)) which sends identity morphisms to identitymorphisms and preserves composition. A contravariant functor from C1 toC2 is a functor from C1 to Cop

2 .

Examples 1.4 Here are some examples of functors.

1. The “most famous” functor is the identity functor idC of any categoryC which leaves all objects and morphisms fixed.

1.. SOME ABSTRACT NONSENSE 5

2. Other basic examples of functors are obtained by fixing an object A ofa category C and considering the covariant functor Hom(A, ) (resp.the contravariant functor Hom( , A)) from C to the category Setswhich 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 composingwith φ.

3. An example of a functor whose definition involves “real mathematics”and not just abstract nonsense is given by the set-valued functor onthe category Top which sends a space into its set of connected com-ponents. Here to see that this is really a functor one has to use thefact that a continuous map between topological spaces sends connectedcomponents to connected components.

Definition 1.5 If F and G are two functors with same domain C1 and targetC2, a morphism of functors Φ between F and G is a collection of morphismsΦA : F (A) → G(A) in C2 for each object A ∈ C1 such that for any morphismφ : A→ B in C1 the diagram

F (A)ΦA−−−→ G(A)

F (φ)

yyG(φ)

F (B)ΦB−−−→ 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” isfrequently used for what we call a morphism of functors. However, we preferthe 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 morphismsof functors. Here the composition rule for some Φ and Ψ is induced by thecomposition of the morphisms ΦA and ΨA for each object A in C1.

Disposing of the definition of an isomorphism between functors, we cannow 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 func-tors F : C1 → C2 and G : C2 → C1 such that F G ∼= idC2 and G F ∼= idC1 .They are anti-equivalent if C1 is equivalent to Cop

2 .


One sees that equivalence of categories has all properties that equivalencerelations on sets have, i.e. it is reflexive, symmetric and transitive. Also,the seemingly asymmetric definition of anti-equivalence is readily seen to besymmetric.

In practice, when one has to establish an equivalence of categories it oftenturns out that the construnction of one functor is easy but that of the onein the reverse direction is rather cumbersome. The following general lemmaenables us to make do with the construction of only one functor in concretesituations. Before stating it, we introduce some terminology.

Definition 1.8 A functor F : C1 → C2 is faithful if for any two objects Aand B of C1 the map of sets FAB : Hom(A,B) → Hom(F (A), F (B)) inducedby 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 isomorphicto some object of the form F (A).

Lemma 1.9 Two categories C1 and C2 are equivalent if and only if thereexists a functor F : C1 → C2 which is fully faithful and essentially surjective.

There is an analogous characterisation of anti-equivalent categories withfully faithful and essentially surjective contravariant functors (defined in theobviuos 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 defineG : C2 → C1 by setting G(V ) = A and G(φ) = F−1

AB(i−1W φ iV ) for

φ ∈ Hom(V,W ), where FAB is the bijection appearing in the definition offully faithfulness. We have G F = idC1 by the particular choice of iF (A). Toconstruct 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 → Wshows that the isomorphism is functorial.

For necessity, assume there exist a functor G : C2 → C1 and isomorphismsof 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. Forfully faithfulness fix any two objects A, B of C1 and consider the sequence ofmaps

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 theidentity 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 abovesequence gives φ = ψ, i.e. F is a faithful functor. Since the situation issymmetric in F and G, we conclude that G is faithful as well. We stillhave to show that any λ ∈ Hom(F (A), F (B)) is of the form F (φ) for someφ ∈ Hom(A,B). For this define φ as the image of λ by the composition ofthe last two maps above. Since the last map is a bijection, chasing this φthrough the above sequence gives G(λ) = G(F (φ)), whence λ = F (φ) byfaithfulness of G.

Note that in the above proof the construction ofG depended on the axiomof choice: we had to pick for each V an element iV of the set of isomorphismsof V with F (A). Different choices define different G’s but the categoriesare equivalent with any choice. This suggests that the notion of equivalenceof categories means that “up to isomorphism the categories have the sameobjects and morphisms” but, as the following example shows, this doesn’tmean at all that there are bijections between objects and morphisms.

Example 1.10 (Linear algebra) Here is an example which despite its fas-tidious appearence contains something very familiar. Consider a field k andthe category FinVectk of finite dimensional k-vector spaces (with linearmaps as morphisms). We show that this category is equivalent to a cate-gory C which we define as follows. Objects of C are to be closed intervalsof the form [1, n] in the ordered set of integers (that is, the objects are1, 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 bymmatrices; here the identitymorphisms are given by the identity matrices and composition by multipli-cation of matrices. There is also a canonical morphism from each [1, n] to0 (the “zero morphism”) and one in the reverse direction (the “inclusion of0”); composition of zero morphisms with matrices gives zero morphisms andsimilarly for the inclusion of 0. (Notice that this is an example of a categorywhere morphisms are not set-theoretic maps.)

To show the asserted equivalence, we introduce an auxiliary subcategoryC′ and show it is equivalent to both categories. This category C′ is to bethe full subcategory of FinVectk spanned by the standard vector spaces kn.The equivalence of C′ and FinVectk is immediate from the criterion of theprevious lemma applied to the inclusion functor of C′ to FinVectk: the firstcondition is tautological as we took C′ to be a full subcategory and the secondholds because any finite dimensional vector space is isomorphic to some kn.

Let’s now show the equivalence of C′ and C. Define a functor F : C′ → Cby associating to kn some fixed numbering of its standard basis vectors andby sending morphisms to their matrices with respect to the standard bases


in the source and target spaces; zero morphisms and inclusions of 0 are sentto the corresponding distinguished morphisms. The fact that F is indeed afunctor hides a non-trivial result of linear algebra, namely that the matrixof the composition of two linear maps φ and ψ is the product of the matrixof φ with that of ψ. One constructs an inverse functor G by reversing thisprocedure: [1, n] is mapped to kn and a matrix to the linear map it defineswith respect to the standard bases. It is immediate to check that in this caseF 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 repre-sentable by an object A of C if there is a canonical isomorphism of functorsF ∼= Hom(A, ). Similarly, a contravariant functor G is representable by Aif G ∼= Hom( , A).

Examples 2.2 To get the flavour of this definition, let us look at someexamples.

1. Consider the forgetful functor Ab → Sets which associates to anabelian group its underlying set. This functor is representable by Zsince A ∼= HomAb(Z, A) as a set and this isomorphism is functorial.

2. The set-valued functor on the category of (commutative) rings givenby R → R×R is representable by the polynomial ring in two variablesZ[T1, T2]. Indeed, each element corresponds to a set-theoretic map fromthe two-element set T = T1, T2 to R; each such map corresponds inturn to a ring homomorphism Z[T1, T2] → R since Z[T1, T2] is the freecommutative ring generated by T . Again this bijection is manifestlyfunctorial. (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 func-tor on the category of fields given by F 7→ Hom(D,F ), the homomor-phisms being ring homomorphisms. This functor is represented by thefraction field of D; indeed, any element of Hom(D,F ) factors throughthe fraction field in a canonical way.

4. We can give a similar example from topology: let M be a fixed metricspace and consider the contravariant set-valued functor on the category

2.. REPRESENTABLE FUNCTORS 9

of complete metric spaces which associates to such a space X the setof continuous maps from M to X. This functor is represented by thecompletion of M .

The most important property of representable functors recieved its attri-bution in reminiscence of a coffee-house conversation between Yoneda andMacLane.

Lemma 2.3 (Yoneda’s Lemma) Suppose F and G are representable func-tors 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 aunique morphism in Hom(B,A). Similarly, if F and G are contravariantfunctors representable by objects A and B, then any morphism between themis induced by a unique morphism in Hom(A,B).

Here perhaps an explanation of the term “induced” is needed. In thecovariant 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 anyobject C of C which is clearly compatible with morphisms C → D, i. e. it isa morphism of functors.

Proof: We treat only the covariant case as the other is similar. Let Φbe a morphism of functors from F ∼= Hom(A, ) to G ∼= Hom(B, ). Theidentity 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 claimthat Φ is induced by φ in the sense explained above. This means that fora fixed object C of C, any element γ ∈ F (C) ∼= Hom(A,C) is mapped toγ φ ∈ Hom(B,C) ∼= G(C) by Φ. But γ is nothing but the image of idA bythe morphism F (γ) : F (A) → F (C) and similarly γ φ is the image of φ byG(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 beproved is the unicity of φ which follows from the fact that any φ inducing Φshould be the image of idA in Hom(B,A).

Corollary 2.4 The object representing a representable functor is unique upto unique isomorphism.


This somewhat succinct assertion means that if A and A′ are two objectsrepresenting the same functor F , then the set Isom(A,A′) consists of exactlyone element. Innocent as it looks, this corollary will be invaluable in thesequel.

3. Tensor Products

In this section we assemble some basic facts concerning tensor products ofmodules. This might be helpful for some readers as the construction, whichis sometimes absent from the basic algebraic curriculum, will be of constantuse in forthcoming chapters. We state only those basic properties that willbe 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 thata 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 theset-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 ′ induces a mapBilinA×B(C) → BilinA×B(C ′) 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 andcall it the tensor product of A and B over R. By the Corollary to the YonedaLemma we see that it is uniquely determined up to unique isomorphism asan object of the category of R-modules. We shall often drop the subscript Rin 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 de-

fined by

MapsA×B(C) = set theoretic maps A× B → C.

3.. TENSOR PRODUCTS 11

All we have to do here is to find some R-module R[A×B] with the propertythat any set-theoretic map A×B → C extends to a unique R-homomophismR[A × B] → C. But this is precisely the property defining the appropriatefree R-module. So we may take R[A × B] as the free R-module whose freegenerating set consists of the elements of A×B (that is, ordered pairs of theform (a, b) with a ∈ A and b ∈ B).

Step 2. Observe that for fixed C any element of BilinA×B(C) is also anelement of MapsA×B(C) so it defines an R-homomorphism f : R[A×B] → Cby Step 1. The extra condition required is that f should satisfy the relations(1). So if we define A⊗RB as the quotient of R[A×B] by the R-submodulegenerated 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 anR-homomorphism A ⊗R B → C. But this is the property required of therepresenting 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 ofA ⊗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) isgenerated over R by a system of elements ai (i ∈ I) (resp. bj (j ∈ J)), thenA ⊗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 shallshortly 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 andZ/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 withnx = 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 abeliangroups such that nA = A and nB = 0 (in this case, A is called n-divisibleand B n-torsion), then A⊗ B = 0.


We now list some basic properties of the tensor product all of whichfollow from the defining property of the tensor product using the (corollaryto the) Yoneda lemma. Readers may work out the proofs as an exercise inrepresentable functors.

Proposition 3.4 Let A, B, C and Ai (i ∈ I) be R-modules. Then there arecanonical isomorphisms

1. A⊗B ∼= B ⊗A;

2. (A⊗B) ⊗ C ∼= A⊗ (B ⊗ C);

3. A⊗R R ∼= A.

4. (⊕i∈IAi) ⊗ B ∼=

⊕i∈I

(Ai ⊗ B).

The two last statements here imply:

Corollary 3.5 If both A and B are free R-modules of rank m and n, respec-tively, 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 inducedby an R-homomorphism φ ⊗ ψ : A1 ⊗ B1 → B1 ⊗R B2. By examining theproof of Theorem 3.1 one checks immediately that φ ⊗ ψ is the map whichassosiates φ(a1)⊗ψ(b1) ∈ A2⊗B2 to a1⊗b1 ∈ A1⊗B1. In the particular caseB1 = 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 Bis a (covariant) functor on the category of R-modules.

On the other hand, the set HomR(A,B) of R-homomorphims from A intoB 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 thatA → HomR(A,B) is a contravariant functor. The above two functors arelinked by the following relation, labelled as “la formule chere a Cartan” byGrothendieck.

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.

3.. TENSOR PRODUCTS 13

Proof: By the corollary to the Yoneda lemma it is enough to construct anisomorphism of functors Ψ between BilinA×B and HomR(A,HomR(B, ). Todo this, observe first that for any f ∈ BiadA×B the map b 7→ f(a, b) is an R-homomorphism 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 amorphism of functors inverse to Ψ. One checks that it is the one defined byassociating the R-bilinear map (a, b) 7→ φa(b) to a homomorphism a 7→ φa inHomR(A,HomR(B,C)).

As a consequence we derive a fundamental property called the right ex-actness 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

A1i→A2

p→A3 → 0 (4)

a sequence of R-homomorphisms. This is an exact sequence if and only if forany 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 leftto the readers. The converse is not much harder: taking C = A3/A2 showsthat injectivity of the second map in (5) implies the surjectivity on the rightin (4), and taking C = A2/im(i) shows that if moreover (5) is exact in themiddle, 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 thatthe sequence

0 → HomR(A3 ⊗R B,C) → HomR(A2 ⊗R B,C) → HomR(A1 ⊗R B,C)


is exact for any R-module C. By Proposition 3.6, this sequence is isomorphicto 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 lemmatogether 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. Forinstance, 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 Propo-sition 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 whentensored with B, then B is said to be flat over R. This notion willcome up in Chapter 9.

In the next chapter, we shall use the tensor product in the followingsituation. Suppose given a ring homomorphism φ : R → S. Via this map Sbecomes an R-module and thus we may consider the tensor product A⊗R Swith any R-module A. On A ⊗R S we can define a canonical S-modulestructure by putting s(a ⊗ s′) = a ⊗ (ss′) for any a ∈ A and s, s′ ∈ S. Weshall speak of the S-module A⊗R S as the S-module obtained by base changefrom A. Note that Proposition 3.4 (3), (4) imply that if A is a free R-modulewith free generators ai (i ∈ I), then A⊗R S is also free with free generatorsai ⊗ 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 asubmodule L generated by “relations” fi (i ∈ I), then A⊗R S is the quotientof 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 relationsover S as A over R”.

Remark 3.11 It is also worth noting that in the above situation given anR-module A as well as an R-module B which is also an S-module such thatthe actions of R and S on B are compatible, A ⊗R B inherits an S-modulestructure from that on B. Then the following refined version of Theorem 3.6holds: 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 anR-algebra structure. Then the tensor product also inherits an R-algebrastructure in a natural way, the multiplication being defined by (a⊗b)(a′⊗b′) =(aa′) ⊗ (bb′) and then extending by linearity. (To see that this indeed givesa well-defined map (A ⊗ B) × (A ⊗ B) → (A ⊗ B) observe that the mapA×B ×A×B → (A⊗B) given by (a, b, a′, b′) 7→ aa′ ⊗ bb′ gives R-bilinearmaps A × B → A ⊗ B when one fixes the first and second (resp. third andfourth) arguments; then apply Theorem 3.1.) We have natural analogues forR-algebras of most of the results proved above for R-modules. We only notethe analogue of the preceding corollary:

Corollary 3.12 If A is a quotient of a polynomial ring R[T1, . . . , Tn] by anR-subalgebra generated by polynomials fi, then A⊗RS can be presented as thequotient of the polynomial ring S[T1, . . . , Tn] by the S-subalgebra generatedby the elements fi ⊗ 1.

Proof: The ring R[T1, . . . , Tn] is a free R-module generated by the mono-mials in T1, . . . , Tn, hence by the previous corollary S ⊗R R[T1, . . . , Tn] is afree S-module on the same generators. The definition of the product oper-ation implies that this latter ring is indeed isomorphic to S[T1, . . . , Tn]. Asimilar 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 notionsof general topology, including subspaces and quotient spaces, connected com-ponents and the theory of compact subsets – the latter will not be assumedto 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 factsfrom 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 isa continuous map f : [0, 1] → X, where [0, 1] is the closed unit interval. Theendpoints of the path are the points f(0) and f(1); if they coincide then thepath 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 mapH : [0, 1]×[0, 1] → X withH(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.


Proof: For reflexivity take H as the composition of the second projectionp2 : [0, 1]× [0, 1] → [0, 1] with f , for symmetry replace H by its compositionwith the map [0, 1]× [0, 1] → [0, 1]× [0, 1], (x, y) 7→ (1−x, y). For transitivitygiven a homotopy H1 between paths f and g and a homotopy H2 between gand h, construct first a continuous map H3 : [0, 2] × [0, 1] → X by piecingtogether 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 theircomposition f • g : [0, 1] → X by setting (f • g)(x) = f(2x) for 0 ≤ x ≤ 1/2and (f • g)(x) = g(2x− 1) for 1/2 ≤ x ≤ 1.

It is an easy exercise to verify that this composition passes to the quotientmodulo 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 tosome 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 wemay define a group law on the set π1(X, x) of homotopy classes of closedpaths f : [0, 1] → X with f(0) and f(1) both equal to some fixed base pointx by mapping the classes of such closed paths f1, f2 : [0, 1] → X to the classof 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] → xand the inverse of a class given by a path f : [0, 1] → X is the class of thepath 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 fundamentalgroup of X with base point x.

Our next goal is to show that for “nice spaces” the isomorphism class ofthe fundamental group does not depend on the choice of base point x. Forthis we need a definition: the space X is path-connected if for any two pointsx, 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 17

Proof: If we can write a space X as a disjoint union of non-empty opensubsets U1 and U2, then no couple of points xi in Ui (i = 1, 2) can be joinedby a path f : [0, 1] → X for otherwise the inverse images of the open subsetsIm (f)∩Ui of Im (f) would give a decomposition of [0, 1] into nonempty opensubintervals which is impossible.

Remark 4.6 Notice that by the same argument we may prove the strongerstatement that a space X is connected if there is some pont x ∈ X that canbe joined to any other point of X by a closed path.

A maximal path-connected subset of a space X is called a path componentof X; we can then write X as a disjoint union of its path components. Wesay that X is locally path-connected if each point of X has a basis of openneighbourhoods consisting of path-connected sets.

Lemma 4.7 The path components of a locally path-connected space X areboth open and closed subsets of X.

Proof: It suffices to show that any path component is open since writing itas the complenent of the disjoint union of union of the other path componentswe get that it is also closed. For openness note that by assumption any pointx ∈ X has a path-connected open neighbourhood U in X. But U must thenbe contained in the path component of x.

Since in a connected space the only nonempty subset which is both openand closed is the space itself, we get the

Corollary 4.8 Any connected and locally path-connected space is path-conn-ected.

Now we can prove:

Lemma 4.9 In a path-connected space the fundamental groups π1(X, x) andπ1(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 withf(0) = x and f(1) = y, the map π1(X, x) → π1(X, y), g 7→ f−1 • g • finduces 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 onthe choice of the path f ; it is unique, however, up to an inner automorphismof π1(X, y).


Definition 4.10 We say that a path-connected space X is simply connectedif 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 pathsf, g : [0, 1] → X with f(0) = g(0) and f(1) = g(1) are homotopic (becauseby 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 abasis 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 continu-ously any closed path to a point), any open subset of Rn is locally simplyconnected. More generally, this holds for spaces that can be covered by opensubsets 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 afunctor in a natural way. For this, define the category of pointed topologicalspaces to be the category whose objects are pairs (X, x) with X a topologicalspace and x a point of X; a morphism (X, x) → (Y, y) in this category is acontinuous 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 isagain a straightforward matter to check that any continuous map φ : X → Ysending x to y induces a group homomorphism φ∗ : π1(X, x) → π1(Y, y) bymapping 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 moreinvolved 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 eachelement α of k is a root of some polynomial with coefficients in k (which wemay as well assume to be monic). If this polynomial is irreducible over k, itis called the minimal polynomial of α. When L is generated as a k-algebraby the (algebraic) elements α1, . . . , αk ∈ L, we write L = k(α1, . . . , αk). Ofcourse, one may find many different sets of such αi’s.

A field is algebraically closed if it has no algebraic extensions other thanitself. An algebraic closure of k is an algebraic extension k which is alge-braically closed; its existence is perhaps the least evident fact in the theoryand can only be proved by means of Zorn’s lemma or some other equivalentform of the axiom of choice. We record it in the following proposition whichalso 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 → kleaving k elementwise fixed.

3. In the previous situation take an algebraic closure L of L. Then theembedding L→ k can be extended to an isomorphism of L onto k.

19

20 CHAPTER 1. GALOIS THEORY OF FIELDS

For the proof, see Lang [1], Chapter V, Corollary 2.6 and Theorem 2.8 orvan der Waerden [1], §72.

Thus henceforth when speaking of algebraic extensions of k we may anddo 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 bya single element with minimal polynomial f , then [L : K] is equal to thedegree 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 minimalpolynomial has no multiple roots. A finite extension L|k is separable if all ofits elements are separable over k. There is another characterisation of thisproperty 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 ndistinct k-homomorphisms to k, with equality if and only if L|k is separable.

Proof: Assume first L is generated over k by a single element α with min-imal polynomial f . Any k-homomorphism L → k is then determined by theimage of α which must be one of the roots of f contained in k. The numberof 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 ofthe 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 sepa-rable 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 usuallyreferred to as the theorem of the primitive element.

Proposition 1.4 Any finite separable extension can be generated by a singleelement.

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-fields of k, their compositum LM is the smallest subfield of k containing bothL and M .

1.. A REVIEW OF CLASSICAL GALOIS THEORY 21

Lemma 1.5 If L, M are finite separable extensions of k, their compositumis 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) fori > 0. Then Mk = LM and each Mi is separable over Mi−1 for αi is separableover k and a fortiori over Mi−1. One concludes by a repeated application ofProposition 1.2.

In view of the above lemma forming the compositum of all finite separablesubextensions of k yields an extension ks of k with the property that eachelement of ks generates a finite separable subextension of k|k. Moreover, bydefinition each finite separable subextension of k|k is contained in ks whichthus merits to be called the separable closure of k in k. When in the sequel weshall refer to “a separable closure of k” we shall mean its separable closurein some chosen algebraic closure. Also, henceforth any (possibly infinite)subextension of ks|k will be called separable.

The field k is perfect if all finite extensions of k are separable. Examples ofperfect fields are fields of characteristic 0, algebraically closed fields and finitefields (see van der Waerden, §45). A typical example of a non-perfect field isa rational function field F(T ) of one variable over a field F of characteristicp: here adjoining a p-th root ξ of the indeterminate T defines an inseparableextension in view of the decomposition Xp −T = (X− ξ)p. For perfect fieldsit is immediate from the definition that their algebraic and separable closurescoincide.

Now we come to the fundamental definition in Galois theory. Given anextension L of k, denote by Aut(L|k) the group of field automorphisms ofL fixing k elementwise. It acts on L and fixes some subextension of L|kelementwise.

Definition 1.6 An algebraic extension L of k is called a Galois extensionof k if it is separable and the elements of L which remain fixed under theaction of Aut(L|k) are exactly those of k. In this case Aut(L|k) is denotedby Gal (L|k) and called the Galois group of L over k.

Example 1.7 The absolute Galois group. The most natural example of aGalois extension of k is that of the separable closure ks (which coincides withthe algebraic closure for perfect fields). It is certainly separable over k, soto check that it is Galois we only have to show that any element α of ks

not contained in k gets moved by an appropriate automorphism of ks overk. Indeed, define an automorphism of the extension k(α) generated by α


over k (itself a subfield of ks) by sending α to another root of its minimalpolynomial. An application of the third part of Proposition 1.1 then showsthat this automorphism can be extended to an automorphism of the algebraicclosure k. To conclude one only has to remark that any automorphism of kmaps ks onto itself since such an automorphism acts on an element β of k bysending it to another root of its minimal polynomial; if β is separable, thispolynomial has no multiple roots.

We shall call the Galois group of ks over k the absolute Galois group of kand denote it by Gal (k).

However, ks is an infinite extension of k whereas in the classical theoryonly finite Galois extensions of k are investigated. The historical reason forthis, as all readers know, is that these are the extensions that arise naturallywhen 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 withcoefficients in k and without multiple roots. Take an algebraic closure k ofk and look at the finite subextension K|k of k|k generated by all roots ofthe 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 isthe compositum of the finitely many separable subextensions k(αi)|k, whereα1, . . . , αk are the roots of f in k. Next, note that any k-automorphism ofk maps K onto itself for it only permutes the roots αi ∈ K of f . Now tosee that Aut(K|k) fixes only k, pick any element α ∈ K \ k. Mapping it toanother root β ∈ k of its minimal polynomial over k defines a k-isomorphismk(α) ∼= k(β) which extends to an automorphism σ of k, again by the thirdpart of Proposition 1.1. But then by the previous observation we must haveσ(K) = K, so that β ∈ K and the restriction of σ to K is an element ofAut(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 isa more natural way of looking at finite Galois extensions: if a finite groupG acts on a field K, then K is a Galois extension of the field of invariantsKG. Of course here it is only separability that needs to be proven, whichfollows from the fact that any element of α ∈ K is a root of the polynomialf =

∏(x− σ(α)), where σ runs over a system of (left) coset representatives

of the stabiliser of α in G. This polynomial lies in KG[x] for it is invariantby 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|KG, we can describeK as the splitting field of f over KG.

Now the main theorem of Galois theory for finite Galois extensions readsas follows.

Theorem 1.10 Consider a finite Galois extension L|k with Galois groupG. Associating to a subfield M of L containing k the subgroup H of Gconsisting of those automorphisms which leave M elementwise fixed gives abijection between subextensions of L|k and subgroups of G. The subfield Mis Galois over k if and only if H is a normal subgroup of G; in this case, theGalois 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 toinfinite Galois extensions. The main difficulty is that for an infinite extensionit will no longer be true that all subgroups of the Galois group arise as thesubgroup fixing some subextension M |k. The first example of a subgroupwhich doesn’t correspond to some subextension has been found by Dedekind,who, according to Wolgang Krull, already had the feeling that “die GaloisscheGruppe gewissermaßen eine stetige Mannigfaltigkeit bilde”. It was Krull whothen cleared up the question in a classic paper (Krull [1]); a modern versionof it will be described in the next section.

2. Infinite Galois Extensions

In the previous section, we defined the separable closure ks of a field k asthe union of all finite separable subextensions of k|k. In fact, it is alreadythe union of the finite Galois subextensions because of the following lemmawhich will be used several times in this section.

Lemma 2.1 Any finite separable subextension of ks|k can be embedded intoa Galois subextension.

Proof: By the theorem of the primitive element (Proposition 1.4), eachfinite separable subextension is of the form k(α) with an appropriate elementα and we may embed k(α) into the splitting field of the minimal polynomialof α which is already Galois over k by Example 1.8.


Now given a tower of finite Galois extensions M |L|k, Theorem 1.10 pro-vides us with a canonical surjection

φML : 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) itselfand just as ks is determined by its finite Galois subextensions, Gal (k) wouldbe determined by its finite quotients. The following definition gives us thekey 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.

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 denotedby 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 candefine the inverse limit of sets, rings, modules, even of topological spaces inan analogous way.

Remark 2.3 Notice that the fact that the index set is directed was notused in the above definition. Thus we may speak more generally of inverselimits of inverse systems with respect to any partially ordered index sets; theword “filtered” in the above definition refers to the fact that the index set isdirected.

While we are at this matter, let us also define the dual notion of directlimits. Here we shall restrict to abelian groups in the main definition (butsee the remark afterwards).

Definition 2.4 A (filtered) direct system of abelian groups (Ai, φij) consistsof:

• a partially ordered set (I,≤) which is directed in the above sense;


• 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 .

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 denotedby lim

→Ai.

Remark 2.5 Note that this notion does not extend to arbitrary groups butit immediately does to abelian groups with additional structure (rings, mod-ules 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 rela-

tion which is generated by pairs (si, sj) ∈ Si × Sj with φik(si) = φjk(sj) forsome 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 systemby putting i ≤ j if Si ⊂ Sj and define φij to be the inclusion map. Now it isstraightforward 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 thatfor any Ri, Rj there is some Rk with Rk ⊃ Ri ∪Rj. Then partially orderingby inclusion turns the system (Ri) into a direct system whose direct limit isR.

Proof: Consider the natural homomorphism⊕

i∈I Ri → R given by thesum of components. Since the φij are now the inclusion maps, the kernel ofthis map is generated by the relations defining the direct limit; surjectivityis equally easy.

Corollary 2.7 The separable closure ks of a field k is a direct limit of itsfinite Galois subextensions.

Proof: It remains only to show that the system of finite Galois subexten-sions is directed which means that the compositum of any two finite Galoissubextensions can be embedded into some finite Galois extension. Since thecompositum is separable, this follows from Lemma 2.1. (In fact, the com-positum is already Galois by Lang [1], Chapter VI, Theorem 1.14.)

The next proposition can now be by no means surprising.


Proposition 2.8 Let k be a field with fixed separable closure ks. Then theGalois groups of finite Galois subextensions of ks|k together with the ho-momorphisms φML of (1.1) form an inverse system whose inverse limit isisomorphic to the absolute Galois group Gal (k).

Proof: Only the last statement needs a proof. For this, define a grouphomomorphism φ : Gal (k) →

∏Gal(L|k) (where the product is over all

finite Galois subextensions L|k) by sending an automorphism σ of ks over kto the direct product of its restrictions to the various subfields L indexingthe product. This map is injective since if an automorphism σ does notfix an element α of ks, then its restriction to any finite Galois subextensioncontaining k(α) is nontrivial (such an extension exists by Lemma 2.1). Onthe other hand, Theorem 1.10 assures that the image of φ is contained inlim←

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

ks 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 theσL 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 byvirtue of the proposition natural surjections Gal (k) → Gal (L|k) for eachfinite Galois subextension L, just as we expected.

Now we come to the Krull-Dedekind idea about the extension of Galoistheory to the infinite case. This is to endow Gal (k) with a suitable topologyfor which the closed subgroups would turn out to be precisely those corre-sponding to some field extension. Of course for any reasonable topology onGal(k) one would like the projections Gal (k) → Gal (L|k) to be continuousso the only judicious choice is to endow the groups Gal (L|k) with the dis-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 bythe discrete topology. Then the inverse limit lim

←Gi is a closed topological

subgroup of the product∏Gi.

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

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


is a suitable choice, being open (by the discreteness of the Gi and by thedefinition of topological product) and containing g but avoiding lim

←Gi.

Definition 2.10 A topological group which is the inverse limit of finitegroups 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 aninverse system: let I be the index set formed by the normal subgroupsof finite index partially ordered by the following relation: Ui ≤ Uj

iff Ui ⊃ Uj . Then if Ui ≤ Uj are such normal subgroups, we have aquotient map φij : G/Uj → G/Ui. The inverse limit of this system is

called the profinite completion of G, customarily denoted by G. Thereis a canonical homomorphism G→ G with dense image; in fact, as thenext proposition will imply, this map is an inclusion.

2. Take G = Z in the previous example. Then I is just Z since each sub-group of finite index is generated by some m ∈ Z. The partial order isinduced by the divisibility relation: m|n iff mZ ⊃ nZ. The completionZ is usually called zed hat, although some people call it the Prufer ring(it indeed has a natural ring structure). One can show that Z is the freeprofinite group generated by one element in the following sense: it hasa canonical topological generator 1 coming from the natural inclusionZ ⊂ Z; mapping it to any element in a profinite group G extends to acontinuous homomorphism Z → G.

3. In the previous example, taking only powers of some prime p in placeof m we get a subsystem of the inverse system considered there; infact it is more convenient to index it by the exponent of p. With thisconvention the partial order becomes the usual (total) order of Z. Theinverse limit is Zp, the (additive) group of p-adic integers; in fact, it isalso a ring, just as in the previous example. One can show using theChinese Remainder Theorem that the ring Z is the direct product ofthe rings Zp for all primes p.

4. By Proposition 2.8, the absolute Galois group of any field F furnishesan example of a profinite group. When F is finite, we get nothingnew: Gal (F ) is canonically isomorphic to Z. Indeed, from the theoryof finite fields (see e.g. Lang [1], Chapter V, §5) we know that foreach positive integer n, F has a unique finite extension Fn of degreen, which is moreover Galois with group Gal (Fn|F ) ∼= Z/nZ, so the


inverse system one gets is exactly that of the second example above.As a contrast, determining Gal (Q) is one of the greatest mysteries inarithmetic.

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 ofneighborhoods 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 finiteindex.

Proof: For the first assertion, one may take as the Ui the kernels of thecontinuous projections G → Gi; these are open normal subgroups by con-struction and their intersection is the unit element, for any nontrivial elementof G has a nontrivial image in some Gi. The second assertion follows fromthe first in view of the following well-known trick: take elements x, y ∈ Gand let Ui be an open neighbourhood of the identity such that xy−1 /∈ Ui;then Uix and Uiy are disjoint open neighbourhoods of x and y. For the thirdassertion, observe first that finite discrete groups are compact; by Tikhonov’stheorem (Chapter 0, Proposition ??) so is their product. To conclude theproof it remains to apply Lemma 2.9 and the fact that a closed subset of acompact topological space is itself compact (Chapter 0, Lemma ??). Finally,note that any open subgroup U is closed since its complement is a disjointunion of cosets gU which are themselves open (the map U 7→ gU being ahomeomorphism in a topological group); by compactness of G, these mustbe finite in number. The converse follows by a similar but simpler argument.

Remark 2.13 In fact, there is a nice topological characterisation of profi-nite groups as being those topological groups which are compact and totallydisconnected (i.e. their only connected subsets are the one-element sets). Seeeg. Gruenberg [1] for a proof.

Now let’s return to Galois theory. Observe first that if L is a subextensionof ks|k, then ks is also the separable closure of L. Hence it is Galois over Land Gal (L) is naturally identified with a subgroup of Gal (k).

Proposition 2.14 Let L be a subextension of ks|k. Then Gal (L) is a closedsubgroup of Gal (k). It is open if and only if L is a finite extension of k.


Proof: Let’s begin with the second assertion. If L is a finite separableextension 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 con-tains 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 con-tinuous and Gal (M |k) has the discrete topology, UL is open. It thus sufficesto 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 fromProposition 2.12 (4) once we have established the first assertion. So assumenow L|k is an arbitary subextension of ks|k and write it as a union of finitesubextensions Li|k. By what we have just proven, each Gal (Li) is an opensubgroup of Gal (k), hence it is also closed by Proposition 2.12 (4). Theirintersection 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 eachsubextension L of ks|k its absolute Galois group Gal (L) gives a bijectionbetween subextensions of ks|k and closed subgroups of G. Here finite exten-sions correspond to open subgroups. The subfield L is Galois over k if andonly if H is a normal subgroup of G; in this case we have an isomorphismGal (L|k) ∼= G/H.

Proof: Associating to a subextension L|k of ks|k the absolute Galois groupGal (L) gives a closed subgroup by the previous proposition. Its fixed fieldis exactly L for ks is Galois over L. Conversely, given a closed subgroupH ⊂ G, it fixes some extension L|k and is hence contained in Gal (L). Toshow equality, let σ be an element of Gal (L), and take any fundamentalopen neighbourhood UM of the identity in Gal (L), corresponding to a Galoisextension M |L. Now H ⊂ Gal (L) surjects onto Gal (M |L) by the naturalprojection since by assumption any element of M \ L is moved by someelement of H ; in particular some element of H must map to the same elementin Gal (M |L) as σ. HenceH 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 precedingproposition and the last assertion is proven exactly as in the finite case (seereferences for Theorem 1.10).

Remarks 2.16 We conclude with the following observations.


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 differentfrom the finite case we must exhibit non-closed subgroups in the Galoisgroup. We have already seen an example in the second and third ex-amples of 2.11: the absolute Galois group of a finite field is isomorphicto Z which in turn contains Z as a non-trivial dense (hence non-closed)subgroup; there are in fact many copies of Z embeddded in Z. The orig-inal example of Dedekind was very similar though it did not involve anabsolute Galois group but the Galois group of an infinite subextensionof Q|Q: he took the field Q(µp∞) obtained by adjoining to Q all rootsof 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 (profi-nite groups have not yet been discovered at the time); he just showedthe existence of a non-closed subgroup. His proof was generalised inKrull[1] to establish the existence of non-closed subgroups in the Galoisgroup of any infinite extension as follows. First one shows that given anon-trivial Galois extension K2|K1, any automorphism of K1 may beextended to an automorphism ofK2 in at least two ways. From this oneinfers by taking an infinite chain of non-trivial Galois subextensions ofan infinite Galois extension L|k that Gal (L|k) is uncountable. By thesame argument (and the first remark above), any infinite closed sub-group of Gal (L|k) is uncountable, hence countably infinite subgroupsare never closed.

3. Finite Etale Algebras

In this section we give a second variant of the Main Theorem which is oftenreferred to as “Grothendieck’s formulation of Galois theory”.

We start again from a base field k of which we fix separable and algebraicclosures ks ⊂ k. Let L be a finite separable extension of k; here we do notconsider L as a subextension of ks. We know that L has only finitely manyk-algebra homomorphisms into k (the number of these is equal to [L : k]by Lemma 1.2); actually the images of these homomorphisms are containedin ks. So we may consider the finite set Homk(L, k

s) which is endowed bya 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, ks) 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 stabiliserof 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 meansprecisely Gal (k) acts continuously on Homk(L, k

s). If L is generated by aprimitive element α with minimal polynomial f , any φ ∈ Homk(L, k

s) isgiven by mapping α to a root of f in ks. Since Gal (k) permutes these rootstransitively, the Gal (k)-action on Homk(L, k

s) is transitive. As the stabiliserof the element gφ is just gUg−1, we find that the stabilisers of the elementsof Homk(L, k

s) are all conjugate open subgroups. Moreover, mapping gφto gU induces an isomorphism of the Gal (k)-set Homk(L, k

s) with the leftcoset space of U . In the case when L is a Galois extension of k, we get fromTheorem 2.15 that all stabilisers are equal to the same open normal subgroupU 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, ks) → Homk(L, k

s) by composition with φ.This map is clearly Gal (k)-equivariant so we have obtained a contravariantfunctor from the category of finite separable extensions of k to the categoryof finite sets with continuous transitive left Gal (k)-action.

Theorem 3.2 Let k be a field with fixed separable closure ks. Then the con-travariant functor which associates to a finite separable extension L|k thefinite Gal (k)-set Homk(L, k

s) gives an anti-equivalence between the categoryof finite separable extensions of k and the category of finite sets with contin-uous and transitive left Gal (k)-action. Here Galois extensions give rise toGal (k)-sets isomorphic to some finite quotient of Gal (k).

Proof: We have already seen the last statement. To prove the rest, wecheck that Homk( , ks) satisfies the conditions of Chapter 0, Lemma 1.9. Webegin 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. Thestabiliser of s is an open subgroup Us of Gal (k) which fixes a finite separableextension L of k. Now define a map of Gal (k)-sets Homk(L, k

s) → S bythe rule g i 7→ gs, where i is the natural inclusion L → ks and g is anyelement 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 becomeisomorphic to the coset space G/Us by the maps sending i (resp. s) to Us.


For the first condition we have to show that given two finite separableextensions L,M of k, the set of k-homomorphisms L → M correspondsbijectively to the set of Gal (k)-maps Homk(M, ks) → Homk(L, k

s). Sinceboth Homk(M, ks) 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, ks).As f is Gal (k)-equivariant, any element of the stabiliser U of φ fixes f(φ) aswell, whence an inclusion U ⊂ V , where V is the stabiliser of f(φ). By whatwe have just seen, taking the fixed subfields of U and V respectively, we get aninclusion of subfields of ks 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 ψf(φ)is the unique element of Homk(L,M) inducing f .

If we wish to extend the previous anti-equivalence to Gal (k)-sets withnot necessarily transitive action, the natural replacement for finite separableextensions of k is the following.

Definition 3.3 A finite dimensional k-algebra A is etale (over k) if it isisomorphic 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-algebraA the finite set Homk(A, k

s) gives an anti-equivalence between the categoryof finite etale k-algebras and the category of finite sets with continuous leftGal (k)-action. Here separable field extensions give rise to sets with transi-tive Gal (k)-action and Galois extensions to Gal (k)-sets isomorphic to finitequotients of Gal (k).

