On an Introduction to Grothendieck's Theory of the Fundamental Group

Home , Fundamental group, Proper morphism

Lectures on An Introduction to Grothendieck’s Theory of the Fundamental Group

By J.P. Murre

Notes by S. Anantharaman

No part of this book may be reproduced in any form by print, microﬁlm or any other means without written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 5

Tata Institute of Fundamental Research, Bombay 1967

Preface

These lectures contain the material presented in a course given at the Tata Institute during the period December 1964 - February 1965. The purpose of these lectures was to give an introduction to Grothendieck’s theory of the fundamental group in algebraic geometry with, as application, the study of the fundamental group of an algebraic curve over an algebraically closed field of arbitrary characteristic. All of the material (and much more) can be found in the “Séminaire de géométrie algébrique” of Grothendieck, 1960-1961 ExposéV, IX and X. I thank Mr. S. Anantharaman for the careful preparation of the notes.

J.P. Murre

iii

Prerequisites

We assume that the reader is somewhat familiar with the notion and elementary properties of preschemes. To give a rough indication: Chap. I, 1-6 and Chap. II, pages 1-14, 100-103 and 110-114 of the EGA § (“Eléments´ de Géometrie Algébrique” of Grothendieck and Dieudon’e). We have even recalled some of these required elementary properties in Chap. I and II of the notes but this is done very concisely. We need also all the fundamental theorems of EGA, Chap. III (first part); these theorems are stated in the text without proof. We do not require the reader to be familiar with them; on the contrary, we hope that the applications which have been made will give some insight into the meaning of, and stimulate the interest in, these theorems.

Contents

Preface iii

Prerequisites v

1 Aﬃne Schemes 1 1.1 ...... 1 1.2 ...... 1 1.4 The Sheaf associated to Spec A ...... 3 1.5 ...... 4 1.6 AﬃneSchemes ...... 6

2 Preschemes 9 2.2 Product of Preschemes ...... 10 2.3 Fibres ...... 12 2.4 Subschemes...... 14 2.5 Some formal properties of morphisms ...... 17 2.6 Aﬃnemorphisms ...... 19 2.7 The ﬁniteness theorem ...... 20

3 Etale´ Morphisms and Etale´ Coverings 23 3.2 Examples and Comments ...... 24 3.3 ...... 25 3.4 ...... 34 3.5 Etale´ coverings ...... 39

vii viii Contents

4 The Fundamental Group 43 4.1 Properties of the category C ...... 43 4.2 ...... 46 4.3 ...... 49 4.4 Construction of the Fundamental group ...... 52

5 Galois Categories and Morphisms of Proﬁnite Groups 69 5.1 ...... 69 5.2 ...... 70

6 Application of the Comparison Theorem.... 75 6.1 ...... 75 6.2 The Stein-factorisation ...... 77 6.3 The ﬁrst homotopy exact sequence ...... 82

7 The Technique of Descents and Applications 89 7.1 ...... 89 7.2 ...... 101 7.3 ...... 112

8 An Application of the Existence Theorem 117 8.1 The second homotopy exact sequence ...... 117

9 The Homomorphism of Specialisation..... 125 9.1 ...... 125 9.2 ...... 126 9.3 ...... 126

Appendix to Chapter IX 127 Chapter 1 Aﬃne Schemes

1.1 1 Let A be a commutative ring with 1 and S a multiplicatively closed a set in A, containing 1. We then form fractions , a A, s S ; two s ∈ ∈ a1 a2 fractions , are considered equal if there is an s3 S such that s1 s2 ∈ s3(a1 s2 a2 s1) = 0. When addition and multiplication are deﬁned in the − 1 obvious way, these fractions form a ring, denoted by S − A and called the ring of fractions of A with respect to the multiplicatively closed set S . There is a natural ring-homomorphism A S 1A given by a a/1. → − 7→ This induces a(1 1) correspondence between prime ideals of A not − 1 intersecting S and prime ideals of S − A, which is lattice-preserving. If 2 f A and S f is the multiplicatively closed set 1, f, f ,... , the ring of ∈ 1 { } = quotients S −f A is denoted by A f . If p is a prime ideal of A and S A p, 1 − the ring of quotients S − A is denoted by Ap; Ap is a local ring.

1.2

Let A be a commutative ring with 1 and X the set of prime ideals of A. For any E A, we deﬁne V(E) as the subset p : p a prime ideal E ⊂ { ⊃ } of X. Then the following properties are easily veriﬁed:

(i) V(Eα) = V( Eα) 2 α α T S 1 2 1. Aﬃne Schemes

(ii) V(E ) V(E ) = V(E E ) 1 ∪ 2 1 · 2 (iii) V(1) = ∅ (iv) V(0) = X.

Thus, the sets V(E) satisfy the axioms for closed sets in a topology on X. The topology thus deﬁned is called the Zariski topology on X; the topological space X is known as Spec A.

Note. Spec A is a generalisation of the classical notion of an aﬃne algebraic variety.