Proof: This follows from the previous theorem in view of the remark thatgiven a decomposition A =

⊕Li into a sum of fields and an element φ ∈

Homk(A, ks), then φ induces the injection of exactly one Li to ks; indeed, if

φ(Li) 6= 0, then being a field, Li injects to ks, and on the other hand, a sumLi⊕Lj cannot inject into ks since ks has no zero-divisors. Thus Homk(A, k

s)decomposes into the disjoint union of the Homk(Li, k

s); this is in fact itsdecomposition into Gal (k)-orbits.

To conclude this section, we wish to give another characterisation of finiteetale k-algebras which ties in with more classical treatments. The reader whoneeds 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 follow-ing 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 condi-tion of the proposition are often called separable k-algebras. The propositionthus provides a structure theorem for these. For the proof we need the fol-lowing lemma which is the commutative version of the Weddernburn-Artintheorem.

Lemma 3.6 Let F be a field, A a finite-dimensional F -algebra. Then A isa direct product of finite field extensions of F if and only if A has no nonzeronilpotent elements.

Proof: (after Frohlich-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 thisrestriction 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 nontrivialdirect sum decomposition since e(1 − e) = e − e2 = 0 by assumption. Thelemma will follow if we show that any nonzero element x ∈ A is invertibleand thus A is a field. Since A is finite dimensional over F , the descendingchain of ideals (x) ⊃ (x2) ⊃ . . . (xn) ⊃ . . . must stabilise and thus for some mwe must have xn = xn+1y with an appropriate y. By iterating this formulawe get xn = xn+iyi for all positive integers i, in particular xn = x2nyn. Thusxnyn = (xnyn)2, i. e. xnyn is an idempotent. By what has been said abovethere are two cases. If xnyn = 0, then xn = (xn)(xnyn) = 0 which is acontradiction since x 6= 0 by assumption and A has no nonzero nilpotents.Otherwise, xnyn = 1 and thus x is invertible.

Remark 3.7 The lemma already implies that a finite-dimensional algebraover a perfect field is etale if and only if it contains no nilpotents. So thesomewhat less manageable third condition of the proposition arises only overnon-perfect fields.

Proof of Proposition 3.5. Let’s begin by proving that the first conditionimplies the second. We may clearly restrict to finite separable extensions Lof k. We then have L = k[X]/(f) with some polynomial f which decomposesas a product of n distinct factors (X − αi) in k. By Chapter 0, Corollary3.12 L⊗k k ∼= k[X]/(f), so we conclude by the chain of isomorphisms

k[X]/(f) = k[X]/(X − α1) . . . (X − αn) ∼=n⊕

i=1

k[X]/(X − αi) ∼=n⊕

i=1

k,


the middle isomorphism holding by the Chinese Remainder Theorem (seee.g. Lang [1], Chapter II, Theorem 2.1).

The derivation of the third condition from the second is immediate; ac-tually, the lemma applied to A ⊗k k shows that they are equivalent. Nowto derive the first condition from the second, let A be the quotient of A byits k-subalgebra of nilpotent elements. The lemma implies that A is a sumof finite extension fields of k. Since k contains no nilpotent elements, anyk-algebra homomorphism A → k factors through A and hence through oneof its decomposition factors L. By lemma 1.2, the number of k-algebra ho-momorphisms L→ k can equal at most the degree of L over k, with equalityif and only if L|k is separable, whence Homk(A, k) has at most dim k(A) el-ements with equality iff A = A and A is etale. On the other hand, we havea canonical bijection of finite sets

Homk(A, k) ∼= Homk(A⊗k k, k).

[To see this, observe that given a k-algebra homomorphism A→ k, tensoringby k and composing by the multiplication map gives a k-homomorphismA ⊗k k → k ⊗k k → k; on the other hand the natural inclusion k → kinduces a k-homomorphism A ∼= A ⊗k k → A ⊗k k which composed byhomomorphisms A⊗k k → k gives a map from the set on the right hand sideto that on the left which is clearly inverse to the previous construction.] Theassumption 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 kA so Ais indeed separable.

Chapter 2

Fundamental Groups inTopology

In the last section we saw that when studying extensions of some field it isplausible 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 pointsmapping to this base point. Galois theory then equips the situation with acontinuous action of the absolute Galois group which leaves the base pointfixed. It is natural to try to extend this situation by taking as a base notjust a point but a more general topological space. The role of field extensionswould then be played by certain continuous surjections, called covers, whosefibres are finite (or, even more generally, arbitrary discrete) spaces. We shallsee in this chapter that under some restrictions on the base space one candevelop a topological analogue of the Galois theory of fields, the part of theabsolute Galois group being taken by the fundamental group of the basespace. We shall also find an analogue of the separable closure, called theuniversal cover, whose group of relative automorphisms over the base spacewill be exactly the fundamental group. However, we point out a notabledifference already at this stage: in contrast to the field-theoretic case thefundamental group will not be profinite in general and the orbits of pointsof the universal cover under its action will not necessarily be finite either.This indicates that when we shall be looking for an algebraic theory thatsubsumes both the Galois theory of the previous chapter and the topologicalconstructions of the present one, we shall have to restrict to covers with finitefibres.

35

36 CHAPTER 2. FUNDAMENTAL GROUPS IN TOPOLOGY

1. Covers

Let us begin with some seemingly uninteresting terminology.

Definition 1.1 LetX be a topological space. A space overX is a topologicalspace Y together with a continuous map p : Y → X. The category of spacesover X is the category whose objects are spaces over X and a morphismbetween to objects pi : Yi → X (i = 1, 2) is given by a continuous mapf : Y1 → Y2 which makes the diagram

Y1f

−−−→ Y2

p1

yyp2

Xid

−−−→ X

commute.

The category of spaces over X, denoted by TopX is naturally a subcat-egory of the category Top of topological spaces. It is readily seen that amorphism f : Y1 → Y2 of spaces over X is an isomorphism in TopX if andonly if f is an isomorphism in Top, i.e. it is a homeomorphism. Now thebasic definition of this section is the following.

Definition 1.2 A covering space of a topological space X, or a cover of Xfor short, is a space Y over X where the projection p : Y → X is subject tothe following condition: each point of X has an open neighbourhood V forwhich p−1(V ) decomposes as a disjoint union of open subsets Ui of Y suchthat 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 morphismof spaces over X. Thus we get a natural full subcategory CovX of thecategory TopX .

Example 1.3 Take a discrete topological space I and form the topologicalproduct X × I. The first projection X × I → X makes X × I into a spaceover 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 propositionshows, every cover is locally a trivial cover.

Proposition 1.4 A space Y over X is a cover if and only if each point ofX has an open neighbourhood V such that the restriction of the projectionp : Y → X to p−1(V ) is isomorphic in the category TopV to a trivial cover.

2.. GROUP ACTIONS AND GALOIS COVERS 37

Proof: One implication follows from the previous example and the other iseasily seen as follows: given a decomposition p−1(V ) ∼=

∐i∈IUi for some index

set I as in the definition of covers, the map associating to ui ∈ Ui the couple(p(ui), i) defines a homeomorphism of

∐i∈IUi onto V × I, where I is endowed

with the discrete topology. It is immediate from the construction that thisis 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 fibreof p over the points of V . The proof shows that the points of X such thatthe fibre of p is I as a discrete set form an open subset of X; for varying Ithey 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 samediscrete set I.

Notice that this does not mean at all that the cover is trivial. Indeed, letus 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 andZW into two equal segments by the points P and Q. Identifying the sidesXY and ZW with opposite orientations we get a Mobius strip on whichthe image of the segment PQ becomes a closed curve C homeomorphic to acircle. The natural projection of the boundary B of the Mobius strip onto Ccoming from the perpendicular projection of the sides XW and Y Z of therectangle onto the segment PQ makes B a space over C which is actually acover since locally it is a product of a segment by a two-point space, i.e. atrivial cover of the segment. However, the cover itself is non-trivial since Bis 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 givena finite Galois extension L|k, all intermediate extensions arise as invariantsunder some subgroup H of Gal (L|k). When one seeks an analogy in thecontext of covers the first difficulty one has to face that it is not completelyevident to find a counterpart of an innocently looking fact hidden in theprevious 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 con-nected (i.e. each point has a basis of neighbourhoods consisting of connected


open subsets). Given a cover p : Y → X, its automorphisms are to beautomorphisms of Y in the category CovX , i.e. topological automorphismscompatible with the projection p. They clearly form a group that we shalldenote by Aut(Y |X). By convention all automorphisms will be assumed toact from the left. Note that for each point x ∈ X, Aut(Y |X) maps the fibrep−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 topologicalautomorphism of Y to be an element of Aut(Y |X).

Lemma 2.1 Any automorphism φ of a connected cover p : Y → X havinga fixed point must be trivial.

Corollary 2.2 Any automorphism of a cover having a fixed point fixes theconnected component of the point elementwise.

Instead of proving the lemma we establish a more general statement whichwill also be needed later. The lemma follows from it by taking Z = Y , f = idand 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 pf = pg. If there is a pointy ∈ Y with f(y) = g(y), then f = g.

Proof: Suppose y ∈ Y is as above, z = f(y) = g(y). Take some connectedopen neighbourhood V of p(z) satisfying the condition in the definition of acover (such a V exists since X is locally connected) and let Ui

∼= V be thecomponent of p−1(V ) containing z. By continuity f and g must both mapsome open neighbourhood W of y into Ui. Since p f = p g and p mapsUi homeomorphically onto V , f and g must agree on W . The same type ofreasoning shows that if f(y′) 6= g(y′) for some point y′ ∈ Y , f and g mustmap a whole open neighbourhood of y′ to different components of p−1(V ).Thus the set y ∈ Y : f(y) = g(y) is nonempty, open and closed in Y soby connectedness it is the whole of Y .

To proceed further, it is convenient to introduce some terminology (adopt-ing the convention of Fulton [1] in contrast with the term “properly discon-tinuous” of classical parlance).

Definition 2.4 Let G be a group acting from the left on a topological spaceY . The action of G is even if each point y ∈ Y has some open neighborhoodU such that the open sets gU are pairwise disjoint for all g ∈ G.


This being done, we can state:

Proposition 2.5 If p : Y → X is a connected cover, then the action ofAut(Y |X) on Y is even.

Proof: Let y be a point of Y and let V be a connected open subset of Xsuch 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 conditionof the above definition. Indeed, since Y is connected, Lemma 2.1 applies andshows that any nontrivial φ ∈ Aut(Y |X) should map Ui isomorphically ontosome Uj, with j ∈ I different from i. Moreover, if φ(Ui) = ψ(Ui) for someφ, ψ ∈ Aut(Y |X), then by composing with p we see that φ should coincidewith ψ pointwise on Ui and thus we conclude by applying Lemma 2.1 toφ ψ−1 that φ = ψ.

Now that we have found a necessary condition for an automorphism of aconnected space Y to be an automorphism of some cover, we shall show thatit is also sufficient, which will furnish the promised topological analogue ofconstructing field extensions by taking invariants under group actions. Recallthat if a group G acts on a topological space Y , one may form the quotientspace Y/G whose underlying set is by definition the set of orbits under theaction of G and the topology is the finest one which makes the projectionY → Y/G continuous.

Proposition 2.6 If G is a group acting evenly on a connected space Y , theprojection pG : Y → Y/G turns Y into a cover of Y/G. The automorphismgroup of this cover is precisely G.

Proof: For the first statement, note that pG is surjective and moreover eachx ∈ Y/G has an open neighbourhood of the form V = pG(U) with a U as inDefinition 2.4. This V is readily seen to satisfy the condition of Definition1.2.

For the second statement, notice first that we may naturally view G as asubgroup of Aut(Y |(Y/G)). Now given an element φ in the latter group, lookat its action on an arbitrary point y ∈ Y . Since the fibres of pG are preciselythe orbits of G we may find g ∈ G with φ(y) = gy. Applying Lemma 2.1 tothe 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 automorphismdefined 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.


2. The previous example can be generalised to any dimension: take anybasis x1, . . . , xn of the vector space Rn and make Zn act on Rn sothat the i-th direct factor of Zn acts by translation by xi. This action isclearly 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 bythe xi is usually called a lattice; thus linear tori are quotients of Rn bylattices.

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→ zn definesa natural homeomorphism of C∗/µn onto C∗ (even an isomorphism oftopological groups) and via this homeomorphism pn becomes identifiedwith the cover C∗ → C∗ given by z 7→ zn. Note that this map does notextend to a cover C → C; this phenomenon will be studied further inChapter 4.

Remark 2.8 It is easy to show that quite generally if a finite group G actswithout fixed points on a Hausdorff space Y , its action is even and thus itfurnishes a cover Y → Y/G for connected Y .

Now given a connected cover p : Y → X, we may form the quotient ofY by the action of Aut(Y |X). It is immediate from the definition of coverautomorphisms that the projection p factors as a composite of continuousmaps

Y → Y/Aut(Y |X)p→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 connectedand the induced map p above is a homeomorphism.

Remark 2.10 Note the similarity of the above definition with that of aGalois extension of fields. This analogy is further confirmed by remarkingthat the cover pG in Proposition 2.6 is Galois.

Proposition 2.11 A connected cover p : Y → X is Galois if and only ifAut(Y |X) acts transitively on the fibres p−1(x) of p.


Proof: Indeed, the underlying set of Y/Aut(Y |X) is by definition the setof orbits of Y under the action of Aut(Y |X) and so the map p is one-to-one precisely when each such orbit is equal to a whole fibre of p, i.e. whenAut(Y |X) acts transitively on the fibres.

Remark 2.12 In fact, for a connected cover p : Y → X to be Galois itsuffices for Aut(Y |X) to act transitively on one fibre. Indeed, in this caseY/Aut(Y |X) is a connected cover of X where one of the fibres consists of asingle 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-mmap 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/Hinto a cover of X.

Conversely, if Z → X is a connected cover fitting into a commutativediagram

Yf

−−−→ Z

py

yq

Xid

−−−→ X

then f : Y → Z is a Galois cover and actually Z ∼= Y/H with some sub-group H of G. In this way we get a bijection between subgroups of G andintermediate covers Z as above.

The cover q : Z → X is Galois if and only if H is a normal subgroup ofG, 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 coverq : Z → X and a continuous map f : Y → Z. If the composite qf : Y → Xis a cover then so is f : Y → Z.


Proof: Let z be a point of Z, x = q(z) and V a connected open neighbour-hood of X satisfying the property of Definition 1.2 for both p = q f andq: 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 withf(Ui) ⊂ Vj. But this is in fact a homeomorphism since both sides get mappedhomeomorphically onto V by q. This implies in particular that the image off is open.

Now to prove the lemma we show first that f is surjective. It is enoughto see by connectedness of Z that the complement of f(Y ) in Z is open, soassume that some point z of Z has no preimage by f . Then the whole com-ponent Vj of q−1(V ) containing z must be disjoint from f(Y ) since otherwiseby the previous argument the whole of Vj would be contained in f(Y ) whichis a contradiction, whence the claim. To conclude the proof of the lemmait remains to notice that the preimage of the above Vj is a disjoint union ofsome Ui’s.

Proof of Theorem 2.14: We have already seen above that pH exists as acontinuous 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 eachpoint of V ) endowed by an H-action. The open set p−1

H (V ) ⊂ Y will thenbe isomorphic to a product of V by the discrete set of H-orbits of F , so byapplying 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 acover. Then H = Aut(Y |Z) is a subgroup of G so to see that the cover isGalois it suffices by Proposition 2.11 to see that H acts transitively on eachfibre 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 , whichis equivalent to saying that the subset S = y ∈ Y : f(y) = f(φ(y)) isequal to the whole of Y . But this follows from Proposition 2.3, applied toour actual Y , Z and f as well as g = f φ.

It is immediate that the two constructions above are inverse to each otherso only the last statement remains. One implication is easy: if H is normalin G, then G/H acts naturally on Z = Y/H and this action preserves theprojection q so we obtain a group homomorphism G/H → Aut(Z|X) whichis 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 thisassumption any element φ of G = Aut(Y |X) induces an automorphism of Zover X. In other words, we need an automorphism ψ : Z → Z which can be

3.. THE MAIN THEOREMS OF GALOIS THEORY FOR COVERS 43

inserted into the commutative diagram:

Yφ

−−−→ Y

f

yyf

Z Z

qy

yq

Xid

−−−→ X

For this take a point y ∈ Y with image x = (qf)(y) inX. By commutativityof the diagram f(y) and f(φ(y)) are in the same fibre q−1(x) of q. SinceAut(Z|X) acts transitively on the fibres of q, there is an automorphismψ ∈ Aut(Z|X) with the property that ψ(f(y)) = f(φ(y)). In fact, ψ isthe unique element of Aut(Z|X) with this property for if λ ∈ Aut(Z|X) isanother one, Lemma 2.1 implies that ψ λ−1 is the identity. We contendthat ψ is the map we are looking for, i.e. the maps ψ f and f φ are thesame. Indeed, both maps are continuous maps from the connected space Yto Z which coincide in the point y and moreover their compositions with qare equal, so the assertion follows from Proposition 2.3. Now it is manifestthat in this way we obtain a homomorphism G → Aut(Z|X). Its kernel isnone 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 theclassical formulation of Galois theory. Now we turn to the analogues of thetwo versions of the Main Theorem of Galois Theory formulated in Chapter1. This is only possible under some restrictions on the base space X.

So assume henceforth that X is a connected and locally simply connectedspace and fix a base point x ∈ X. Consider the set-valued functor Fibx onthe category CovX of covers of X which associates to a cover p : Y → X thefibre p−1(x). This is indeed a functor since any morphism of covers respectsby definition the fibre over x.

Theorem 3.1 Let (X, x) be a pointed topological space as above.

1. The functor Fibx is representable by a Galois cover π : X → X.

2. The group Aut(X|X) is naturally isomorphic to the fundamental groupπ1(X, x).


Definition 3.2 A cover π : X → X as in the theorem is called a universalcover of X.

By definition, the universal cover has the property that cover maps fromπ : X → X to a fixed cover p : Y → X correspond bijectively (and in afunctorial way) to points of the fibre p−1(x) ⊂ Y . In particular, since Xitself 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 acanonical element x of the fibre π−1(x) called the universal element. Nowfor an arbitrary cover p : Y → X and element y ∈ π−1(x) the cover mapπy : X → Y corresponding to y via the isomorphism Fibx(Y ) ∼= Hom(X, Y )maps x to y by commutativity of the diagram

Hom(X, X)∼=−−−→ Fibx(X)

yy

Hom(X, Y )∼=−−−→ Fibx(Y )

where the vertical maps are induced by πy.Since X is the representing object of a fuctor, it is unique up to unique

isomorphism according to the Yoneda Lemma; note, however, that it a pri-ori depends on the base point x. In fact, Corollary 3.8 below will showthat changing the base point induces a (generally non-unique) isomorphismbetween the two corresponding universal covers.

We postpone the proof of the theorem to the next section. It is there thatwe shall also show the following a priori non-obvious statement:

Lemma 3.3 Under the assumptions of the theorem, assume that p : Y → Xis a cover of X and q : Z → Y is a cover of Y . Then p q : Z → X is acover of X.

In the rest of the section we take the statements of Theorem 3.1 andLemma 3.3 for granted and discuss some important corollaries and comple-ments. For these we need the following easy lemma whose proof is left to thereaders.

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;


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 thetheorem, then so does any connected cover p : Y → X.

Proposition 3.5 Take (X, x) as in the previous theorem, and let p : Y → Xbe a connected cover. Then the universal cover X of X is a universal coverof Y , i.e. it represents Fiby for some y ∈ p−1(x).

Proof: Since X represents Fibx, the point y corresponds to a canonicalmap πy : X → Y which turns X into a cover of Y by virtue ofLemma 2.15.Our task is to show that this cover represents the functor Fiby. So take acover q : Z → Y and pick a point z ∈ q−1(y). Since by Lemma 3.3 thecomposition p q turns Z into a cover of X with z ∈ (p q)−1(x), the point zcorresponds to a morphism πz : X → Z of covers of X mapping the universalpoint x of X to z. It is now enough to see that πz is also a morphism ofcovers of Y , i.e. πy = q πz. But p πy = p q πz by construction andmoreover both πy and q πz map the universal point x to y so the assertionfollows from Lemma 2.3.

Corollary 3.6 The space X is its own universal cover; hence it is simplyconnected.

Proof: Only the simply connectedness needs a proof but it follows fromthe second statement of the theorem in view of π1(X) ∼= Aut(X|X) ∼= 1.

Combining the above results with Theorem 2.14, we get the followinganalogue of the first version of the main theorem of Galois theory discussedin Chapter 1.

Theorem 3.7 Let X be a connected, locally connected and locally simplyconnected topological space and x ∈ X a fixed base point. Then for eachconnected cover p : Y → X and base point y ∈ p−1(x) there is a naturalinjective homomorphism π1(Y, y) → π1(X, x) which endows the universalcover X with an action of π1(Y, y) so that we have X/π1(Y, y) ∼= Y . Inthis way we get a bijection between connected covers of Y and subgroups ofπ1(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 mapinduced by the projection p. See Remark 4.8 (1).


The theorem allows one to give a simple criterion for a cover to be uni-versal which also settles the question of independence of the choice of basepoint.

Corollary 3.8 For X as in the theorem, a connected cover π : X → X is auniversal cover of X (i.e. it represents Fibx for some x ∈ X) if and only ifX is simply connected.

Proof: We have already seen one implication in Corollary 3.6. The otherimplication follows from the Theorem since it implies the chain of isomor-phisms X ′ ∼= X ′/π1(X, x) ∼= X with some universal cover X ′ and pointx ∈ X.

Of course, what is really hidden behind this corollary is the independenceof the fundamental group of base points. This will become more transparentin 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 firsttwo examples in 2.7 actually give universal covers of the circle and of lineartori, respectively. On the other hand, the third example there does not givea universal cover since C∗ is not simply connected. However, the complexplane C, being a 2-dimensional R-vector space, is simply connected and theexponential map C → C∗, z 7→ exp(z) is readily seen to be a cover. HenceC is the universal cover of C∗.

Example 3.10 Let us investigate in more detail an example that will servein 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 tothe left half plane L = z ∈ C : Re z ≤ 0 furnishes a universal coverof D. This map is in fact a homomorphism from the additive group L tothe multiplicative group D; its kernel is isomorphic to Z, being given byinteger 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 anyconnected 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; thecorresponding covers of D are L and D itself via the map z 7→ zk.

We next state the second form of the main theorem of Galois theoryfor covers. So that it can be completely analogous to the field case, it isconvenient to make some convention about left and right actions which isenabled by the following definition.


Definition 3.11 Let G be any group. The opposite group of G, denoted byGop, is the group having the same elements as G but which has the productrule (α, β) 7→ βα.

The help of this definition is that any right action by a group on some setor space defines canonically a left action of the opposite group, simply by therule (α, x) → xα. Now in our present situation, in which the fundamentalgroup plays the role of the absolute Galois group, it is a right action ofπ1(X, x) on the fibres which naturally arises, in contrast with the left actionin the previous chapter.

Indeed, observe that if p : Y → X is a cover, then the fibre p−1(x) isequipped with a natural right action by π1(X, x) as follows: a point y ∈p−1(x) corresponds by Theorem 3.1 to a homomorphism πy : X → Y ; on theother hand, the natural left action of π1(X, x) on X via the isomorphism inTheorem 3.1 (2) induces a right action on cover maps to Y : here the actionof α ∈ π1(X, x) maps πy to πy α. This action is called the monodromyaction on the fibre p−1(x). (Again we shall see in the next section how toobtain 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 acanonical 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 func-tor Fibx induces an equivalence of the category CovX with the category ofleft 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 andq : Z → X, any map f : Fibx(Y ) → Fibx(Z) of π1(X, x)

op-sets comes froma unique map Y → Z of covers of X. For this we may assume Y , Z areconnected and that actually Y is a quotient of X via the map πy : X → Ycorresponding to some fixed element y ∈ Fibx(Y ). In fact, πy makes Yisomorphic to the quotient of X by the stabiliser Uφ of φ; let ψy : Y → X/Uφ

be the inverse map. Since Uφ injects into the stabiliser of f(φ) via f , thenatural map πz : X → Z corresponding to f(φ) induces a map X/Uφ → Zby passing to the quotient; composing it with ψy gives the required mapY → Z.

For the second condition we have to show that any right π1(X, x)-set Sis isomorphic to the fibre of some cover of X. For S transitive we may takethe quotient of X by the action of the stabiliser of some point; in the general


case we decompose S into its π1(X, x)-orbits and take the disjoint union ofthe corresponding covers.

Remark 3.13 Notice the discrepancy with Chapter 1, Theorem 3.4: here weobtain an equivalence of categories whereas there we had an anti-equivalence.This problem will be resolved in Chapter 5 where we introduce the categoryof finite etale k-schemes which is the opposite category to that of finite etalek-algebras; for the k-schemes we will thus indeed get an equivalence of cate-gories with the category of Gal (k)-sets.

We now state a corollary of the above theorem that is even closer toChapter 1, Theorem 3.4 in its formulation and which will be invoked insubsequent chapters. First a definition: call a cover Y → X finite if it hasfinite 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

of finite continuous left π1(X, x)op-sets. Connected covers correspond to finiteπ1(X, x)op-sets with transitive action and Galois covers to coset spaces of open

normal subgroups.

Here π1(X, x)op denotes the profinite completion of π1(X, x)op (Chapter

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 byπ1(X, x)op factoring through a finite quotient. To show it is continuous, it

suffices to show that the stabiliser of any point, which is a subgroup of finiteindex, contains a normal subgroup of finite index, for then these are both

open subgroups of π1(X, x)op by definition. But the latter property holds forany group (take the intersection of conjugates of a subgroup of finite index).

Conversely, any continuous action of π1(X, x)op on a finite set factorsthrough a finite quotient, which is also a quotient of π1(X, x)

op, whence theresult.

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 havetwo canonical left group actions on the fibre p−1(x): one is the action byAut(Y |X) and the other is the monodromy action by π1(X, x)

op. We contendthat 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 againTheorem 3.1 to identify y with a morphism πy : X → Y of covers; φ(y) thenbecomes just φ πy and yα becomes πy α where α is again viewed as anautomorphism of X. Thus the desired compatibility is simply a consequenceof 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 statementsuggests, the universal cover must have something to do with path homo-topies, so we shall make use of their theory and refer the readers in need toChapter 0 for background.

The crucial lemma to be used in construction is the following importantresult 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 pathf : [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 thesame endpoint as f , i.e. we have f(1) = g(1).

Actually, the proof will show that in the second situation the liftings fand g are homotopic but this will not be needed later.

Proof: For the first statement, note first that uniqueness follows fromProposition 2.3 applied with X, Y and Z replaced by our actual X, [0, 1] andY . The existence is immediate in the case of a trivial cover. To reduce thegeneral case to this, for each x ∈ f([0, 1]) choose some open neighbourhoodVx satisfying the condition in the definition of a cover. The sets f−1(Vx) forman open covering of the interval [0, 1] from which we may extract a finite sub-covering since the interval is compact. We may then choose a subdivision0 = t0 ≤ t1 ≤ . . . ≤ tn = 1 of [0, 1] such that each closed interval [ti, ti+1] iscontained in some f−1(Vx) (simply put a ti where two open intervals of theform f−1(Vx) overlap); hence the cover is trivial over each f([ti, ti+1]). Wecan now construct f inductively: given a lifting fi of the path f restrictedto [t0, ti] (the case i = 0 being trivial), we may construct a lifting of therestriction of f to [ti, ti+1] starting from fi(ti); piecing this together with fi

gives fi+1.


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 liftingh : [0, 1] × [0, 1] → Y with p h = h, h(0, t) = f(t) and h(1, t) = g(t). Theconstruction is similar to that for f : first choose a sufficiently fine subdivisionof [0, 1] × [0, 1] into small subsquares Sij so that over each h(Sij) the coveris trivial. That this may be done is assured by a well-known fact fromthe 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). Thenproceed by piecing together liftings over each subsquare, moving “serpent-wise” from the point (0, 0) (for which we put h(0, 0) = y) towards the point(1, 1). Note that by uniqueness of path lifting it is sufficient to find a locallifting which coincides with the previous one at the left corner of the sidewhere two squares meet; they will then coincide over the whole of the commonside. Again by uniqueness of path lifting we get successively that the patht 7→ h(0, t) is f (since both are liftings of f starting from y), the paths 7→ h(s, 0) is the constant path [0, 1] → y and that t 7→ h(1, t) is none butg. Finally, s 7→ h(s, 1) is a path joining f(1) and g(1) which lifts the constantpath [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-connectedspace is trivial.

Proof: It is enough to show that given a connected cover p : Y → X of aspace 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 soby Chapter 0, Corollary 4.8 it is actually path-connected. Now assume thereexist points y0 6= y1 in Y with p(y0) = p(y1). By path-connectedness of ythere is a path f : [0, 1] → Y with f(0) = y0 and f(1) = y1. The path fis the unique lifting starting from y0 of the path f = p f which is a closedpath around x = p(y0) = p(y1). Since X is simply connected, this pathis 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 acontradiction with the second statement of the previous lemma according towhich 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 coversp : Y →X and q : Z→Y , their composite p q : Z→X is also a cover of X.


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 trivialcover of U . Repeating this argument for q over each of the connected com-ponents of p−1(U) (which are simply connected and locally path-connectedthemselves, 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 thespace X as follows. The points of X are to be homotopy classes of pathsstarting from x. The projection π is defined in the following straightforwardfashion: any point y ∈ X is represented by a path f : [0, 1] → X withf(0) = x and we put π(x) = f(1) = y. Note that this gives a well-definedmap by the second statement of Lemma 4.1. We define next the topologyon X by taking as a basis of open neighbourhoods of a point y the followingsets U : we start from a simply connected neighbourhood U of π(y) and iff : [0, 1] → X is a path representing y, we define Uy to be the set of homotopyclasses of paths obtained by composing the homotopy class of f with thehomotopy 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 havingthe same endpoints have the same homotopy class. Thus in more picturesqueterms, U is obtained by “continuing homotopy classes of paths arriving at yto other points of U”. This gives indeed a basis of open neighbourhoods of yfor given two neighbourhoods Uy and Vy, their intersection Uy ∩ Vy containsWy with some simply connected neighbourhood W of y contained in U ∩ V ;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 inverseimage by π of any simply connected neighbourhood of a point y will be thedisjoint union of the open sets Uy for all inverse images y of y. Finally notethat there is a “universal element” x of the fibre π−1(x) corresponding to thehomotopy class of the constant path.

Lemma 4.4 The space X is connected.

Proof: It is enough to see that X is path-connected, for which we showthat there is a path in X connecting the universal point x to any other pointx′. Indeed, let f : [0, 1] → X be a path representing x′. The multiplicationmap m : [0, 1]× [0, 1] → [0, 1], (s, t) 7→ st is continuous, hence so is f m andthe restriction of f m to any subset of the form s×[0, 1]; such a restrictiondefines a path fs from x to f(s), with f0 the constant path [0, 1] → x andf1 = f . The definition of the topology of X implies that the map associatingto s ∈ [0, 1] the homotopy class of fs is continuous and thus defines the


path we need; in fact, it is the unique lifting of f to X beginning at x.Alternatively, one may start by taking this unique lifting and check by goingthrough the construction that its endpoint is indeed x′.

Lemma 4.5 The cover X → X defined above represents the functor Fibx.

Proof: We have to show that for any cover p : Y → X any point y of thefibre p−1(x) corresponds in a canonical and functorial manner to a morphismπy of covers from π : X → X to p : Y → X. Now if πy exists, the fact that itshould be a morphism of covers implies that for any point x′ ∈ X, its imageπy(x

′) should map by p to the endpoint f(1) of any path f : [0, 1] → Xrepresenting x′. So there is only one reasonable choice for πy: given a pointx′ ∈ X represented by a path f : [0, 1] → X, we map it to f(1) wheref : [0, 1] → Y is the unique path lifting f with f(0) = y whose existenceis guaranteed by the first part of lemma 4.1. The second part of the lemmaimplies that this map is well defined; there is no difficulty in checking thatthis is indeed a map of covers. The map y 7→ πy is a bijection betweenp−1(x) and the set of morphisms from π : X → X to p : Y → X foran inverse is given by mapping a morphism f to the image f(x) of thedistinguished element x. Finally, we have obtained an isomorphism betweenthe functors Y → Fibx(Y ) and Y → Hom(X, Y ). Indeed, for any morphismof p : Y → X to some cover p′ : Y ′ → X mapping y to some y′ ∈ Y ′ theinduced map Hom(X, Y ) → Hom(X, Y ′) maps πy to πy′ since these are themaps mapping x to y and y′, respectively.

Lemma 4.6 The cover π : X → X is Galois.

Proof: By Remark 2.12 it is enough to show that Aut(X|X) acts transi-tively on the fibre π−1(x). By Lemma 4.5 for any point y of the fibre π−1(x)we have a continuous map πy : X → X compatible with π and mappingthe universal element x to y. We show that πy is an automorphism. SinceX is connected, by Lemma 2.15 πy endows X with a structure of a coverover itself – in particular it is surjective. Take an element z ∈ π−1

y (x). Since

π πy : X → X is also a cover of X according to Lemma 3.3 (proven inCorollary 4.3), we may apply Lemma 4.5 to this cover to get a continuousand surjective map πz : X → X with πz(x) = z and π πy πz = π. Butπy πz(x) = x, hence πy πz is the identity map of X by Lemma 2.3. Bysurjectivity 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 ina separate lemma:


Lemma 4.7 There is a natural isomorphism Aut(X|X) ∼= π1(X, x).

Proof: First observe that X is endowed by a natural left action of π1(X, x)defined as follows: given a point x′ ∈ X and an element α ∈ π1(X, x) withrespective path representatives f and fα, we may take the composition fα •fand then take the homotopy class of the product. It is straightforward tocheck that the map φα : X → X thus obtained is continuous and compatiblewith π, i.e. it is a cover automorphism. Moreover, we have obtained a grouphomomorphism π1(X, x) → Aut(X|X) which is injective since any nontrivialα moves the distinguished element x. It remains to prove the surjectivity ofthis homomorphism. For this purpose, take an arbitrary φ ∈ Aut(X|X) anda point x′ ∈ X represented by some path f : [0, 1] → X. The point φ(x′) isthen represented by some g : [0, 1] → X. Now g•f−1 is a closed path aroundx in X with (g • f−1) • f = g. Let α be the class of g • f−1 in π1(X, x); weshow that φα = φ. Indeed, the automorphism φ φ−1

α fixes x′ and thus it isthe identity by connectedness of X and Lemma 2.1.

Remarks 4.8 We conclude this section by showing how the above construc-tion makes some statements of the previous section more explicit.

1. First we clear up the issue invoked after the statement of Theorem3.7. Namely, if p : Y → X is a connected cover, then by virtue ofthe theorem it is a quotient of X by some subgroup H of π1(X, x).On the other hand, since X is also a universal cover of Y , we haveH ∼= 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. Nowif we choose y in the fibre over x, we claim that iy is nothing but thenatural inclusion induced by mapping closed paths passing through yto paths through x. Indeed, take a point x′ ∈ X represented by thepath f : [0, 1] → X. We know by construction that x′ as a point ofthe universal cover of Y is represented by the homotopy class of somepath in Y starting from y. This path, however, must map to f by theprojection 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 ofx′ under the action of π1(Y, y) are given by composing f with closedpaths through y; the projection p maps the situation to composing fwith paths through x which is precisely what we had to show.

2. Secondly we give a more concrete description of the monodromy actionof π1(X, x)

op, i.e. the canonical right action of π1(X, x) on the fibrep−1(x) of some connected cover p : Y → X. By Lemma 4.5 any point


y of the fibre corresponds to a morphism of covers πy : X → Y andthe proof shows that πy maps points of X represented by paths f topoints of the form f(1) where f : [0, 1] → Y is the lifting of f withf(0) = y; in particular y, being the image of the class of the constantpath 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 anelement of π1(X, x) is represented by a path fα, it acts on c by mappingit to the class of fα. Hence the action of α on p−1(x) maps y to fα(1),where fα : [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 maintheorem of Galois Theory for covers in terms of locally constant sheaves.Esoteric as these objects may seem to a novice, they stem from reformulatingin a modern language very classical considerations from analysis, such as thestudy of local solutions of holomorphic differential equations. These theoriespredate the invention of covers themselves, so what we present here as areformulation of the theory of the previous chapter in fact gives a naturalframework for the study of older concepts, as the example of the last sectionwill demonstrate.

1. Sheaves

In this section we introduce the notion of a sheaf, to which the first step isthe following definition.

Definition 1.1 Let X be a topological space. A presheaf of sets F on X isa rule which associates to each nonempty open subset U ⊂ X a set F(U) andeach inclusion of open sets V ⊂ U a map ρUV : F(U) → F(V ), the maps ρUU

being identity maps and the identity ρUW = ρV W ρUV holding for a towerof inclusions W ⊂ V ⊂ U . Similarly, one defines a presheaf of groups (orabelian groups, or rings, etc.) by requiring that the F(U) be groups (abeliangroups, rings...) and the ρUV homomorphisms. Elements of F(U) are calledsections of the presheaf over U .

Remark 1.2 Here is a more fancy formulation. Let’s associate a categoryXTop 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 thenatural inclusion V → U whenever V ⊂ U and to be empty otherwise.

55

56 CHAPTER 3. LOCALLY CONSTANT SHEAVES

Then a presheaf of sets is just a contravariant functor XTop → Sets andsimilarly a presheaf of abelian groups, for instance, is a contravariant functorXTop → Ab.

With this interpretation, we see immeadiately that presheaves of sets(abelian groups, etc.) on a fixed topological space X form a category: formorphisms 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 ⊂ Uthe diagram

F(V )ΦV−−−→ G(V )

ρFUV

yyρG

UV

F(U)ΦU−−−→ G(U)

commutes.

Example 1.3 The basic example to bear in mind is that of continuous real-valued functions defined locally on open subsets of X; in this case, the mapsρUV are given by restriction of functions to some open subset.

Motivated by the example, given an inclusion V ⊂ U , we shall also usethe more suggestive notation s|V instead of ρUV (s) for a section s ∈ F(U).

The presheaf in the above example has a particular property, namely thatcontinuous functions may be patched together over open sets. More precisely,given two open subsets U1 and U2 and continuous functions fi : Ui → R fori = 1, 2 with the property that f1(x) = f2(x) for all x ∈ U1 ∩ U2, we mayunambiguously define a function f : U1 ∪ U2 → R by setting f(x) = fi(x) ifx ∈ Ui. In fact, most presheaves we shall encounter in the sequel will sharethis 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 itsatisfies the following two axioms:

1. Given any nonempty open set U and any covering Ui : i ∈ I of U bynonempty open sets, if two sections s, t ∈ F(U) satisfy s|Ui

= t|Uifor

all i ∈ I, then s = t.

2. For any open covering Ui : i ∈ I of U as above, given a system ofsections 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 aspace 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 asheaf. Take a set S containing at least two elements s1 6= s2. Given atopological space X, define a presheaf FS by setting FS(U) = S for allnonempty open sets U ⊂ X and ρUV = idS for all open inclusions V ⊂ U .This is indeed a preshaf; in fact, it is the unique contravariant functor fromXTop to the category constisting of the single object S and single morphismidS. Now if X is not an irreducible topological space, i.e. if there exist twononempty disjoint open subsets U1, U2 in U , then FS is not a sheaf sincethere 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 complexvalued continuous functions defined locally on some open subset of Xform a sheaf of abelian groups, and even a sheaf of rings with the usualring operations for functions. More generally, if Y is any topologicalspace (or abelian group, ring, etc.), continuous functions U → Y de-fined on some open U ⊂ X with the natural restriction operators definea 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 ofholomorphic functions on D to be the sheaf of rings whose sectionsover some open subset U ⊂ D are complex functions holomorphic onU . This construction carries over to any complex manifold. Similarlyone can define the sheaf of analitic functions on some real analyticmanifold, or the sheaf of C∞ functions on a C∞ manifold, and so on.We shall encounter concrete examples of these in the next chapter onRiemann surfaces.