Suppose k is an algebraically closed field and let k[X1,..., Xn] = k[X] be the polynomial ring in n variables over k. Let a be an ideal of k[X] and V be the set in kn defined by V = (α ,...,α ) : f (α ,...,α ) { 1 n 1 n = 0 f a . Then V is said to be an affine algebraic variety and the ∀ ∈ } Hilbert’s zero theorem says that the elements of V are in (1 1) corre- − spondence with the maximal ideals of k[X]/a.

Remarks 1.3. (a) If a(E) is the ideal generated by E in A, then we have: V(E) = V(a(E))

(b) For f A, deﬁne X = X V( f ); then the X form a basis for the ∈ f − f Zariski topology on X. In fact, X V(E) = X f by (i). − f E S∈ 3 (c) X is not in general Hausdorﬀ; however, it is T0.

+ n (d) X f X fα an n Z such that f is in the ideal generated ⊂ α ⇐⇒ ∃ ∈ S by the fα′ s. For any ideal a of A, deﬁne √a = a A : an a for some + { ∈ ∈ n Z . √a is an ideal of A and we assert that √a = p : ∈ } ∩{ p a prime ideal a . To prove this, we assume, (as we may, by ⊃ } passing to A/a) that a = (0). Clearly, √(0) p. On the other ⊂ ∩ hand, if a A is such that an , 0 n Z+, then S 1A = A is a ∈ ∀ ∈ a− a non-zero ring and so contains a proper prime ideal; the lift p of this prime ideal in A is such that a < p. 1.4. The Sheaf associated to Spec A 3

It is thus seen that V(a) = V( √a) and V( f ) V(a) f √a. ⊃ ⇐⇒ ∈ Hence:

= X f X fα V( f ) V( fα) V( fα ) ⊂ [α ⇐⇒ ⊃ \α [α { }

and this proves (d).

(e) The open sets X f are quasi-compact.

In view of (b), it is enough to consider coverings by Xg′ s; thus, if r n = X f X fα , then by (d), we have f ai fαi (say); again by ⊂ α i=1 S r P = n (d), we obtain X f X f X fα . ⊂ i=1 i S (f) There is a (1 1) correspondence between closed sets of X and 4 − roots of ideals of A; in this correspondence, closed irreducible sets of X go to prime ideals of A and conversely. Every closed irreducible set of X is of the form (x) for some x X; such an x is ∈ called a generic point of that set, and is uniquely determined.

(g) If A is noetherian, Spec A is a noetherian space (i.e. satisﬁes the minimum condition for closed sets).

1.4 The Sheaf associated to Spec A

We shall define a presheaf of rings on Spec A. It is enough to define the presheaf on a basis for the topology on X, namely, on the X′f s; we n set F (X f ) = A f . If Xg X f , V(g) V( f ) and so g = a0 f for some ⊂ ⊃ · a aq n Z+ and a A. The homomorphism A A given by a q · 0 ∈ 0 ∈ f → g f 7→ gqn is independent of the way g is expressed in terms of f and thus defines f a natural map ρ : F (X ) F (X ) for X X . The transitivity g f → g g ⊂ f conditions are readily verified and we have a presheaf of rings on X. This defines a sheaf A = OX of rings on X. It is easy to check that the stalk O of O at a point p of X is the local ring A . p,X X e p 4 1. Affine Schemes

If M is an A-module and if we define the presheaf X M = f 7→ f M A , we get a sheaf M of A-modules (in short, an A-Module M), ⊗A f whose stalk at p X is M A = M . ∈ ⊗eA pe P e e 5 Remark. The presheaf X M is a sheaf, i.e. it satisfies the axioms f 7→ f (F1) and (F2) of Godement, Théorie des faisceaux, p. 109.

1.5

In this section, we brieﬂy recall certain sheaf-theoretic notions.

1.5.1 Let f : X Y be a continuous map of topological spaces. Sup- → pose F is a sheaf of abelian groups on X; we deﬁne a presheaf of abelian groups on Y by U Γ( f 1(U), F ) for any open U Y; for V U, 7→ − ⊂ ⊂ open in Y, the restriction maps of this presheaf will be the restriction homomorphisms Γ( f 1(U), F ) Γ( f 1(V), F ). This presheaf is already − → − a sheaf. The sheaf deﬁned by this presheaf is called the direct image f (F ) of F under f . ∗ If U is any neighbourhood of f (x) in Y, the natural homomorphism 1 Γ( f − (U), F ) Fx given a homomorphism Γ(U, f (F )) Fx; by → ∗ → passing to the inductive limit as “U shrinks down to f (x)”, we obtain a natural homomorphism:

fx : f (F ) Fx. ∗ −−−→f (x)

If OX is a sheaf of rings on X, f (OX) has a natural structure of a ∗ sheaf of rings on Y. If F is an OX-Module, f (F ) has a natural structure ∗ of an f (OX)-Module. ∗ The direct image f (F ) is a covariant functor on F . ∗

6 1.5.2 Let f : X Y be a continuous map of topological spaces and g → be a sheaf of abelian groups on Y. Then it can be shown that there is a unique sheaf F of abelian groups on X such that: (a) there is a natural homomorphism of sheaves of abelian groups

ρ = ρg : g f (F ) → ∗ 1.5. 5

and

(b) for any sheaf H of abelian groups on X, the homomorphism HomX(F , H ) HomY (g, f (H )) given by ρ f (ϕ) ρg is → ∗ 7→ ∗ ◦ an isomorphism. The unique sheaf F of abelian groups on X 1 with these properties is called the inverse image f − (g) of g under f .

It can be shown that canonical homomorphism

1 (*) f ρ : g f − (g) x ◦ f (x) f (x) → x is an isomorphism, for every x X. 1 ∈ The inverse image f − (g) is a covariant functor on g and the isomorphism ( ) shows that it is an exact functor. ∗ 1 If OY is a sheaf of rings on Y, f − (OY ) has a natural structure of a 1 sheaf of rings on X. If g is an OY -Module, f − (g) has a natural structure 1 of an f − (OY )-Module.

1.5.3 A ringed space is a pair (X, OX) where X is a topological space 7 and OX is a sheaf of rings on X, called the structure sheaf of (X, OX). A morphism Φ : (X, O ) (Y, O ) of ringed spaces is a pair ( f,ϕ) such X → Y that (i) f : X Y is a continuous map of topological spaces, and → (ii) ϕ : OY f (OX) is a morphism of sheaves of rings on Y. → ∗ Ringed spaces, with morphisms so defined, form a category. Ob- serve that condition (ii) is equivalent to giving a morphism f 1(O ) − Y → OX of sheaves of rings on X (see (1.5.2)). If F is a sheaf of OX-modules, we denote by Φ (F ) the sheaf ∗ f (F ), considered as an OY -Module through ϕ. If g is an OY -Module, ∗ 1 1 1 f − (g) is an f − (OY )-Module and the morphism f − (OY ) OX, de- 1 → fined by ϕ, gives an OX-Module f − (g) 1 ( ) OX; the stalks of this f − ⊗ OY O -Module are isomorphic to g O , under the identification X f (x) ⊗O f (x) x f 1(g) g . We denote this O -Module by Φ (g). In general, Φ is − x ≃ f (x) X ∗ ∗ not an exact functor on g. 6 1. Affine Schemes

1.6 Aﬃne Schemes

A ringed space of the form (Spec A, A), A a ring, deﬁned in (1.4) is called an aﬃne scheme. e

8 1.6.1 Let ϕ : B A be a ring-homomorphism; ϕ deﬁnes a map → f = aϕ : X = Spec A Spec B = Y → 1 p ϕ− (p). 7→ Since aϕ 1(V(E)) = V(ϕ(E)) for any E B, aϕ is a continuous map. − ⊂ Let s B; ϕ deﬁnes, in a natural way, a homomorphism ∈ ϕ : B Aϕ . s s → (s) In view of the remark at the end of (1.4), this gives us a homomorphism:

A Bs =Γ(Ys, B) ϕ(s) =Γ(Xϕ(s), A) =Γ(Ys, f (A)) → ∗ and hence a homomorphisme ϕ : B f (Ae). If x X thee stalk map → ∗ ∈ deﬁned by ϕ, namely e e e e ϕx : O f (x) B f (x) Ox Ax ≃ → ≃ is a local homomorphisme (i.e. the image of the maximal ideal in B f (x) is contained in the maximal ideal of Ax).

9 Definition 1.6.2. A morphism Φ : (Spec A, A) (Spec B, B) of two → affine schemes, is a morphism of ringed spaces with the additional prop- e e erty that Φ is of the form (aϕ, ϕ) for a homomorphism ϕ : B A of → rings. e It can be shown that a morphism Φ = ( f,ϕ) of ringed spaces is a morphism of affine schemes (spec A, A)(spec B, B) if and only if the stalk-maps e e O O defined by Φ(rather, by ϕ) f (x) → x are local homomorphisms. 1.6. Affine Schemes 7

Remarks 1.6.4. (a) If M is an A-module, M is an exact covariant functor on M. e

(b) For any A-modules M, N, HomA(M, N) is canonically isomorphic to Hom (M, N). A e e e

(c) If (X, OX) = (spec A, A), (Y, OY ) = (spec B, B) are affine, there is a natural bijection from the set Hom(X, Y) of morphisms of affine e e schemes X Y onto the set Hom(B, A) of ring-homomorphisms → B A. → (d) Let (X, OX) = (Spec A, A) be an affine scheme and F an OX- module. Then one can show that F is quasi-coherent (i.e. for e every x X, an open neighbourhood U of x and an exact se- ∈ ∃ (I) (J) quence (O U) (O U) F U 0) F M for an X| −−→ X| −−→ | → ⇔ ≃ A-module M. If we assume that A is noetherian, one sees that F e is coherent F is to M for a finite type A-module M. ⇔ ≃ Φ e (e) Let X Y be a morphism of affine schemes and F (resp. g) 10 −→ be a quasicoherent OX-Module (resp. OY -Module). Then one can define a quasi-coherent OY -Module (resp. OX-Module) denoted by Φ (F ) (resp. Φ∗(g)) just as in (1.5.3); if X = Spec A, Y = ∗ Spec B and Φ = (aϕ, ϕ) for a ϕ : B A and if F = M, M an A- → module (resp. g = N, N a B-module) then Φ (F ) (resp. Φ∗(g)) is ∗ e canonically identifiede with [ϕ]M, where [ϕ]M is the abelian group e M considered as a B-module through ϕ (resp. M]A). g ⊗B (For proofs see EGA Ch. I)

Chapter 2 Preschemes

Deﬁnition 2.1. A ringed space (X, OX) is called a prescheme if every 11 point x X has an open neighbourhood U such that (U, O U) is an ∈ X| aﬃne scheme.