3. Constant sheaves. A special case of the first example is when we takeY to be a discrete topological space (or abelian group, etc.) In thiscase the corresponding sheaves are called constant sheaves: the namecomes from the fact that over a connected open subset U the sectionsare 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: theanomaly encountered there does not arise for the sheaf associates Y ×Yto a disjoint union of nonempty open subsets.


4. Skyscraper sheaves. Here is a more eccentric example. Fix an abeliangroup A and a point x of a given topological space X. Define a presheafof abelian groups Fx on X by the rule Fx(U) = A if x ∈ U andFx(U) = 0 otherwise, the restriction morphisms being the obviousones. This presheaf is easily seen to be a sheaf, called the skyscrapersheaf over x.

2. Locally Constant Sheaves and Their Clas-

sification

Given an open subset U of a topological space X, there is an obvious notionof the restriction of a presheaf F from X to U : one simply considers thesections of F only over those open sets which are contained in U . Thisremark 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 eachpoint of X has an open neighbourhood U such that the restriction of F toU 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. Assumehenceforth that all spaces are locally connected. First a definition which willultimately 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 thatp s = idU.

Now let p : Y → X be a space over X. To this space we can associate apresheaf 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 therestriction 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 locallyconstant. 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 sectionsof Y . (Note that FY depends on Y as a space over X so there is actually aslight abuse of notation here.)

2.. LOCALLY CONSTANT SHEAVES AND THEIR CLASSIFICATION59

Proof: The sheaf axioms follow from the fact that the sections over U arecontinuous functions U → Y and hence satisfy the patching properties. Nowassume p : Y → X is a cover. Given a point x ∈ X, take a connectedopen neighbourhood V of x over which the cover is trivial, i.e. isomorphicto V × F where F is the fibre over x. The image of any section V → Yis a connected open subset mapped isomorphically onto V by p, hence itmust be one of the connected components of p−1(V ). Thus sections over Vcorrespond bijectively to points of the fibre F and the restriction of FY to Vis isomorphic to the constant sheaf defined by F . We get the constant sheafif and only if we may take V to be connected component containing x in theabove argument.

Thus for instance Example 1.6 which shows a simple example of a non-trivial cover also gives, via the preceding proposition, a simple example of alocally constant but non-constant sheaf.

Now we shall show that conversely, given any locally constant shaf F ona space X, one can cook up a cover pF : XF → X whose sheaf of localsections will be exactly F . However, just as before, we shall first present aconstruction working for any presheaf F and then show that in the case oflocally constant sheaves we obtain covers.

To begin with, notice that in the previous construction when we startfrom a cover p : Y → X, then for any point x ∈ X the set FY (U) is just thefibre p−1(x) over x for all connected open neighbourhoods of x over whichthe cover is trivial; thus, roughly speaking, FY (U) is “equal to p−1(x) for allsufficiently small U”. In fact, this statement can be made precise for moregeneral spaces over X as follows. Define a partial order on the system ofopen neighbourhoods of x by setting U ≤ V whenever V ⊂ U . This partiallyordered set is directed since for any U , V we have U, V ≤ U ∩ V . Now thesets FY (U) form a direct system indexed by this partially ordered set withrespect to the restriction maps of the sheaf FY .

Before stating the next lemma, recall that a continuous map p : Y → Xis called a local homeomorphism if any y ∈ Y has an open neighbourhoodmapped homeomorphically onto its image by p. This is the same as sayingthat for any y the image p(y) has an open neighbourhood over which p admitsa continuous section with open image.

Lemma 2.4 Assume that p : Y → X is a local homeomorphism. Thenmapping 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.


Note that covers satisfy the assumption of the lemma.

Proof: Recall that by definition elements of the direct limit on the left aresections of F over some open neighbourhood of x modulo the equivalencerelation that identifies two sections if they coincide over some smaller openneighbourhood. This shows that the map of the lemma is well defined andalso that it is injective since if two sections si : Ui → Y (i = 1, 2) bothmap x to the same point y ∈ p−1(x), then s1(U1) ∩ s2(U2) must be an openneighbourhood of y mapped homeomorphically onto U1 ∩ U2 by p and thesi must coincide over U1 ∩ U2. Finally surjectivity is immediate from theassumption that p is a local homeomorphism, for it implies that any elementy ∈ p−1(x) has an open neighnourhood over which p has a section mappingx to y.

This prompts the following definition.

Definition 2.5 Let F be a presheaf of sets (abelian groups, etc.) on atopological space X and let x be a point of X. The stalk Fx of F at x isthe direct limit of the direct system of sets (abelian groups, etc.) formedby the F(U) for the open sets U containing x and by the natural restrictionmorphisms F(U) → F(V ) attached to the inclusions V ⊂ U .

Notice that any morphism of presheaves F → G induces a natural mor-phism on the stalks Fx → Gx. Hence taking the stalk of a presheaf atsome point defines a functor from the category of presheaves of sets (abeliangroups, 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 apresheaf 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. Thenatural projection pF is then induced by the constant maps Fx → x. Todefine the topology on XF , note first that given an open set U ∈ X, anysection s ∈ F(U) defines a map is : U → XF which maps x ∈ U to theimage of s in the stalk Fx. This being said, we may take on XF the topologyfor which the sets is(U) for all open U ⊂ X and all s ∈ F(U) form a basis ofopen sets. The construction shows that the projection pF is continuous forthis topology; it is even a local homeomorphism as all the is are continuoussections of pF over the appropriate U .

Remark 2.7 Note that pF may have continuous sections which are not ofthe 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 patchtogether to give a section of F over U if F is not a sheaf but the continuousmaps isj

do path together to give a continuous section of pF over U .

We can infer something positive from the previous remark, namely thatany section s ∈ F(U) is naturally a section of the sheaf of local sections ofpF : XF → X over U . The latter sheaf is usually denoted F ] and is calledthe sheaf associated to the presheaf F . Moreover, the above argument givesa natural morphism of presheaves F → F ].

Proposition 2.8 If F is a sheaf, then the sheaf of local sections of the spacepF : XF → X constructed above is isomorphic F : in fact, the natural mapF → F ] is an isomorphism. If moreover F is locally constant, then pFendows XF with the structure of a cover over X.

Proof: The first statement is immediate from the above considerations. Toprove the second one, take a point x ∈ X and a connected open neighbour-hood U of x over which F is constant. Then, as we have seen above, we haveFx

∼= F(U) ∼= Fy for all y ∈ U , hence p−1F (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 otherwords, the sheaf F ] represents the functor on the category of sheaves thatassociates to each G the set of presheaf morphisms F → G.

For any (locally connected) topological space denote by LocX the fullsubcategory of the category TopX of spaces over X whose objects are spacesp : Y → X for which p is a local homeomorphism. Note that the ruleY 7→ FY is a functor from the category LocX to the category of sheaves ofsets: any morphism φ : Y → Z in LocX induces a morphism FY → FZ ofsheaves by composing sections with the map φ (if p : Y → X and q : Z → Xare the respective structure maps, then composing a section s of p with φgives 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 cat-egory LocX with the category of sheaves of sets on X. In this equivalencecovers of X correspond to locally constant sheaves of sets on X.


Proof: It suffices to prove the first statement and again it is convenientto apply the criterion of chapter 0, Lemma 1.9. The second condition fol-lows from the previous proposition. For fully faithfulness, assume given amorphism of sheaves of the form φF : FY → FZ , with p : Y → X andq : Z → X two objects of LocX . For any point x ∈ X, φ induces a mapon 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), wherex = p(y). To see that φ is continuous, choose an open neighbourhood U of ysuch that q|U is a homeomorphism. Then φ−1(U) = p−1(q(U)) is an open setover which φ is equal to the composition of continuous maps q−1 p. Thus φis a map of spaces over X and one checks by going through the constructionsthat it is indeed the unique map Y → Z inducing φF .

In the rest of this section we consider only locally constant sheaves. Com-bining Proposition 2.10 with Theorem 2.14, we get:

Theorem 2.11 Assume X is a connected and locally simply connected topo-logical space and fix a point x ∈ X. Then the category of locally constantsheaves of sets on X is equivalent to the category of sets endowed with a leftaction of π1(X, x)

op. This equivalence is induced by the functor mapping asheaf 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 bya 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 givinga 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 con-stant sheaves of abelian groups on X is equivalent to the category of leftπ1(X, x)

op-modules. This equivalence is induced by the functor mapping asheaf F to its stalk Fx at x.

More generally, for any commutative ring R the category of locally con-stant sheaves of R-modules on X is equivalent to the category of left modulesover 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 sheafof abelian groups is a left π1(X, x)

op-module, i.e. the action of π1(X, x) iscompatible 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 immediatelythat the morphism Fx × Fx → Fx obtained by applying the stalk functoris 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 andσ ∈ 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 byapplying 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 precedingconstruction, which is also the one that first arose historically.

Definition 3.1 A complex local system on X is a locally constant sheaf offinite dimensional complex vector spaces. If X is connected, all stalks musthave 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 topo-logical space and fix a point x ∈ X. Then the category of complex localsystems on X is equivalent to the category of finite dimensional left repre-sentations of π1(X, x)

op.

Thus to give a local system on X is the same as giving a homomorphismπ1(X, x)

op → GL(n,C) for some n. This representation is called the mon-odromy representation of the local system.

The following example shows where to find local systems “in nature”. Ituses the straightforward notion of a subsheaf of a sheaf F : it is a sheaf whosesections over each open set U form a subset (subgroup, subspace etc.) ofF(U).

Example 3.3 Let D be a complex domain (i.e. a connected open subsetof the complex plane). Consider over D a homogenous n-th order lineardifferential equation

y(n) + an−1y(n−1) + . . .+ a1y

′ + y = 0


where the ai are holomorphic functions over D. We can employ a classi-cal trick which consists of considering the y(i) as indeterminates and adding(n − 1) new equations of the form (y(i))′ = y(i−1) for 2 ≤ i ≤ n. Therebysolving the above equation becomes equivalent to solving a system of n ho-mogenous first order linear equations which may be written in the form

x′ = Ax (3.1)

where the variable x is a holomorphic function D → Cn and A is a non-singular 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-valuedholomorphic functions. This sheaf is the n-fold direct product of the sheafO of Example 1.6 (2) with itself and is clearly a sheaf of complex vectorspaces. Now over each U linear combinations of solutions of (3.1) are alsosolutions of (3.1) and given an open covering Ui : i ∈ I of U , any sectionx ∈ 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 ofOn. 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 Dhas an open neighbourhood U ⊂ D for which S(U) is an n-dimensional vec-tor space over C or, in more down-to-earth terms, there exist n holomorphicsolutions x1, . . . , xn ∈ S(U) such that any solution of (3.1) over U can bewritten uniquely as a linear combination of the xi. Since the restrictions ofthe xi to smaller open sets obviously have the same property, we see that therestriction of S to U is a constant sheaf of complex vector spaces. Thus thesheaf S is a local system of dimension n.

Remark 3.4 According to Corollary 3.2, the local system S of the aboveexample is uniquely determined by an n-dimensional left representation ρ ofπ1(X, x)

op, where x is some fixed point of D (notice that D is locally simplyconnected). Let us first describe ρ explicitly. We know that it is given by thecanonical right action of π1(X, x) on the stalk Sx which is an n-dimensionalvector space over C. Take a closed path f : [0, 1] → D representing anelement α ∈ π1(X, x) and take an element s ∈ Sx which is, in classicalterminology, a germ of a (vector-valued) holomorphic function satisfying theequation (3.1). Now s is naturally a point of the fibre over x of the coverpS : DS → D associated to S by Proposition 2.8. The action of α on s isthen 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 lookingat the construction of the unique lifting in the proof of Lemma 4.1, one canmake 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

3.. LOCAL SYSTEMS 65

left to right”(in particular, x ∈ U1 ∩ Uk) and moreover S is constant overeach Ui and there are sections si ∈ S(Ui) such that the restrictions of si andsi+1 to Ui ∩ Ui+1 coincide for all 1 ≤ i ≤ k − 1 and s1 (resp. sk) maps tos (resp. sα) in Sx. Classically this is expressed by saying that sα is theanalytic continuation of the holomorphic function germ s along the the pathf representing α. (Notice that the existence and the uniqueness of sα areguaranteed by the fact that S is a locally constant sheaf and hence DS → Dis a cover. Had we worked with the bigger sheaf On instead of S, the analyticcontinuation of an arbitrary germ may not have been possible.)

Example 3.5 Let us illustrate the above theory by working out the simplestnontrivial case in detail. Take as D an open disc in the complex planecentered around 0, of radius 1 < R ≤ ∞, with the point 0 removed. Wechoose 1 as base point for the fundamental group of D. We study the localsystem of solutions of the first order differental equation

x′ = fx (3.2)

where f is a holomorphic function on D which will be assumed to extendmeromorphically into 0. It is well known that the solutions to (3.2) in someneighbourhood of a point x ∈ D are constant multiples of functions of theform exp F where F is a primitive of f . Thus the solution sheaf is alocally constant sheaf of 1-dimensional complex vector spaces. The reasonwhy it is locally constant but not constant is that, as we learn from complexanalysis, the primitive F exists locally but not globally on the whole ofD. For instance, a well defined primitive F1 of f exists, for instance, overU1 = D \ (0,−iR) and another primitive F2 over U2 = D \ (0, iR); we mayassume that F1 and F2 coincide over U1 ∩ U2, so in particular at -1 (theymay only differ by a constant anyway). Thus the local system of solutionsto (3.2) is isomorphic over each Ui (i = 1, 2) to the constant sheaf defined bythe 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 exam-ple in 3.9) that π1(D, 1) ∼= Z; a generator α is given by the class of the pathg : [0, 1] → D, t 7→ e2πit which “goes counterclockwise around the unit cir-cle”. 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 bythe 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 functiongerm φ defined in a neighbourhood of 1 and satisfying (3.2) with x = φ, theanalytic 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


along g, we obtain exp F2 (since we have to switch from F1 to F2 when thepath “goes out of U1”). Thus

m = exp (F1(1))(exp (F2(1)))−1 = exp (F1(1) − F1(−1) + F2(−1) − F2(1))

= exp (∫ 1

0f 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 occuringin the equation (3.2). Conversely, one sees that for any one-dimensionalmonodromy 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 maytake, for example, x′(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 moredetails.)

Remark 3.6 The above example is the solution of the so-called Riemann-Hilbert problem in the simplest case. This question, a special form of whichwas first raised by Riemann in 1857 and which later became the 21st ofthe famous Hilbert problems, asks the following: given a finite set of pointsx1, . . . , xk ∈ C, does any n-dimensional right representation ρ of π1(C \x1, . . . , xk, x) (with some x 6= xi) arise as the monodromy representationof the local system of solutions to a system of n homogenous linear differentialequations with coefficients having poles only at the xi? Moreover, it is re-quired that the system should be of Fuchsian type (which means roughly thatthe local behaviour of the solution sheaf in the neighbourhood of points con-tained in x1, . . . , xn∪∞ should be like in the example above; see Forster[1] for a precise formulation). Riemann and Hilbert themselves showed thatthe answer was positive in some particular cases. The affirmative answer ingeneral came as early as 1908 by work of Plemelj [1]. However, the area hasremained a very active field of research up to the present day, for far-reachinggeneralisations have been obtained since. As a milestone we may mentionDeligne’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 tocome. Indeed, Riemann surfaces furnish the best examples for the theory ofcovers which we have developed so far and at the same time motivate thealgebraic theory of the forthcoming chapters. In particular, as we shall see,the theory of Riemann surfaces subsumes the theory of complex algebraiccurves, so the study of the former yields sometimes motivation, sometimesingredients 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 calledholomorphic if it is holomorphic in each variable; a map f : U → Cm isholomorphic if all of its coordinate functions fi : U → C are holomorphicfunctions for i = 1, . . . , m.

Let now X be a connected Hausdorff space. A complex atlas of dimensionn on X is an open covering U = Ui : i ∈ I of X together with mapsfi : Ui → Cn mapping Ui homeomorphically onto an open subset of Cn

such that for each pair (i, j) ∈ I2 the map fj f−1i : fi(Ui ∩ Uj) → Cn is

holomorphic. The maps fi are called coordinate charts.Two complex atlases U = Ui : i ∈ I and U ′ = U ′i : i ∈ I ′ on X are

equivalent if their union (defined by taking all the Ui and U ′i as a coveringof X together with all coordinate charts) is a also complex atlas. Note thatthe extra condition to be satisfied here is that the maps f ′j f

−1i should be

holomorphic on fi(Ui ∩ U′j) for all Ui ∈ U and Uj ∈ U ′.

Definition 1.1 An n-dimensional complex manifold is a connected Haus-

67

68 CHAPTER 4. RIEMANN SURFACES

dorff space X together with an equivalence class of complex atlases. A one-dimensional complex manifold is called a Riemann surface.

We shall often refer to the equivalence class of atlases occuring in theabove 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 complexmanifold 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 withthe structure of a Riemann surface; one may similarly turn arbitraryeven-dimensional linear tori into complex manifolds. Consider C as a2-dimensional real vector space and let c1, c2 ∈ C be a basis. The cigenerate a discrete subgroup Λ of C isomorphic to Z × Z; let T bethe topological quotient. Now cover C by sufficiently small open discsDi 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 ofT by definition of the quotient topology and the projection maps Di

homeomorphically onto its image. The images of the Di thus form anopen covering of T . (Since T is compact, being the continuous image ofthe fundamental parallelogram spanned by the ci, one can even extracta finite subcovering out of the above one.) It is straightforward to checkthat we have thus obtained a complex atlas.

3. Projective spaces. Consider Cn+1 \ (0, . . . , 0) as a topological spaceand 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 aconnected Hausdorff space; it is called the n-dimensional complex pro-jective 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∗; wecall the (n + 1)-tuple (x0, . . . xn) the homogenous coordinates of thepoint. Thus points of Pn(C) correspond bijectively to 1-dimensionalsubspaces of the complex vector space Cn. Now define a complex atlason Pn(C) as follows. Let Ui be the open subset whose points havehomogenous 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 givenby mapping the point (t1, . . . , tn) ∈ Cn to the point of Pn(C) havinghomogenous coordinates (t0, . . . , ti−1, 1, ti, . . . tn). One checks immedi-ately that the maps (fj f

−1i )|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 twoantipodal points cut out from S by the line corresponding to x. SinceS is known to be compact, Pn(C) as its continuous image by p iscompact as well.

The one-dimensional case is the projective line P1(C): as a topologicalspace, it is just C plus a point ∞ having homogenous coordinates(0, 1); the complex atlas consists of the two open sets U0 = C andU1 = P1(C) \ 0; the holomorphic map f1 f

−10 on C \ 0 is the

function z 7→ 1/z.

4. Smooth plane curves. Let X be the closed subset of C2 defined as thelocus of zeros of a polinomial f ∈ C[x, y]. Assume that there is nopoint of X where the partial derivatives ∂xf and ∂yf both vanish. Wecan then endow X with the structure of a Riemann surface as follows.In the neighbourhood of a point where ∂yf is nonzero, define a complexchart by mapping a point to its x-coordinate and similarly for pointswhere ∂xf is nonzero we take the y-coordinate. By the inverse functiontheorem for holomorphic functions, in a sufficiently small neighbour-hood the above mappings are indeed homeomorphisms. Secondly, theholomorphic version of the implicit function theorem implies that in theneighbourhood of points where both x and y define a complex chart,the transition function from x to y is holomorphic, i.e. when ∂yf doesnot vanish at some point, we may express y as a holomorphic functionof 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 3variables, i.e. F (tx, ty, tz) = tdF (x, y, z) for all t 6= 0, then F induces awell defined function on P2(C), we may thus regard its locus of zeroesY in P2(C) which is a closed, hence compact subset. As above, if thereis no point in P2(C) where all partial derivatives of F vanish, then Ycan be endowed with the structure of a (compact) Riemann surface. Infact, since this is a local task, we may carry out the above construction


on the three canonical copies of C2 covering P2(C); on the intersectionswe of course get the same result.

Of course, one can find lots of polynomials satisfying the condition onpartial derivatives: the Fermat curves defined by xm + ym + zm are onefamily of examples.

Remark 1.3 The last example above may be generalised to a system ofm (homogenous) polynomial equations in Cn (or Pn(C)): if their Jacobianmatrix has rank n−m in each point, we get a complex manifold.

Next we define mappings of complex manifolds, beginning with mappingsto C.

Let X be a complex manifold. A holomorphic function on an open subsetU ⊂ X is a continuous map f : U → C such that for any open set Ui belong-ing to an atlas defining the complex structure on X the map (f f−1

i )|fi(U∩Ui)

is holomorphic. By the definition of a complex atlas this notion is indepen-dent of the atlas chosen. The holomorphic functions on U form a ring O(U)with respect to the natural addition and multiplication of functions. Thusthe rule U 7→ O(U) together with the natural restriction maps for functionsdefines 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 thereexist open sets U1 ⊂ X1 and U2 ⊂ X2 containing respectively x1 and f(x1)and belonging to some atlas defining the complex structures onX1 (resp. X2)such that the induced map f2 φ (f1)

−1|f1(U1) is holomorphic. Again onechecks 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 eachpoint of X1. Complex manifolds together with holomorphic maps thus forma 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 holomorphicmap U → C (with C endowed by its canonical complex structure as in thefirst example above).

Now we give a second description of the category of complex manifoldsusing 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 settingφ∗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 holomorphicfunction 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 holo-morphic if and only if the induced morphism O2 → φ∗C1 takes its values inthe subsheaf φ∗O1 of φ∗C1.

Proof: For necessity we need to note merely that if φ is holomorphic, thenf φ is holomorphic for each f ∈ O2(U). Sufficiency follows from the factthat if a neighbourhood U2 ⊂ X2 of a point φ(x1) ∈ φ(X1) belongs to acomplex atlas, the coordinate chart f2 : U2 → Cn is given by n holomorphicfunctions on U2. The assumption means that these holomorphic functionsgive 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 abovefunctions with f−1

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 connectedHausdorff 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 Λare all homeomorphic, but the complex structures we defined in the aboveexample may differ for different Λ. Thus the statement of the followingcorollary is nonvacuous.

Corollary 1.6 The complex structure on a given complex manifold X isuniquely determined by the underlying topological space and by the sheaf Oof holomorphic functions.

Proof: By unwinding the definitions, one sees that two complex atlases Uand U ′ on the same topological space X are equivalent (and thus define thesame complex structure) precisely if the identity map of X is a holomorphicmap between the manifolds (X,U) and (X,U ′). By the previous lemma, thiscondition may be checked by looking at the sheaves of holomorphic functionswith respect to the two complex structures.

Remark 1.7 The above corollary motivates the general definition of com-plex analytic spaces as found in Gunning [1], for instance.


2. Branched covers of Riemann Surfaces

In this section we study holomorphic maps between Riemann surfaces. Hence-forth we shall always tacitly assume that the maps under consideration arenon-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→ zk

defines a cover of C× by itself but its extension to C does not. But in bothcases we get not only a continuous map but a holomorphic map of Riemannsurfaces. The next proposition shows that locally any holomorphic map ofRiemann surfaces looks like this, from which we shall be able to deduce thetopological properties of holomorphic mappings.

Proposition 2.1 Let φ : Y → X be a holomorphic map of Riemann sur-faces. Denote by D the unit disc in C. Then for any point y ∈ Y with imagex 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

Uyφ

−−−→ Ux

fy

yyfx

Dz 7→zk

−−−→ D

commutes.

Proof: By performing affine linear transformations in C and by shrinkingUx and Uy if necessary, one may find charts fy and fx satisfying all conditionsof the proposition except perhaps the last one. In particular, fx φ f−1

y

is a holomorphic function in a neighbourhood of 0 which vanishes at 0. Assuch, it must be of the form z 7→ zkH(z) where H is a holomorphic functionwith H(0) 6= 0. It is a known fact from complex analysis that by shrinkingUy 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) logH)), where log is a branch ofthe logarithm in a neighbourhood of H(0) 6= 0). Thus replacing fy by itscomposition 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).

2.. BRANCHED COVERS OF RIEMANN SURFACES 73

Proof: Indeed, the map z → zk is open.

Definition 2.3 The integer k of the proposition is called the ramificationindex or branching order of φ at y and is denoted by ey. The points y withey > 1 are called branch points. We denote the set of branch points of φ bySφ.

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 puncturedneighbourhood in which φ is one-to-one and any branch point has an openneighbourhood 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 sur-faces. 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

∑

y∈φ−1(x)

ey = n,

where n is the cardinality of the fibres of the previous cover.

Proof: The first statement follows from the previous corollary since discretesubsets of a compact space are finite. The second statement holds becauseφ(X) is open in Y (by Corollary 2.2), but it is also closed (being the contin-uous image of X which is compact) and Y is connected. For (3) note that byProposition 2.1 any of the finitely many preimages of x ∈ X \ φ(Sφ) has anopen neighbourhood mapping homeomorphically onto some open neighbour-hood of x; the intersection of these is a distinguished open neighbourhood ofx as in the definition of a cover. For (4), given x ∈ X, by the same argumentwe find an open neighbourhood Uy of any preimage y of x that surjects ontoa suitably small fixed open neighbourhood U of x and such that φ behavesin Uy as in Proposition 2.1. Over x′ ∈ U \ x the map φ behaves as a coverand has n distinct preimages by statement (3). But for each preimage y of


X, the open neighbourhood Uy contains exactly ey preimages of x′, as themap z 7→ zey has the same property in a sufficiently small open disc aroundthe origin in C. The statement follows from this by summing over y.

Continuous maps Y → X with the topological properties stated in theproposition are usually called finite branched covers or finite ramified coversof compact Riemann surfaces.

We now would like to prove a converse, and show that if we throw awayfinitely many points (or none) from a compact Riemann surface X, any finitecover of the remaining space can be embedded into a compact Riemannsurface Y mapping holomorphically onto X such that the map Y → Xextends 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 connectedcover of X as a topological space. Then Y can be endowed with a uniquecomplex structure for which p becomes a holomorphic mapping.

In fact, the proof will show that it is enough to require that p is a localhomeomorphism.

Proof: Any point y ∈ Y has a neighbourhood V that projects homeomor-phically onto a neighbourhood U of p(y). Take a complex chart f : U ′ → Cwith U ′ ⊂ 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 followsfrom the fact that for any complex structure on Y the restriction of p to Umust be an analytic isomorphism.

Notice that the lemma solves our problem in the case where we throwaway no points. Now comes the general case.

Proposition 2.7 Assume given a compact Riemann surface X, a finite setx1, . . . , xn of points of X and a finite connected cover φ′ : Y ′ → X ′, whereX ′ = X \ x1, . . . , xn. Then there exists a compact Riemann surface Ycontaining Y ′ as an open subset and a holomorphic map φ : Y → X withφ|Y ′ = φ′.

Proof: We assume for simplicity that n = 1; for n > 1 one performs thesame construction as below simultaneously for all points x1, . . . , xn and getsa similar result.

By performing an affine linear transform in C if necessary, we may finda complex chart mapping a connected open neighbourhood U of x = x1

2.. BRANCHED COVERS OF RIEMANN SURFACES 75

homeomorphically onto the open unit disc D ⊂ C. The restriction of φ′ toφ′−1(U \ x) is a finite cover, hence φ′−1(U \ x) decomposes as a finitedisjoint 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 discD = D \ 0 via the complex chart and hence by Chapter 2, Example 3.10it must be isomorphic to the cover D → D given by z 7→ zk for somek > 1 (in fact, we considered the closed disc there but the open case followsimmediately from it). Now choose “abstract” points yi for each i and defineY as the disjoint union of Y ′ with the yi. Define an extension φ of φ′ toY by mapping the yi to x and a topology on Y by taking as a basis ofopen neighbourhoods of each yi the inverse image by φ|Vi∪yi of a basisof open neighbourhoods of x contained in U . Finally, endow Y ′ with thecomplex structure defined in the previous lemma and define a complex chartcontaining each yi by extending the isomorphism of Vi with D discussed abovesuch that yi maps to 0. It follows from the proof of the previous lemma thatthese “new charts” are holomorphically compatible with those of the lemma,so that Y becomes a Riemann surface with this structure. Moreover, φ is aholomorphic map as it is holomorphic away from the yi by the lemma and inthe neighbourhood of these looks like z 7→ zk. We also see that Y is compact,for it is mapped onto the compact space X by the continuous surjection φwhich is an open mapping as well (according to Corollary 2.2).

In fact, the Riemann surface Y is unique up to isomorphism, as thefollowing result will imply.

Proposition 2.8 In the situation of the previous proposition, assume givena holomorphic map of compact Riemann surfaces ψ : Z → X having branchpoints only over the xi, such that there is an isomorphism of covers λ :

Y ′∼=→Z ′, where Z ′ = 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 previousproof. Let Wi be a connected component of ψ−1(U). Then we may find aconnected component of φ−1(U \ x), say Vi, such that λ(Vi) ⊂ Wi. Wecontend that Wi \ λ(Vi) consists of a single element zi ∈ ψ−1(x). Indeed, ifit contained another element z′i, then the ramification index of ψ at both zi

and z′i would be equal to the cardinality of the fibres of the connected coverof U \ x obtained by restricting ψ to λ(Vi). Choosing points in the fibreψ−1(x) lying in other components and applying the same argument, we seethat 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


the other components. It is immediate that in this way we obtain a mapof spaces over X which is compatible with complex structures (for complexcharts which map both Vi and λ(Vi) to a punctured disc, the extension justmaps 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 ′ → X ′ extends uniquely to an auto-morphism of Y over X.

Remark 2.10 By the second statement of the corollary, the automorphismgroup of Y as a space over X is the same as that of Aut(Y ′|X ′). Therefore itmakes sense to call Y a finite Galois branched cover of X if Y ′ is Galois overX ′. 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 onX is a holomorphic function f on an open subset X \ S, where S ⊂ X is adiscrete set of points, such that for any complex chart φ : U → C containingsome 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 ringwith respect to the usual addition and multiplication of functions; we denotethis ring by M(X). In fact, it is none but the ring of global sections of thesheaf of meromorphic functions MX on X; this sheaf associates to an opensubset U ⊂ X the direct sum of the rings MX(Ui), where the Ui are theconnected 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 givean 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 complexchart containing x we get a holomorphic function on some complex domainwhose set of zeros has a limit point. By the Identity Principle of complex

3.. RELATION WITH FIELD THEORY 77

analysis (Rudin [1], Theorem 10.18) this implies that the function is iden-tically 0, so f is 0 in some neighbourhood of s. Now consider the set ofthose points y of X for which f vanishes identically in a neighbourhood ofy. This set is open by definition but it is also closed for it contains any ofits boundary points by the previous argument. Since X is connected, thisimplies f = 0, a contradiction.

Nonconstant meromorphic functions play a crucial role in the theory ofRiemann surfaces for the following reason.

Lemma 3.3 Any nonconstant meromorphic function f on a Riemann sur-face X defines a holomorphic map X → P1(C) by mapping x to f(x) if x isnot 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 onU \ 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. Thereis 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 RemovableSingularity Theorem (see e.g. Rudin [1], Theorem 10.20).

Thus if X is a compact Riemann surface, then by virtue of Proposition2.5 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 carriesnon-constant meromorphic functions, for this would imply that any compactRiemann surface is a branched cover of P1(C). Surprisingly, this is a deeptheorem of Riemann which we state without proof.

Theorem 3.4 (Riemann’s Existence Theorem) Any compact Riemannsurface X admits a nonconstant meromorphic function. Moreover, for anypoint x we may find a meromorphic function on X that is holomorphic onX \ x.

The theorem can be deduced from the Riemann-Roch theorem for com-pact Riemann surfaces; see e.g. Forster [1], Theorem 14.12 where it is provenusing a related cohomological result. We shall use the second statementthrough the following corollary:

Corollary 3.5 Let x1, . . . , xn be a finite set of points on a compact Riemannsurface X. Then for any sequence a1, . . . , an of complex numbers there existsf ∈ M(X) such that f is holomorphic at all the xi and f(xi) = ai.


Proof: It is enough to find fi with fi(xi) = 1 and fi(xj) = 0 for i 6= jas then we may define f as a linear combination of the fi. To find fi takefirst n − 1 meromorphic functions gj (1 ≤ j ≤ n, j 6= i) such that gj has apole only at xj ; for some constant c ∈ C the sum c +

∑gj takes a non-zero

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 Riemannsurfaces. The map φ induces a map of sheaves of meromorphic functionsMX → φ∗MY ; in particular, taking global sections we get an inclusionM(X) ⊂ M(Y ) of fields of global meromorphic functions.

Proposition 3.6 If φ : Y → X is a holomorphic map of compact Riemannsurfaces which has degree d as a branched cover, then the induced field ex-tension M(Y )|M(X) has degree d.

Proof: We first show that the extension has degree at most d. For thisit suffices to see that any meromorphic function in M(Y ) has degree atmost 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 d′ ≤ 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 primitiveelement. In particular we have M(X)(f) ⊂ M(X)(h), but since h shouldalso have degree at most d′ 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 (notnecessarily irreducible) polynomial equation of degree d over M(X). Let S bethe set of branch points of φ. Any x /∈ φ(S) has some open neighbourhood Usuch that φ−1(U) decomposes as a finite disjoint union of open sets V1, . . . , Vd

homeomorphic to U . Let si be the (holomorphic) section of φ mapping Uonto Vi and put fi = f si. The function fi is then a meromorphic functionon U . Put

F (T ) =∏

(T − fi) = T n + an−1Tn−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 distinguishedopen neighbourhood U1, then on U ∩U1 the coefficients of the polynomial F1

corresponding to the similar construction over U1 must coincide with those ofF since the roots of the two polynomials are the same meromorphic functionsover 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, oneconsiders coordinate charts of the form fy : Uy → D in the neighbourhoods


of branch points y as in Proposition 2.1. If f is holomorphic at y, f f−1y is a

bounded function on f(Uy \ y). Choosing similar charts for all preimagesof x = φ(y), we infer from Proposition 2.1 that there is a complex chartfx : Ux → C such that all the aj f

−1x are bounded functions on fx(Ux \x)

and as such extend to holomorphic functions on the whole disc by Riemann’sremovable singularity theorem. The case of f having a pole reduces to above,by arguing about some function zkf f−1

y which is already holomorphic onD.

To see that the field extension is of exact degree d, take a point x ∈ Xwhich is not the image of a branch point and let y1, . . . , yd be its inverseimages in Y . By Corollary 3.5 we may find f ∈ M(Y ) that takes distinctnonzero values at the yi. Now by what we have seen f satisfies a polynomialequation (φ∗an)fn + . . .+(φ∗a0) = 0, with ai ∈ M(Y ) and n ≤ d. Note thatsince 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 theequation could not hold. Hence evaluating the ai at x we get a polynomialwith coefficients in C that has d distinct roots (namely the f(yj)), whencen = d.

Using Riemann’s existence theorem and the well-known fact that the fieldof meromorphic functions on P1(C) is isomorphic to the rational functionfield C(T ), we get an important corollary:

Corollary 3.7 The field of meromorphic functions on a compact Riemannsurface is a finite extension of C(T ), that is, an algebraic function field ofone variable over C.

Another important consequence of the proposition is that associating toa compact Riemann surface Y the field M(Y ) gives a contravariant functorfrom the category of compact Riemann surfaces mapping holomorphicallyonto a fixed Riemann surface X to the category of finite field extensions ofM(X). For example, for X = P1(C) we get a contravariant functor definedon the whole category of compact Riemann surfaces, by preceding discussion.Our next goal is to prove that this functor induces an anti-equivalence ofcategories.

We first prove that it is essentially surjective.

Proposition 3.8 Let X be a compact Riemann surface and let L|M(X) bea finite extension of its field of meromorphic functions. Then there exists acompact Riemann surface Y mapping holomorphically onto X such that theinduced field extension M(Y )|M(X) is isomorphic to L|M(X).


Proof: Let n be the degree of the extension L|M(X), let α be a primitiveelement and let F be the minimal polynomial of α. The derived polynomialF ′ does not vanish identically, so since F is irreducible the ideal (F ′, F ) ofthe polynomial ring M(X)[T ] (which is a principal ideal domain) must bethe whole ring. Therefore there are functions a, b ∈ M(X) such that

aF + bF ′ = 1.

Denote by Fz the complex polynomial obtained from F by evaluating itscoefficients at a point z ∈ X where all of them are holomorphic. From theabove equation we infer that Fz and F ′z may have a common root only atthose points z ∈ X where one of the functions a, b has a pole. Therefore ifwe denote by T ⊂ X the discrete set consisting of poles of the coefficients ofF as well as those of a and b, we get that on X ′ = X \ T all coefficients of Fare holomorphic and Fz(x) = 0 implies F ′z(x) 6= 0.

Now denote by F the subsheaf of the sheaf of those holomorphic functionson X ′ consisting over U ⊂ X ′ of those φ ∈ OX′(U) for which F (φ) = 0.We contend that F is a locally constant sheaf of sets each fibre of whichconsists of n elements. Indeed, by the holomorphic version of the implicitfunction theorem (see e.g. Griffiths-Harris [1], p. 19) for each point x ∈ Xwith Fz(x) = 0 for some z the condition F ′z(x) = 0 implies that there is aholomorphic function φx defined in a neighbourhood of z with F (φx) = 0and φx(z) = x. For each of the n distinct roots of Fz we thus find n differentfunctions 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 sectionssince the product of the polynomials (T − φx) already gives a factorisationof F in the polynomial ring M(V )[T ].

Now from Chapter 3, Proposition 2.8 we get a cover pF : X ′F → X ′.Next we can apply Proposition 2.7 to each of the connected components Y ′iof X ′F and get compact Riemann surfaces Yi mapping holomorphically ontoX. We now show that i = 1, i.e. XF is connected. Indeed, define a functionf on XF by putting f(φx) = φx(pF(φx)). One sees by the method of proof ofProposition 3.6 that f extends to a meromorphic function on each Yi and byapplying Proposition 3.6 we see that f as an element of M(Yi) has a minimalpolynomial G over M(X) of degree at most di, where di is the cardinalityof the fibres of the cover Y ′i → X ′. But since manifestly F (f) = 0, G mustdivide F in the polynomial ring M(X)[T ], whence F = G by irreducibilityof F and finally d = n. This proves that XF is connected and we may denotethe single Yi by Y .

Finally, mapping f to α induces an inclusion of fields M(Y ) ⊂ L whichmust be an equality by counting degrees.


Remark 3.9 One may replace the use of the implicit function theorem byusing ideas from residue calculus in complex analysis as in Forster [1], Corol-lary 8.8.

Theorem 3.10 Let X be a compact Riemann surface. Then the functorwhich associates to each pair (Y, φ) with Y a compact Riemann surface andφ : Y → X a (non-constant) holomorphic map the induced field extensionM(Y )|M(X) induces an anti-equivalence of categories from the category ofcompact 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. Fullfaithfulness follows from the following two facts.

The first is that given a pair (Y, φ) as in the theorem and a generator fof the extension M(Y )|M(X) with minimal polynomial F , then the coverof X given by the restriction of φ to the complement of branch points andinverse images of poles of coefficients of F is canonically isomorphic to thecover XF defined in the previous proof. This isomorphism is best definedon the associated sheaf of local sections: just map a local section si to theholomorphic function f si. The isomorphism extends to an isomorphism ofY with the compactification of XF defined in the previous proof; in partic-ular, morphisms Y → Z of compact Riemann surfaces over X correspondsbijectively 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, re-spectively. But their composite is a holomorphic map YM → X and the mapYM → YL is thus a canonical holomorphic map over X inducing the extensionM |L.

For the last statement we remark first that any automorphism of Y overX 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 overM(X) by classical Galois theory.

Remark 3.11 Let X be a compact Riemann surface such that M(X) canbe written in the form C[x, y]/(f) where the “homogenisation” F (z) =zdf(x/z, y/z) of the degree d polynomial f is such that its partial derivativesdon’t vanish simultaneously at any point of the complex projective plane. Itcan be shown using the theorem that M(X) is isomorphic to the complexplane curve defined by the equation F = 0.