An open set U such that (U, O U) is an aﬃne scheme is called an X| aﬃne open set of X; such sets form a basis for the topology on X.

Deﬁnition 2.1.1. A morphism Φ : (X, O ) (Y, O ) of preschemes X → Y is a morphism ( f,ϕ) of ringed spaces such that for every x X, the ∈ stalk-map ϕ : O O deﬁned by Φ is a local homomorphism. x f (x) → x Preschemes then form a category (Sch). In referring to a prescheme, we will often suppress the structure sheaf from notation and denote (X, OX) simply by X.

2.1.2 Suppose C is any category and S Ob C . We consider the pairs ∈ (T, f ) where T Ob C and f Hom (T, S ). ∈ · ∈ C If (T1, f1), (T2, f2) are two such pairs, we deﬁne Hom((T1, f1), (T , f )) to be the set of C -morphisms ϕ : T T , making the dia- 2 2 1 → 2 gram ϕ T1 / T2 @ @@ ~~ @@ ~~ f1 @ ~ f2 @ ~~ S

9 10 2. Preschemes

commutative. 12 This way we obtain a category, denoted by C S . In the special case | C = (Sch), the category (Sch /S ) = (Sch) S is called the category of S - | preschemes; its morphisms are called S -morphisms. S itself is known as the base prescheme of the category.

Remark 2.1.3. Let Spec A be an aﬃne scheme and Y any prescheme. Then Hom(A, Γ(Y, OY)) is naturally isomorphic to Hom(Y, Spec A).

In fact, let (U ) be an aﬃne open covering of Y and ϕ Hom i ∈ (A, Γ(Y, OY )). The composite maps

ϕ restriction ϕ : A Γ(Y, O ) Γ(U , O ) i −→ Y −−−−−−−→ i Y give morphisms aϕ : U Spec A, for every i, since the U are aﬃne. i i → i It is easily checked that aϕ = aϕ on U U , i, j. We then get i j i ∩ j ∀ a morphism aϕ : Y Spec A; the map ϕ aϕ is a bijection from → 7→ Hom(A, Γ(Y, OY)) onto Hom(Y, Spec A) (cf. (1.6.4)(c)). It follows that every prescheme X can be considered as a Spec Z- prescheme in a natural way:

(Sch) = (Sch / Spec Z) = (Sch /Z).

Remark 2.1.4. Let (X, OX) be a prescheme and F an OX-module. Then 13 it follows from (1.6.4)(d) that F is quasi-coherent for every x X ⇔ ∈ and any aﬃne open neighbourhood U of x, F U M , for a Γ(U, O ) | ≃ U X - module M . U e

We may take this as our deﬁnition of a quasi-coherent OX-module.

2.2 Product of Preschemes

2.2.0 Suppose (X, f ), (Y, g) are S -preschemes. We say that a triple (Z, p, q) is a product of X and Y over S if: (i) Z is an S -prescheme

(ii) p : Z X, q : Z Y are S -morphisms and → → 2.2. Product of Preschemes 11

(iii) for any T (Sch /S ), the natural map: ∈ Hom (T, Z) Hom (T, X) Hom (T, Y) S → S × S f (p. f, q. f ) 7→ is a bijection.

The product of X and Y, being a solution to a universal problem, is obviously unique upto an isomorphism in the category. We denote the product (Z, p, q), if it exists by X Y and call it the ﬁbre-product of X ×S and Y over S ; p, q are called projection morphisms.

Theorem 2.2.1. If X,Y (Sch /S ), the ﬁbre-product of X and Y over S ∈ always exists.

We shall not prove the theorem here. However, we observe that if 14 X = Spec A, Y = Spec B and S = Spec C are all affine, then Spec(A B) ⊗C is a solution for our problem. In the general case, local fibre-products are obtained from the affine case and are glued together in a suitable manner to yield a fibre product X Y. ×S (For details see EGA, Ch. I, Theorem (3.2.6)).

Remarks. (1) The underlying set of X Y is not the ﬁbre-product of ×S the underlying sets of X and Y over that of S . However if x X, ∈ y Y lie over the same s S , then there is a z X Y lying over ∈ ∈ ∈ ×S x and y. (For a proof see Lemma (2.3.1)).

(2) An open subset U of a prescheme X can be considered as a prescheme in a natural way. Suppose S S , U X, V Y are open ′ ⊂ ⊂ ⊂ sets such that f (U) S , g(V) S ; we may consider U, V as ⊂ ′ ⊂ ′ S ′-preschemes. When this is done, the ﬁbre-product U V is iso- S×′ 1 1 morphic to the open set p− (U) q− (V) in Z = X Y, considered ∩ ×S as a prescheme.

This follows easily from the universal property of the ﬁbre-product. 12 2. Preschemes

2.2.2 Change of base: Let X, S ′ be S -preschemes. Then the ﬁbe- product X S ′ can be considered as an S ′-prescheme in a natural way: ×S

X S ′ X o ×S

S o S ′

15 When this is done, we say that X S ′ is obtained from X by the base- ×S change S S and denote it by X . Note that, in the aﬃne case, this ′ → (S ′) corresponds to the extension of scalars. If X is any prescheme, by its reduction mod p, p Z+, (resp. ∈ mod p2 and so on) we mean the base-change corresponding to Z → Z/ (resp. Z Z 2 and so on). (p) → /(p ) If f : X X ., g : Y Y are S -morphisms f and g deﬁne, in → ′ → ′ a natural way an S -morphism: X Y X′ Y′, which we denote by ×S → ×S f g or by ( f, g)S . When g = IS : S ′ S ′, we get a morphism: ×S ′ → f IS = f(S ) : X(S ) X′ . ×S ′ ′ ′ → (S ′)

2.3 Fibres

Let (X, f ) be an S -prescheme and s S be any point. Let U S be ∈ ⊂ an aﬃne open neighbourhood of s and A =Γ(U, OS ). If ps is the prime

ideal of A corresponding to s, Os,S is identified with Aps . Denote by k(s) the residue field of O , = A . The composite A A k(s) defines s S ps → ps → a morphism Spec k(s) Spec A = U S ; i.e. to say, Spec k(s) is an S - → ⊂ prescheme in a natural way. Consider now the base-change Spec k(s) → S : p X′ = X Spec k(s) X o ×S f q S o Spec k(s)

16 The ﬁrst projection p clearly maps X into the set f 1(s) S . We ′ − ⊂ 2.3. Fibres 13

1 1 claim that p(X′) = f − (s) and further that, when we provide f − (s) with the topology induced from X, p is a homomorphism between X′ and 1 f − (s). To prove this, it suffices to show that for every open set U in a covering of X, p is a homeomorphism from p 1(U) onto U f 1(s). In view − ∩ − of the remark (2) after Theorem (2.2.1) we may then assume that X, S 1 are affine say X = Spec A, S = Spec C. That p(X′) = f − (s) will follow as a corollary to the following more general result. Lemma 2.3.1. Let X = Spec A,Y = Spec B be affine schemes over S = Spec C. Suppose that x X, y Y lie over the same element s S. ∈ ∈ ∈ Then the set E of elements z Z = X Y lying over x, y is isomorphic to ∈ ×S Spec(k(x) k(y)) (as a set). k(⊗S ) Proof. One has a homomorphism ψ : A B k(x) k(y) which gives ⊗C → k⊗(s) a morphism aψ : Spec k(x) k(y) Z; clearly the image of aψ is k⊗(s) → 1 1 contained in the set E = p (x) q (y). That aψ is injective is seen − ∩ − by factoring ψ as follows: A B Ax By k(x) k(y). In order ⊗C → C⊗s → k⊗(s) to see that aψ is surjective one remarks that for z E the homomor- ∈ phism A A B k(z) factors through k(x), similarly for B, therefore → ⊗C → we have for A B k(z) a factorisation A B k(x) k(y) k(z). ⊗C → ⊗C → k⊗(s) → Q.E.D.

In the above lemma if we take B = k(s) it follows that in the diagram 17

X′ = X Spec k(s) = Spec(A k(s)) ×S ⊗C qq MMM qqq MMM qqq MMM qqp q MM qqq MMM qqq MMM xqq M& X = Spec A Y = Spec B NN p NNN ppp NNN f g ppp NNN ppp NNN ppp NNN ppp NN' wppp s S = Spec C ∈ 14 2. Preschemes

the map p : X f 1(s) is a bijection. ′ → −

2.3.2 Returning to the assertion that X′ is homeomorphic to the fibre f 1(s) (with the induced topology from X) we note that if ϕ : C k(s) − → is the natural map, then p : X X is the morphism corresponding to ′ → 1 ϕ : A A k(s). To show that p carries the topology over, it is ⊗ → ⊗C enough to show that any closed set of X′, of the form V(E′), is also of the form V((1 ϕ)E) for some E A. ⊗ ⊂ ci Now, any element of A k(s) can be written in the form ai = ⊗C i ⊗ t ! 1 P1 18 (ai ci) 1 with ai A, ci, t C. Since 1 is a unit i ⊗ ! · ⊗ t ! ∈ ∈ ⊗ t ! ofPA k(s), we can take for E A, the set of elements aici where ⊗C ⊂ i P ai ci/t is an element of E′. Q.E.D. i ⊗ P 1 Note. The fibre f − (s) can be given a prescheme structure through this 1 homeomorphism p : X′ f − (s). If, in the above proof, we had taken n+1 →= Os/Ms , instead of k(s) Os/Ms we would still have obtained home- n+1 1 omorphisms pn : X Spec Os/Ms f − (s). The prescheme struc- ×S → 1 th ture on f − (s) defined by means of pn, is known as the n -infinitesimal neighbourhood of the fibre.