Note that by virtue of Theorem 3.4 (and the lemma preceding it) anycompact Riemann surface X admits a non-constant holomorphic map φ :X → P1(C) and furthermore given any holomorphic map Y → X we mayregard it as a map of spaces over P1(C) by composing with φ. Hence thetheorem immediately implies

Corollary 3.12 Associating to a compact Riemann surface its field of mero-morphic functions defines an anti-equivalence of the category of compact Rie-mann surfaces with non-constant holomorphic maps with that of finite fieldextensions of C(T ).

Before stating the next corollary, define a non-connected compact Rie-mann surface to be a finite disjoint union of compact Riemann surfaces andextend the notion of a holomorphic map to this larger category in the obviousway. Then the theorem implies that the category of not necessarily connectedcompact Riemann surfaces equipped with a holomorphic map onto a fixedcompact 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 ofnot necessarily connected compact Riemann surfaces admitting a holomorphicmap 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 thathere we allow branch points as well. But the category of finite covers ofX is a subcategory of that of finite branched covers, so by comparing the

two corollaries one expects that π1(X, x)op is a quotient of Gal (M(X )) in anatural 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 overthe same topological (Hausdorff) space X. The fibre product of Y and Z overX 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 exerciseto the readers.


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 thenatural inclusion).

3. A basic property of Y ×X Z is that it represents the set-valued con-travariant 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 backφ and ψ by composition with the natural projections of Y ×X Z ontoY and Z; moreover, the construction is functorial in S.

Lemma 3.16 Let X be a connected Hausdorff space, p : Y → X be acontinuous 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 andX 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 neigh-bourhood of p(y) such that q−1(V ) decomposes as a disjoint union of opensubsets Vi. Then the inverse image of U = q−1(V ) by p1 decomposes as thedisjoint union of the open subsets U ×V Vi.

The second and third statements follow from the first by noting thatunder the assumptions any composition of covers is a cover. For X locallysimply connected, this was shown in Chapter 2, Lemma 3.3; in the case offinite covers the claim follows by direct checking from the definition of acover.

Note that even if Y and Z are connected, their fibre product will not beconnected in general.

Now we can prove:

Theorem 3.17 Let X be a compact Riemann surface and let X ′ be the com-plement of a finite (possibly empty) set of points in X. Let K be the compositein a fixed separable (=algebraic) closure M(X) of M(X) of all finite subex-tensions which arise from holomorphic maps of compact Riemann surfacesY → X that restrict to a cover over X ′. Then K is a Galois extension ofM(X) with Galois group isomorphic to the profinite completion of π1(X

′, x)op

(for some base point x ∈ X ′).


Proof: We first show that any finite subsextension of K|M(X) is the fieldof meromorphic functions of some Y which restricts to a cover over X ′. Forthis, 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 holomorphicmap Y → X corresponding to K|M(X) had a branch point mapping tosome x ∈ X ′, then using the formula in Proposition 2.5 (4) one would getthat 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 toL|M(X), so that Z → X would not restrict to a cover over X ′.

This being said, for our claim it suffices to show that given two subexten-sions Li|M(X) (i = 1, 2) coming from Riemann surfaces Yi → X which re-strict to covers over X ′, their composite in M(X) comes from some Y12 → Xwhich is also a cover over X ′. For this, take the fibre product Y1×X Y2. Sinceit may not be connected, its ring of meromorphic functions is a finite etaleM(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 homomor-phisms L1 × L2 → R; the anti-equivalence of Theorem 3.10 (extended toetale M(X)-algebras) transforms this exactly to the defining property of fi-bre 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 tothe factorisation of a minimal polynomial of a generator of L1|M(X) intoirreducible factors over L2. But when we look at the fixed embeddings ofthe Li into M(X), the component coming from one of these factors becomesexactly the composite L1L2, and we are done.

Next, K is Galois over M(X) for if a finite extension L|M(X) comesfrom a Riemann surface that restricts to a cover over X ′, then so does theone inducing any conjugate of L, as one immediately sees by looking at theproof of Theorem 3.10.

Now any finite quotient of π1(X ′, x) corresponds to a finite Galois cover ofX ′, which corresponds to a finite Galois branched cover of X by Proposition2.7 and finally to a finite Galois subextension of K|M(X). By Theorem3.10 and the first part of the proof we get in this way a bijection between

isomorphic finite quotients of π1(X ′, x) and Gal (K|M(X)), respectively, andmoreover this bijection is seen to be compatible with taking towers of coverson the one side and field extensions on the other. We get the statement ofthe proposition by passing to the inverse limit.

Remark 3.18 The groups π1(X′, x) are known from topology (see the next

section). This makes it possible to determine the absolute Galois group ofM(X) as follows. Any finite subextension of M(X) is contained in some ex-


tensionKX′ corresponding to anX ′ ⊂ X by the proposition, hence the Galoisgroup Gal (M(X)) is isomorphic to the inverse limit of the natural inversesystem of Galois groups formed by the Gal (KX′ |M(X)). (Note that this in-verse limit is filtered, for any two groups Gal (KX′ |M(X)), Gal (KX′′ |M(X))are targets of a map from the group Gal (KX′∩X′′ |M(X)).) But this result isinteresting 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. Nowany finite group G is a quotient of some Gn: indeed, if G is generated by nelements 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)) equalsC(T ), we get the corollary:

Corollary 3.19 Any finite group G is the Galois group of some finite ex-tension L|C(T ).

There is no currently known proof of this fact that uses purely algebraicmethods.

Remark 3.20 The question whether any finite group G is the Galois groupof a finite regular Galois extension L|Q(T ) (regular meaning that Q has noalgebraic extensions contained in L) is one of the most famous open problemsin algebra. It would imply in particular that any finite group is a Galois groupover Q: this follows from Hilbert’s irreducibility theorem which implies thatfor any irreducible polynomial F ∈ Q(T )[X] defining a Galois extension asabove we may find infinitely many values t0 ∈ Q such that the polynomialobtained from F by substituting t0 defines a Galois extension of Q with thesame group G (see e.g. Serre [3], Chapter 9).

It can be shown by methods of algebraic geometry (see more on this inChapter 9) that any finite Galois extension L|C(T ) comes by tensoring withC(T ) from a finite Galois extension L′|Q(T ) with the same Galois group. Toperform a “descent” of this extension from Q(T ) to Q(T ) is known to be pos-sible in many concrete cases, including almost all finite simple groups. Thisis done by the (purely algebraic) rigidity method developed by Belyi, Fried,Matzat, Thompson and others to which a nice introduction can be foundin Serre [4]; see also Malle/Matzat [1] and Volklein [1] for more extensivetreatments.


Remark 3.21 There is the following interesting analogue of the last corol-lary due to D. Harbater [1]: any finite group G arises as the Galois group ofa 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-Thelene [1] and Kollar [1] for other proofsand generalisations.

4. Topology of Riemann Surfaces

In order to give a reasonable complete treatment of the theory of coversof Riemann surfaces, we have to mention several topological results whichare proven by methods different from those encountered above. Since thismaterial is well documented in several introductory textbooks in topology,we shall mostly state the results without proof, the book of Fulton [1] beingour main reference.

Our first topic is the topological classification of Riemann surfaces. Thisis a very classical theorem stemming from the early days of topology and isproven 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 toruswith 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 2-dimensional) sphere and to attach g handles on it like on some mug. Anotheris to take what is called in topology a connected sum of g tori which is doneas follows. Take first two copies of the usual torus (which are homeomorphicto the sphere with “one handle attached”), cut out a piece homeomorphic toa closed disc from each and glue the two pieces together by identifying theboundaries of the two holes just cut out. In this way we obtain a torus withtwo holes and the process may be continued g times.

A third way generalises the fact that we may obtain the usual torus byidentifying opposite sides of a square (with the same orientation, i.e. withoutsuch twists as in the construction of the Moebius band). To generalise this,take a regular 4g-gon and label its sides clockwise by a1, b1, a

−11 , b−1

1 , . . . , ag,bg, a

−1g , b−1

g . Here the notation means that we consider the ai, bi with clock-

wise orientation and the a−1i , b−1

i with counterclockwise orientation. Nowidentify each ai with a−1

i and bi with b−1i taking care of the chosen orienta-

tions (see the very suggestive drawings on pp. 240-241 of Fulton [1]). In this

4.. TOPOLOGY OF RIEMANN SURFACES 87

way we get a sphere with g handles and the sides ai, bi, a−1i , b−1

i of our initialpolygon get mapped to closed paths all of which go through a common pointx.

Remark 4.2 The proof of the theorem uses several facts about the topologyof the compact Riemann surfaces under consideration. The first is that theyare topological manifolds of dimension 2, which means that the underlyingcontinuous maps of the complex charts induce homeomorphisms of someneighbourhood of each point with some open subset of R2. The secondis that they are orientable manifolds, which can be expressed in this caseby remarking that if fi : Ui → C (i = 1, 2) are some complex charts onour Riemann surface, then the R2-valued differentiable real function comingfrom the real and complex parts of the map f−1

1 f2 regarded as functionsof two real variables has an everywhere positive Jacobian determinant; thisis a consequence of the Cauchy-Riemann equations. Finally, one also has touse the fact that our compact Riemann surfaces can be triangulated; this weshall discuss below.

The third representation of tori with g holes described above makes itpossible to compute the fundamental group of a compact Riemann surfaceof genus g. Here it is convenient to take as a base point the point x whereall the closed paths coming from the ai, bi, a

−1i , b−1

i meet; in fact, it can beshown that these paths generate the fundamental group. More precisely, oneproves (cf. Fulton [1], Proposition 17.6):

Theorem 4.3 The fundamental group of a compact Riemann surface X ofgenus g has a presentation of the form

π1(X, x) =< a1, b1, . . . , ag, bg|[a1, b1] . . . [ag, bg] = 1 >,

where the brackets [ai, bi] denote the commutators a−1i b−1

i aibi.

The proof uses the definition of the fundamental group in terms of closedpaths and in particular the van Kampen theorem.

Remark 4.4 In fact, by the same method that proves the theorem one canalso determine the fundamental group of the complement of n + 1 pointsx0, . . . , xn in a compact Riemann surface X of genus g. Here one has toadd one generator γi for each xi given by a closed path going through x andturning 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 inthe previous section.


Remark 4.5 Realising the groups described in the theorem as automor-phism groups of the universal cover gives rise to a fascinating classical theoryknown as the theory of uniformisation; see Chapter IX of Shafarevich [1] fora nice introduction.

The main result here, originating in work by Riemann and proven com-pletely by Poincare and Koebe, is that any simply connected Riemann surfaceis isomorphic as a complex manifold to the projective line P1(C), the com-plex 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 aboveas follows. In the case of P1(C) there is no quotient other than itself, forany automorphism of P1(C) is known to have a fixed point. For C, one canprove that the only even action on it with compact quotient is the one byZ2 as in the second example of Chapter 2, Example 2.7, so the quotient isa torus C/Λ; this is in accordance with the case g = 1 of the theorem. Allother compact Riemann surfaces are thus quotients of the open unit disc Dby some even group action. Poincare studied such actions and showed thatthey come exactly from transformations mapping ai to a−1

i and bi to b−1i in

a 4g-gon with sides labelled as above; the only difference is that in this casethe sides of the polygon are not usual segments but circular arcs inscribedinto the unit disc, for he worked in the model of the hyperbolic plane namedafter him.

We finally discuss triangulations of Riemann surfaces; we restrict our-selves to the compact case. So let X be a compact topological manifold ofdimension 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 embed-dings 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 containpoints from S0;

• the sets κ(∆), where ∆ is the interior of ∆, are pairwise disjoint anddo not meet the images of the edges in S1;

4.. TOPOLOGY OF RIEMANN SURFACES 89

• 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 topologicalspace of P1

C) has several natural triangulations; one of them is cut out bythe equator and two meridians.

Proposition 4.7 Any compact Riemann surface has a triangulation.

The proposition is an immediate consequence of the above example, ofRiemann’s Existence Theorem and the following lemma.

Lemma 4.8 Let φ : Y → X be a branched cover of compact Riemannsurfaces (eg. a holomorphic map). Than any triangulation of X can be liftedcanonically to a triangulation of Y .

Before giving the proof note the obvious fact that given a triangulationof a compact topological surface X and a point x ∈ X \S0, the triangulationcan be refined in a canonical way to a triangulation whose set of verticesis 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 correspondingelements of S1 by the restrictions of κ to the edges and faces of the smallertriangles arising from the subdivision.

Proof: By refining the triangulation as above if necessary, we may assumethat in the given triangulation of X the set S0 contains all images of branchpoints. Hence the restriction of φ to X \ φ−1(S0) is a cover. As the openinterval (0, 1) (resp. the subset ∆′ ⊂ ∆ obtained by omitting the vertices)are simply connected, taking their image by an edge ι (resp. a face κ) weobtain subsets of X over which the cover is trivial, so the restriction of ιto (0, 1) (resp. that of κ to ∆′) can be canonically lifted to each sheet ofthe 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 atriangulation 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 triangulationby the process described above. Hence by choosing common refinements oftwo triangulations we see that χ does not depend on the triangulation butis an invariant of the surface itself.


Proposition 4.9 Let φ : Y → X be a holomorphic map of compact Rie-mann surfaces inducing a field extension of degree d. Then the Euler char-acteristics χX and χY of X and Y are related by the formula

χY = dχX −∑

y

(ey − 1)

where the sum is over the branch points of φ and ey is the ramification indexat the branch point y ∈ Y .

Proof: In the process of lifting a triangulation to a branched cover as in thelemma the number of edges of the lifted triangulation is ds1 and the numberof 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 butwith each branch point the number of preimages diminishes by ey − 1.

Now it is a known topological fact that the Euler characteristic of a toruswith 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) +∑

y

(ey − 1)

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 tostudy the analogues of the theory developed in the previous chapters withinthe realm of arithmetic and algebraic geometry. The convenient category towork in is Grothendieck’s category of schemes, so we shall discuss as muchof their theory as we shall need for subsequent applications.

As always, the word “ring” in this chapter will mean commutative ringwith unit. Also, when referring to compact topological spaces, we do notassume 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 fg ∈ S. A prime ideal P is an idealsuch that the set A \ P is multiplicatively closed. An equivalent formulationof 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 thatany maximal ideal of A (i.e. an ideal contained in no other proper ideal ofA than itself) is always a prime ideal since in this case the quotient ring is afield.

We now turn the set of prime ideals of an arbitrary ring A into a topo-logical space.

Definition 1.1 The prime spectrum SpecA of A is the topological spacewhose points are prime ideals of A and a basis of open sets is given by thesets

D(f) := P : P is a prime ideal withf /∈ P

for all f ∈ A.

91

92 CHAPTER 5. ENTER SCHEMES

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

D(f) ∩D(g) = D(fg) (5.1)

for by definition a prime ideal avoids fg if and only if it avoids f and g.It follows from the definition that a closed subset in the topology of

SpecA can be described as the set of prime ideals containing some fixedideal I (generated by a system of elements fi : i ∈ J of A). Thus one-pointsets given by maximal ideals are closed; in fact, maximal ideals give the onlyclosed points of the prime spectrum since for any prime ideal P a closedsubset containing P contains the maximal ideals containing P as well. Thisshows that in general the prime spectrum does not satisfy even the weakestof the separation axioms in topology. However, it enjoys a nice topologicalproperty:

Proposition 1.2 For any ring A the prime spectrum SpecA 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 ifthe sets D(fi) give an open covering of SpecA.

Proof: Indeed, if the fi generate A, there can be no prime ideal of Acontaining all of them, which is equivalent to the D(fi) covering SpecA. If,however, they do not generate A, then they are all contained (by Zorn’sLemma) in some maximal ideal M which thus gives an element of SpecAnot contained in any of the D(fi).

Proof of Proposition 1.2: Let Ui : i ∈ I be an open covering of SpecA;we may assume that each Ui is in fact some basic open set D(fi). By theabove lemma the fi generate A. In particular, there is a relation of the form

a1f1 + a2f2 + . . .+ anfn = 1 (5.2)

with ai ∈ A and f1, . . . , fn chosen among the fi above. This means, how-ever, that already f1, . . . , fn generate A, i.e. the sets D(f1), . . . , D(fn) coverSpecA.

Equations of type (5.2) are sometimes referred to as algebraic analoguesof partitions of unity.

Examples 1.4 We conclude this section by some easy examples.

2.. SCHEMES – MOSTLY AFFINE 93

1. The prime spectrum of a field consists of a single point, correspondingto the ideal (0).

2. The prime spectrum of Z is the space consisting of a closed point foreach prime p, and a non-closed point corresponding to (0), called thegeneric point, whose closure is the whole space. Other closed subsetsare only finite sets of primes; indeed, any ideal in I ⊂ Z is of theform mZ for some positive integer m; the prime ideals containing I aregenerated 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 subsetsare again finite sets of closed points. Indeed, C[x] is a principal idealring with prime elements the polynomials x− a (a ∈ C), and we mayargue as in the previous case.

4. If A is isomorphic to a finite direct sumn⊕

i=1Ai, then SpecA is a dis-

joint union of clopen sets each of which is homeomorphic to one of theSpecAi. To see this, observe first that if one writes ei for the idempo-tent given by putting 1 at the i-th component and 0 elsewhere, any pair-wise product eiej 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 theseis 1. Thus we conclude that P concludes all of the ej except one, sayei, 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 SpecA isthen induced by the map P 7→ Pi.

In particular, the prime spectrum of a finite etale algebra over somefield k is a finite discrete set of points. This confirms our intuition thata finite etale k-algebra is like a cover of the one-point space arising asthe prime spectrum of k.

2. Schemes – Mostly Affine

The prime spectrum of a ring is a rather coarse invariant: for instance, itcannot even distinguish between two fields. We shall remedy this by definingsome additional structure on the prime spectrum. To motivate the construc-tion 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 functionson C having at worst a pole at infinity. The prime spectrum of this ring can


be identified to C with a generic point (0) added; closed sets are finite setsnot containing (0). Obviously one cannot recover C[x] from these data; wecannot even distinguish between constant functions. Remember, however,that we have seen in the previous chapter that a Riemann surface is uniquelydetermined by the underlying topological space plus the sheaf of holomorphicfunctions on it. If we restrict to the sheaf of holomorphic functions on Chaving at worst a pole at infinity, we can easily describe its sections over aset of the form D(f) with the generic point removed (this is an open set inthe complex topology). For instance, over D(x) (which with the generic pointthrown away identifies to C∗) the sections are the rational functions whosedenominator is a power of x, for thesections are meromorphic functions onP1(C) and hence elements of C(T ); moreover, any denominator other thanthe 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 thesheaf 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 firsthave to extend the notion of a rational function, i.e. give a meaning tofractions of elements in an arbitrary ring A. So let S be a multiplicativelyclosed subset of A. We would like to define a ring AS which is to be the “ringof 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 thiscase A admits a fraction field K. Elements of K can be represented byfractions f/g with f, g ∈ A, g 6= 0, where f/g = f1/g1 whenever fg1 = f1g.We may then take AS to be the subring of those elements which can bewritten as fractions with denominators in S; this is indeed a subring as S ismultiplicatively closed.

Now to treat the general case, observe first that just as the fraction fieldK can be defined as the object representing a certain functor (see Chapter0, Section 2), the ring AS of the previous example is easily seen to representthe 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 ringA and multiplicatively closed subset S.


The ring AS is called the localisation of A with respect to S. By theYoneda lemma, it is determined up to unique isomorphism. Moreover, it isequipped with a canonical homomorphism φS : A→ AS sending elements ofS 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 equivalencerelation:

(f, s) ∼ (f ′, s′) iff there is a t ∈ S with (fs′ − f ′s)t = 0.

One sees that this is indeed an equivalence relation; for transitivity, note thatthe equations (fs′−f ′s)t= 0 and (f ′s′′−f ′′s′)u= 0 imply (fs′′−f ′′s)s′tu= 0(multiply the first equation by s′′u and the second by st). Denote by f/s theimage of (f, s) in AS and define the addition and multiplication laws as forfractions; 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 thatunits are never zero-divisors, so φ(s)−1 is a well-defined element of R). Thisis a well-defined map, for if (f ′, s′) is another representative for f/s, we have

0 = φ((fs′ − f ′s)t) = (φ(f)φ(s′) − φ(f ′)φ(s))φ(t),

whence φ(f)φ(s′) = φ(f ′)φ(s) as φ(t) is a unit. Conversely, as any elementof 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 isimmediately 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 betweenprime 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 isdifferent from AS. Moreover, it is a prime ideal, for if (f/s)(g/t) ∈ φS(P )AS,then ufg ∈ 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 idealQ of AS the ideal φ−1

S (Q) is a prime ideal of A avoiding S; the assertion thenfollows from the equality φS(φ−1

S (Q))AS = Q. Similarly, injectivity followsfrom φ−1

S (φS(P )AS) = P ; the verification of these relations is left to thereader.


Examples 2.5 The two key examples of localisation to be used in the sequelare the following.

1. Let S be the set 1, f, f 2, f 3, . . . of all powers of f for some f ∈ A. Inthis case elements of AS are represented by fractions with numerator inA and denominator a power of f ; we shall use the notation Af for thisparticular AS. The previous lemma implies that SpecAf is naturallyhomeomorphic 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 commonabuse of notation from the literature, we shall denote the localisationof A with respect to S by AP instead of AA\P . The points of SpecAP

correspond to prime ideals of A contained in P ; in particular, AP hasa unique maximal ideal generated by the image of P . Rings having aunique 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 spectrumX 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 inAf is a unit.

Proof: Indeed, if g did not give a unit in Af , it would be contained in amaximal ideal Q. By Lemma 2.4 there is a unique prime ideal P of A whoseimage in Af generates Q. This P contains g but not f , a contradiction.

Combining the lemma with Proposition 2.3, we get for any inclusionD(f) ⊂ D(g) of basic open sets a canonical restriction homomorphismAg → Af . Clearly for a tower of inclusions D(f) ⊂ D(g) ⊂ D(h) themap Ah → Af thus obtained is the composition of the intermediate mapsAh → Ag and Ag → Af . So putting OX(D(f)) = Af , we have obtained“something which behaves like a presheaf on basic open sets”. That thisindeed extends to a presheaf on X follows from the first statement of thefollowing formal lemma (of which we advise the readers to skip the proof ina first reading).


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 ′ ⊂ V of elements of V a map (resp. homomorphism)ρV V ′ : F(V ) → F(V ′) satisfying ρV V = idF(V) and ρV V ′′ = ρV ′V ′′ ρV V ′ foreach tower V ′′ ⊂ V ′ ⊂ V of elements of V.

1. There exists a presheaf of sets (resp. abelian groups, rings, etc.) Fon X whose sections and restriction maps over elements of V can becanonically identified to those given above.

2. Assume moreover that the F(V ) above satisfy the sheaf axioms for allcoverings of elements of V by elements of V. Then there is a uniquesheaf F on X whose sections and restriction maps over elements of Vare those given above.

3. Finally assume given two sheaves F , G on X and for each V ∈ V a mapφV : F(V ) → G(V ) such that for each inclusion V ′ ⊂ V of elements ofV the diagram

F(V )φV−−−→ G(V )

ρFV V ′

yyρG

V V ′

F(V ′)φV ′

−−−→ G(V ′)

commutes. Then there is a unique morphism of sheaves φ : F → Gwith the φV given as above.

Proof: For the first statement, consider for a given open set U ⊂ X the setVU of elements of V contained in U ; this set is partially ordered by inclusion.The restriction maps φV V ′ for V ′ ⊂ V ⊂ U turn the system of F(V ) withV ∈ VU into an inverse system. Note that this is a non-filtered inverse systemin the sense of Chapter 1, Remark 2.3. (Note also that here we are workingwith the ordering opposite to that used in the definition of stalks in Chapter2, 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 ′ = φV V ′(fV ) whenever V ′ ⊂ V . If U ′ ⊂ U ,define a restriction map ρUU ′ by mapping the sequence (fV ) above to thesequence of those fV for which V ⊂ U ′. There is no difficulty in checkingthat we have thus defined a presheaf. Moreover, for W ∈ V, the sections ofF over W can be canonically identified with the elements of the prescribedset F(W ) as in this case the sequences (fV ) defining the inverse limit aregiven by restrictions of elements of the prescribed F(W ) to all elements ofV contained in W . Thus for any U containing W , the restriction map ρUW


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 firstsheaf axiom since each open U ⊂ X can be covered by elements of V. So itsuffices to show that the presheaf F we have just defined satisfies the sheafaxioms. By construction of F , for the first sheaf axiom it is enough to seethat 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 overeach W ∈ VU . Since V is a basis of open sets (hence in particular closedunder 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 thesame section over W by the assumption. The verification of the second sheafaxiom is similar and is left to the readers.

Finally, the last statement follows from the fact that the maps φV inducea morphism of the inverse systems defining F(U) and G(U) for a general Uas above. The map φU : F(U) → G(U) is then obtained by passing to thelimit: 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 =SpecA for which OX(D(f)) = Af for all f ∈ A and the restriction mapsOX(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 SpecAf

is naturally homeomorphic to D(f) and OX(D(f)) = Af , we may replaceA by Af and assume f = 1. Then for the first sheaf axiom we are given acovering of X by basic open sets D(fi); by compactness of X we may assumethe covering is finite, say X = D(f1) ∪ . . . ∪ D(fn). To give a section ofOX(X) = A restricting to 0 over each D(fi) is to give an element g ∈ Asatisfying

fkii g = 0 (5.3)

for all 1 ≤ i ≤ n with appropriate positive integers ki. Now by the definitionof prime ideals we have D(fki

i ) = D(fi) for all i, so the D(fkii ) cover X as

well and hence by Lemma 1.3 there exist gi ∈ A with

g1fk11 + . . .+ gnf

knn = 1 (5.4)


Thus if we multiply each equation in (5.3) by gi and take the sum we getg = 0, as desired.

For the second sheaf axiom, assume again given a covering of X by basicopen sets D(fi) (1 ≤ i ≤ n) and elements ai/f

kii ∈ Afi

(viewed as sectionsof a would-be sheaf over D(fi)) whose restrictions to the pairwise intersec-tions D(fi) ∩D(fj) = D(fifj) coincide. This latter property can be written

explicitly as (aifkj

j − ajfkii )(fifj)

kij = 0 with some positive integer kij . Bychanging the ai if necessary we may assume ki = k for all i and kij = m forall i, j, where m is large enough. Thus

aifkj (fifj)

m = ajfki (fifj)

m (5.5)

for all 1 ≤ i, j ≤ n. Now apply (5.4) with ki = k + m for all i to get somegi with

∑igif

k+mi = 1 and define a =

∑igiaif

mi . Using equation (5.5) we get

for all j a chain of equations

fk+mj a =

n∑

i=1

giaifkj (fifj)

m =n∑

i=1

giajfki (fifj)

m = ajfmj

∑

i

gifk+mi = ajf

mj

which means that the image of a in Afjcoincides with aj/f

kj , as required.

Definition 2.9 An affine scheme is a pair (X,OX) consisting of a topolog-ical space X and a sheaf of rings OX on X such that X = SpecA for somering A and OX is the sheaf occuring in the above theorem. We call OX thestructure sheaf of X.

By abuse of notation, we shall frequently write X or SpecA instead ofthe pair (X,OX). Next an important fact:

Proposition 2.10 If X = SpecA 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 filtereddirect system of the rings OX(U), for U containing P . Since basic open setsD(f) are cofinal in the index set of this direct system, we may restrict to therings Af . Then the proposition follows from the fact that the direct limit ofthese rings is obtained by “dividing out by all f /∈ P”. More precisely, itfollows 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 fng = 0 for some f /∈ P and n ≥ 0 and thus g = 0 inAP ; surjectivity is equally obvious.

Now some definitions.


Definition 2.11 A ringed space is a pair (X,F) where X is a topologicalspace 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 amorphism 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 additionalproperty that the stalks of the structure sheaf are all local rings.

Next notice that given a morphism of ringed spaces (φ, φ]) as above, forany x ∈ X the morphism φ] induces a ring homomorphism Gφ(x) → Fx onthe stalks, for by definition Gφ(x) is the (filtered) direct limit of G(U) for theopen sets U containing φ(x), whereas φ∗F(U) = F(φ−1(U)) and there is anatural map from the direct limit of the latter sets to Fx, for Fx is definedas the direct limit of all open neighbourhoods containing x.

Definition 2.12 A locally ringed space is a ringed space (X,F) such thatthe stalk Fx is a local ring for all x ∈ X. A morphism of locally ringed spacesis 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 thepreimage of the maximal ideal of Fx is the maximal ideal of Gφ(x). Thus thecategory 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 opencovering 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 locallyringed spaces whose objects are schemes.

Construction 2.13 As we have already remarked, for any commutativering A the affine scheme X = SpecA is naturally an object of the cate-gory of schemes. We now show that the map A 7→ SpecA is in fact acontravariant functor from the category of rings to that of schemes. Forthis we have to assign to any ring homomorphism φ : A → B a mor-phism (Spec (φ), Spec (φ)]) : SpecB → SpecA of schemes. The definitionof Spec (φ) is obvious: it maps a prime ideal P ∈ SpecB to φ−1(P ) whichis immediately seen to be a prime ideal of A. The map is continuous sincethe inverse image of a basic open set D(f) is just D(φ(f)). Now for definingSpec (φ)] note that by the third statement of Lemma 2.7 it suffices to con-sider sections over the basic open sets D(f). By construction of the structuresheaves, 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: itcorresponds to the composite A→ B → Bφ(f).


Now consider the natural question: given an affine scheme X = SpecA,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 amorphism φ : X → Y of affine schemes, we have in particular a morphism ofsheaves φ] : OY → φ∗OX , whence a homomorphism OY (Y ) → φ∗OX(Y ) =OX(X).

Theorem 2.14 The functors A 7→ SpecA and X → OX(X) are inverse toeach other. Thus the category of affine schemes is isomorphic to the oppositecategory of the category of commutative rings with unit.

Proof: If Y = SpecB and the scheme morphism X → Y comes from ahomomorphism λ : A → B, by construction the map OY (Y ) → OX(X) isnone but λ.

We are left to prove that given a morphism (φ, φ]) : SpecB → SpecA ofschemes, if λ : A → B is the ring homomorphism induced by taking globalsections, then (φ, φ]) = (Spec (λ), Spec (λ)]). For this, we have to show firstthat for P ∈ SpecB we have φ(P ) = λ−1(P ). Indeed, φ] induces a map onthe stalks φ]

P : Aφ(P ) → BP which by definition makes the diagram

Aλ

−−−→ By

y

Aφ(P )

φ]P−−−→ BP

commute. But φ]P is a local homomorphism, i.e. φ(P )Aφ(P ) = (φ]

P )−1(PBP );on the other hand, the vertical maps are localisation maps, whence the as-sertion. The equality φ] = Spec (λ)] follows from the analogous commutativediagram

Aλ

−−−→ By

y

Af

φ]

D(f)−−−→ Bλ(f)

for sections over basic open sets.

This result enables us to reformulate once more the main theorem ofGalois theory. Given a field k, define a finite etale k-scheme to be the affinescheme SpecA for A a finite etale k-algebra. By the theorem, the categoryof finite etale k-schemes (viewed as a full subcategory of the category ofaffine schemes) is isomorphic to the opposite category of that of finite etalek-algebras. Thus Chapter 1, Theorem 3.4 yields:


Theorem 2.15 For any field k the category of finite etale k-schemes isequivalent 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. Butwhile we are at finite dimensional k-algebras, we close this section by a purelyalgebraic result about them to be used in forthcoming chapters, which canbe given an elegant proof via the theory of affine schemes.

Proposition 2.16 Let k be a field. Then any finite dimensional k-algebradecomposes as a finite direct sum of local rings whose maximal ideals consistof 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 betweenidempotents of A and clopen subsets of the underlying space of SpecA.

Proof: As we have, the basic open set D(e) is closed for D(1−e) is its opencomplement. Conversely, given a decomposition of the underlying space ofX into the disjoint union of two open subsets U and V , consider the sections1 ∈ OX(U) and 0 ∈ OX(V ). Since U and V are disjoint, the sheaf axiomsimply 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 easilythat the two constructions are inverse to each other.

Corollary 2.18 Let A be a ring and I an ideal consisting of nilpotent ele-ments. Given an idempotent e ∈ A/I, there is a unique idempotent e ∈ Amapping onto e by the projection A→ A/I.

Proof: Since I consists of nilpotent elements, it is contained in all primeideals of A. Hence the map Spec (A/I) → SpecA induced by the projectionis the identity on the underlying topological spaces and as such induces abijection of clopen subsets (and of functions which are 1 on the subset and0 on the complement).

Proof of Proposition 2.16: It is enough to show that if A is indecompos-able and I is its ideal of nilpotent elements, then A/I is a field. If it werenot, then by Chapter 1, Lemma 3.6 it would decompose as a direct sum offields so in particular it would contain a nontrivial idempotent. But then bythe previous corollary A would contain one as well, which is a contradictionwith A being indecomposable.

3.. FIRST EXAMPLES OF SCHEMES 103

3. First Examples of Schemes

It is now time for some examples. Let us first take a new look of those of thefirst section, but this time considering the structure sheaves as well.

Examples 3.1 1. For k a field, the underlying topological space of Spec kis a single point. The stalk of the structure sheaf at this point is k.

2. The generic stalk of SpecZ, i.e. the stalk of OSpec Z at the generic point(0) is Q. The inclusion Z → Q corresponds to a morphism SpecQ →SpecZ, identifiable as the inclusion of the generic point into SpecZ. Ata closed point corresponding to the prime ideal (p) the stalk OSpec Z,(p) isisomorphic to the subring of Q formed by fractions whose denominatoris 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 SpecFp → Spec Z, the inclusion of the closedpoint (p).

3. The generic stalk of SpecC[x] is the rational function field C(x). At theclosed point (x− a) the stalk consists of those elements of C[x] whosedenominator 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 maxi-mal ideal is isomorphic to C; the image of a function by the projec-tion 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 Qas functions on the space Spec Z. If the denominator of f ∈ Q is not divisibleby 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 beconsiderably refined later.

Example 3.2 Affine spaces. For a field k we define affine n-space over k asthe affine scheme An

k = Spec k[x1, . . . , xn] (with k[x1, . . . , xn] the polynomialring in n variables over k). An explanation for this name is provided by aform 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 anymaximal ideal of k[x1, . . . , xn] is of the form (x1 −a1, . . . , xn −an) with someai ∈ k. Thus in this case we may identify the set of closed points of An

k withelements of kn. Note that the above statement is false even for n = 1 if kis not algebraically closed: for instance, the polynomial x2 + 1 generates amaximal ideal of R[x] not of the above form.


We next give the basic example of a non-affine scheme. Before discussingit, an easy definition.

Definition 3.3 An open subscheme of a scheme X is the ringed space con-sisting of an open subset U and the restriction OX |U of the structure sheafof X to U .

Indeed, one checks that U admits an open covering by affine schemes (usebasic open sets, for instance).

Example 3.4 Projective spaces. For k an infinite field, we construct ascheme Pn

k by a similar procedure to that in the complex analytic case. So de-note 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 accordancewith 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 cor-responds to a canonical action of Gm on An+1

k ; moreover, each t ∈ Gm fixesthe 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 andthe structure sheaf OPn

kis given by V 7→ OU(p−1(V ))Gm, with p : U → Pn

k

the natural projection and the superscript Gm denoting invariants by theaction of Gm. It is immediate that OPn

kis indeed a sheaf of rings, and that

the ringed space thus defined is a locally ringed space. Finally we have todefine 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 Pn

k foreach 0 ≤ i ≤ n. By definition, we have OPn

k(D+(xi)) = k[x0, . . . , xn]Gm

xi. But

since k is infinite, each element of this ring can be represented by a fractionf/xd

i with f a homogenous polynomial of degree d (indeed, these elementsare manifestly invariant under Gm; otherwise decompose a polynomial f intohomogenous components fk of degree k and look at what happens for fk/x

di

for k 6= d). Thus the k-homomorphism

k[t0, . . . , ti−1, ti+1, . . . , tn] → k[x0, . . . , xn]Gmxi

, tj 7→ xj/xi

is an isomorphism. We leave it to the reader to check that this induces anisomorphism of the ringed space (D+(xi),OPn

k|D+(xi)) with An

k .

Remark 3.5 In general, it is possible to define projective spaces over anycommutative ring A (not to mention any scheme...) The preceding construc-tion may break down, however, so one should use another construction. The

3.. FIRST EXAMPLES OF SCHEMES 105

most down-to-earth method among several available ones is to define PnA by

“patching” together the affine schemes

D+(xi) ∼= SpecA[(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) ofD+(xj). How this patching can be done precisely will be explained in thenext section (see Construction 5.3).

The next definition enables us to define the basic objects of study inalgebraic geometry, namely loci of zeros of systems of polynomials in affineor projective space.

Definition 3.6 A morphism φ : Y → X of schemes is a closed immersionif the underlying continuous map is the inclusion of a closed subset of X andmoreover there is a covering of X by affine open subschemes Ui = SpecAi

such that for all i the open subscheme of Y defined by φ−1(Ui) is isomorphicto an affine scheme SpecBi with the induced maps Ai → Bi surjections. Wesay that Y is a closed subscheme of X if there is a closed immersion of Yinto X.

Remark 3.7 It can be shown that any closed subscheme of an affine schemeX = SpecA is of the form SpecA/I with some ideal I. However, the readershould be warned that in general it is possible to give several different closedsubscheme structures on a given closed subset of the underlying topologicalspace of a scheme.

Example 3.8 An (irreducible) affine hypersurface of dimension n−1 over afield k is the closed subscheme of An

k given by the quotient of the polynomialring k[x1, . . . , xn] by the principal ideal generated by an irreducible polyno-mial f (here the covering of An

k is just the one-element covering by the wholespace). Affine hypersurfaces of dimension 1 are also called plane curves. Forinstance, the quotient of k[x1, x2] modulo the principal ideal generated bythe polynomial x1x2 − 1 defines an affine plane curve: the conic of equationx1x2 = 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) viathe isomorphisms D+(xi) ∼= An

k . As above, in dimension 1 we get projectiveplane curves. For instance, the locus of zeros of the homogenous polynomialX0X1 − X2

2 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 = x2

2

and on D+(X2) that of equation x0x1 = 1 (notice that different types ofaffine conics arise from the same projective conic).


Remark 3.9 If k = C, a plane curve defined by a polynomial whose partialderivatives do not vanish simultaneously carries the structure of a schemeand 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 = SpecA.Here we study how to associate sheaves on X to modules over the ring A, aconstruction that will be very useful in the next chapter.

As the construction of affine schemes makes one expect, sheaves associ-ated to A-modules should be, in some sense, modules over the structure sheafOX . 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 eachopen U ⊂ X the group F(U) is equipped with an OX(U)-module structureOX(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 whenF(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 thelevel 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 onY . This is particularly interesting when X is a closed subscheme of Yand φ is the natural inclusion. In this case we call I the sheaf of idealsdefining X.

3. More generally, given any OX -module F one can define its annihilatoras the ideal sheaf whose sections over an open set U consist of thosef ∈ OX(U) for which fs = 0 for all s ∈ F(U). For instance, theannihilator of the OX -module 0 is OX .

4.. QUASI-COHERENT SHEAVES 107

We may now proceed to construct OX -modules over affine schemes frommodules over rings. For this we first need an algebraic concept.

Definition 4.3 Let A be a ring, S ⊂ A a multiplicatively closed subset andM an A-module. The localisation of M by S is the AS-module MS given byM ⊗A AS.

As in the case of rings, given an element f ∈ A or a prime ideal P , weshall 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 mul-tiplicatively closed S ⊂ A there is a natural map M → MS obtained bytensoring the natural map A → AS by M and similarly there is a naturalmap 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 thatLemma 2.7 may be applied to get a sheaf M over X which is an OX -moduleby construction.

The rule M → M is naturally a functor from the category of A-modulesto 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 categoriesarises, 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 = SpecAconsists only of two points: a closed point coming from (x) and a so-calledgeneric point η coming from the ideal (0). The stalks of OX are A in theclosed point and k(x) in the generic point. Now define an OX -module F onX = SpecA 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 andX itself, these data indeed define an OX -module whose A-module of globalsections is A. But this OX -module is not isomorphic to A as the stalks at ηare 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 thatthe restriction of F to each Ui = SpecAi is isomorphic to an OUi

-moduleof the form Mi with some Ai-module Mi. If moreover each Mi is finitelygenerated over Ai, then F is called a coherent sheaf.

Remark 4.7 It can be shown that for an affine scheme X = SpecA the func-tor M → M establishes an equivalence between the category of A-modulesand that of quasi-coherent sheaves; since we shall not need this, we omit theproof and refer the interested reader to Hartshorne [1], Corollary II.5.5.


We now return to the first example in 4.2 and investigate the question ofdetermining whether a morphism φ : X → Y yields a quasi-coherent sheafφ∗OX in Y . Unfortunately, this is not true in general but Section II.5 ofHartshorne [1] contains several sufficient conditions. For our purposes thefollowing easy condition on φ will suffice.

Definition 4.8 A morphism φ : X → Y of schemes is affine if Y has ancovering by affine open subsets Ui = SpecAi such that for each i the opensubscheme φ−1(Ui) of X is affine as well.

Any morphism of affine schemes is obviously affine. We shall see otherexamples of affine morphisms in the next chapter.

Lemma 4.9 If φ : X → Y is an affine morphism, then φ∗OX and the idealsheaf defined by the kernel of φ] are quasi-coherent sheaves on Y .

Proof: Assume first X = SpecB and Y = SpecA are affine schemes. Thenφ∗OX is just B with B regarded as an A-module via the map λ : A → Binducing φ. Indeed, it is enough to check this over basic open sets D(f) forwhich we may argue in the same way as in the second half of the proof ofTheorem 2.14. Moreover, a similar reasong shows that the ideal sheaf on Ydefined by the kernel of φ] is just I with I = ker(λ). Once we have theseresults at hand, the general case of the lemma follows from the definition ofaffine morphisms and quasi-coherent sheaves.

The lemma applies in particular to a closed immersion i : X → Y ofschemes which is affine by definition. Thus to any closed subscheme of Y wemay associate a quasi-coherent sheaf of ideals. We conclude this section byproving the converse.

Proposition 4.10 The above construction gives a bijection between closedsubschemes 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 cov-ering of Y by affine open subschemes Uj = SpecAj as in the definition ofquasi-coherence. Define for each j a closed immersion ij : Xj → Uj as themap induced by the projection Aj → Aj/Ij , where Ij is the ideal for whichI|Uj

∼= Ij . 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 theintersections Ui ∩ Uj). But then any point of Uj ∩ (Y \ X) has an openneighbourhood contained in Uj \ (X ∩ Uj), whence the claim. Finally, the ijendow X with the structure of a closed subscheme. It is manifest that thetwo constructions are inverse to each other.

5.. FIBRES OF A MORPHISM 109

5. Fibres of a Morphism

In our study of covers in topology, the examination of fibres played a pre-eminent role. Since our ultimate goal is to study covers in the context ofschemes, an unavoidable task is to define quite generally the fibre of a mor-phism of schemes as a scheme and not just a point set. Motivated by thediscussion in Chapter 4, Section 3, we introduce more generally the notionof 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 thespace representing the functor.


on the category of topological spaces over X. We can adopt the same defini-tion in the context of schemes if we show that the similarly defined functoron the category of schemes equipped with morphisms to a fixed base schemeX is representable. We first prove representability in the category of affineschemes.

Proposition 5.1 Assume given affine schemes Y = SpecA and Z = SpecBequipped with morphisms p : Y → X, q : Z → X into an affine schemeX = SpecR. Then the contravariant functor


on the category of affine schemes is representable by Y×XZ := Spec (A⊗RB).

Proof: Indeed, by Theorem 2.14 the statement of the proposition is equiv-alent 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 thedefining property of the tensor product (of R-algebras).

Remark 5.2 It is not true that the underlying topological space of a fibreproduct of affine schemes is the topological fibre product of the underlyingtopological spaces of the schemes. As an easy example, take X = Spec kwith some field k, Y = SpecL, with L|k a finite separable extension of k,Z = Spec k. Then the topological fibre product of Y and Z over X is aone-element set, whereas L ⊗k k is a direct sum of [L : k] copies of k andhence its prime spectrum consists of [L : k] points.


To extend this construction to arbitrary schemes, the idea is of courseto cover them with open affine subschemes and then to “patch” the fibreproducts of these affine schemes together. How this can be done precisely isexplained next.

Construction 5.3 Assume given a family of schemes Xi : i ∈ I and foreach (i, j) ∈ I2 an open subscheme Uij ⊂ Xi. Assume moreover we are givenisomorphisms φij : Uij → Uji satisfying the compatibility conditions

φij = φ−1ji

andφij(Uij ∩ Uik) = Uji ∩ Ujk, φik = φjk φij on Uij ∩ Uik.

We now construct a scheme X having an open covering Ui : i ∈ I suchthat each Ui is isomorphic to Xi as a scheme and via these isomorphisms theUij correspond to the intersections Ui∩Uj. The above compatibility relationsfor 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 ofUji via φij. The compatibility conditions for the φij ensure that this is indeedan 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 byLemma 2.7 to define its sections over a basis of open sets in X in a compatiblefashion. The open sets U which are contained in one of the Ui clearly form abasis. For such a U one is tempted to define OX(U) as OXi

(p−1i (U)), but the

problem is that U may be contained in the intersection of several Ui. How-ever, the rings obtained for each choice of Ui are all isomorphic via the φij,so to remedy this one defines OX(U) to be the subring of

∏U⊂Ui

OXi(p−1

i (U))

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−1j (U))). One defines restriction

maps for subsets V ⊂ U as induced by the product of the restriction mapsof the OXi

; in fact, any element of OX(V ) is determined by its componentsindexed by the sets Ui containing U . It is now straightforward to check thesheaf axioms over U as well as the fact that the ringed space thus obtainedis a scheme.

Armed with this patching construction, we may now construct fibre prod-ucts of arbitrary schemes. Just as in topology, let us refer to a morphismp : Y → X of schemes as a scheme over X.

5.. FIBRES OF A MORPHISM 111

Proposition 5.4 Given two schemes p : Y → X, q : Z → X over the samescheme 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 isequipped with two canonical morphisms into Y and Z making the diagram

Y ×X Zπ2−−−→ Z

π1

yyq

Yp

−−−→ 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 schemeY ×X Z defined in Proposition 5.1 is indeed a fibre product in the categoryof schemes. For this we have to see that for an arbitrary scheme S anyelement of FY Z(S) factors as a composite (π1, π2) φ with some morphismφ : S → Y ×S Z. Choosing an affine open cover Si : i ∈ I of S, foreach i we dispose of a morphism φi : Si → Y ×X Z with the above propertyaccording to Proposition 5.1. Since by definition for any affine open subsetU ⊂ 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 arethe 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 andZj : j ∈ J of Y and Z, respectively. First fix some l ∈ J . We then disposeof affine shemes Yi ×X Yl for each i ∈ I. Now note that quite generally ifY ×X Z represents the functor FY Z and U ⊂ Y is an open subscheme, thenthe open subscheme π−1

1 (U) ⊂ Y represents the functor FUZ and as suchis unique up to unique isomorphism by the Yoneda lemma. Applying thisremark with Zl in place of Z, Yi (resp. Yj) in place of Y and Yi ∩ Yj in placeof 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×XZl → Yi (resp.Yj ×X Zl → Yj). The uniqueness of the φij implies that the compatibilityconditions in Construction 5.3 are satisfied, so we may patch the Yi ×X Zl

together along the Uij to obtain a scheme Y ×XYl. The projections Y ×XZl →Y and Y ×X Zl → Zl are defined by patching the projections from the


elements of the open covering Yi ×X Zl : i ∈ I of Y ×X Zl as in theprevious paragraph. To show that Y ×X Zl represents FY Zl

one considers fora pair (φ, ψ) ∈ FY Zl

(S) the restrictions (φ|φ−1(Yi), ψ) ∈ FYiZland patches the

corresponding morphisms S → Yi ×X Zl together, again arguing as in theprevious paragraph.

Now by exactly the same method one shows that the schemes Y ×X Zl

patch together to give a scheme Y×XZ representing FY Z . Finally one extendsthe construction to arbitary X by choosing a covering of X by affine opensubschemes Xk and noting that the open subschemes Yk = p−1(Xk) form anopen covering of Y such that the fibre products Yk ×Xk

q−1(Xk) representthe functors FYkZ where the Yk are viewed as schemes over X (indeed, given(φ, ψ) ∈ FYkZ(S) we must have ψ(S) ⊂ q−1(Xk)), so one may repeat theprevious procedure to patch the schemes Yk×XZ = Yk×Xk

q−1(Xk) together.

Now if we imitate the situation in topology, to define the fibre of a mor-phism Y → X at some point P of X we first need to define the inclusionmorphism P → X. This is achieved as follows. Take an affine openneighbourhood U = SpecA. Then P is identified with a prime ideal ofA and we dispose of a morphism A → AP which we may compose withthe natural projection AP → AP/PAP =: κ(P ). We get a morphismSpecκ(P ) → U , whence by composition with the inclusion map U → Xa morphism iP : Specκ(P ) → X.

Lemma 5.5 The morphism iP : Specκ(P ) → X just defined does not de-pend on the choice of U .

Proof: If V = SpecB is another affine open neighbourhood of P , thenthere is some affine open subscheme W ⊂ V ∩ U . We may assume that Was a subscheme of U is of the form D(f) with some f ∈ A \ P . But thelocalisation map A→ AP factors through Af (since f is a unit in AP ), whichmeans that the map SpecAP → U factors through W . By symmetry, we getthe same conclusion for V .

Definition 5.6 Given a morphism φ : Y → X and a point P of X, the fibreof φ at P is the scheme YP := Y ×X Specκ(P ), the fibre product being takenwith respect to the maps φ and iP .

We saw in Remark 5.2 that the underlying topological space of a fibreproduct is not a topological fibre product in general. However, the good newsis:

6.. SPECIAL PROPERTIES OF SCHEMES 113

Proposition 5.7 Given a morphism φ : Y → X and a point P of X, theunderlying topological space of the fibre YP is homeomorphic to the subspaceφ−1(P ) of the underlying space of Y .

Proof: We may assume we are dealing with affine schemes Y = SpecBand X = SpecA. We first show there is a bijection between φ−1(P ) andSpecB⊗Aκ(P ) as sets, the homomorphism λ : A→ B defining the A-modulestructure of B being the one corresponding to φ by Theorem 2.14. Now theabove λ induces a map λ : A/P → B/λ(P )B and a point Q ∈ SpecB is in

φ−1(P ) if and only if its image Q in B/λ(P )B satisfies λ−1

(Q) = (0). Thisis 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 thelatter ring is none but B ⊗A κ(P ). To see this, note first the isomorphismB/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 multiplicationmap b⊗a 7→ ba and so the image of B⊗AP in B is exactly λ(P )B (since B isan A-module via λ). Now the natural map A/P → B⊗A (A/P ) = B/λ(P )Bis given by a 7→ 1 ⊗ a and the localisation of B ⊗A (A/P ) by the subset1 ⊗ a : a ∈ (A/P ) \ 0 is exactly B ⊗A κ(P ).

In the above procedure we identified YP = SpecB⊗Aκ(P ) with a subset ofSpecB; by looking at basic open sets D(f) one sees easily that the topologyof YP corresponds to the subspace topology.

6. Special Properties of Schemes

In this section we have assembled some technical notions concerning schemesthat are to be used in the sequel. The reader is advised to take a brief glanceat 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 ⊂ Xthe ring OX(U) is an integral domain.

This algebraic notion has a strong consequence for the underlying topo-logical space of the scheme. Namely, call a topological space X irreducible if


it cannot be written as a union of two closed subsets properly contained inX, 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 irre-ducible.

Proof: Indeed, if U1 and U2 are nonempty disjoint open subsets of a schemeX, then the sheaf axioms imply that OX(U1 ∪U2) is isomorphic to the directsum 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 spaceis irreducible and if the rings OX(U) contain no nilpotent elements. SeeHartshorne [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 ofX.

2. The stalk OX,η is a field K which is naturally isomorphic to the fractionfield 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 = SpecAcontains both η1 and η2: they correspond to prime ideals P and Q of A withthe property V (P ) = V (Q) = U . Since A is an integral domain, this is onlypossible 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 closurein 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 butthe fraction field of A which is the common fraction field of all local rings ofU ; for the points of X \ U we work with other affine open subsets which allhave a non-empty intersection with U by irreducibility of X and hence havea local ring in common.

Definition 6.5 The point η of the proposition is called the generic point ofX 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 iff ∈ OX,P ; it has a zero at P if it is contained in the maximal ideal of OX,P .


Lemma 6.6 ?? Let X be an integral scheme The sets of points

Df = P ∈ X : f is regular atP

andD0

f = P ∈ X : f is regular and does not have a zero atP

are open in X.

Proof: Let f be regular at P and take an affine open neighbourhood U =SpecA of P . We may write f = x/y with x, y ∈ A; thus f is regular in theopen neighbourhood D(y) of P , which proves openness of Df . As for D0

f , itis the intersection Df ∩Df−1 .

Definition 6.7 An integral closed subscheme of some affine (resp. projec-tive) n-space over a field k is called an affine (resp. projective) variety overk.

Remark 6.8 In the literature one often finds a stronger condition imposedon affine and projective varieties X, namely that they should also be geo-metrically integral, which means integrality of the scheme X ×Spec k Spec kfor an algebraic closure k of k. But in some texts no integrality condition isrequired at all.

Next we state a finiteness condition which is always satisfied for affineand projective varieties. Recall that a ring is noetherian if all of its idealsare finitely generated.

Definition 6.9 A scheme X is noetherian if it is compact and has a coveringby 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 noetherianscheme 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, wewould like affine and projective n-space to be n-dimensional, a point to be 0-dimensional, a plane curve 1-dimensional, a surface 2-dimensional, etc. Oneheuristic approach is the following inductive “argument”: a curve should beof dimension 1 because its irreducible proper closed subsets are only points,a surface should have dimension 2 as it contains only curves and points asproper closed subsets etc. This approach is summarised in the followingdefinition.


Definition 6.11 The dimension of a scheme X is the supremum of the in-tegers 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 ismainly interesting for noetherian schemes because noetherian rings haveno infinite ascending chains of prime ideals. However, there exist noethe-rian rings whose associated affine scheme has infinite dimension; see Atiyah-Macdonald [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 aneasy lemma.

Lemma 6.13 Let X = SpecA be an affine scheme. Then any irreducibleclosed 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 assumethat I is the intersection of the prime ideals corresponding to the points ofZ. Assume fg ∈ I for some f, g ∈ A. Then any prime ideal containing Imust contain f or g, hence the union of the closed subsets V (I + (f)) andV (I + (g)) is Z. Therefore Z is irreducible if and only if one of them, sayV (I + (f)) equals Z. By our assumption on I this is equivalent to f ∈ I,whence the claim.

By the lemma, the dimension of SpecA is the supremum of the lengths ofchains of prime ideals in A. In ring theory this is called the Krull dimensionof A and is usually denoted by dimA.

Thus for instance, the Krull dimension of a field is 0, that of Z is 1. Butin general with the above definition the dimension is hard to determine inpractice. It is not even clear that affine or projective spaces have the dimen-sion we expect. Fortunately, this can be remedied by means of a criterion forwich we need to recall a definition first.

Definition 6.14 The transcendence degree of a field extension K|k is themaximal number of elements of K algebraically independent over k; thetranscendence degree of an integral domain A containing k is defined as thetranscendence 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 afinitely generated k-algebra. Then the Krull dimension of A is equal to itstranscendence degree over k.


For a proof, see e.g. Matsumura [1], Theorem 5.6.

Example 6.16 As immediate applications of the proposition, we get thatAn

k and Pnk both have dimension n as expected, and that affine and projective

plane curves have dimension 1. In general, affine or projective varieties ofdimension 1 are called curves, those of dimension 1 surfaces.

Chapter 6

Dedekind Schemes

In this chapter we introduce the main protagonists of the following two chap-ters, namely Dedekind schemes. These will be schemes characterised bycertain special properties that are common to smooth algebraic curves andspectra of rings of integers in number fields. Analogies between algebraicnumbers and functions on algebraic curves have already been noticed in the19th century; since then, several axiomatisations of the common featureshave been proposed of which the notion of a Dedekind scheme seems to beparticularly satisfactory.

1. Integral Extensions

In this section we review the basic theory of integral extensions of rings. Asthis 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 tobe integral over A if it is a root of a monic polynomial xn+an−1x

n−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 equiv-alent 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 asan A-module.

119

120 CHAPTER 6. DEDEKIND SCHEMES

Proof: For the implication 1) ⇒ 2) note that if x satisfies a monic polyno-mial of degree n, then 1, x, . . . , xn−1 is a basis of A[x] over A. The implication2) ⇒ 3) being trivial, only 3) ⇒ 1) remains. For this consider the A-moduleendomorphism of C given by multiplication by x. Its characteristic polyno-mial 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 subringin B.

Proof: Indeed, given two elements x, y ∈ B integral over A, the elementsx−y and xy are both contained in the subring A[x, y] of B which is a finitelygenerated A-module by assumption.

If all elements of B are integral over A, we say that the extension A ⊂ Bis integral.

Corollary 1.3 Given a tower extensions A ⊂ B ⊂ C with A ⊂ B andB ⊂ 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, theintegral closure of A in L is the subring of L formed by elements integral overA. We say that A is integrally closed if its integral closure in the fractionfield K is just A. By the corollary above, the integral closure of a domain Ain some extension L of its fraction field is integrally closed.

Example 1.4 Any unique factorisation domain A is integrally closed. In-deed, if an element a/b ∈ K (with a, b coprime) satisfies a monic polynomialequation of degree n, then by multiplying with bn we see that an should bedivisible 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. Wedenote by OK the integral closure of Z in K and call it the ring of integersof K. Of course, OK is integrally closed.

We next collect some easy results that will be needed in subsequent sec-tions.

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.

1.. INTEGRAL EXTENSIONS 121

Proof: For the first statement note that if a nonzero u ∈ I satisfies anequation un + an−1u

n−1 + . . . + a1u + a0 = 0 with ai ∈ k, then a0 ∈ I ∩ A.Since B is a domain, we may assume a0 6= 0, whence the assertion. Thesecond statement follows from this for if I is a nonzero ideal of B, we musthave 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 Bis 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 primeidal P of A the localisation AP is integrally closed.

Proof: Denote by K the fraction field of A. One implication is easy: if anelement x ∈ K satisfies a monic equation over AP , then by multiplying witha suitable common multiple s of the denominators of the coefficients one getsthat sx is integral over A, hence sx ∈ A. For the converse, take an elementx = a/b ∈ K integral over A. If b is not a unit in A, there is some maximalideal 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 proofshows:

Lemma 1.8 Let A be a domain with fraction field K ans let S ⊂ A be amultiplicatively closed subset. Then for any field extension L|K, the integralclosure of the localisation AS in L is BS, where B is the integral closure ofA in L.

The last topic to be treated in this section is the question whether theintegral closure of an integral domain A in a finite extension of its fractionfield is a finitely generated A-module. Unfortunately, this property does nothold for arbitrary domains, even under the assumption that A is noetherian.But there are two classical sufficient conditions which we now quote from theliterature.

Proposition 1.9 Let A be an integrally closed noetherian domain with frac-tion field K and let L be a finite separable extension of K. Then the integralclosure 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], Chap-ter I, Proposition 6.

The other sufficient condition is the following.


Proposition 1.10 Let k be a field and A an integral domain which is afinitely generated k-algebra. If L is any finite extension of the fraction fieldK 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 ofcharacteristic 0. In the case of positive characteristic, there is a simple prooffor polynomial rings in Shafarevich [1], Appendix, Section 8. The generalcase 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 ringswhich enjoy very similar properties to rings of germs of meromorphic func-tions in a neighbourhood of some point of a Riemann surface. The first ofthe 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 principalideal domain which is not a field.

Before stating the first equivalent characterisations, observe that if A is alocal ring with maximal ideal P , then the A-module P/P 2 is in fact a vectorspace over the field κ(P ) = A/P , simply because multiplication by P mapsP into P 2.

Proposition 2.2 For a local domain A with maximal ideal P the followingconditions 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 overκ(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 beextremely useful in other situations as well:

Lemma 2.3 (Nakayama’s Lemma) Let A be a local ring with maximalideal P and M a finitely generated A-module. If PM = M , then M = 0.

2.. DEDEKIND SCHEMES 123

Proof: Assume M 6= 0 and let m0, . . . , mn be a minimal system of genera-tors of M over A. By assumption m0 is contained in PM and hence we havea relation m0 = p0m0 + . . . , pnmn with all the pi elements of P . But here1 − p0 is a unit in A (as otherwise it would generate an ideal contained inP ) and hence by multiplying the equation by (1− p0)

−1 we may write m0 asa linear combination of the other terms, which is in contradiction with theminimality 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

⋂

i

P i = (0).

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 isfinitely generated. Moreover, PQ = Q and the lemma applies. The secondstatement 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 elementst1, . . . , tm ∈ M whose images in the A/P -vector space M/PM form a gen-erating system. Then they generate M over A.

Proof: Let T be the A-submodule generated by the ti; we have M =T + PM by assumption. Hence M/T = P (M/T ) and the lemma givesM/T = 0.

Proof of Proposition 2.2: Assume A is a discrete valuation ring and P isgenerated by t. Then by Corollary 2.4 any nonzero prime ideal Q ⊂ P mustcontain a power of t. But being a prime ideal, it must then contain t itself, sothat Q = P and A is of Krull dimension one. Also, the image of t generatesthe vector space P/P 2, whence the second condition. The third conditionfollows from the second by applying Corollary 2.5 with M = P . Finally, toshow that the third condition implies the first, assume the maximal ideal Pof A is generated by some element t. We first show that any element a ∈ Acan 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 acan be written in the required form. If a = utn = vtn, then u = v since A isa domain. Now take an ideal I of A. As A is Noetherian, I can be generated


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. Eachai 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 valuationring A can be written uniquely in the form x = utn with u a unit in A, t agenerator of the maximal ideal and n a (possibly negative) integer.

The second condition of the lemma may seem a bit technical, but it is veryuseful for it is a special case of a more general notion coming from algebraicgeometry.

Definition 2.7 A noetherian local ring A with maximal ideal P is regularif 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) ≥ minv(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 valuationring of v.

Proposition 2.9 A domain A is a discrete valuation ring if and only if itis the valuation ring of some discrete valuation v : K → Z ∪ ∞, where Kis 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 bythe previous corollary. It is immediate to check that v is a discrete valuationwith valuation ring A. Conversely, given a discrete valuation v on K, themaximal ideal of its valuation ring is generated by any t with v(t) = 1 andwe may apply the previous proposition.

We can now discuss the example mentioned at the beginning of this sec-tion.


Example 2.10 Let M be the sheaf of meromorphic functions on some Rie-mann surface X and x a point of X. Define v(f) = m if f is holomorphicat x and has a zero of order m and define v(f) = −v(1/f) otherwise. Thenv is a discrete valuation on K = M(X) whose discrete valuation ring is thestalk of M at x.

The last characterisation of discrete valuation rings we shall need is thefollowing.

Proposition 2.11 A domain A is a discrete valuation ring if and only if Ais noetherian, integrally closed and has a unique nonzero prime ideal.

For the proof, which we have taken from Serre [1], we need a technicallemma.

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 withxA ⊂ P .

Proof: Observe first that for any f ∈ P the localisation Af is a field andhence equals K itself. Indeed, for a maximal ideal M of Af the prime idealM ∩ A doesn’t contain f , hence it can be only (0). But for any nonzeroelement y/fn ∈ M we would have 0 6= y ∈ M ∩ A, whence M = 0, asdesired. Take now another x ∈ P ; by the above we may write x−1 = a/fm

with some m, so fm = ax, and thus fm ∈ (x). Letting f vary in a finite set ofgenerators of P we conclude that a sufficiently high power of P is containedin (x). Let PN be the least such power; we may thus find y ∈ PN−1 withy /∈ (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 factthat any nonzero ideal of A is of the form (tn) with t a generator of themaximal 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 thatit is principal. Evidently A ⊂ P−1, so P ⊂ P−1P , the latter being the idealof A generated by elements of the form xy with x ∈ P−1 and y ∈ P . SinceP is maximal, there are two cases: either P−1P = A or P−1P = P . Wenow show that if the first case holds, then P is principal. Indeed, in thiscase there is a relation of the form x1y1 + . . .+ xnyn = 1 with xi ∈ P−1 andyi ∈ P . Here there is at least one i for which xiyi /∈ P , so there is some unit


u with uxiyi = 1. We contend that P = (yi). Indeed, if z ∈ P , we havez = uzxiyi, but uzxi ∈ A since xi ∈ P−1.

To finish the proof we show that the case P−1P = P cannot occur. We dothis by showing that this assumption implies P−1 = A, in contradiction withlemma 2.12. So take x ∈ P−1. By assumption xP ⊂ P ; iterating this we getxnP ⊂ P for all n, so xn ∈ P−1 for all n. In particular, for any f ∈ P allpowers of x are contained in the A-submodule of K generated by f−1, whichis a finitely generated A-module. But A is noetherian and a submodule of afinitely 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, thisimplies x ∈ A, as desired.

Remark 2.13 The affine scheme SpecA is particularly simple for a discretevaluation ring. It consists only two points, a closed point x (correspondingto the maximal ideal) and a non-closed generic point η (corresponding to theideal (0)). The stalk of the structure sheaf at η is the fraction field K of Aand 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 inte-grally closed domains. A Dedekind scheme is an integral noetherian normalscheme of dimension 1. A Dedekind ring is a domain A such that SpecA isa Dedekind scheme.

Remarks 2.15 Here some remarks are in order.

1. There is another restriction that is convenient to impose on the schemeswe shall be looking at, namely that the local rings of X should bedistinct when viewed as subrings of the function field K; in this caseone says X is separated (over Z). This condition plainly holds in theaffine case where the local rings are all localisations at different primeideals; 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 butwhich is not separated. This is constructed by taking two copies of theaffine line A1

k over some field k and patching them over the open subsetD(x) using the isomorphism given by the identity map. In this way weget two closed points coming from the two origins whose local rings arethe same.

Henceforth we shall tacitly assume that all integral schemes under con-sideration are separated.


2. Call a noetherian scheme regular if all of its local rings are regular.Then by Proposition 2.2 Dedekind schemes are precisely regular inte-gral schemes of dimension 1.

Now let us draw some immediate consequences from the definition ofDedekind 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 withfraction field K.

3. Any affine open subset of X is of the form SpecA, with A an integrallyclosed domain.

Proof: Since X is noetherian, it is compact, so for the first statement wemay assume X = SpecA with a Noetherian ring A. In SpecA any closedsubset is a finite union of irreducible closed subsets. (To see this, decomposeany reducible closed subset Z as a union of nonempty closed subsets Z1∪Z2; ifthese are not irreducible, decompose them again - the process must terminatein finitely many steps as otherwise we would get an infinite strictly increasingchain of ideals of A which is impossible in a noetherian ring.) But anyirreducible closed subset of X is a closed point by Chapter 5, Lemma 6.13and the fact that all prime ideals of A are maximal. This proves the firststatement; the second follows from Proposition 2.11 in view of the easy factthat any localisation of a noetherian ring A is noetherian (as the ideals of thelocalisation are all generated by ideals of A according to Chapter 4, Lemma2.4). 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 SpecOK

is a Dedekind scheme.

Indeed, A is a domain, so SpecA is integral. That it is noetherianfollows from Proposition 1.9. To prove that it is of dimension 1, we showthat any nonzero prime ideal P of OK is maximal. Indeed, by the firstpart of Lemma 1.5, the intersection P ∩Z is nonzero, hence generatedby some prime number p. Now the induced extension Z/pZ ⊂ OK/P isstill integral, so we may apply the second statement of the same lemmato conclude that OK/P is a field. Finally the normality of SpecAfollows from Lemma 1.7.


2. The second basic example of a Dedekind scheme is given a by one-dimensional normal integral closed subscheme of affine (resp. projec-tive) n-space over some field k. These we shall call smooth affine (resp.projective) curves over k. For the moment, this definition is tautolo-gous but one can still give some concrete examples. For instance, theaffine line A1

k is a smooth affine curve (being the spectrum of the one-dimensional unique factorisation domain k[t]) and the projective lineP1

k is a smooth projective curve over k for it is integral and can becovered by two copies of A1

k.

However, for a general closed subscheme of affine or projective spacethis ring-theoretic definition in the second example is rather hard to check.What is much more preferable is the smoothness condition encountered inour discussion of Riemann surfaces; for a plane curve this said that the partialderivatives of its defining equation should not simultaneously vanish at somepoint. In the next section we introduce an algebraic formalisation of thenotion 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 varietyis defined to consist of so-called linear derivarions, i.e. linear maps thatassociate 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 intoM 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 classicalderivation rules. If moreover B is an A-algebra for some ring A (for exampleA = Z), an A-linear derivation is called an A-derivation. The set of A-derivations of B to M is equipped with a natural B-module structure via therules (d1 + d2)x = d1x + d2x and bdx = dbx. This B-module is denoted byDerA(B,M).

Note that applying the Leibniz rule to the equality 1 ·1 = 1 gives d(1) = 0for all derivations; hence all A-derivations are trivial on the image of A in B.

3.. MODULES AND SHEAVES OF DIFFERENTIALS 129

In the example one encounters in (say) real differential geometry we haveA = M = R, and B is the ring of germs of differentiable functions at somepoint; R is a B-module via evaluation of functions. Now comes a purelyalgebraic example.

Example 3.2 Assume given an A-algebra B which decomposes as an A-module into a direct sum B ∼= A ⊕ I, where I is an ideal of B with I2 = 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 ∈ Band write xi = ai + dxi with ai ∈ k for i = 1, 2. Now we have

d(x1x2) = d[(a1 + dx1)(a2 + dx2)] = d(a1a2 + a2dx1 + a1dx2) = x2dx1 +x1dx2

where we used several times the facts that I2 = 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 therule (a1, i1)(a2, i2) = (a1a2, a1i2 +a2i1). So the above method yields plenty ofexamples of derivations.

Now notice that for fixed A and B the rule M → DerA(B,M) definesa 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 B-module ΩB/A.

Proof: The construction is done in a similar way to that of the tensorproduct of two modules. Define ΩB/A to be the quotient of the free B-module generated by symbols dx for each x ∈ B modulo the relations givenby the additivity and Leibniz rules as in Definition 3.1 as well as the relationsd(λ(a)) = 0 for all a ∈ A, where λ : A → B is the map defining the A-module structure on B. The map x → dx is an A-derivation of B intoΩB/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 whosecomposition with d is just δ. This implies that ΩB/A represents the functorM → DerA(B,M); in particular, d is the universal derivation correspondingto 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 pre-sented A-algebra.


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 theB-submodule generated by the elements

∑j(∂jfi)dxj (i = 1, . . . , m), where

∂jfi denotes the j-th (formal) partial derivative of fi.

Proof: First consider the case B = A[x1, . . . , xn]. As B is the free A-algebra generated by the xi, one sees that for any B-module M there is abijection between DerA(B,M) and maps of the set x1, . . . , xn into B. Thisimplies that ΩB/A is the free A-module generated by the dxi.

The general case follows from this in view of the easy observation thatgiven any M , composition by the projection A[x1, . . . , xn] → B induces anisomorphism 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′

Ω(B⊕B′)/A∼= ΩB/A ⊕ ΩB′/A.

2. (Exact sequence) Given a map of A-algebras φ : B → C, there is anexact 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⊗BC → ΩC/A.

3. (Base change) Given a ring homomorphism A→ A′, denote by B′ theA′-algebra B ⊗A A

′. There is a natural isomorphism

ΩB/A ⊗B B′ ∼= ΩB′/A′ .

4. (Localisation) For any multiplicatively closed subset S ⊂ B there is anatural isomorphism

ΩBS/A∼= ΩB/A ⊗B BS.


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 0and the isomorphism HomB(ΩB/A,M) ∼= HomC(ΩB/A ⊗B C,M). This iso-morphism is obtained by mapping a homomorphism ΩB/A → M to the com-posite 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⊗BC.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 isan A-module homomorphism and so tensoring it by A′ we get a map

d′ : B′ → ΩB/A ⊗A A′ ∼= ΩB/A ⊗B B ⊗A A

′ ∼= ΩB/A ⊗B B′

which is easily seen to be an A′-derivation. Now any A′-derivation δ′ : B′ →M ′ induces an A-derivation δ : B → M ′ by composition with the naturalmap B → B′. But δ factors as δ = φ d, with a B-module homomorphismφ : ΩB/A →M ′, whence a map φ′ : ΩB/A ⊗B B

′ →M ′ constructed as above.Now one checks that δ′ = φ′ d′ which means that ΩB/A ⊗B B′ representsthe functor M ′ 7→ DerA′(B′,M ′).

For the localisation property, given an A-derivation δ : B → M , wemay extend it uniquely to an A-derivation δS : BS → M ⊗B BS by settingδS(b/s) = (δ(b)s − bδ(s)) ⊗ (1/s2). (We leave it to the reader to check thatfor b′/s′ = b/s we get the same result – this is much simpler in the casewhen 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, andone 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 characteri-sation 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. ThenA is etale if and only if ΩA/k = 0.


Proof: For necessity we may assume by compatibility of ΩA/k with directsums that A is a finite separable field extension L of k. Then by the theoremof the primitive element A ∼= k[x]/(f) with some polynomial f ∈ k[x] andso by Proposition 3.4 the L-module ΩL/k can be presented with a singlegenerator dx and relation f ′dx = 0. But since the extension is separable,the polynomials f and f ′ are relatively prime and hence the image of f ′ inL ∼= 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, Proposition1.2 that A⊗k k is etale over k with k an algebraic closure of k. So using thebase change property of differentials we may assume k is algebraicaly closed.Moreover, using the direct sum property as above we may even assume thatA is indecomposable. Denoting by I its ideal of nilpotent elements, Chapter5, 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 ⊕ Iof A as a k-module. Now to finish the proof we show that assuming I 6= 0implies ΩA/k 6= 0. For this it is enough to show by the surjectivity propertyin Lemma 3.5 (2) that Ω(A/I2)/k 6= 0, so we may as well assume I2 = 0. Butthen we are (up to change of notation) in the situation of Example 3.2 whichshows 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 one-dimensional closed subscheme of affine or projective space to be a smoothcurve. For this it is enough to check that all local rings at closed points arediscrete valuation rings. Since the proof works more generally for regularlocal rings, we state the result in this context.

Proposition 3.7 Let k be a perfect field and let A be a localisation of afinitely generated n-dimensional k-algebra at some closed point P . Then Ais a regular local ring if and only if ΩA/k is a free A-module of rank n. Inparticular, if n = 1, A is a discrete valuation ring if and only if ΩA/k is freeof 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 thatthe proposition amounts to saying that among the relations

∑j(∂jfi)dxj = 0

there should be exactly d − n linearly independent ones, which in turn isequivalent by linear algebra to the fact that the k × m “Jacobian” matrixJ = [∂jfi] should have rank d− n. In fact, for k = C reducing the entries of


J modulo the maximal ideal of A gives just the classical Jacobian matrix ofthe closed subscheme of Cd defined by the equations fi = 0 at the point Pcorresponding to A and the condition says that some open neighbourhood ofP should be a complex manifold of dimension n.

For the proof of the proposition we need two lemmas from algebra. Thefirst of these is a form of Hilbert’s Nullstellensatz (which implies the one usedin the previous chapter).

Lemma 3.9 Let k be a field and let P be a maximal ideal in a finitely gen-erated 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 Atiyah-Macdonald [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 fieldextension of transcendence degree n. Then there exist algebraically indepen-dent 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 isof dimension n, a basis being given by the dxi.

Proof: We may write the field K as the fraction field of the quotient A ofthe polynomial ring k[x1, . . . , xn, x] by a single polynomial relation f . Heref is the minimal polynomial of a generator of the extension K|k(x1, . . . , xn)multiplied with a common denominator of its coefficients. Now accordingto Proposition 3.4 the A-module ΩA/k has a presentation with generatorsdx1, . . . , dxn, dx and a relation in which dx has a nontrivial coefficient becausef ′ 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 assump-tion that there exists a subfield k ⊂ k′ ⊂ A that maps isomorphically ontothe residue field κ(P ) = A/P by the projection A → A/P . (Lemma 3.9implies 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 tothis one.

Notice that since k is perfect and k′|k is a finite extension by Lemma 3.9,we have Ωk′|k = 0 by Proposition 3.6 (or the previous corollary). Hence by


applying Lemma 3.5 (2) (with our k in place of A, k′ in place of B and A inplace of C) we get ΩA/k

∼= ΩA/k′, so we may as well assume k = k′ ∼= κ(P ).In this case the k-module P/P 2 is canonically isomorphic to ΩA/k/PΩA/k.

Indeed, the latter k-vector space is immediately seen to represent the functorM → Derk(A,M) for any k-vector space M viewed as an A-module via thequotient map A → A/P ∼= k. On the other hand, the above functor is alsorepresented by P/P 2. To see this, note first that the Leibniz rule impliesthat any k-derivation δ : A → M is trivial on P 2, hence we may as wellassume P 2 = 0. But then we are in the situation of Example 3.2 and we mayobserve that δ factors uniquely as δ = φ d, with d as in the quoted exampleand φ ∈ Homk(P,N).

Now if ΩA/k is free of rank n, then ΩA/k/PΩA/k∼= P/P 2 has dimension n.

For the converse, observe first that the previous isomorphism and the corol-lary to Nakayama’s lemma (Corollary 2.5) gives that ΩA/k can be generatedas an A-module by n elements dt1, . . . , dtn. Were there a nontrivial relation∑fidti = 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 dis-cussed above it is convenient to use the completion A of A. This is theinverse limit of of the natural inverse system formed by the quotients A/P n

of A. There is a natural map A → A which is injective for A noethe-rian by Corollary 2.4. The image of P gives a maximal ideal P of A withP i/P i+1 ∼= P i/P i+1 for all i > 0. If A is of dimension 1, the case i = 1of this isomorphism together with Corollary 2.5 implies that A is a discretevaluation ring if and only if A is. In general, we get that A is regular if andonly if A is regular, for one can prove (see Atiyah-Macdonald [1], Corollary11.19) that the Krull dimension of A is the same as that of A. Also, the basechange property of differentials implies that ΩA/k is free of rank n if and onlyif ΩA/k is.

Therefore it remains to see that A satisfies the condition at the beginningof 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 aroot of f in A. This can be done by means of Hensel’s lemma (see Chapter7, Section 4).

In the remaining of this section we discuss quasi-coherent sheaves associ-ated to modules of differentials. Namely, we shall define sheaves of relativedifferentials ΩY/X for certain classes of morphisms of schemes Y → X. Infact, one may define these for any morphism Y → X but since we did not


develop the necessary background we refer the interested readers to the ex-cellent 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 specialcases we shall need.