2.4 Subschemes

2.4.0 Let X be a prescheme and J a quasi-coherent sheaf of ideals of OX. Then the support Y of the OX-Module OX/J is closed in X and (Y, O / Y) has a natural structure of a prescheme. In fact, the question X J | is purely local and we may assume X = Spec A. Then J is defined by an ideal I of A and Y corresponds to V(I) which is surely closed. The ringed space (Y, O / Y) has then a natural structure of an affine X J | scheme, namely, that of Spec(A/I). Such a prescheme is called a closed subscheme of X. An open subscheme is, by definition, the prescheme induced by X on an open subset 2.4. Subschemes 15 in a natural way. A subscheme of X is a closed subscheme of an open subscheme of X.

2.4.1 A subscheme may have the same base-space as X. For example, 19 one can show that there is a quasi-coherent sheaf N of ideals of OX such that Nx = nil-radical of Ox. N deﬁnes a closed subscheme Y of X, which we denote by Xred and which is reduced, in the sense that the stalks Oy,Y of Y have no nilpotent elements. X and Y have the same base space (A and Ared have the same prime ideals). Consider a morphism f : X Y of preschemes. Suppose ϕ : → X X X, ϕ : Y Y are the natural morphisms. Then a morphism red → Y red → ∃ f : X Y making the diagram red red → red f X / Y O O

ϕX ϕY

Xred / Yred fred commutative. This corresponds to the fact that a homomorphism ϕ : A B of rings deﬁnes a homomorphism ϕ : A B such that → red red → red ϕ A / B

ηA ηB

ϕred Ared / Bred

2.4.2 A morphism f : Z X is called an immersion if it admits a 20 f j → factorization Z ′ Y X where Y is a sub-scheme of X, j : Y X −→ −→ → is the canonical inclusion and f : Z Y is an isomorphism. The ′ → immersion f is said to be closed (resp. open) if Y is a closed subscheme (resp. open subscheme) of X. 16 2. Preschemes

f Example . Let X S be an S -prescheme. Then, there is a natural −→ S -morphism ∆ : X X X such that the diagram → ×S X I =id lll X Xlll lll lll X ull `B ∆ BB BB I p1 B Ö X X X ×S B f BB p2 BB B X lll lll lll lll f S ull is commutative. ∆ is called the diagonal of f . It is an immersion.

Deﬁnition 2.4.2.1. A morphism f : X S is said to be separated (or X → is said to be an S -scheme) if the diagonal ∆ : X X X of f is a closed → ×S immersion.

A prescheme X is called a scheme if the natural map X Spec Z is → separated.

Remark . Let Y be an affine scheme, X any prescheme and (Uα)α an 21 affine open c over for X. One can then show that a morphism f : X Y → is separated if and only if α, β, ∀ (i) Uα Uβ is also affine ∩ (ii) Γ(Uα Uβ, ) is generated as a ring by the canonical images of ∩ OX Γ(Uα, OX) and Γ(Uβ, OX). (For a proof see EGA, Ch. I, Proposition (5.5.6)).

2.4.3 Example of a prescheme which is not a scheme. Let B = k[X], C = k[Y] be polynomial rings over a field k. Then Spec BX and Spec CY are affine open sets of Spec B and Spec C respectively; the isomorphism 2.5. Some formal properties of morphisms 17 f (X) f (Y) of B onto C defines an isomorphism Spec C ∼ Xm 7→ Ym X Y Y −→ Spec BX. By recollement of Spec B and Spec C through this isomorphism, one gets a prescheme S , which is not a scheme; in fact, condition (ii) of the proceding remark does not hold: for, Γ(Spec B, O ) S ≃ B = k[X] and Γ(Spec C, O ) C = k[Y]; the canonical maps from these S ≃ into Γ(Spec B Spec C, O ) k[u, u 1] are given by X u, Y u and ∩ S ≃ − 7→ 7→ the image in each case is precisely = k[u].

2.5 Some formal properties of morphisms

(i) every immersion is separated (ii) f : X Y, g : Y Z separated g f : X Z separated. → → ⇒ ◦ → (iii) f : X Y a separated S -morphism f : X Y is → ⇒ (S ′) (S ′) → (S ′) separated for every base-change S S . ′ → (iv) f : X Y, f ′ : X′ Y′ are separated S -morphisms f f ′ : 22 → → ⇒ ×S X X′ Y Y′ is separated. ×S → ×S (v) g f separated f is separated ◦ ⇒ (vi) f separated f separated. ⇔ red The above properties are not all independent. In fact, the following more general situation holds: Let P be a property of morphisms of preschemes. Consider the following propositions: (i) every closed immersion has P (ii) f : X Y has P, g : Y Z has P g f has p → → ⇒ ◦ (iii) f : X Y is an S -morphism having P f : X Y → ⇒ (S ′) (S ′) → (S ′) has P for any base-change S S . ′ → (iv) f : X Y has P, f ′ : X′ Y′ has P that f f ′ : X X′ → → ⇒ ×S ×S → Y Y′ has P ×S 18 2. Preschemes

(v) g f has P, g separated f has P. ◦ ⇒ (vi) f has P f has P. ⇒ red If we suppose that (i) and (ii) hold then (iii) (iv). Also, (v), (vi) ⇔ are consequences of (i), (ii) and (iii) (or (iv)).