Construction 3.13 First, if Y = SpecB and X = SpecA are both affine,we define ΩY/X as the quasi-coherent sheaf ΩB/A. Notice that according tothe localisation property of differentials, over a basic open setD(g) = SpecBg

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 anarbitrary scheme X; we shall use the abusive notation ΩX/k for the corre-sponding sheaf of differentials which we now construct. For any affine opencovering of X by subsets Ui = SpecAi the rings Ai are all k-algebras and thesheaf ΩUi/k = ΩAi/k is defined on Ui. Moreover, any basic open subset con-tained in Ui ∩Uj is canonically isomorphic to both (Ai)fi

and (Aj)fj, whence

an isomorphism Ω(Ai)fi/k

∼= Ω(Aj )fj/k. These isomorphisms are compatible

for inclusions of basic open sets, so the third statement of Chapter 5, Lemma2.7 applies to give an isomorphism (ΩUi/k)|Ui∩Uj

∼= (ΩUj/k)|Ui∩Uj. These lat-

ter 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, Construc-tion 5.3 (which adapts to the construction of quasi-coherent sheaves) to getΩX/k. Finally one checks that if we use a different open covering we get anOX -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 schemeif and only if the stalk of the sheaf ΩX/k at each point is free of rank n (forthe generic point this follows by localisation). From the next section on, weshall call such sheaves locally free (see Lemma 4.3 below). Also, those X forwhich ΩX/k is locally free are usually called smooth (over k).

In particular, an affine or projective variety of dimension 1 is a Dedekindscheme if and only if it is a smooth curve.

Construction 3.16 Finally, the other case where we shall use relative dif-ferentials is that of an affine morphism φ : Y → X. In this case X is coveredby affine open subsets Ui = SpecAi whose inverse images Vi = SpecBi forman 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 ofthe basic open set D(fi) = Spec (Ai)fi

is none but D(gi) which in turn is iso-morphic to Spec (Bi⊗Ai

(Ai)fi); indeed, one checks easily that (Bi⊗Ai

(Ai)fi)


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/Uitogether over inverse images of basic affine

open subsets contained in Ui ∩ Uj by the same method as in the previouscase.

4. Invertible Sheaves on Dedekind Schemes

In this section we shall study some special coherent sheaves of fundamentalimportance 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 forwhich there exists an open covering U = Ui : i ∈ I of X such that therestriction of F to each Ui is isomorphic to Oni

Uifor some positive integer ni.

A trivialisation of F is a covering U as above and a system of isomorphismsOni

Ui

∼= F|Ui.

If X is connected, then the ni are all equal to the same number n calledthe rank of F . A locally free sheaf of rank 1 is called an invertible sheaf or aline bundle.

Remark 4.2 For any locally free sheaf F and point P ∈ X with residuefield κ(P ) the group FP ⊗κ(P ) is a finite dimensional κ(P )-vector space. Sowe may think of a locally free sheaf as a family of κ(P )-vector spaces whichis “locally trivial”. In fact, locally free sheaves correspond to vector bundlesin 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 noetherianand connected, a coherent sheaf F on X is locally free of rank n if and onlyif 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 = SpecAcontained in one of the Ui as in the definition. Then by the assumption therestriction of F to V is isomorphic to the coherent sheaf defined by the freeA-module A⊕ . . .⊕ A (with A repeated ni times).

In the second statement necessity follows from the definitions by takingthe direct limit. For sufficiency, assume FP is freely generated over OX,P bysome generators t1, . . . , tn. We may view the ti as sections generating F(U)

4.. INVERTIBLE SHEAVES ON DEDEKIND SCHEMES 137

for some suficiently small open neighbourhood U of P . By shrinking U ifnecessary we may assume U = SpecA and F = M for some A-module Mgenerated by the ti. Since X is noetherian, M is the quotient of the freeA-module of rank n by a submodule generated by finitely many relationsamong the ti. By assumption, any of the finitely many coefficients occuringin these relations vanishes when restricted to some open neighbourhood ofP 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 theabove proof shows that if F is a coherent sheaf on any scheme X and P is apoint for which FP = 0 then there is some open neighbourhood V of P withF|V = 0.

The crucial importance of invertible sheaves for the study of Dedekidschemes is shown by the following proposition.

Proposition 4.5 Any nonzero coherent sheaf of ideals I on a Dedekindscheme X is invertible.

Proof: By the criterion of Lemma 4.3, it is enough to check that IP is afree OX,P -module of rank 1 for each P ∈ X. If P is the generic point, thisis obviously true since any nonzero ideal in a ring generates the unit ideal inits fraction field. If P is a closed point, we are done by the fact that OX,P isa principal ideal ring.

Up to now, the notion of an invertible sheaf may well have seemed tobe rather abstract, but now we explain a method for constructing invertiblesheaves on Dedekind schemes. First a definition.

Definition 4.6 A (Weil) divisor on a Dedekind scheme X is an element ofthe free abelian group Div(X) generated by the closed points of X.

Thus a divisor is just a formal linear combination D =∑miPi of finitely

many closed points of X. If P is a closed point, we define vP (D) to beequal to mi if P = Pi for some i and 0 otherwise. On the other hand, letK be the function field of X; it is the common fraction field of all localrings of X. Since the local ring OX,P is a discrete valuation ring, there is anassociated discrete valuation vP on K which takes finite values on nonzeroelements of K. By analogy with the case of meromorphic functions on aRiemann surface, for a nonzero element f 6= 0 of K we say that f has a zeroof order m at P if m = vP (f) > 0 and that f has a pole of order m at P


if m = vP (f) < 0. The following lemma shows that elements of K behavein 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 nonzerofunction f ∈ K has only finitely many zeros and poles.

Proof: This follows from Chapter 5, Lemma ?? and the first statement ofProposition 2.16.

Thanks to the lemma, we may define the divisor of a nonzero functionf ∈ K as the divisor D with vP (D) = vP (f) for all closed points P . In thisway, we obtain a homomorphism

div : K× → Div(X)

where K× is the multiplicative group of K. Elements of the image of div aretraditionally called principal divisors.

Now denote by K the constant abelian sheaf on X defined by the addi-tive group of K. It has an OX -module structure coming from the naturalembedding of OX into K but is not a quasi-coherent sheaf. However, givenany divisor D ∈ Div(X) we may define a subsheaf of K which is not onlyquasi-coherent but, as we shall see shortly, even invertible. Namely, definefor 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 bythose 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 ofD to U”: they should be regular except perhaps at the points P ∈ U withvP (D) > 0 where a pole of order at most vP (D) is allowed; furthermore, atpoints 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 naturalembedding. Moreover, each L(D) becomes an OX-submodule of K with itsnatural OX -module structure.

Proposition 4.8 Each L(D) is an invertible sheaf on the Dedekind schemeX. Moreover the rule D 7→ L(D) induces a bijection between divisors andinvertible subsheaves of K.

Here the term “invertible subsheaf” means that we consider invertiblesheaves which are OX-submodules of K with its canonical OX -module struc-ture.


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 themaximal ideal of OX,P ). Denote by S the set of closed points Q of X withvQ(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) arefunctions g which are regular outside P and vP (g) + vP (f) ≥ 0. Hence themap of OU -modules induced by 1 7→ f−1 gives an isomorphism of OU ontoL(D)|U . Since P was arbitrary, we get a covering of X by open subsets overeach of which L(D) is isomorphic to the structure sheaf.

Conversely, given an invertible subsheaf L of K there is a trivialisationof L over some open covering U which we may choose finite by compactnessof X. For each Ui ∈ U denote by fi the image of 1 ∈ OX(Ui) under theisomorphism OX(Ui) ∼= L(Ui) arising from the trivialisation. Now define adivisor 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 tothe previous lemma, D is indeed a divisor. We still have to check that thedefinition 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 functionsu, v ∈ OX,P with fi = ufj and fj = vfi, whence both u and v are units inOX,P and vP (fi) = vP (fj). Secondly, the definition of D does not dependon the choice of the trivialisation for passing to another one induces anautomorphism of the stalk LP viewed as a free OX,P -module of rank 1, andsuch 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 sheafL(−D) is a subsheaf not only of K but also of OX and hence is an ideal sheaf.Now assume X = SpecA is affine and take a nonzero ideal I of A. Accordingto Proposition 4.5, the coherent ideal sheaf I is an invertible subsheaf of K, sothe proposition applies and shows that I can be identified with an invertiblesheaf of the form L(−D), where D =

∑miPi is a divisor with all mi > 0. By

definition, a global section of the latter sheaf is an element of A contained inthe intersection ∩Pmi

i , where the Pi are viewed as prime ideals of A. But asthe ideals Pmi

i are pairwise coprime, their intersection is the same as theirproduct, so by taking global sections of I = L(−D) we get:

Corollary 4.9 In a Dedekind ring any ideal decomposes uniquely as a prod-uct of prime ideals.

This applies in particular to rings of integers in number fields. For thering Z it is none but the Fundamental Theorem of Arithmetic.


Remark 4.10 Of course, we could have obtained this result without makingall this detour amidst schemes and invertible sheaves. But having done so,we get as a bonus a geometric interpretation of the situation, namely thatideals in a Dedekind ring can be regarded as compatible systems of localsolutions for a problem of finding functions with restricted behaviour at afinite set of points. This problem can be regarded as an analogue of theclassical Mittag-Leffler and Weierstrass problems in complex analysis.

Remark 4.11 The preceding arguments also enable one to prove directlythat on X = SpecA all quasi-coherent ideal sheaves (in fect, all of themare also coherent for A is noetherian) are isomorphic to some I. Indeed,we already know that nonzero ideal sheaves are of the form L(−D), withvP (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 iso-morphisms I(U) ∼= L(−D)(U) over basic open sets U = D(f). But wehave I(U) = I ⊗A Af and by the same argument as for U = X we getL(D)(U) = IAf ; the isomorphism is then given by the natural multiplica-tion map I ⊗A Af → IAf .

Returning to Proposition 4.8, one might be under the impression that theinvertible 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 topologicalspace is compact. Then any invertible sheaf L on X is isomorphic to asubsheaf of the constant sheaf K associated to the function field of X.

Proof: By compactness we may choose a trivialisation of L over a finitecovering of X by open subsets U1, . . . , Un; by irreducibility of X the intersec-tion U0 = ∩Ui is a nonempty open subset in X. For each integer 0 ≤ i ≤ nlet si be the image of 1 by the isomorphism OX(Ui) ∼= L(Ui) coming from thechosen trivialisation. For each i > 0 there exists a section fi ∈ OX(U0) withfis0 = si|U0 . Now we define an injective morphism L → K. According tothe third part of Chapter 4, Lemma 2.7 for this it is enough to define a com-patible system of embeddings L(U) → K(U) over those open subsets U withU ⊂ Ui for some i. These we define as the OX(U)-module homomorphismsinduced by the maps si|U → fi|U , viewing fi as a global section of K. Thedefinition does not depend on the choice of i, for if U ⊂ Ui∩Uj and s ∈ L(U)arises both as s = aisi|U and s = ajsj |U , we have s|U0 = aifis0 = ajfjs0,whence the function aifi − ajfj vanishes over the dense open subset U , soaifi = ajfj in K = K(X) by Chapter 5, Lemma ??. Injectivity of themorphisms L(U) → K(U) is obvious.


Remark 4.13 In fact, the compactness assumption in the proposition issuperfluous. See Hartshorne [1], Proposition II.6.15.

Next, note that Proposition 4.8 implies that over a Dedekind scheme it ispossible to introduce an abelian group law on the set of invertible subsheavesof 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 onemay define a natural abelian group law on the set of isomorphism classes ofinvertible sheaves on any scheme X. For this we first need the notion of thetensor product F ⊗G of two OX -modules F and G: this is none but the sheafassociated to the presheaf U → F(U) ⊗OX(U) G(U).

Remark 4.14 One checks easily that F⊗G represents the set-valued functoron the category of OX -modules which maps an OX -module M to the set ofOX -bilinear maps F × G → M. (Use the representability for each U of thefunctor 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 in-duces an abelian group structure on the set of isomorphism classes of invert-ible sheaves on X. The unit element of this group is OX and the inverse ofa class represented by an invertible sheaf L is the class of the sheaf L∨ givenby 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 apoint 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 sincethe stalks of the sheaf associated to a presheaf are the same as that ofthe 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 topologicalspace is an isomorphism if and only if for each point P the induced grouphomomorphisms on the stalks φP : FP → GP are.


Proof: One implication is easy. For the other, assume that the φP areall isomorphisms. We have to show that the maps φU : F(U) → G(U) areall bijective. For injectivity, assume s ∈ F(U) maps to 0 by φU . Then byinjectivity of the maps φP for P ∈ U we get that the image of s in the stalksFP is 0 for all P , whence a covering of U by open subsets over each of which srestricts to 0. Hence s = 0 by the first sheaf axiom. The proof of surjectivityis similar, using the second sheaf axiom (and the injectivity just proven).

Proof of Proposition 4.15: The group law is well defined since the tensorproduct of modules respects isomorphisms. The abelian group axioms con-cerning commutativity, associativity and the unit element follow from thecorresponding properties of the tensor product (of modules, which are clearlyinherited 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 com-patible with restriction maps for open inclusions V ⊂ U and moreover it isOX(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 mapsLP ⊗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,PHomOX,P

(LP ,OX,P ) ∼= OX,P ⊗OX,PHomOX,P

(OX,P ,OX,P ) ∼= OX,P

and we are done.

Definition 4.18 The group of isomorphism classes of invertible sheaves ona scheme X is called the Picard group of X and is denoted by Pic(X).

We finally establish the link between the group Div(X) of divisors on aDedekind scheme and the group Pic(X). For this denote by [L] the class ofan invertible sheaf L in the Picard group. By what we have seen so far, wedispose of a map Div(X) → Pic(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) fortwo 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 pointP is a free OX,P -module generated by some fi; that of L(D1) ⊗ L(D2) is


generated by f1⊗f2 and that of L(D1 +D2) by f1f2. Hence mapping f1 ⊗f2

to f1f2 indeed induces an isomorphism.

Proposition 4.20 The sequence

0 → OX(X)× → K×div−→Div(X)

D 7→[L(D)]−→ Pic(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 thedefinitions and surjectivity on the right follows from Propositions 4.8 and4.12. For exactness at the third term note first that for any divisor of theform D = div(f) the invertible sheaf L(D) maps to the unit element of thePicard group as multiplying sections of L(D) over an open set U by f |Uinduces an isomorphism L(D) ∼= OX . Conversely, if such an isomorphism isknown, let f be the image of 1 ∈ OX(X) by this isomorphism. Then onechecks easily that D = div(f−1).

Remarks 4.21

1. Traditionally when X = SpecA, the cokernel of the map div is calledthe (ideal) class group of the Dedekind ring A. When A is the ringof integers in a number field, a classical theorem of the arithmetic ofnumber fields asserts that this is a finite. See e.g. Lang [2], p. 100 orNeukirch [1], Chapter I, Theorem 6.3 for a proof of this fundamentalfact.

2. The results of this section generalise to integral schemes that are locallyfactorial, i.e. their local rings are unique factorisation domains. In thegeneral context closed points have to be replaced by closed subschemesof codimension one. See Hartshorne [1], Section II.6 for details.

Chapter 7

Finite Covers of DedekindSchemes

In this chapter we study finite (branched) covers of Dedekind schemes, whichwill turn out to behave in a strongly analogous way to finite branched coversof compact Riemann surfaces. On the way, we also prove classical number-theoretic results in a geometric manner which emphasises their analogy withthe 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 ofintegers of the algebraic number field Q(i). The natural inclusion correspondsto a morphism of Dedekind schemes Spec Z[i] → SpecZ: we now describe itsfibres.

The fibre over the generic point (0) is

Spec (Z[i]⊗Z Q) ∼= Spec ((Z[x]/(x2 +1))⊗Z Q) ∼= SpecQ[x]/(x2 +1) ∼= Q(i).

Similarly, the fibre over a closed point (p) of Spec Z is Spec (Z[i] ⊗Z Fp) =Spec (Fp[x]/(x

2 + 1)). Now there are three cases:

• If p ≡ 1 mod 4, then the polynomial x2 + 1 factors as the product oftwo distinct linear terms over Fp, hence (by the Chinese RemainderTheorem) the fibre is isomorphic to Spec (Fp ⊕ Fp).

• If p ≡ −1 mod 4, then the polynomial x2 +1 is irreducible over Fp andhence the fibre is isomorphic to Spec Fp2.

145

146 CHAPTER 7. FINITE COVERS OF DEDEKIND SCHEMES

• For p = 2 we have x2 + 1 = (x + 1)2 over Fp and hence the fibre isSpec (Fp[x]/(x+ 1)2). The underlying topological space of this schemeis a single point (the maximal ideal (x+ 1)) and the local ring at thispoint contains nilpotent elements (for example x+ 1).

One is thus tempted to regard the point in the fibre over (2) ∈ Spec Z asa kind of a branch point for this is the only point contained in a fibre whichis degenerate in the sense that there are nilpotent functions on it. (We shallsee shortly that though the fibres of the second type consist only of a singlepoint, it is not reasonable to consider them as degenerate.) The followingexample confirms this intuition.

Example 1.2 For Riemann surfaces, the basic example of a branched coverwas the cover C → C, z 7→ zn (indeed, we saw that any branched cover isanalytically isomorphic to this one in the neighbourhood of a branch point).Algebraically, this corresponds to the morphism SpecC[z] → SpecC[z] com-ing from the C-algebra homomorphism C[z] → C[z] induced by z 7→ zn. In-troduce the variable y = zn in the second ring: we thus have an isomorphismC[z] ∼= C[z, y]/(y − zn) and the above homomorphism corresponds to map-ping z to y. Now first look at the fibre over the generic point (0) ∈ Spec C[z]:it is

Spec (C[z, y]/(y − zn) ⊗C[z] C(z)) ∼= Spec (C(y)[z]/(y − zn))

(don’t forget that the C[z]-module structure on C[z, y]/(y − zn) occuring inthe tensor product is given by z 7→ y). Hence the fibre is the spectrum of adegree n Galois field extension of the rational function field C(y).

Now a closed point of the Dedekind scheme SpecC[z] is given by a max-imal ideal (z − a) with some a ∈ C; the residue field of the local ring at thispoint is C[z]/(z − a) ∼= C. Hence the fibre over this point is

Spec (C[z, y]/(y − zn) ⊗C[z] C) ∼= Spec (C[z]/(a− zn))

since the C[z]-modules on the two terms of the tensor product are givenrespectively by z 7→ y and z 7→ a. Now there are two cases:

• For a 6= 0 the polynomial zn − a splits into a product of n distinctlinear terms over C and thus the fibre is Spec (C⊕ . . .⊕C) (n copies).

• For a = 0 the fibre is Spec C[z]/(zn), a one-point scheme with nilpotentelements in its unique local ring.

1.. LOCAL BEHAVIOUR OF FINITE MORPHISMS 147

Our intuition is thus confirmed and it can be made more precise: in thefirst two cases of the first example and in the first case of the second, thefibre is a finite etale k-scheme, whereas in the remaining cases of the twoexamples (the “branch points”) it is not. This also explains why there wereonly two cases in the second example but three in the first: over C a finiteetale algebra can only be a finite direct sum of copies of C, whereas overa non-separably closed field we may have non-trivial finite separable fieldextensions as well.

The essence of the phenomenon encountered above is distilled in the defi-nitions we are to give. First a natural restriction on morphisms which ensuresin particular that they have finite fibres.

Definition 1.3 A morphism of schemes φ : Y → X is called finite if X hasa covering by open affine subsets Ui = SpecBi such that for each i the opensubscheme Vi = φ−1(Ui) of Y is an affine scheme Vi = SpecAi and the ringhomomorphism λi : Ai → Bi corresponding to φ|Vi

turns Bi into a finitelygenerated Ai-module.

In particular, any finite morphism is affine. This implies that for anyP ∈ X the fibre YP is affine as well; the additional property of finite mor-phisms 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 inducedmap OX,η → (φ∗OY )η at the generic point η of X is nonzero (and hence isan injection, for OX,η is the function field of X). This seemingly innocentassumption has an important consequence (valid for any noetherian integralscheme 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 fibreis vacuous, i.e. is the spectrum of the zero ring. But then by Nakayama’slemma (Chapter 6, Lemma 2.3), the stalk of φ∗OY at P would be zero aswell, so since φ∗OY is a coherent sheaf, it would be 0 in an neighbourhoodof 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 morphismSpecOL → SpecOK is finite as OL is a finitely generated OK-moduleaccording to Chapter 6, Proposition 1.9.


2. We shall prove in the next chapter that any nonconstant morphism ofsmooth projective curves is finite. For affine plane curves, this is notalways true: consider, for instance, the curve Spec C[x, y]/(xy − 1) ∼=SpecC[x, x−1]; this is the complex affine line with the point 0 removed.The natural inclusion map SpecC[x, x−1] → SpecC[x] is not finite; itis not even surjective.

3. One may nevertheless construct many examples of finite maps of affinecurves: for instance, Spec C[x, y]/(yn − f) → Spec C[x] is such for anyf ∈ C[x].

Remark 1.6 It can be shown that if φ : Y → X is a finite morphism thenany affine open cover of X satisfies the property of the definition. For basicopen sets this is easy to check: if D(fi) = Spec (Ai)fi

⊂ Ai is such, then thefact 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 afinite morphism φ : Y → X, the fibre YP over any point P of X is thespectrum of a finite dimensional κ(P )-algebra, so by Chapter 5, Proposition2.16 decomposes as a finite disjoint union of schemes each of which has asingle 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. thegerms of functions vanishing at Q) are nilpotent.

Definition 1.7 Let X be a Dedekind scheme and φ : Y → X a finitemorphism. We say that φ is etale at a point Q ∈ Y if the component of thefibre YP corresponding to Q is etale over Spec κ(P ), i.e. if it is the spectrumof a finite separable field extension of κ(P ). The morphism φ is a finite etalemorphism if it is etale at all points of Y .

In particular, all fibres of a finite etale morphism are finite etale schemesover residue fields of points of X.

Remark 1.8 Though not used explicitly in the above definition of finiteetale morphisms, the assumption that X is a Dedekind scheme is importanthere. When defining finite etale morphisms of general schemes, one needs anadditional 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 ofrelative differentials ΩY/X at Q is 0.


Proof: Take an affine open neighbourhood U = SpecA of P whose inverseimage in Y is of the form V = SpecB and identify P and Q with thecorresponding prime ideals of B and A as usual. Then by the localisationproperty of differentials the stalk of ΩY/X at Q is ΩBQ/A, with BQ regardedas an A-algebra via the composite map A→ B → BQ.

Next observe that the local component of the fibre YP corresponding toQ is precisely Spec (BQ⊗Aκ(P )). (Indeed, by definition the local componentis the spectrum of the localisation (B⊗A κ(P ))Q, where Q is the image of Qin B ⊗A κ(P ). This localisation is obtained by localising first B/PB by theimage of (A/P ) \ 0 and then by the complement of Q which is the same aslocalising B by Q first, then passing to the quotient by the image of P andfinally localising by the image of (A/P ) \ 0.) By the base change propertyof differentials we have an isomorphism

Ω(BQ⊗Aκ(P ))/κ(P )∼= ΩBQ/A ⊗BQ

(BQ ⊗A κ(P )).

Now assume ΩBQ/A = 0. Then the left hand side vanishes and so by Chapter6, 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, thenby applying the argument backwards we get that ΩBQ/A ⊗BQ

κ(Q) ∼= 0. Butthis latter ring is isomorphic to ΩBQ/A/QΩBQ/A and the assertion followsfrom 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 . Inparticular, if the fibre of φ at the generic point is etale, then φ is etale at allbut finitely many closed points of Y .

Example 1.11 It may very well happen that the generic fibre is not etaleand hence φ is nowhere etale. An example is given by the map SpecFp[t] →SpecFp[t] induced by the Fp-algebra homomorphism t 7→ tp (an analogue ofExample 1.2 in characteristic p > 0). As already noted in Chapter 1, on thegeneric stalks this induces an inseparable field extension.

Remark 1.12 The theorem shows that when X and Y are smooth complexcurves, then φ is etale precisely over those closed points P ∈ X for which theassociated holomorphic map gives a cover in some complex open neighbour-hood of P . We illustrate this by the example when Y is an affine plane curveof equation f(x, y) = 0 and φ is the map which projects Y onto the x-axis;


the general case is based on the same principle. In our case ΩY/X is the coher-ent OX -module associated to the C[x, y]/(f)-module with single generatordy and relation ∂yf . Hence its stalk is 0 precisely at those points (x, y) ofX where ∂yf(x, y) 6= 0. But by the implicit function theorem, these are thepoints where the holomorphic map associated to φ is a local isomorphism;the points with ∂yf(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 schemeswhich is etale at the generic point. Then the we define the different DY/X asthe nonzero ideal sheaf on Y which is the annihilator of ΩY/X .

Putting Theorem 1.9 together with the theory of the previous section weget:

Corollary 1.14 For φ : Y → X as in the above definition, the differentDY/X is an invertible sheaf of the form L(D), where D =

∑imiQi 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 thebranch points of φ or those points at which φ is ramified.

Remark 1.16 In the case when Y = SpecOL and X = SpecOK are spectraof rings of integers in number fields we get using Chapter 6, Remark 4.11that the different is of the form I, with I an ideal of OL. Thus I is a productof powers of those prime ideals of OL at which the map is not etale. Thisgives the link to the classical concept of the different in algebraic numbertheory.

In the remaining of this section we investigate fibres of finite morphismsmore closely, especially those containing branch points. For this the followingobservation 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 locallyfree OX-module of rank |L : K|.


Proof: Let P be a point of X. As usual, we consider an affine open neigh-bourhood U = SpecA of P over which φ comes from a ring homomorphismλ : A→ B. Here B is the integral closure of A in L, being integrally closedand 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 ofB 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 ofrelative differentials that Bλ(f) is canonically isomorphic to B ⊗A Af ; thestatement follows from this by passing to the direct limit (a union in thiscase). Taking Chapter 6, Lemma 1.8 into account, this shows that BP isthe integral closure of AP in L. In particular, since any element of L canbe multiplied with an appropriate element of K to become integral over AP ,any generating system of the finitely generated AP -module BP generates theK-vector space L. According to (the corollary to) Nakayama’s lemma we getsuch a generating system by choosing elements t1, . . . , tn ∈ BP whose imagesmodulo PBP form a basis of the κ(P )-vector space BP/PBP (the spectrumof this κ(P )-algebra is none but the fibre over P ). It remains to be seenthat the ti are linearly independent over K, for this implies n = |L : K| aswell. So assume there is a nontrivial relation

∑aiti = 0 with ai ∈ K. By

multiplying with a suitable power of a generator of P (viewed as the maximalideal of AP ) we may assume that all ai lie in AP and not all of them are inP . But then reducing modulo P we obtain a nontrivial relation among theti in BP/PBP , a contradiction.

To derive the next result we need to introduce some notation and ter-minology. Recall that the fibre YP of φ over a point P ∈ Y decomposes asa finite disjoint union of spectra of local κ(P )-algebras each of which corre-sponds to a point Q in the topological fibre. We have already seen duringthe proof of Theorem 1.9 that if V = SpecB is an affine open set contain-ing YP , then the local component corresponding to Q is the spectrum ofBQ ⊗A κ(P ) ∼= BQ/PBQ. Here BQ is a discrete valuation ring whose max-imal ideal QBQ induces the maximal ideal Q of BQ/PBQ. By Chapter 5,Proposition 2.16 Q consists of nilpotent elements, so being finitely generated,it is actually a nilpotent ideal (i.e. some power of it is 0).

Definition 1.18 With the above notations, the smallest nonnegative integern for which Qn = 0 is denoted by e(Q|P ) and is called the ramification indexof φ at Q. The degree f(Q|P ) of the residue field extension κ(Q)|κ(P ) iscalled its residue class degree.

In particular, φ is etale at Q if and only if e(Q|P ) = 1 and κ(Q) isseparable over κ(P ). In the case where X = SpecB and Y = SpecA are


affine there is a more classical definition of the ramification index: it is themultiplicity of Q in the product decomposition of the ideal PB in B (cf.Chapter 6, Corollary 4.9).

Remark 1.19 Applying the above definition to Example 1.2 corroboratesthat the above definition of the ramification index is compatible with the oneused for Riemann surfaces. The extension of non-trivial residue class degreesis, however, a phenomenon which does not arise over the complex numbers.

Now we can state a fundamental equality of the arithmetic of Dedekindschemes which is the analogue of Chapter 4, Proposition 2.5 (4).

Proposition 1.20 In the situation of the previous proposition, let P be apoint of Y . Then we have the equality

∑

Q

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

where Q runs over the points of the topological fibre over P .

Proof: During the previous proof we have seen that the dimension of theκ(P )-space BP/PBP is |L : K|. Since BP/PBP decomposes as a direct sumof its components BQ/PBQ it will be enough to show that the dimension ofsuch a component over κ(P ) is precisely e(Q|P )f(Q|P ). For this, notice thatsince BQ is a discrete valuation ring, multiplication by the k-th power of agenerator 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

BQ/PBQ ⊃ Q ⊃ Q2 ⊃ . . . ⊃ Qe(Q|P ) = 0

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 etalecovers of a fixed Dedekind scheme just as the absolute Galois group classi-fies finite etale algebras over a field or as the profinite completion of thetopological fundamental group classifies finite covers of a compact Riemannsurface. The method we shall follow will be the exact analogue of the proce-dure for Riemann surfaces in Chapter 4, Section 3; the technical details willbe different, though.

2.. FUNDAMENTAL GROUPS OF DEDEKIND SCHEMES 153

So to start the procedure, we have to study function fields of Dedekindschemes. Consider the functor which associates to a Dedekind scheme X itsfunction field K. This is indeed a functor, for given another Dedekind schemeY with function field L and a morphism φ : Y → X, we have an inducedmorphism K → L on generic stalks of structure sheaves. If the morphism φis finite, we get in this way a finite field extension L|K. Call the morphismφ separable if the extension L|K is separable.

Proposition 2.1 Let X be a Dedekind scheme with function field K. Thenfor any finite separable extension L|K there is a Dedekind scheme Y withfunction 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 φ′ : Y ′ → Xare two normalisations of X in L. Choose some affine open subset SpecA ⊂X. By Remark 1.6, over any basic open set D(f) = SpecAf ⊂ SpecA bothφ and φ′ satisfy the condition for a finite map. Let D(g) (resp. D(g′)) bethe basic open set in Y (resp. Y ′) which is the preimage of D(f). Then bothOY (D(g)) and OY ′(D(g′)) are finitely generated Af -modules. Moreover, theyare integrally closed with fraction field L (since their localisations at closedpoints are), so they are both isomorphic to the integral closure of Af in Lvia their embeddings in L. This yields an isomorphism D(g) ∼= D(g′). UsingChapter 6, Lemma 1.8 we see that these isomorphisms are compatible overintersections of basic open sets, therefore they define an isomorphism of Ywith Y ′ over X.

Now assume X = SpecA is affine with fraction field K and let B be theintegral closure of A in L. To prove that SpecB is a Dedekind scheme oneemploys exactly the same argument as in the special case A = Z treated be-fore in Chapter 6, Example 2.17 (note that it is here that we use separabilityof L|K). By Chapter 6, Proposition 1.9, B is a finitely generated A-module,so the morphism φA : SpecB → SpecA is finite. Before going over to thegeneral case, notice that if U = SpecAf 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 sufficeto show that there exist isomorphisms φij : φ−1

i (Ui ∩ Uj) → φ−1j (Ui ∩ Uj)

compatible over triple intersections, for then the Vi may be patched togetherusing the construction of Chapter 5, Construction 5.3. To do this, cover


Ui∩Uj by basic affine open sets Wk; their inverse images by φi and φj (whichcover φ−1

i (Ui∩Uj) and φ−1j (Ui∩Uj), respectively) both give a normalisation of

Wk in L by the remark in the previous paragraph and hence are canonicallyisomorphic by the uniqueness statement. Since moreover these isomorphismsare readily seen to be compatible over the intersections of the Wk (whichare themselves basic open sets), we can conclude that they can be patchedtogether to define the required φij: for the underlying continuous maps thisis immediate and for the maps on the structure sheaves one uses the thirdpart of Chapter 5, Lemma 2.7.

For a fixed Dedekind scheme X define the category DedsX of finite sepa-

rable Dedekind schemes over X as the category whose objects are Dedekindschemes Y equipped with a finite separable morphism Y → X and whosemorphisms are finite morphisms compatible with the projections onto X.

Corollary 2.2 The functor mapping an object Y → X to the induced ex-tension of function fields induces an anti-equivalence between the categoryDeds

X and the category of finite separable extensions of the function field Kof X (with morphisms the inclusion maps).

Proof: The proposition shows that the functor is essentially surjective; itremains to check that it is fully faithful. For this, note first that given afinite morphism Y → X inducing a finite separable extension L|K, there isan isomorphism of Y with the normalisation of X in L constructed in theabove proof. Fixing such an isomorphism for each object of Deds

X we geta canonical bijection between morphisms Y → Z in Deds

X and morphismsbetween normalisations of X in finite separable extensions of K. But these inturn correspond bijectively to towers of extensions K ⊂ L ⊂ M for the nor-malisation in M of the normalisation of X in L is none but the normalisationof X in M .

Now let X be a Dedekind scheme. A finite etale X-scheme is a scheme Yequipped with a finite etale morphism Y → X. The following result showsthat 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 ofDedekind schemes.

Proof: Let Q be a closed point of Y and let BQ be the local ring of Yat Q whose maximal ideal we also denote by Q. Denote by P the imageof Q in X and by AP its local ring, which is a discrete valuation ring withmaximal ideal P generated by a single nonzero element t. The spectrum


of BQ/PBQ = 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 amaximal ideal in BQ which is only possible for Q = (t). Furthermore, BQ is anoetherian local ring, being a localisation of a finitely generated AP -module.We now show that BQ is a discrete valuation ring. Combining what we knowso far with Chapter 6, Lemma 2.2, it is enough to see that B is an integraldomain. For this, let M ∈ SpecBQ be an inverse image of the generic point(0) of SpecAP . Since M as an ideal of BQ is properly contained in Q, wehave t /∈M and m = bt for any m ∈M with some b ∈ BQ. As M is a primeideal, this forces b ∈M , so that QM = M and finally M = 0 by Nakayama’slemma.

Now since X is noetherian and Y is finite over X, we conclude from thedefinitions that Y is noetherian as well. Furthermore, now that we knowthat the local rings of Y at closed points are discrete valuation rings, we mayconclude that Y is normal and of dimension 1 (since all other local ringsare localisations of those at closed points). So it remains to be seen thatY is a finite disjoint union of integral schemes. For this, let η1, . . . , ηn bethe finitely many points in the generic fibre Yη and let Yi be the closure ofηi in Y . We have Yi 6= Yj for i 6= j as the generic point of an irreducibleclosed 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 SpecA ⊂ Y1 containing Q theirreducible closed subset Y2 ∩ SpecA ⊂ SpecA would define a nonzero primeideal of A properly contained in Q, which is impossible as the local ring atQ is a domain of dimension 1. Hence the Yi are the finitely many connectedcomponents of Y ; endowing them with their open subscheme structure weget 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. Thenp 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 isthe composite Y ×X Z → X.

Note that by our convention (Remark 1.4) all finite morphisms underconsideration are surjective.


Proof: For the first statement, we may assume (using Remark 1.6 if neces-sary) that X = SpecA, Y = SpecB, Z = SpecC are all affine. For a pointP ∈ SpecA 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 ofseparable extensions. For the converse we argue as in the beginning of theproof of Chapter 4, Theorem 3.17: if K ⊂ L ⊂M are the respective functionfields of X, Y and Z, it follows from Proposition 1.20 (in fact, already fromProposition ??) that C ⊗A κ(P ) is etale if and only if all of its residue fieldsare separable over κ(P ) and the sum of residue class degrees is [M : K]. Nowif one of the residue fields of B ⊗A κ(P ) is inseparable over κ(P ) or if thesum of its residue class degrees is less than [L : K], then a counting showsthat C ⊗A κ(P ) cannot be etale. This not being the case by assumption, Yis etale over X, and since in this case by formula (7.1) C ⊗A κ(P ) is just thedirect 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 ×XZ → Y followsfrom the definitions. For etaleness again we may assume X = SpecA, Y =SpecB, Z = SpecC are all affine. Then for Q ∈ SpecB, 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 Pis the image of Q in X. Now since by assumption C ⊗A κ(P ) is etale overκ(P ), the algebra C⊗A κ(Q) ∼= C⊗A κ(P )⊗κ(P ) κ(Q) is etale over κ(Q) (onemay 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 morphismof Dedekind schemes and q : Z → Y any morphism of Dedekind schemessuch that the composite p q is finite and etale, then q is also finite andetale. This follows immediately from the first statement of the lemma andthe definition of finite morphisms once we know that q is surjective. Theproof of this innocent-looking fact, however, requires more technique thanwe 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 → Xa Dedekind scheme finite and etale over X. By Proposition 2.3, the fibreof φ over the generic point of X is the spectrum of a finite etale K-algebrawhich defines via the functor of Chapter 1, Theorem 3.4 a finite continuous


Gal (K)-set SY . Moreover, the rule Y 7→ SY defines a functor from thecategory of schemes finite and etale overX to the category of finite continuousGal (K)-sets. (The perceptive reader will have noticed that in order to makeeverything fit here we should consider only those morphisms between schemesover X which are finite and separable but if we admit the above remark thiscondition is satisfied by all morphisms of finite etale X-schemes.)

Theorem 2.6 Let X be a Dedekind scheme with function field K. Fix aseparable closure Ks of K and let Ket be the composite in Ks of all finitesubextensions L|K which correspond by Corollary 2.2 to Dedekind schemesetale over X. Then for each finite etale X-scheme φ : Y → X the action ofGal (K) on the set SY defined above factors through the quotient Gal (Ket|K)and in this way we obtain an equivalence between the category of schemesfinite and etale over X and the category of finite continuous left Gal (Ket|K)-sets.

To be consistent with the notations in previous chapters, we call the op-posite group of Gal (Ket|K) the algebraic fundamental group of the Dedekindscheme X.

Proof: The first statement of the theorem is immediate from the construc-tion and implies via Chapter 1, Theorem 3.4 that the functor we are inves-tigating is fully faithful. To see that any finite continuous Gal (Ket|K)-set Sis isomorphic to some SY , one first produces from Chapter 1, Theorem 3.4a finite direct sum of finite subextensions of Ket|K giving rise to S. Thenit remains to see that any finite subextension of Ket|K is the function fieldof some Dedekind scheme finite and etale over X. But this can be provenby 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 thenext section; here we briefly discuss spectra of rings of integers in numberfields.

We know several classical facts concerning these from algebraic numbertheory. Firstly, a theorem of Minkowski states that π1(Spec Z) is trivial.Secondly, a theorem of Hermite and Minkowski states that for any numberfield K the group π1(SpecOK) has only finitely many finite quotients of givenorder. Thirdly, one of the main results of class field theory (usually attributedto Hilbert) states that the maximal abelian quotient of π1(SpecOK) is finitefor all K and isomorphic (up to a finite group of exponent 2 coming from so-called real places of K) to Pic(SpecOK) (see the next chapter for a geometric


analogue). Proofs of these classical theorems can be found in the books ofLang [2] and Neukirch [1], for example.

But as for the groups π1(SpecOK) themselves and not just their finitequotients, our current knowledge is far from being ample. For several struc-tural results, including the fact that they are topologically finitely generated(in contrast to the groups Gal (K)), see chapter X of the monograph ofNeukirch-Schmidt-Wingberg [1], where the results are stated more generallyfor open subschemes of SpecOK .

3. Galois Branched Covers and Henselisation

The perceptive reader has noted that though the main theorem of the lastsection was completely analogous to the topological situation, there was onepoint missing from the presentation, namely the analogue of Galois branchedcovers. In this section we first repair this sin of omission.

We define (rather tautologically) a finite morphism φ : Y → X ofDedekind schemes to be Galois if the induced inclusion K ⊂ L of functionfields 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 ofX byUi = SpecAi whose inverse image consists of affine open sets Vi = SpecBi.Then B is the integral closure of Ai in L, so that σ(Bi) = Bi for all i andσ ∈ G and if Q is a prime ideal in Bi, then σ(Q) is also a prime ideal ofBi. From this we deduce an action of G on X as a topological space butwe have actually more: from the automorphism σ|Bi

: Bi → Bi we deducean automorphism of Vi by Chapter 5, Theorem 2.14. These automorphismsare easily seen to be compatible over the intersections Vi ∩ Vj, so we get anaction of G on X as a scheme.

The next proposition shows that topologically this action by G has theproperty of a Galois cover.

Proposition 3.1 The group G acts transitively on the fibres of φ.

Proof: We may assume X = SpecA and Y = SpecB are affine. Let Pbe 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. Assumethere is some prime ideal Q lying over P which is not among the Qi. SinceQ and the Qi are all maximal ideals, we may apply the Chinese RemainderTheorem (Lang [1], Chapter II, Theorem 2.1) which gives an isomorphismB/(QQ1 . . . Qr) ∼= B/Q⊕B/Q1 ⊕ . . .⊕B/Qr, hence we may find x ∈ Q notcontained in any of the Qi (we may even find an x mapping to 1 modulo

3.. GALOIS BRANCHED COVERS AND HENSELISATION 159

each Qi and to 0 modulo Q). Now for any σ ∈ G the element σ(x) is stilloutside 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 haveN(x) ∈ Q ∩ A = P , a contradiction.

Remark 3.2 The proposition implies that X as a scheme is the quotient ofY by the action of G defined above. To make this precise, note that we maytake the quotient of Y by the G-action in the category of locally ringed spacesjust as in the construction of projective spaces: we take the quotient of theunderlying space by the action of G, whence a canonical topological projec-tion p : Y → Y/G; then for each open set U ⊂ Y/G we define OY/G(U) as thering 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→ Xwith φ = ψ p. By the proposition ψ is an isomorphism on the underlyingtopological spaces. Now observe that if we choose an affine open covering ofX by subsets SpecAi whose inverse images in Y are of the form SpecBi, thenby construction the G-orbit of any point of Y is entirely contained in one ofthe SpecBi. This implies that we may define the structure of a scheme onY/G by choosing as an affine open covering the schemes SpecBG

i , where BGi

is the ring of G-invariant elements of Bi. By construction we have BGi∼= Ai

and hence Y/G ∼= X as schemes.

Now let us draw some immediate consequences from the proposition con-cerning 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 . Accordingto Proposition 3.1, any other point of the fibre YP over P is of the formσ(Q) with some σ ∈ G and by construction e(σ(Q)|P ) = e(Q|P ) andf(σ(Q)|P ) = f(Q|P ). Hence we may denote simply be e end f the com-mon ramification index and residue class degree of the points in YP . Now letDQ be the stabiliser of Q with respect to the action of G on Y . Again byProposition 3.1, the cosets of G mod DQ are in bijection with the points inYP . Hence Proposition 1.20 implies that the order of DQ is exatly ef .

Now let LQ be the fixed field of DQ and X ′ the normalisation of X in LQ.Denote by Q′ the image of Q by the map φ′ : Y → X ′.

Lemma 3.3 The topological fibre of φ′ over Q′ consists only of Q. We havethe equalities

e(Q|Q′) = e, f(Q|Q′) = f and e(Q′|P ) = f(Q′|P ) = 1.

In particular, the finite morphism X ′ → X is etale at Q′.


Proof: According to Proposition 3.1, the group DQ acts transitively on thefibre of φ′ over Q′. On the other hand, it fixes Q, whence the first statement.The second statement follows from the already proven fact that the order ofDQ is ef in view of the equalities

e(Q|Q′)e(Q′|P ) = e and f(Q|Q′)f(Q′|P ) = f

which hold quite generally and follow from the definitions.

Remark 3.4 In arithmetic terminology, the group DQ is called the decom-position group of Q and LQ is called its decomposition field.

Thus the local ring OX′,Q of X ′ at Q has the property that its integralclosure in L is a discrete valuation ring (the local ring of Q) and hencethe ramification index and the residue class degree are easy to compute; inparticular, the formula of Proposition 1.20 reduces to ef = [L : LQ]. Thisproperty is not shared by the local ring OX,P of X at P : its integral closurein 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 discretevaluation rings but then we lose the finiteness property of the correspondingmorphism of affine schemes.

So we see that in order to study the local behaviour of φ at Q it is muchbetter to work with OX′,Q than with OX,P ; moreover, by the corollary weget the same ramification index and residue class degree. This procedure isa priori only available in the Galois case but if φ : Y → X is only seperablewith function field extension L|K, we may embed L into a Galois extensionM of K and study the corresponding morphism of Dedekind schemes. Thisprompts the idea that if we choose M to be the biggest Galois extensionavailable, namely the separable closure, then by the generalising the aboveprocedure we may reduce the study of the local behaviour of finite morphismsnear branch points to a problem about finite extensions of discrete valuationrings.

We now make this idea precise. First an easy lemma which could havefigured earlier.

Lemma 3.5 Let A be a discrete valuation ring with maximal ideal P andB an integral extension of A. Then there is a prime ideal Q of B withQ ∩ A = P .

Note that here the morphism SpecB → SpecA is not assumed to befinite.

3.. GALOIS BRANCHED COVERS AND HENSELISATION 161

Proof: It is enough to show that PB 6= B, for then PB is contained insome maximal ideal of B whose intersection with B is nonzero according toChapter 6, Lemma 1.5 (1) and hence cannot be but P . So assume PB = B.Then there are b1, . . . , br ∈ B and p1, . . . , pr ∈ P with

∑i pibi = 1. Hence the

subring B′ = A[b1, . . . , br] also satisfies PB′ = B′. But B′ is integral overA and hence a finitely generated A-module, so Nakayama’s lemma impliesB′ = 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 Pand fraction field K. Fix a separable closure Ks of K. Denoting by B theintegral closure of A in Ks, apply the lemma to find some Q ∈ SpecB lyingabove P . Let DQ be the stabiliser of Q with respect to the natural action ofGal (K) on SpecB (defined in the same way as for finite Galois extensions).Let K ′ be the fixed field of DQ and put B′ = B ∩ K ′, Q′ = Q ∩ B′. Thelocalisation of B′ at Q′ 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 asA. Its maximal ideal is generated by any generator of P .

2. The isomorphism class of Ah does not depend on the choice of the primeideal Q.

For the proof we need the following generalisation to Proposition 3.1 toinfinite 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 Galoissubextension 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 areprime ideals of the integral closure of A in L, Proposition 3.1 implies thatXL 6= ∅ for any L. Moreover, each XL is a closed subset of G, for if someσ /∈ XL, then the whole left coset σGal (L) of the open subgroup Gal (L) iscontained in G \ XL. But G is compact, so we have X =

⋂LXL 6= ∅. Any

element of X maps Q1 onto Q2.

Proof of Proposition 3.7: For the first statement, note that by Corollary3.3 for any finite separable extension L|K the ring Ah ∩ L is a discrete val-uation ring whose spectrum is etale over SpecA. Hence Ah is the union of


an increasing chain of such discrete valuation rings, which shows that it islocal (its maximal ideal being the union of that of the L∩Ah, and the sameholding for its units) and its maximal ideal is generated by any generator ofP . The same type of argument shows that the residue field of Ah is the sameof that of A. The second statement follows from the lemma which impliesthat performing the above construction with a maximal ideal above P otherthan Q yields the ring σ(Ah) for some σ ∈ Gal (K).

We conclude this section by the following proposition which shows thathenselisations do serve for the purpose they were constructed for.

Proposition 3.9 Let Y → X be a finite separable morphism of Dedekindschemes. Let L (resp. K) be the function field of Y (resp. of X), and embedL|K into a separable closure Ks|K. Fix a closed point Q of Y mapping to apoint 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

Kh ⊂ Ks.

1. The integral closure of AhP in the composite field LKh is isomorphic to

the henselisation BhQ of BQ.

2. The finite map SpecBhQ → SpecAh

P thus obtained has ramificationindex e(Q|P ) and residue class degree f(Q|P ).

3. The stalk of the different DSpec BhQ

/Spec AhP

of the above map at the closed

point of SpecBhQ is the ideal of Bh

Q generated by the stalk of the differentDY/X at Q. (Here BQ is viewed as a subring of Bh

Q.)

Proof: For the first statement, note that by construction BhQ is a dis-

crete valuation ring with fraction field LKh integral over AhP . The sec-

ond statement follows from Corollary 3.3 by taking the intersection of LKh

with each finite Galois extension of K containing L. For the proof of thethird statement, take an affine open neighbourhood SpecA of P with in-verse image SpecB in Y . Then the stalk of DY/X at Q is the annihilator ofΩBQ/A

∼= ΩB/A ⊗B BQ by the localisation property of differentials, whereasDSpec Bh

Q/Spec Ah

Pis the annihilator of ΩBh

Q/Ah

P

∼= ΩB/A ⊗AAhP by the first state-

ment 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 finitemorphisms of Dedekind schemes can be reduced to the study of the induced

4.. HENSELIAN DISCRETE VALUATION RINGS 163

morphisms on the henselisations. In this section we study the henselisationsin general and determine their fundamental groups. First a general definition.

Definition 4.1 A discrete valuation ring is called henselian if its integralclosure 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 henselianvaluations as in Neukirch [1]. For its relation to the more generalconcept of henselian local rings, see Remark 4.8 below.

2. An immediate consequence of the definition is that the integral clo-sure of a henselian discrete valuation ring in any finite extension of itsfraction field is again henselian.

The following proposition shows that the above definition is not out ofplace here.

Proposition 4.3 The henselisation Ah of any discrete valuation ring A ishenselian.

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 > 0and let K ⊂ L′ ⊂ L be the maximal separable subextension (i.e. the com-positum of all separable extensions of K contained in L). Then there existsa positive integer m such that xpm

∈ L′ for all x ∈ L.

For a proof, see Lang [1], Chapter V, Section 6. The extension L|L′ iscalled purely inseparable. We now have the following general lemma.

Lemma 4.5 Let A be a discrete valuation ring with fraction field K of char-acteristic p > 0 and let L be a purely inseparable finite extension of K. Thenthe integral closure B of A in L is a discrete valuation ring.

Proof: Let m be a positive integer for which xpm

∈ K for all x ∈ L. Thenif v denotes the discrete valuation associated to A, the map x 7→ v(xpm

) is ahomomorphism from the multiplicative group of L to Z. Moreover, denotingby b a positive generator of its image in Z and setting w(0) = ∞, the formulaw(x) = (1/b)v(xpm

) defines a discrete valuation w : L → Z ∪ ∞. Thevaluation ring of w is precisely B, for an element x ∈ L is integral over A ifand only if xpm

∈ A. (Indeed, xpm

is always an element of K and is integralover A if and only if x is; now use the fact that A is integrally closed.)


Remark 4.6 Quite generally, the integral closure B of any integrally closedlocal domain A in a finite purely inseparable extension L|K of its fractionfield is always local. To see this, one shows that the elements x ∈ L suchthat xpm

lies in the maximal ideal of A form a unique maximal ideal in B.

Proof of Proposition 4.3: Let L|Kh be a finite extension. The construc-tion of Ah (and the arguments preceding it) imply that the integral closure A′

of Ah in the maximal separable subextension L′ ⊂ L is a discrete valuationring. The proposition then follows by applying the above lemma with L′ inplace of K and A′ in place of A.

Now we have the following important characterisation of henselian dis-crete 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 localring.

3. Given a monic polynomial f ∈ A[x] whose reduction f modulo P fac-tors as f = f1f2 with f1 and f2 relatively prime monic polynomialsin κ(P )[x], there exists a factorisation f = f1f2 of f into the productof two relatively prime monic polynomials in A[x] such that fi = fi

modulo P for i = 1, 2.

4. If f ∈ A[x] is a monic polynomial such that its reduction f modulo Phas a simple root α in κ(P ), then there is α ∈ A with f(α) = 0 andα = α modulo P .

Proof: To show that (1) implies (2), assume there is an integral domainB ⊃ A with fraction field L that is finitely generated over A and has at leasttwo maximal ideals P1 6= P2. But then the common integral closure C of Aand B in L has two different prime ideals lying above the Pi in contradictionwith (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] pro-vides a counterexample to the property (3). We may assume f is irreduciblein 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 factorisationdomain (in fact, to see this it would suffice to use that A is interally closed),


and so B = A[x]/(f) is an integral domain integral over A. But then byassumption

B/PB ∼= B ⊗A κ(P ) ∼= κ(P )[x]/(f1) + κ(P )[x]/(f2),

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 theintegral closure B of A in some finite extension L|K has at least two distinctmaximal ideals Q1 and Q2. We then concoct a polynomial f ∈ A[x] which isa counterexample to (4). Thanks to Lemma 4.5 there is no harm in supposingL|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′ 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 ofB, namely Qi = σi(Q1). Denote by Q′i the image of each Qi in SpecB′; byLemma 3.3 we have Q′i 6= Q′1 for any i 6= 1. Using the Chinese RemainderTheorem as in the proof of Proposition 3.1 we may thus find an element αlying in the intersection of the Q′i for i > 1 but not in Q′1 (and hence not inK). Since α ∈ L1, the σi(α) for 1 ≤ i ≤ n are exactly the distinct conjugatesof α in L. Again using Proposition 3.1 and the fact that the σi form a two-sided 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 theminimal polynomial f = xn + an−1x

n−1 + . . . a0 ∈ A[x] of α over K. Herean−1 is up to sign the sum of the σi(α), so it does not lie in Q1 ∩A = P . Butfor s < n− 1 the coefficient as is (still up to sign) a higher order symmetricpolynomial 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 + an−1xn−1. This contradicts (4)

as f is irreducible over K.

Remark 4.8 An analysis of the above proof shows that in proving the equiv-alence of statements (2)–(4) we did not use the assumption that A wasnoetherian of dimension 1, hence these are equivalent conditions for any inte-grally closed local domain. The construction of the henselisation also worksin this generality. It gives an integrally closed local domain Ah with thesame 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 localring. It can then be shown (see Nagata [1], Theorem 4.11.7) that the henseli-sation Ah of an integrally closed local domain A represents the contravariantfunctor on the category of henselian local rings which associates to an ob-ject B the set of local homomorphisms A → B. Thus Ah is the “smallest”


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 adifferent construction. See Milne [1], Section I.4 for more details concerningthis point.

Example 4.9 Besides the henselisation, there is a classical example for ahenselian discrete valuation ring, namely that of a complete discrete valuationring which we now explain.

Quite generally, one calls a local ring A complete if the natural mapA → A into its completion (see Chapter 6, Remark ??) is an isomorphism.Obviously A is complete for any local ring A, so as concrete examples wemay mention the ring Zp of p-adic integers encountered in Chapter 1 (whichis the completion of the localisation of Z at (p)) and the ring of formal powerseries k[[t]] over a field k (which is the completion of any discrete valuationring with residue field k that contains a subring isomorphic to k).

Now we show that any complete local ring A satisfies condition (4) ofProposition 4.7; this assertion is classically known as Hensel’s lemma, whencethe term “henselian”. So let P be the maximal ideal of A and assume thereduction f of f ∈ A[x] modulo P has a root a1 ∈ A/P which is simple, i.e.f ′(a1) 6= 0. We construct a lifting a ∈ A of a1 with f(a) = 0 by Newton’smethod of successive approximation. Represent a by a coherent sequence(ai), with ai ∈ A/P i and assume ai is already determined (this being thecase for i = 1). Lift ai arbitrarily to an element bi ∈ A/P i+1; the ai+1 weare 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 ′(bi)is a unit in the local ring A/P i+1 for its image f ′(a1) modulo P is nonzero.By the Taylor Formula of order 2 (which is quite formal for polynomials), wehave

f(bi + p) = f(bi) + f ′(bi)p+ cp2

with some c ∈ A/P i+1, but anyway we have p2 = 0, so since we are aimingat f(bi + p) = 0, we only have to choose p = −f(bi)/f

′(bi).

The next proposition shows that finite etale covers of spectra of henseliandiscrete valuation rings have a simple description.

Proposition 4.10 Let X = SpecA, where A is a henselian discrete valua-tion ring with maximal ideal P .

1. Let f be a polynomial whose reduction f modulo P is irreducible and de-fines a finite separable extension of κ(P ). Then the ring B = A[x]/(f)is a discrete valuation ring and the canonical morphism SpecB → Xis finite and etale.


2. Any finite etale X-scheme is a finite disjoint union of affine schemesSpecB, with B of the above type.

3. The functor SpecB 7→ Spec (B⊗Aκ(P )) defines an equivalence betweenthe 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 A-module, and hence is noetherian. Since f is irreducible it is also an integraldomain (see the proof of Proposition 4.7), and hence it is a local ring byProposition 4.7 (2). By Chapter 6, Proposition 3.4, ΩB/A = 0 and so by basechange ΩB⊗Aκ(P )/κ(P ) = 0, hence SpecB is etale over X and P generates themaximal 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 ofX is by X itself, any finite X-scheme is necessarily affine. If it is moreoveretale, decomposing it into a finite disjoint union of components, we mayassume it is integral and hence the spectrum of some integral domain B. Byexactly the same argument as above, we see that B is a discrete valuationring. Now to prove that B is of the required form, let f be the minimalpolynomial 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 rootof f in A, whence an injective morphism A[x]/(f) → B. Here both ringsare discrete valuation rings with the same residue field and their spectra arefinite and etale over X, so by Proposition 1.20 their fraction field K must bethe 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 fi-nite 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 min-imal polynomial of a generator of L|κ(P ). For fully faithfulness, assumeB = A[x]/(f), C = A[x]/(g) are such that SpecB, SpecC are etale overX and assume given a morphism B ⊗A κ(P ) → C ⊗A κ(P ). It is given bymapping a generator of the field extension B ⊗A κ(P )|κ(P ) to a root α of fin C⊗A κ(P ). Lifting α to a root α of f in C gives a homomorphism B → Cinducing the above one by tensoring with κ(P ). To see that this morphism isunique, it is enough to see that α is the unique root of f in C lifting α. Forthis, by enlarging C if necessary we may assume that C ⊗A κ(P ) is Galoisover κ(P ) (embed C ⊗A κ(P )|κ(P ) in a finite Galois extension and lift itsdefining polynomial to A[x]). Then f decomposes as a product of distinctlinear factors in C ⊗A κ(P ) and each of its roots lifts to a different root of fin C.


Corollary 4.11 For X as in the proposition there is a canonical isomor-phism π1(X)op ∼= Gal (κ(P )).

Proof: The group π1(X)op is the Galois group of the Galois extensionKet|K defined in Theorem 2.6. Let Aet be the integral closure of A in Ket;as A is henselian, it is a local ring with maximal ideal P et. Any element ofGal (Ket|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 ofκ(P ) by the third statement of the proposition. Whence a homomorphismπ1(X)op → Gal (κ(P )) of which the proposition implies the bijectivity.

Remark 4.12 Let X = SpecA be as above, and denote by K the fractionfield of A. In this case the closed normal subgroup of Gal (K) which is thekernel of the canonical map Gal (K) → Gal (Ket|K) (and hence of the mapGal (K) → Gal (κ(P )) defined by composing with the above isomorphism) isusually called the inertia subgroup.

5. Dedekind’s Different Formula

In this section we harvest the fruits of our efforts in the two previous ones andcomplete our study of finite morphisms of Dedekind schemes; in particularwe 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 fractionfield K and maximal ideal P , let B be its integral closure in a finite separableextension L|K and let Q be the maximal ideal of B. Assume further that theresidual extension κ(Q)|κ(P ) is separable.

Then there is a unique discrete valuation ring A ⊂ C ⊂ B with residuefield κ(Q) and such that the map SpecC → SpecA is etale.

Proof: Embed L in a separable closure Ks and put M = L ∩ Ket, withKet as in Theorem 2.6. Let C be the integral closure of A in M . The affinescheme SpecC is integral, finite and etale over SpecA by construction andhence C is a discrete valuation ring by Proposition 4.10 (2). The residue fieldof C is κ(Q), for otherwise, B being henselian by Remark 4.2 (2), arguingas in the proof of Proposition 4.10 (2) we would get a subring C ′ ⊂ Bproperly containing C with residue field κ(Q) and SpecC ′ → SpecA etale,contradicting the construction of C.

5.. DEDEKIND’S DIFFERENT FORMULA 169

Using Proposition 1.20 we get that in the situation of the propositionthe finite morphism SpecB → SpecC has ramification index e(Q|P ) at Qand residue class degree 1. It is convenient to separate this property in adefinition.

Definition 5.2 A finite morphism φ : Y → X is called totally ramified overa closed point P of X if the underlying space of the fibre YP consists of asingle point Q with f(Q|P ) = 1. In the case when Y is the spectrum of adiscrete valuation ring, we simply say that φ is totally ramified if it is totallyramified over the closed point of X.

Totally ramified extensions of discrete valuation rings have the followingcharacterisation, somewhat analogous to Proposition 4.10 (1), (2) but validfor not necessarily henselian rings as well.

Proposition 5.3 Let A ⊂ B be an extension of discrete valuation rings withmaximal ideals P ⊂ Q, and fraction fields K ⊂ L, respectively. If the inducedmap φ : SpecB → SpecA is finite and totally ramified, then B = A[t] witha generator t of Q and the minimal polynomial of t over K is of the formf = xe + ae−1x

e−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 SpecB →SpecA is totally ramified.

A polynomial f as in the statement of the proposition is called an Eisen-stein 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−1t

e−1 + . . .+a1t+a0 with ai ∈ A. Sincethe ramification index is e, for all a ∈ A the valuation v(a) is divisible bye, and hence the integers v(ait

i) are all distinct modulo e. From this we seethat v(

∑aiti) = min v(aiti) 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) andB = A[t] = A[x]/(f) with some monic polynomial f . To see that f is of theabove type, remark that by the above argument, f(t) = te+ae−1t

e−1+. . .+a0

can only be 0 if two of the terms with the smallest valuation have equalvaluation. But for 0 < i < e the v(ait

i) are distinct and nonzero modulo e;on the other hand v(te) = e. Since v(a0) is divisible by e the only possibilitythat 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 irreduciblein A[x] (same proof as over Z), so B is a domain that is finitely generated as


an A-module and hence noetherian. The fibre over P is Specκ(P )[x]/(xe),from which we see that the last statement holds, but also that B is a localring whose maximal ideal is generated by t.

We can now prove the promised formula of Dedekind. To state it we needone 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. Wesay that φ is tamely ramified at Q if the ramification index e(Q|P ) is notdivisible 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 afinite morphism of Dedekind schemes such that the corresponding extensionL|K of function fields is separable. Let Q1, . . . , Qn be the branch points ofφ with (not necessarily distinct) images P1, . . . , Pn in Y and let DY/X =L(

∑imiQi) 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 = SpecA and Y = SpecB with A ⊂ Bhenselian discrete valuation rings with maximal ideals P ⊂ Q. Consider themaximal etale subextension A ⊂ C ⊂ B. Since SpecC → SpecA is etale, weget from the exact sequence in Chapter 6, Lemma 3.5 (2) an isomorphismΩB/A

∼= ΩB/C . Thus we may assume A = C and so by Proposition 5.3 wehave B = A[t] ∼= A[x]/(f) with f ∈ A[x] an Eisenstein polynomial. In thiscase ΩB/A is generated by dt and has f ′ = 0 as its single relation, so thedifferent is the ideal generated by f ′ = ete−1 + (e− 1)ae−1t

e−2 + . . .+ a1. Ifv is the valuation associated to B, here we have v(ai) ≥ e for all i, henceall 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 themap φ : SpecOK → Spec Z. For a prime number p (viewed as a closed pointof SpecZ), denote by Zh

p the henselisation of Z at p and Qhp its fraction field.

Let Bp be the integral closure of Zhp in Qh

p(ζ).

5.. DEDEKIND’S DIFFERENT FORMULA 171

• 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 theequation 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 discretevaluation ring finitely generated as a Zh

P -module, it must equal Bp, soSpecBp is etale over SpecZh

p and a fortiori φ is etale at all points ofOK not lying above `.

• For ` = p, the maximal etale subextension is just Zhp so the map

SpecBp → SpecZhp is totally ramified. We now show that the de-

gree of the field extension Qhp(ζ)|Q is exactly p − 1, which, together

with Proposition 1.20, will imply that the map SpecOK → SpecZ istotally 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 bysubstituting 1 + y in place of x in xp−1 + xp−2 + . . . + 1. One seesimmediately that F is an Eisenstein polynomial, so it is irreducible inQh

p [y].

• By Proposition 5.5 we get that the different DSpecOK/Spec Z is the idealsheaf associated to the divisor (` − 2)S, where S is the unique pointlying above (`). We could have obtained this result immediately if weknew that OK = Z[ζ ] but this fact is not obvious; to prove it onecommonly uses nearly all the information obtained above.

Remark 5.7 As an amusing application of the previous example we showthat any finite abelian group occurs as the Galois group of a finite Galoisextension K|Q. Indeed, let A be a finite abelian group of order m anddecomposing as a direct sum A ∼= A1 ⊕ . . . ⊕ An with each Ai cyclic. ByDirichlet’s theorem of prime numbers in an arithmetic progression we mayfind n different prime numbers `1, . . . , `n each congruent to 1 modulo m.Choose a primitive ì-th root of unity ζi for each i. The Galois group Gi ofthe Galois extension Q(ζi)|Q is cyclic of order ì − 1, hence divisible by mand as such has a quotient isomorphic to Ai. Denote by Ki the correspondingGalois extension of Q. For each i the map SpecOKi

→ SpecZ is etale at thepoints not mapping to ì but totally ramified at the unique point lying overì; a degree count shows that this implies that we must have Ki ∩Kj = Qfor i 6= j. Hence the composite K of the Ki is Galois over Q with group A.

Chapter 8

Finite Covers of AlgebraicCurves

1. Smooth Proper Curves

In Chapter 6 we have already defined smooth affine (resp. projective) curvesas 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 butpreserving 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 kif it admits an affine open covering by affine schemes associated to finitelygenerated k-algebras. Note that this implies the existence of a canonicalmorphism X → Spec k.

A smooth curve over k is a Dedekind scheme of finite type over k. Amorphism of smooth k-curves is a morphism compatible with the projectionsonto Spec k.

We see that smooth affine and projective curves defined previously aresmmoth curves in the above sense.

Let now X be a smooth curve and let K be the function field of X. Weview k as a subfield of K and identify the local rings at closed points of Xwith 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 valuationring containing k whose fraction field is K is in fact a local ring at someclosed point of K.

173

174 CHAPTER 8. FINITE COVERS OF ALGEBRAIC CURVES

Example 1.3 Clearly, affine curves are not proper, for the “points at infin-ity” are missing. For instance, if we view the affine line A1

k = Spec k[t] asthe projective line with a point ∞ deleted, then the local ring of ∞ is nota local ring of the affine line but it is a discrete valuation ring containing kwhose 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 thatX is embedded as a closed subscheme into

some Pnk by a map i : X → Pn

k does not map X into any hyperplane of Pnk .

Then from the defintion of closed subschemes we infer that the homogeneouscoordinate functions x0, . . . , xn on Pn

k induce nonzero functions x0, . . . , xn

on X. Let now K be the function field of X and A a discrete valuationring containing k with fraction field K. Denoting by v the discrete valuationassociated with A, choose a pair (i, j) for which the valuation v(xi/xj) ismaximal. For any k 6= j we have v(xk/xj) = v(xi/xj) − v(xi/xk) ≥ 0, henceall xk/xj are contained in A. But X ∩ D+(xj) is an affine scheme which isprecisely the spectrum of the k-algebra R generated by the xk/xj for k 6= j,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 ′

is maximal and the localisation RP ′ 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 frac-tion field K. Then A = B.

Proof: The proof is based on the basic property of discrete valuation ringsthat any element of their fraction fields is either contained by them, or is theinverse of an element of their maximal ideal.

So let t be a generator of the maximal ideal of B. Then t ∈ A, forotherwise we would have t−1 ∈ A ⊂ B, which is impossible for t is not a unitin B. Similarly, any unit u of B is contained in A, for otherwise u−1 wouldbe in the maximal ideal of A, hence in that of B.

We next prove a statement that can be regarded as the algebraic analogueof Riemann’s existence theorem. Recall that the main statement of the latteris the existence of a non-constant meromorphic function on a compact Rie-mann surface, which then endows it with the structure of a branched coverof the projective line. The existence of non-constant elements in the functionfield of a proper smooth curve is evident, so what remains to be proven isthe following statement.

1.. SMOOTH PROPER CURVES 175

Proposition 1.6 Let X be a smooth proper k-curve with function field Kand f ∈ K a transcendental element over k. Then f induces a finite mor-phism φf : X → P1

k.

Before the proof, recall that given a finite separable extension L of thefunction field K of X, we constructed in Chapter 7, Proposition 2.1 the nor-malisation of X in L, i.e. a Dedekind scheme with function field L equippedwith a finite morphism onto X. Now for smooth curves exactly the sameargument 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 finiteextension 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 definedfollows from the fact that it maps onto X by a finite morphism. To seethat Y is proper, let A be a discrete valuation ring with fraction field L andmaximal ideal P . Then A′ = A ∩ L is a discrete valuation ring of fractionfield K, hence a local ring of X. The integral closure B′ of A′ in L has finitelymany maximal ideals one of which is P ′ = P ∩ B′. The localisation of B′P ′

is then a local ring of X contained in A; according to Lemma 1.5, we haveA = B′P ′ .

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 ofthe set of poles of f and V ⊂ X that of its zeros. Take an affine open coveringof U by subsets Ui = SpecAi of U (in fact, we shall see later that U itself isaffine). 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 containedin the intersections Ui ∩ Uj . Whence a morphism φU

f : U → Spec k[f ]; ina similar way, one constructs φV

f : V → k[f−1]. Again it can be readilychecked that φU

f 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 nor-malisation Y of P1

k in the field extension K|k(f) is obtained by patching thenormalisations of Spec k[f ] and Spec k[f−1] together, so the above construc-

tion shows that φf factors as a composite Xλ→Y → P1

k, where the secondmap is the finite map coming from the normalisation. So it is enough toshow that λ is an isomorphism. But for all closed points P ∈ X we have aninclusion OX,P ⊃ OY,λ(P ) (for the latter ring is a localisation of the integralclosure of k[f ] or k[f−1] in L which must be contained in the integrally closedring OX,P ). By Lemma 1.5 we have equality here, whence the claim.


Next we may draw the analogy between compact Riemann surfaces andproper smooth curves even closer. Recall that when studying branched coversof compact Riemann surfaces we always took non-constant holomorphic mapsas morphisms in our category whereas in the analogous theory for Dedekindschemes we regarded only finite morphisms. Now we can show (as alreadypromised in the last chapter) that for proper smooth curves non-constantand finite morphisms are the same.

To be precise, call a morphism Y → X of schemes constant if it factorsas Y → Specκ(P ) → X for some closed point P of X. Obviously, such amorphism cannot be finite when X is of dimension at least 1.

Proposition 1.8 Let φ : Y → X be a morphism of proper smooth curvesover a field k. If φ is not constant, then φ is finite.

Recall from the last chapter that the statement is false without assumingX proper. For the proof, we need the following easy but crucial lemma.

Lemma 1.9 Any non-constant morphism φ : Y → X of proper smoothcurves over a field k is surjective.

Proof: Let K (resp. L) be the function field of X (resp. of Y ). Since φis not constant, the point of Y underlying the image of the composite mapSpecL → Y → X must be the generic point of Y . Whence a map K → Lwhich cannot be 0 since φ is a morphism over k. Now let OX,P be the localring of a closed point P of X. Its integral closure in L has finitely manymaximal ideals; localising by one of them we get a discrete valuation ring Awith fraction field L. Since Y is proper, we must have A = OY,Q for someclosed point Q; moreover, Q must map to P for if we view OY,Q as a subringof L, the morphism of local rings induced by φ must be the inclusion ofOY,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 particularany transcendental element f ∈ K may be viewed as an element of L. Bythe previous proposition, we have a finite morphism φf : X → P1

k attachedto f ; its composition with φ is the finite morphism Y → P1

k correspondingto 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 Chapter7, Section 2 in the case of proper smooth curves by considering non-constantmorphisms and arbitrary finite field extensions in our categories. We onlynote here the corollary:

1.. SMOOTH PROPER CURVES 177

Corollary 1.10 Let k be a field. Then associating to a proper smooth curveover k its function field induces an anti-equivalence between the category ofproper smooth curves over k with nonconstant morphisms and the category offinitely generated extensions of k of transcendence degree 1, with the inclusionmaps as morphisms.

Comparing this result for k = C with those of Chapter 4 we get anequivalence of categories between smooth proper complex curves and com-pact Riemann surfaces. We conclude this section by making this equivalenceexplicit and show that over C the set of closed points of any smooth propercurve (in fact, any smooth curve) can be endowed with the structure of aRiemann surface. If we knew that any smooth proper curve admits a closedimmersion into some projective space (which we shall not prove), this wouldfollow easily for then we could simply take the complex manifold structurecoming from the embedding. We take another approach here which has theadvantage of describing explicitly a complex atlas. It is based on the factthat “locally any smooth curve is ismorphic to a plane curve”.

More precisely, let k be any algebraically closed field, X a smooth curveover k with function field K and P a closed point of X. Let x be a generatorof the maximal ideal P of OX,P . Then we may write K = k(x, u) with someu integral over the polynomial ring k[x]. If f is the minimal polynomial ofu over k(x), then K is the fraction field of the ring A = k[x, y]/(f) whichcorresponds to an affine curve of equation f = 0 in the (x, y)-plane. As u isintegral over k[x], we have A ⊂ OX,P . Therefore P ∩ A is a nonzero, hencemaximal 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 ofChapter 6, Proposition 3.7, this is the case if and only if ΩA∩P/k/PΩA∩P/k

has dimension 1 over k. But ΩA∩P/k is generated by dx and dy with a singlerelation ∂1fdx + ∂2fdy; since ∂1f gives an element of P ∩ A, we see thatOX,P = AP∩A if and only if ∂2f(0, a) 6= 0.

Lemma 1.11 In the above situation we may always choose u such that∂2f(0, a) 6= 0 and hence AP∩A = OX,P .

The proof will implicitly use the fact that K is separable over k(x). Thereader may prove this as an exercise or assume k is of characteristic 0 (sincewe are mostly interested in the case k = C anyway).


Proof: (after Lang [3]) If u does not satisfy already the condition of thelemma, we modify it as follows. Let u = a0 + a1x + a2x + ... be the powerseries expansion of u at P . (Recall that this is obtained as follows: reduceu modulo P to get a0, so that u − a0 = b1x; now reduce b1 modulo P toget a1 and so on.) For a fixed positive integer r write Pr for the polynomiala0 + . . . + arx

r. Let L ⊃ K be a finite Galois extension of k(x) and letu = u1, u2, . . . , un be the roots of f (viewed as a polynomial with coefficientsin k(x)) in L. Let B be the integral closure of A in L and let Q be a point ofSpecB above P . The localisation BQ is a discrete valuation ring with residuefield 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 enoughwe may assume that vi has a pole at Q for i > 1 and that v1 has neither apole nor a zero. The vi are permuted by the elements of Gal (L|k(x)) andhence are the roots of an irreducible polynomial F with coefficients in k(x);by clearing denominators we can actually assume F = φny

n + . . .+ φ0 withφj ∈ k[x] and having no common divisor. This F will play the role of f andv1 the role of u before. Now write

F = φnv2 . . . vn(y − v1)(y/v2 − 1) . . . (y/vn − 1).

Here w = φnv2 . . . vn lies in A for it equals v−11 φ0. Hence by reducing modulo

Q we get F (0, y) = w(−1)n−1(y − v1), where w (resp. v1) is the image of w(resp. v1) modulo Q. Note that F (0, y) 6= 0, for otherwise the φj would allbe divisible by x. This implies w 6= 0, whence 0 6= (−1)n−1w = ∂2F (0, v1),which was to be seen.

Using the lemma we may construct the Riemann surface structure onsmooth complex curves as follows.

Construction 1.12 In the situation above assume k = C. The plane curveSpecA is not necessarily smooth but both differentials of f can vanish at onlyfinitely many points (as their loci of zeros define proper closed subsets). Awayfrom these singular points the local rings of SpecA may be identified withlocal rings of X by the above discussion. Also, by the condition ∂2f(0, a) 6= 0furnished by the lemma the construction of the Riemann surface structurefor smooth plane curves in Chapter 4 may be applied to the complementof the singular points and we get that x defines a complex chart centeredaround (0, a) in a sufficiently small open neighbourhood U not containingsingular points. Identifying U with a subset of closed points of X we maydeclare U open in the complex topology of X and get a bijection of U ontosome open subset in C via x; we declare this map to be a complex chart andin particular a homeomorphism.

2.. INVERTIBLE SHEAVES ON CURVES 179

Now assume x′ is another generator of the maximal ideal of OX,P andperform the above construction for x′ in place of x. We have to show that inthis way we get compatible complex charts; this will imply in particular thatx and x′ define the same topology in a sufficiently small neighbourhood of P .To see this, observe that (again by the condition ∂2f(0, a) 6= 0) we may applythe holomorphic implicit function theorem to the polynomial f around (0, a)and express u as a holomorphic function (i.e. a convergent power series) inx. As any element of OX,P = AP∩A is a rational function in x and u it mustbe holomorphic as well. This applies in particular to x′ and shows that thetwo charts are compatible.

Carrying out the above construction around each closed point of X thusgives a complex topology and a complex atlas on X, as required.

We have just seen that local rings of X consist of holomorphic functionsin the complex structure, from which it follows that any element f of thefunction field K is meromorphic and also (via Chapter 4, Lemma 1.5) thatany morphism of smooth curves induces a holomorphic map of Riemannsurfaces. To complete our discussion, we still need to show:

Corollary 1.13 If X is a proper smooth curve, then the associated Riemannsurface is compact.

Proof: By what we have just observed, the finite, hence surjective mor-phism φf : X → P1

C associated to a nonconstant function f ∈ K induces aholomorphic map. But any holomorphic map is open (Chapter 4, Corollary2.2), 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 Dedekindschemes. For the special class of proper smooth curves they play an evenmore important part, as the presence of a base field enables one to establishseveral very useful results about them which are not available in the generalcase.

Throughout this section (and the next), X will be a smooth proper curveover a perfect field k. We shall also make the additional assumption that kis algebraically closed in the function field K of X, i.e. any element of K \ kis transcendental over k; we shall call such elements non-constant.

Remark 2.1 Curves (or more generally, varieties) satisfying the above addi-tional assumption are usually called geometrically integral, for one may prove


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 4that any such L is isomorphic to some sheaf the form L(D), with D a divisoron X. The presence of the base field k enables us to define a very importantinvariant of D.

Definition 2.2 The degree of a divisor D =∑

imiPi on X is the integerdeg(D) =

∑imi[κ(Pi) : k].

In this way we obtain a homomorphism deg : Div(X) → Z. Now recallthat in Chapter 5, loc. cit. we defined the divisor Div (f) of a functionf ∈ 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) isthe divisor for which vP (Div 0(f)) = vP (f) if vP (f) > 0 and 0 otherwise andsimilarly Div∞(f) is the divisor with vP (Div∞(f)) = −vP (f) if vP (f) < 0and 0 otherwise. These are called the divisor of zeros and the divisor of polesof 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 hencedeg(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 isviewed as a coordinate function on P1

k, it generates the maximal ideal of thelocal ring at 0 and its inverse generates that at ∞. This implies that for Pa 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 theassertion 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 theproposition.

Corollary 2.5 If D1, D2 ∈ Div(X) are two divisors with L(D1) ∼= L(D2),then degD1 = degD2.


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 Lon X as degD for any D ∈ Div(X) with L(D) ∼= L (such a D exists byChapter 5, Propositions 4.8 and 4.12). Moreover, the corollary implies thatthe 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 issuch a sheaf, then L(X) is naturally a vector space over k. Now we have thefollowing basic finiteness result.

Lemma 2.6 For any invertible sheaf L on X, the k-vector space L(X) isfinite dimensional.