Proof. Assume (ii) and (iii). The morphism f f ′ admits a factorization: ×S

f f ′ ×S Y Y′ X X′ / S × EE × EE z= EE zz EE zz f IX EE zz IY f ′ ×S ′ " z ×S Y X′ ×S

23 By (iii), the morphisms f TX and IY f ′ have P and so by (ii) f f ′ ×S ′ ×S ×S also has P.

On the other hand, assume (i) and (iv). IS ′ being a closed immersion, has P by (i) and so f(S ) = f IS has P by (iv). ′ ×S ′ ∆ Now assume (i), (ii) and (iii). If g : Y Z is separated, Y Y Y → −→ ×Z f is a closed immersion and has P by (i); by making a base-change X Y ∆Y IX −→ we get a morphism X Y X × Y Y X X Y which, by (iii) has ≃ ×Y −−−−−→ ×Z ×Y ≃ ×Z property p. The projection p2 : X Y Y satisﬁes the diagram: ×Z →

ϕ X Y X / ×Z

g f p2 ◦

g Z o Y

i.e. to say, p is obtained from g f by the base-change Y Z and so, 2 ◦ → by (iii) has P. 2.6. Aﬃne morphisms 19

∆Y IX Finally, f : X Y is the composite of X × X Y and p2 : → −−−−−→ ×Z X Y Y and so by (ii) has P. ×Z → To prove (vi) from (i), (ii), (iii) use the diagram 24

fred Xred / Yred

ϕX ϕY ' f X / Y and the facts that the canonical morphisms ϕX, ϕY are closed immersions and so have P, that ϕ f = f g has P and that a closed immersion Y ◦ red ◦ is separated, then use (v). Q.E.D. We now remark that if we replace (i) of the above propositions by (i) every immersion has P, then (i), (ii), (iii) imply (v) g f has P, ′ ′ ◦ g has P f has P. ⇒

2.6 Aﬃne morphisms

Definition 2.6.1. A morphism f : X S of preschemes is said to be → affine (or X affine over S ) if, for every affine open U S , f 1(U) is ⊂ − affine in X.

It is enough to check that for an aﬃne open cover (Uα) of S , the 1 f − (Uα) are aﬃne.

2.6.2 Suppose that B is a quasi-coherent OS -Algebra. Let (Uα) be an affine open cover of S ; set Aα = Γ(Uα, OS ), Bα = Γ(Uα, B) and Xα = Spec Bα. The homomorphism Aα Bα defines a morphism 25 → fα : Xα Uα; the Xα′ s then patch up together to give an S -prescheme f → X S ; this prescheme X is affine over S , is such that fα(O ) B, −→ X ≃ and is determined, by this property, uniquely upto an isomorphism. We denote it by Spec B. conversely, every affine S -prescheme is obtained 20 2. Preschemes

as Spec B, for some quasi-coherent OS -Algebra B. (For details, see EGA Ch. II, Proposition (1.4.3)). Remarks. (a) Any aﬃne morphism is separated. (Recall the remark at the end of (2.4.2).)

(b) If S is an affine scheme, a morphism f : X S is affine X is → ⇔ an affine scheme.

h (d) Suppose X Y is an S -morphism. If f , g are the structural mor- −→ phisms of X, Y resply, the homomorphism OY h (OX) deﬁned → ∗ by h, given an OS -morphism

A (h) : g (OY ) g (h (OX)) = f (OX); ∗ → ∗ ∗ ∗

then we have a natural map: HomS (X, Y) HomOS (g (OY ) → ∗ → f (OX)) (the latter in the sense of OS -Algebras) defined by h ∗ 7→ A (h). If Y is affine over S , it can be shown that this natural map is a bijection. (EGA Ch II, Proposition (1.2.7)). (Also, compare with remark (1.6.4) (c) for affine schemes, and with remark (2.1.3)).

2.7 The finiteness theorem 26 Definition 2.7.0. A morphism f : X Y of preschemes is said to be of → 1 finite type, if, for every affine open set U of Y, f − (U) can be written as n 1 f − (U) = Vα, with each Vα affine open in X and each Γ(Vα, OX) a α=1 finite type ΓS(U, OY )-algebra.

It is again enough to check this for an affine open cover of Y. An affine morphism f : X Y is of finite type the quasi-coherent → ⇐⇒ OY -Algebra fα(OX) is an OY -Algebra of finite type. In particular, a morphism f : Spec B Spec A is of finite type B is a finite type → ⇐⇒ A-algebra (i.e. is finitely generated as A-algebra). 2.7. The finiteness theorem 21

Deﬁnition 2.7.1. A morphism f : X S is universally closed if, for → every base-change S ′ S , the morphism f(S ) : X S ′ S ′ is a closed → ′ ×S → map in the topological sense.