Before giving the proof it is convenient to introduce a partial order onthe group Div(X): we shall write D1 ≥ D2 if vP (D1) ≥ vP (D2) for all closedpoints P andD1 > D2 ifD1 ≥ D2 and there is some P with vP (D1) > vP (D2).

Proof: The results from Chapter 5, Section 4 enable us to choose someD 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 inductionon deg(D). For D = 0 we have dim kL(0)(X) = 1 by Corollary 2.4. For theinductive step we show that if D ≥ 0 and P is a closed point of X, then thequotient 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 ) bymapping a function f ∈ L(D)(X) to (tm+1f)(P ), where m = vP (D) and t isa generator of the maximal ideal of OX,P . (By assumption, tm+1f ∈ OX,P ; itsevaluation at P means its reduction modulo the maximal ideal.) The kernelof this homomorphism is exactly L(D)(X), whence the claim.

Remark 2.7 The lemma is a special case of much more general finitenessresults about coherent sheaves. See Hartshorne [1], Section 8.8 and the moregeneral theorem of Grothendieck cited there.

The dimension whose finiteness is asserted by the lemma is a very im-portant invariant of the invertible sheaf L; roughly speaking, it measuresthe number of independent solutions to the Mittag-Leffler type problem offinding functions on X with zero and pole behaviour restricted by any D forwhich L = L(D).

Also, the lemma applies in particular to ΩX/k which is an invertible sheafon X since k is perfect, so we may define a very important invariant of X.


Definition 2.8 The dimension g of the finite dimensional k-vector spaceΩX/k(X) is called the genus of the curve X.

See the next section for the relation over C with the topological genusdefined 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 kL(D) ≤ degD + 1

for D ≥ 0, but this is insufficient for most applications; the following theoremis much more precise.

Theorem 2.9 (Riemann-Roch Formula) For any invertible sheaf L onX we have the formula

dim k L(X) − dim k (ΩX/k ⊗OXL∨)(X) = degL − g + 1,

where g is the genus of X.

We shall not prove the theorem here for developing the necessary tech-niques would lead us to far astray. For modern proofs using the cohomologyof coherent sheaves, see Hartshorne [1], Section IV.1 or the first chapter ofSerre [2] (these authors assume k algebraically closed but the proof workswith minor modifications in general). For a more traditional proof due toWeil, see e.g. Lang [3], Stichtenoth [1] or van der Waerden [1].

We content ourselves here by deriving some consequences, of which hereare the first ones.

Corollary 2.10

1. deg ΩX/k = 2g − 2.

2. If degL > 2g − 2, then dim k L(X) = degL − g + 1.

Proof: The first assertion follows by substituting L = ΩX/k into the the-orem. For the second, assume degL > 2g − 2. We show that this impliesdim k (ΩX/k ⊗OX

L∨)(X) = 0, for which by the first part it is enough to seethat for any invertible sheaf L with degL < 0 we must have dim kL(X) = 0.So write L = L(D) and assume dim kL(D)(X) > 0. This means there is afunction f ∈ K with vP (f) + vP (D) ≥ 0 for all closed points P . Multiplyingby [κ(P ) : k] and taking the sum over P we get degD = degL ≥ 0 usingLemma 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.


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 largepositive integer m we have

dim kL((m+ 1)Pi)(X) − dim kL(mPi)(X) > 0

for all 1 ≤ i ≤ n, so for each i there exists a function fi ∈ K having a poleat Pi and regular elsewhere. Hence the set of poles of f = f1 + . . . + fn isexactly S. Now consider the morphism φf : X → P1

k defined by f . Theinverse image of the complement of ∞ (i.e. the normalisation of Spec k[f ] inthe extension K|k(f)) is precisely U , and we are done.

The Riemann-Roch formula is an invaluable tool in the classification ofcurves. 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 sheafL = L(P ) and computes the dimension of L(X) to be 2, hence this spacemust contain besides the constant functions some function f with a simplepole at P and regular elsewhere. The morphism φf : X → P1

k associated tof 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 propercurves of genus 0 with a k-rational point are isomorphic to P1

k.

Remark 2.13 The assumption on the existence of a k-rational point is im-portant if we wish to get an isomorphism defined over k (i.e. compatiblewith the projections to Spec k). Indeed, the projective plane curve of equa-tion x2

0 + x21 + x2

2 = 0 can be shown to be smooth of genus 0 over Q but isnot isomorphic over Q (or even R) to P1.

Example 2.14 Let now X be a smooth proper curve of genus 1 with ak-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 hasdimension 2; pick a basis consisting of the constant function 1 and somefunction x with a single pole of order 2 at P (it cannot have a simple polefor then X would be isomorphic to P1

k by the previous example, which isimpossible for P1

k 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 atP . 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 bothdegrees here, it must be 1.


Now apply the corollary to the space L(6P )(X) to see that that it is6-dimensional. Since it contains the seven elements 1, x, x2, x3, xy, y, y2, wemust have a linear relation of the form

a6y2 + a5y + a4xy = a3x

3 + a2x2 + a1x+ 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, assumeit has a point where both partial derivatives vanish; by performing a linearchange of coordinates we may assume this is the point (0, 0). Then theequation must have the form

a6y2 + a4xy = a3x

3 + a2x2

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 inthe previous example, this implies that the morphism X → P1

k associated toy/x is an isomorphism, which is again impossible. So we have shown that theaffine scheme V = Spec k[s, t]/(a6t

2 + a5t+ a4st− a3s3 − a2s

2 − a1s− a0) is asmooth plane curve. Now let U be the complement of P in X; by Corollary2.11 U is the form SpecA, with A containing x and y, so sending s to xand t to y induces a morphism of smooth affine curves U → V ; arguing in asimilar way as in the proof of Proposition 1.6, we see that this is actually anisomorphism. Finally, consider the projective plane curve Y with equation

a6x0x22 + a5x

20x2 + a4x0x1x2 = a3x

31 + a2x0x

21 + a1x

20x1 + a0x

30.

One checks easily that there is a single point ∞ on this curve for whichx0 = 0, that the complement of ∞ is isomorphic to V above and that thecurve is smooth at ∞ as well. Now define a map X → Y by sending U to Vby the above isomorphism and mapping P to ∞. The local rings OX,P andOY,P are then the same as subrings of K (being “the missing local rings fromX and Y ”) and we get an isomorphism X ∼= Y .

We have thus proven that any elliptic curve is isomorphic to a cubicprojective plane curve with an equation as above.

Remark 2.15 In the above two examples we have constructed closed im-mersions of our curve X into some projective space. Quite generally, it ispossible to find invertible sheaves on X using the Riemann-Roch formulafrom which one can construct closed embeddings by a similar method. SeeHartshorne [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 eitheraffine or projective.

3.. THE HURWITZ GENUS FORMULA 185

3. The Hurwitz Genus Formula

In this section we prove a classical formula due to Hurwitz relating the dif-ferent of a finite separable morphism φ : Y → X of smooth proper curves tothe genera of the curves X and Y . Since both the different and the genusare defined using differentials, the existence of such a relation is by no meanssurprising.

In order to derive the precise formula, we need some technical preliminar-ies. The first of these is the concept of the pullback of a quasi-coherent sheafby a morphism, which we did not need up till now. As in the constructionof sheaves of differentials in Chapter 6, we only define this concept for anaffine morphism and refer the readers to Hartshorne [1], Section II.5 for thegeneral case.

Construction 3.1 Let X = SpecA be an affine scheme and F = M a quasi-coherent sheaf associated to an A-module M . Given a morphism φ : Y → Xwith Y = SpecB an affine scheme, we define the sheaf φ∗F on Y as thequasi-coherent-sheaf associated to the B-module M ⊗A B.

In general, given an affine morphism of arbitrary schemes φ : Y → Xand a quasi-coherent sheaf F on X, we may find an affine open covering ofX by subsets Ui = SpecAi whose inverse images are of the form SpecBi.By refining the covering of X if necessary, we may assume that over each Ui

the sheaf F is of the form Mi with some Ai-module Mi and the definitionof quasi-coherent sheaves shows that over basic open subsets Uij containedin the intersections Ui ∩ Uj there exist canoinical isomorphisms Mi|Uij

∼=

Mj |Uij. This shows that we may define a quasi-coherent sheaf φ∗F on Y by

performing the above construction over each Ui. One also checks by choosinga common refinement of two coverings of X that this definition does notdepend on the covering chosen.

Proposition 3.2 Let φ : Y → X be a finite separable morphism of smoothproper curves over a field k. Then we have an exact sequence

0 → φ∗ΩX/kλ→ΩY/k → ΩY/X → 0

of invertible sheaves on Y .

Proof: Choosing suitable affine open coverings of X and Y , it follows fromChapter 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. Forthis by the injectivity part of Chapter 6, Lemma 4. it is enough to check


that the induced map λP is injective on all stalks at points P of Y . If P isthe 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 thatλP is a surjective morphism of 1-dimensional L-vector spaces (where L isthe 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 homo-morphism, 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 overan open subset U of Y . But since we have just seen that φ is etale over thegeneric point of X, by Chapter 7, Corollary ?? ΩY/X = 0 over a nonemptyopen 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 thedifferent.

Proposition 3.3 In the situation of the previos proposition we have an iso-morphism

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 aboutsheaves 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 andfor each Pi choose an open neighbourhood Ui over which L is trivial. Considerthose open subsets V ⊂ Y for which either (i) V ⊂ Ui for some i or (ii) V doesnot contain any of the Pi. These subsets form a basis of open sets of Y , soby Chapter 5, Lemma 2.7 if we want to define a morphism φ : L⊗OY

F → Fit 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 isomorphismsL(V ) ⊗ F(V ) ∼= OY (V ) ⊗ F(V ) ∼= F(V ) coming from the trivialisation andover a V of type (ii) we define φV to be 0. It is straightforward to check thatthe φV so defined are compatible and thus define a φ we are looking for; bylooking at the stalks we see moreover that φ is an isomorphism.


Proof of Proposition 3.3: Tensoring the exact sequence of Proposition 3.2by 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 correspond-ing fact about tensor products of modules from Chapter 0. Furthermore, thesequence is also exact on the left, for as above we need only to check this onstalks where we are tensoring with a free module of rank one. So applyinglemma 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 theproposition by tensoring with ΩY/k.

As a fairly easy consequence we get:

Theorem 3.5 (Hurwitz Genus Formula) Let φ : Y → X be a finiteseparable morphism of smooth proper geometrically integral curves over afield k, of genus gY and gX , respectively. Let L|K be the correspondingextension of function fields. Then

2gY − 2 = [L : K](2gX − 2) − degDY/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) +∑

Q

(e(Q|P ) − 1)[κ(Q) : k].

For the proof we need the following lemma.

Lemma 3.6 In the situation of the theorem, for any invertible sheaf L onX we have deg φ∗L = [L : K] degL.

Proof: The lemma follows using Chapter 7, Proposition 1.20 once we showthat writing L ∼= L(D) with some divisor D =

∑imiPi in Div(X), we have

φ∗L ∼= L(∑

i

mi

∑

Q∈YPi

e(Q|Pi)Q).

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


K and writing mi = vPi(f−1) with a generator f of the stalk LPi

viewedas 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 asa function on Y via the embedding K ⊂ L. But for any point Q lying overPi we have vQ(f−1) = e(Q|Pi)vPi

(f−1).

Proof of Theorem 3.5: The first formula follows by taking the degrees ofthe invertible sheaves that figure in the previous proposition and using thelemma. The second statement follows from the first by Dedekind’s DifferentFormula (Chapter 7, Theorem 5.5).

The Hurwitz formula is very useful for computing genera of curves. Atypical 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 equa-tion of the form

y2 = (x− a1) . . . (x− a2m+1)

with some integer m > 1 and all ai distinct. Making the usual substitutionx = x0/x2, y = x1/x2 we get a smooth projective plane curve Y which has asingle 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 atinfinity and the points above the ai in the (x, y)-plane. Writing the Hurwitzformula gives 2gY − 2 = −4 + 2m+ 2, i.e. gY = m.

We now turn to applications to the theory of covers. It is well knownfrom topology that the projective line and the affine line over C are simplyconnected. Using the Hurwitz formula, we can prove the following generali-sation.

Corollary 3.8

1. Any finite etale morphism φ : Y → P1k with Y a smooth proper geo-

metrically integral curve is an isomorphism.

2. Let φ : Y → P1k be a finite morphism with Y a smooth proper geomet-

rically integral curve. Assume that all of its branch points lie over thepoint at infinity and that φ is tamely ramified at these points. Then φis an isomorphism.

The tameness assumption in the second statement is essential: there aremany non-trivial covers of P1

k in positive characteristic that are etale overA1

k and are wildly ramified at infinity.


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 happenfor gY = 0 and L = K. For the second statement, we see by comparingChapter 7, Proposition 1.20 with the Hurwitz formula in the tamely ramifiedcase that the maximal contribution of the different can be [L : K]−1 (in thecase of a single branch point that is rational over k). Hence the right handside of the formula can be at most −[L : K]−1 < 0, which again only allowsgY = 0 and L = K.

Remark 3.9 The first part of the corollary implies that for k a perfectfield the group π1(P

1k) is isomorphic to Gal (k). This follows from results

of the next section but the reader may check it as an exercise. A similarresult holds for the so-called tame fundamental group of the affine line bythe 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 impliesthat we must have −2 = [L : K](2gX −2)−deg DP1/X which is only possiblefor gX = 0 and L = K as deg DP1/X is always non-positive.

In general let K ⊂ L′ ⊂ L be the maximal separable subextension andwrite L (which is now the function field of P1

k) as k(x) with some function x.Then L = L′(x) and we know from Chapter 7, Lemma ?? that the minimalpolynomial of x over L′ is of the form xpn

−a with some a ∈ L′. In particular,L′ contains the field k(xpn

) and since [k(x) : k(xpn

)] = [k(x) : L′] = pn weactually have L′ = k(xpn

). Thus L′ is a rational function field, so that thenormalisation of X in L′ is isomorphic to P1

k and we may conclude by theseparable case.

Remark 3.11 We have the following algebraic reformulation of the abovecorollary: any transcendental subextension of a purely transcendental exten-sion k(x)|k is itself purely transcendental. This result is known as Luroth’stheorem. For degree 2 transcendental extensions of C the analogous state-ment remains true but in degree 3 counterexamples have been found inde-pendently by Artin/Mumford, Clemens/Griffiths and Iskovskih/Manin in thebeginning 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


(Chapter 4, Corollary 4.10) for branched covers of Riemann surfaces. As bothformulae 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 propercurves over C.

1. If Q is a closed point of Y mapping to a closed point P of X, theramification index e(Q|P ) is the same as the topological ramificationindex at Q defined in Chapter 4.

2. The genus gX of X (resp. gY of Y ) is equal to the topological genus ofX (resp. Y) defined in Chapter 4.

Proof: For the first statement, let tP be a generator of the maximal idealof OX,P . Then φ induces a natural inclusion OX,P ⊂ OY,Q and by definitionof the ramification index tP generates the e(Q|P )-th power of the maximalideal 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 chartof 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 zeroof order e(Q|P ) at 0, which is the topological definition of the ramificationindex.

For the second statement, note that because any complex curve admits afinite morphism onto P1

C and because of the same shape of the formulae ofTheorem 3.5 and Chapter IV, Corollary 4.10 it is enough to prove that P1

C

has genus 0 in both senses. This is tautological in the topological case andwe have seen the algebro-geometric case in the previous section.

Remark 3.13 One can give a more direct proof of the second statementabove by using some difficult theorems about complex varieties. Indeed, oneknows that for X a compact Riemann surface of genus g (in the topologicalsense), the singular cohomology group H1(X,C) is of dimension 2g over C.(One may prove this directly or deduce it from the structure of the fundamen-tal group of X discussed in Chapter 4 via the Hurewicz isomorphism.) Thenthe Hodge decomposition theorem (Griffiths-Harris [1], p. 116) and Serreduality (ibid., p. 102 or Hartshorne [1], Corollary 7.13) imply that H1(X,C)decomposes as a direct sum of ΩX/C(X) and of its dual vector space, whencethe result. This proof is not only more direct but is also more conceptual forit reduces the equality of the two notions of genera to fundamental relationsbetween the topology and the differentiable structure of X.

4.. ABELIAN COVERS OF CURVES 191

4. Abelian Covers of Curves

5. Fundamental Groups of Curves

In this section we give an overview of some of the major results currentlyknown about the structure of fundamental groups of smooth curves overalgebraically closed fields. We begin by discussing smooth proper curves.

To state the fundamental theorem, we first need to introduce some ter-minology: given a prime p, the maximal prime-to-p quotient (resp. maximalpro-p quotient) of a profinite group G is defined as the inverse limit of allfinite 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 ofthe natural inverse system of its finite quotients of order prime to p (resp. apower of p) is the profinite p′-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 ofcharacteristic p ≥ 0 and let X be a smooth proper connected curve of genusg over k. Then π1(X)(p′) is isomorphic to the profinite p′-completion of thegroup 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 resultof Chapter 4, Theorem 4.3. To be able to apply this result, one uses thefact that any topological cover of X is in fact a smooth proper curve havingthe structure of an etale cover of X; this follows from Riemann’s ExistenceTheorem. Thus the algebraic fundamental group is the profinite completionof the topological one.

Step 2. From this transcendental result one deduces the theorem for anyalgebraically closed field of characteristic 0 by applying a general theorem(to be stated precisely in the next section) according to which in charac-teristic 0 the algebraic fundamental group does not change by extensions ofalgebraically closed fields. This immediately applies to curves defined overalgebraically closed subfields of C. But in fact any curve in characteristic zerocan be defined over such a subfield: since its definition involves only finitelymany coefficients of finitely many polynomial equations, the algebraic closureof the field generated over Q by these coefficients can be embedded in C asthe latter has infinite transcendence degree over Q.


Thus in characteristic 0 the theorem reduces to the topological result ofChapter 4. But to treat positive characteristic (which is the main contribu-tion of Grothendieck) one needs further arguments.

Step 3. One begins by showing that it is possible to “lift curves from positivecharacteristic to characteristic 0”. More precisely, one first shows that givenany algebraically closed k of characteristic p > 0 there exists a discretevaluation ring A of with fraction field K and maximal ideal P such that Kis 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). Onethen proves that there exists a scheme X → SpecA smooth and proper ofrelative dimension 1 over SpecA with closed fibre X (this amounts basicallyto the fact that the fibre over the closed point of SpecA is isomorphic to Xand that over the generic point is a smooth proper curve). Intuitively, thegeneric fibre XK “reduces modulo P to X”.

The existence of X is by no means easy to prove. Grothendieck originallyused techniques of formal geometry to construct a so-called formal schemeof 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 ringsof X and only afterwards on the local rings themselves. Nowadays one wouldprefer to construct X by applying a form of Hensel’s Lemma to a suitablemoduli stack parametrising smooth proper curves. But one then has to give arigorous meaning to the preceding sentence which requires some hard work.In the notes of Popp [1] one finds an elementary version of this argumentusing only the classification of plane curves (which is elementary) but theprice 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-calledspecialisation theorem: denoting by XK the base-change of XK to an algebraicclosure of K there is a natural homomorphism π1(XK) → π1(X) inducing anisomorphism on maximal prime-to-p quotients. The theorem then follows byapplying the characteristic 0 result to XK .

The proof of the specialisation theorem is of more elementary nature thanthat of the previous step (it basically reposes on the so-called Zariski-Nagatapurity theorem and Abhyankar’s Lemma) but still it requires some delicatearguments. 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 mayask the similar question about its maximal pro-p quotient; of course thisquestion only arises in characteristic p > 0.

5.. FUNDAMENTAL GROUPS OF CURVES 193

Before stating the corresponding result, we need a definition: the p-rankof an abelian variety over an algebraically closed field is the dimension of theFp-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 alge-braically closed field of characteristic p > 0. Then π1(X)(p) is a free profinitegroup whose rank equals the p-rank of the jacobian variety of X.

One may infer that the previous two theorems elucidate the structure offundamental groups of smooth proper curves over algebraically closed fields.This is indeed the case in characteristic 0 but in positive characteristic evenif we know the maximal pro-p and prime-to-p quotients, the structure ofthe group itself remains a mystery. The theorems give, however, a gooddescription of its maximal abelian quotient: this group is the direct sumof its maximal prime-to-p and pro-p quotients and hence the previous twotheorems together suffice to describe it.

To emphasize further the difference between the characteristic 0 and char-acteristic p > 0 case, let us mention a striking recent result of Tamagawa’s(building upon earlier work by Raynaud, Pop and Saıdi). Observe that incharacteristic 0 the fundamental group is completely determined by the topol-ogy of the underlying complex curve and hence by the genus. Thus there aremany curves having the same fundamental group. However, the positivecharacteristic case turns out to be completely different:

Theorem 5.3 (Tamagawa) Let k be an algebraically closed field of char-acteristic p > 0 and let G be a profinite group. Then there are only finitelymany smooth proper curves of genus g ≥ 2 over k whose profinite group isisomorphic to G.

Of course, the theorem is interesting only for those G which actually ariseas the fundamental group of some curve as above. We leave it to the readersas an easy exercise to treat the cases g = 0, 1.

Now we turn to affine curves. It turns out that Grothendieck’s methodfor proving Theorem 5.1 can be refined to handle this case as well. Thereofreone has:

Theorem 5.4 (Grothendieck [1]) Let k be an algebraically closed field ofcharacteristic p ≥ 0 and let X be a smooth affine connected curve over karising from a smooth proper curve X ′ ⊃ X of genus g by deleting n closedpoints.


Then π1(X)(p′) is isomorphic to the profinite p′-completion of the groupΠg given by 2g+n generators a1, b1, . . . , ag, bg, γ1, . . . , γn subject to the singlerelation

[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 thefollowing 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 Gdenote 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 closedof characteristic p > 0 and let X be a smooth affine connected curve over karising from a smooth proper curve X ′ ⊃ X of genus g by deleting n points.

Then any group G for which G/p(G) can be generated by 2g + n − 1elements arises as a quotient of π1(X).

Raynaud proved the crucial caseX = A1k by explicitly constructing Galois

covers using two radically different methods, based respectively on techniquesfrom rigid analytic geometry and the theory of semi-stable curves. The proofalso uses some previous results of Serre [5]. Harbater then observed how toreduce the general case to the case of the affine line; this reduction was latersimplified by Pop [1]. See also the chapters by Chambert-Loir and Saıdi inBost/Loeser/Raynaud [1].

Chapter 9

The Algebraic FundamentalGroup

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 eachfibre XP of φ is the spectrum of a finite etale κ(P )-algebra and if the directimage sheaf φ∗OX is locally free (of finite rank). We shall speak of finiteetale covers if in addition φ is surjective.

In the case when Y is a Dedekind scheme, we have seen in Setion 8of Chapter 5 that the second condition is automatically satisfied, so thisdefinition is a generalisation of the previous one.

As immediate consequences of this definition, we get that any compositionof finite etale morphisms is etale and that any base change of a finite etalemorphism 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 withφ s the identity of S) is clopen.

3. The image of the diagonal morphism X → X ×S X (corresponding tothe identity maps of X the two components) is clopen.

195

196 CHAPTER 9. THE ALGEBRAIC FUNDAMENTAL GROUP

We omit the proof.

Corollary 1.3 If Z → S is a connected S-scheme and f, g : Z → X aretwo S-morphisms into a finite etale S-scheme which coincide at some pointof Z, then φ1 = φ2.

Here the term “coincide” should be understood in the scheme-theoreticsense, i.e. we require not only that there should be a point P ∈ Z such thatf(P ) = g(P ) = Q ∈ X, but also that the maps κ(Q) → κ(P ) induced by fand g should be the same.

Proof: By taking the fibre product of Z and Y over S and using the basechange property of etale morphisms we may assume S = Z. Then we haveto prove that if two sections of a finite etale cover X of a connected schemeS coincide at a point, then they are equal. This follows from the secondstatement of the lemma, for each such section being injective, it must bean isomorphism of S onto a connected component of X and hence it isdetermined 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 connectedfinite etale cover is finite.

We have the following statement essentially proved in Chapter 5, Section7:

Lemma 1.5 If S is a locally noetherian scheme, φ : X → S is a finite etalecover and G is a finite group of S-automorphisms of X, then the quotientX/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 amorphism s : Spec Ω → S, where Ω is some algebraically closed field. Thetopological image of s is a point s of S such that Ω is an algebraically closedextension of κ(s).

Now consider the category Fet|S of schemes finite and etale over S, withmorphisms 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 ΩS induced by s. Regarding Fibs(X) onlyas a set, we get a set-valued functor Fibs on Fet|S called the fibre functor.

1.. THE GENERAL NOTION OF THE FUNDAMENTAL GROUP 197

Theorem 1.6 Assume further S is connected and locally noetherian. Thenthere is a profinite group π1(S, s) such that for each X finite and etale overS the set Fibs(X) is canonically equipped with a continuous left action ofπ1(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 cat-egory 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 andlocally simply connected topological space. If we restrict to covers withfinite fibres only, then the analogy is even more explicit: the categoryof these is easily seen to be equivalent to the category of finite setsequipped with a continuous action of the profinite completion of thefundamental group. (Note, however, an analogue of the above theoremcan be proved by the method we shall employ for finite covers of anytopological 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 ks, X),where ks is the separable closure of k in Ω via the embedding given bys and π1(S, s) = Gal (ks|k). Note that this does not mean that the fibrefunctor is representable, for ks is not a finite etale k-algebra. However,its spectrum is the inverse limit of finite (Galois) subextensions whichalready are. This motivates the following definition.

Definition 1.9 Let C be a category and F a set-valued functor on C. Wesay 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 elementpj 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 byF (φ). (Notice that the inverse limit of the Pi may not exist but thedirect limit of their Hom’s does in the category of sets.)


Thus in the last example the fibre functor is strictly prorepresentable bythe inverse system of spectra of finite (Galois) extensions conatined in thefixed separable closure. We shall prove that quite generally the fibre functoris strictly pro-representable, from which the theorem will follow by formalarguments.

As in topology, we define a connected finite etale cover X → S to beGalois if its S-automorphism group acts transitively on the fibres. Nowthe key lemma is the generalisation of the well-known fact that any finiteseparable field extension can be embedded in a finite Galois extension andthere is a smallest such extension, namely the Galois closure.

Lemma 1.10 Let φ : X → S be a connected finite etale cover. Considerthe full subcategory Gal|Sof Fet|S spanned by Galois covers. Then the set-valued 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 S-morphism π : P → X such that any S-morphism from a Galois cover toX factors through p.

Proof: Let Fibs(X) be the finite set F = x1, . . . , xn. The xi induce acanonical geometric point x of the n-fold fibre product Xn = X ×S . . .×S Xgiving xi in the i-th component. Let P be the connected component of Xn

containing the image of x and let π : P → X be the map induced by the firstprojection of Xn to X. Using the composition and base change properties ofX 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 repre-sented by an n-tuple (xσ(1), . . . , xσ(n)) for some permutation σ ∈ Sn. Indeed,any point of Fibs(X

n)) corresponds to an element of F n so we only haveto show that those points which are concentrated on P have distinct coordi-nates. But by the third statement of Lemma 1.2, if ∆ denotes the diagonalimage of X in X ×S X, its inverse image π−1

ij (∆) by any of the projectionsπij mapping Xn to two of its components is a clopen subset. Since P is con-nected, π−1

ij (∆) ∩ P 6= ∅ would imply π−1ij (∆) ⊃ P , which is impossible for x

hits Xn away from any of the π−1ij (∆). Now to show that P is Galois over S,

simply remark that any permutation σ ∈ Sn induces an S-automorphism φσ

of Xn by permuting the components. If φσx ∈ Fibs(P ), then φσ(P )∩P 6= ∅and so φσ ∈ Aut(P/S) by connectedness of P . So Aut(P/S) acts transitivelyon one geometric fibre, from which we conclude as in topology (using Lemma1.5) that P is Galois.

1.. THE GENERAL NOTION OF THE FUNDAMENTAL GROUP 199

Now if q : Q→ X is an S-morphism with Q Galois, choose a preimage yof x1. By composing with appropriate elements of Aut(Q/S) we get n mapsq = q1, . . . , qn : Q → X such that qi y = xi. Whence an S-morphismQ → Xn 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 functorFibs is strictly prorepresentable.

Proof: Take the index set I to be the set of all finite etale Galois coversPi → S and define Pi ≤ Pj if there is a morphism Pj → Pi. This partiallyordered set is directed (apply the previous lemma to any connected compo-nent of a fibre product Pi ×S Pj). For each Pi ∈ I fix an arbitrary elementpi ∈ FibS(Pi). Then since the objects are Galois, if there exists an S-mapPj → Pi then there is also an S-map mapping pj to pi; such a map is evenunique by Corollary 1.3. So we take the inverse system formed by the Pi ofthe above particular maps. To conclude the proof we only have to construct afunctorial map Fibs(X) → lim

→Hom(Pi, X) inverse to the natural map. For

this, we may assume X is connected (otherwise take disjoint unions). Givenan element x ∈ Fibs(X), consider the Galois closure π : P → X given bythe previous lemma. Since P is Galois, we may assume that F (π) maps thedistinguished element p ∈ F (P ) to x. Now mapping x to the class of p inlim→

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 auto-morphism group of the inverse system constructed in the above proof. Bydefinition, this automorphism group is the inverse limit of the Aut(Pi) bythe maps induced by those in the inverse system; it is profinite by Corollary1.4. Finally, the opposite group simply means that we change the multipli-cation law from (x, y) 7→ xy to (x, y) 7→ yx; this ensures that if we agreethat automorphisms of finite etale covers act from the left, then π1(S, s) alsoacts from the left on the geometric fibres. The statements in the theorem arethen proved as in the special case of fields, if we replace the arguments whichinvolve taking the fixed field of an open subgroup of the Galois group bytaking first an open normal subgroup V contained in a given open subgroupU of π1(S, s) and then taking the appropriate quotient of the Galois covercorresponding to V .

Remark 1.12 It can be checked by looking at the construction that, justas in the topological case, changing the base point induces an isomorphismof the corresponding fundamental groups. This isomorphism is unique up toan inner automorphism of one of the groups in question.


Just as the topological fundamental group, the algebraic fundamentalgroup is a functor on the category of pointed schemes.

Proposition 1.13 Let φ : (S ′, s′) → (S, s) be a map of pointed schemes,i.e. a morphism of schemes with φ s′ = s.

1. The map φ induces a canonical functor Fet|S → Fet|S′ by mappingan object X → S to the fibre product X ×S S

′ → S ′ and a canonicalmorphism φ∗ : π1(S

′, s′) → π1(S, s) of profinite groups.

2. If any finite etale cover X ′ → S is of the form X ×S S′, 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 etalecover X → S the cover X ×S S

′ is connected as well.

Proof: To show the existence of φ∗, it will suffice to construct a compati-ble system of homomorphisms π1(S/, s

′) → AutS(Pi)op for each Pi ∈ i, where

the Pi prorepresent the fibre functor associated to (S, s). Now to give anS-automorphism of Pi is the same according to the theorem as giving anautomorphism of the π1(S, s)-set Fibs(Pi). But by definition of the fibreproduct we have a bijection of sets Fibs(Pi) ∼= Fibs′(Pi ×S S

′), so via itsaction on the right hand side each element of π1(S

′s′) induces an automor-phism of the left hand side set. To see that it commutes with the actionof elements of π1(S, s) notice first that each such element comes from an S-automorphism of Pi. Whence an S ′-automorphism of Pi ×S S

′, inducing viathe theorem an automorphism of Fibs′(Pi ×S S

′) commuting with the actionof π1(S

′, s′). This is what we wanted to see.The second statement follows from the observation, implied by the theo-

rem, that an element of π1(S, s) is the identity if and only if it acts triviallyon the geometric fibres of each finite etale cover of S. For the third, use thefact that connected covers map via the fibre functor to sets with transitiveπ1(S, s)-action. This immediately gives one implication. For the converse,notice that if connected covers pull back to connected covers, then Galoiscovers pull back to Galois covers, so that each automorphism of a Galoiscover of S comes from a corresponding one over S ′.

2. The Outer Galois Action

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

2.. THE OUTER GALOIS ACTION 201

induces a map π1(X, x) → Gal (k) once a geometric point x is chosen. Foran algebraic closure k of k we also dispose of a geometric point of X =X ×Spec k Spec k induced by x, whence another map π1(X, x) → π1(X, x).These two maps are related by the following important proposition.

Proposition 2.1 Assume X is a noetherian and geometrically connectedscheme over a perfect field k. (The latter condition means that X is con-nected.) Then once a geometric point x is chosen, we have an exact sequenceof profinite groups

1 → π1(X, x) → π1(X, x) → Gal (k) → 1.

The proposition also holds over non-perfect fields. For this one showsthat the groups under consideration are not affected by purely inseparableextensions of the base field. For the proof we need a lemma.

Lemma 2.2 Any finite etale cover Y → X arises by base change to k froma cover YL → XL, where L is a suitable finite extension of k. The same holdsfor automorphisms of covers.

Proof: We prove the first statement, the proof of the second being sim-ilar. Since X is compact by assumption, it has a finite covering by opensubschemes Ui = SpecAi such that the restrictions Y Ui

of the cover Y toeach Ui are defined by finitely many equations with coefficients in Ai; thesegenerate subalgebras A′i ⊂ Ai which in turn arise as quotients of some poly-nomial ring k[x1, . . . , xm] by an ideal generated by finitely many polynomials.The coefficients of all these polynomials are contained in some finite Galoisextension L|k. Similarly, one sees that the isomorphisms showing the com-patibility of the Y Ui

over the overlaps Ui ∩ Uj can be defined by equationsinvolving only finitely many coefficients so we conclude by taking a suitablylarge 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 separableextension L|k, so its pullback is XL = X ×Spec k SpecL which is connectedby the assumption, being the continuous image of X.

Next observe that π1(X, x) is isomorphic to the projective limit of the nat-ural projective system formed by the groups π1(XL, x) with L running overthe set of finite Galois extensions of k. Indeed, there are natural compatiblemaps π1(X, x) → π1(XL, x) induced by functoriality, whence a map in thelimit. To show its injectivity, notice that a nontrival element of π1(X, x) can


be represented by a nontrivial automorphism of some Galois cover P → Xand by the lemma this automorphism must come from an automorphism ofsome cover PL → XL for L large enough. For surjectivity, one representsan element of the inverse limit by a compatible system of automorphisms ofGalois covers Pi,L → XL. Keeping those terms which correspond to coversremaining connected after base change to k, we get a compatible system ofautomorphisms of Galois covers P → X by base change, whence an elementof π1(X, x).

On the other hand, for each finite Galois extension L|k there is an exactsequence 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 groupis naturally a quotient of π1(X, x); moreover, one sees using Lemma 1.10 thatthe open kernel of the corresponding projection is exactly π1(XL, x). Notethat 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 mapsan element of Gal (L|k) to the automorphism of XL it induces via its actionon L; this is an injective map. Now passing to the projective limit and usingthe observation of the previous paragraph we get an exact sequence

1 → π1(X, x) → π1(X, x) → lim←

Aut(XL/X)op,

which immediately yields that the sequence of the proposition is exact onthe left. Exactness in the middle follows from the fact that the last mapin 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 forthen the following results apply.

Theorem 2.3 Let K|k be an extension of algebraically closed fields, X anoetherian connected k-scheme. Put XK = X ×Spec k SpecK as above andchoose 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.


In fact, the second case can be deduced from the first using Abhyankar’slemma.

Theorem 2.4 The algebraic fundamental group of X a scheme of finite typeover C is isomorphic to the profinite completion of the associated complexanalytic space.

We shall not define the general notion of a complex analytic case. In thecase 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 smoothcompactification and delete the extra points from the associated Riemannsurface. Thus we see that in these cases the theorem follows if one knowsthat any finite (branched) topological cover of a compact Riemann surface isin 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 alge-braic fundamental group is an extension of the absolute Galois group of thebase field by the profinite completion of the topological fundamental group.

Remark 2.6 The latter group is known from topology: if the associatedRiemann surface is topologically a torus with g holes and there are n deletedpoints, 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 onn−1 generators. For g ≥ 2 it has a quotient isomorphic to the free group on 2generators. Since by the classification of finite simple groups any such groupcan be generated by 2 elements, they are all quotients of the topological, andhence the algebraic, fundamental group. So in these cases we get a groupthat is monstruously big.

Now returning to the situation of Proposition 2.1, observe that the groupπ1(X, x) acts on its closed normal subgroup π1(X, x) by conjugation, whencea representation π1(X, x) → Aut(π1(X, x)). The restriction of this map tothe π1(X, x) is a homomorphism π1(X, x) → Inn (π1(X, x)), where the lattergroup denotes the group of inner automorphisms. Thus by passing to thequotient we get a representation ρ : Gal (k) → Out (π1(X, x)), where thelatter group is the group of outer automorphisms of π1(X, x) defined simplyas the quotient of Aut(π1(X, x)) by its subgroup Inn (π1(X, x)). If k ⊂ C,this representation thus maps Gal (k) into the outer automorphism group ofa “transcendental object”. So the following fact may be surprising.


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 sointeresting is explained by the following proposition, also due to Belyi (whichin 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 pointsclassifies all smooth proper curves defineable over an algebraic number field(but the isomorphism class of a curve may be counted several times). Thisfact is so surprising that Grothendieck didn’t even venture to conjectureit. Note also that by virtue of Theorem 2.3 the converse is also true in thefollowing form: any ramified cover of the complex projective line that is etaleoutside 0, 1,∞ can be defined over Q.

Proof: In any case there is is a morphism p : X → P1 defined over Q andetale above the complement of a finite set S of closed points (any rationalfunction induces such a map, as we well know).

In a first step, we reduce to the case where S consists of points definedover 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 agenerator 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 ′ = φf(S)∪ ∞∪ φf(Sf )where Sf is the branch locus of φf , i.e. the support of Div 0(f

′). Now ∞ hasdegree 1 over Q; the points of Sf , and hence of φf(Sf ) have degree at mostn − 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 ′ than in S. Replacingp by φf p and S by S ′ we may continue this procedure until we arrive atn = 1.

So assume all points in S are defined over Q. If S consists of at most threepoints, we are done by composing with an automorphism of P1; otherwisewe 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


well-chosen polynomial f . This time we seek f in the form zA(z−1)B . Thenφf will map both 0 and 1 to 0 and fix ∞. However, φf p will be alsoramified over the images of those points where the derivative f ′ vanishes.These are given by the equation AzA−1(z − 1)B + BzA(z − 1)B−1 = 0, orelse A(z − 1) + Bz = 0 which gives z = A/(A + B). So if we choose A andB 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,∞. Assumethat ρ has a nontrivial kernel, fixing a (possibly infinite) extension L|k. Thenthe representation ρL : Gal (L) → Out (π1(XL, x))) is trivial which meansthat any automorphism of π1(X, x) induced by conjugating with an elementx ∈ π1(XL, x) is in fact conjugation by an element y ∈ π1(X, x). This impliesthat y−1x is in the centraliser C of π1(X, x) in π1(XL, x), so that the lattergroup is generated by C and π1(X, x). But as we have remarked above, inthis case π1(X, x) is isomorphic to the profinite completion of the free groupon two generators which is known to have a trivial center. Hence C andπ1(X, x) have trivial intersection, which implies that π1(XL, x) is actuallytheir direct product. Whence a quotient map π1(XL, x) → π1(X, x) whichgives the identity by composing with the natural inclusion in the reversedirection. Applying this to finite continuous π1(X, x)-sets we get that anyfinite etale cover of X comes by base change from a cover of XL. Combiningthis with the above proposition implies that any smooth proper curve definedover Q can in fact be defined over L. But there are counter-examples to thislatter assertion even for curves of genus 1 (take an elliptic curve with j-invariant an algebraic number outside L).

Notes on Galois Theory, Riemann Surfaces and Schemes (T. Szamuely)

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