Deﬁnition 2.7.2. A morphism f is proper if

(i) f is separated

(ii) f is of ﬁnite type and

(iii) f is universally closed.

The formal properties of (2.5) hold then P is the property of being proper.

Definition 2.7.3. A prescheme Y is locally noetherian, if every y Y ∈ has an affine open neighbourhood Spec B, with B noetherian. It is said n to be noetherian, if it can be written as Y = Yi where the Yi are affine 27 i=1 open sets such that the Γ(Yi, OY ) are noetherianS rings.

If f : X Y is a morphism of ﬁnite type and Y is locally noetherian, → then X is also locally noetherian.

2.7.4 Let (X, OX) and (Y, OY) be ringed spaces and f a morphism from + X to Y. Let F be an O -Module. We then define, for every q Z , a X ∈ presheaf of modules on Y by defining: U Hq( f 1(U), F ) for every 7→ − open U Y (See EGA Ch 0, III 12). The sheaf that this presheaf de- ⊂ § fines on Y is called the qth-direct image of F and is denoted by Rq f (F ). ∗ Theorem 2.7.5. Let X, Y be preschemes, Y locally noetherian, and f a proper morphism from X to Y. Then, if F is any coherent OX-Module, q the direct images R f (F ) are all coherent OY -Modules. ∗ (For a proof see EGA Ch. III, Theorem (3.2.1)). This is the theorem of finiteness for proper morphisms.

Chapter 3 Etale´ Morphisms and Etale´ Coverings

Throughout this chapter, by a prescheme we will mean a locally noethe- 28 rian prescheme and by a morphism, a morphism of finite type (unless it is clear from the context that the morphism is not of finite type, e.g., S Spec O , , Spec A Spec A, A a noetherian local ring). ← s S ← Definition 3.1.0. A morphism f b: X S is said to be unramified at a → point x X if (i) M O = M , (ii) k(x)/k( f (x)) is a finite separable ∈ f (x) x x extension.

Definition 3.1.1. A morphism f : X S is said to be flat at a point x → ∈ X if the local homomorphism O O is flat (i.e., O , considered as f (x) → x x an O -module is flat; note that since the homomorphism O O f (x) f (x) → x is local, Ox will be faithfully O f (x)-flat).

Definition 3.1.2. A morphism f : X S is said to be étale at a point → x X if it is both unramified and flat at x. ∈ We say that f : X S is unramified (resp. flat, étale) if it is unram- → ified (resp. flat, étale) at every x X. ∈ f Remarks 3.1.3. (1) An unramified morphism X S is étale at x −→ ∈ X O O is flat. ⇔ f (x) → x b b 23 24 3. Etale´ Morphisms and Etale´ Coverings

29 (2) A morphism f : X S is unramiﬁed at →

x X fSpec k( f (x)) : X Spec k( f (x)) Spec k( f (x)) ∈ ⇐⇒ ×S → is unramified at the corresponding point; in other words, it is enough to look at the fibre for nonramification.

֒ (3) If f : X S is unramiﬁed at x X and k( f (x))=→k(x) then → ∈ O O is surjective; if f is ´etale in addition, then O ∼ f (x) → x f (x) −→ Obx. b b b 3.2 Examples and Comments

(1) A morphism f : Spec A Spec k (k a field) is étale it is r → ⇔ unramified A = K , K / finite separable extensions. ⇔ i i k Li=1 (2) Let X and S be irreducible algebraic varieties, S normal. Then it can be shown that a dominant morphism X S is étale = it is → ⇒ unramified.

(3) If S is non-normal, an unramified dominant morphism X S → need not be étale; for instance, let c be an irreducible curve over an algebraically closed filed with an ordinary double point x, c its normalisation and p : c c the natural map. → e (i) p is unramified.eOne has to prove this only at the points a, b c sitting over x. Now M is generated in O by ∈ a a a function with a simple zero at a, but such a function we e 30 can find already in Ox, for instance a function induced by a straight line through x not tangent to c at x. (ii) p is not étale: otherwise, O ∼ O but we know that O is x −→ a x not a domain, while O is. a b b b (4) A connected étale varietyb over an irreducible algebraic variety need not be irreducible. 3.3. 25

For instance, in the previous example we may take two copies c and c′ of the normalisation of c and fuse them together in such a e way that the points a, b on c are identified with the points b′, a′ e on c′. We then get a connected but reducible variety X and the morphism p : X c definede in the obvious manner is surely → étale.e

(5) Let X and S be irreducible algebraic varieties and suppose that 31 S is normal and f : X S a dominant morphism. Assume, in → addition, that the function field R(X) of X is a finite extension of degree n of the function field R(S ) of S . If x X is such that the 1 ∈ number of points in the fibre f − ( f (x)) equals n, then it can be shown that f is étale in a neighbourhood of x.

3.3

Our aim now is to give a necessary and suﬃcient condition for a morphism to be unramiﬁed.

ϕ 3.3.0 Some algebraic preliminaries. Let A B be a homomorphism −→ of rings deﬁning an A-algebra structure on B. Then B B is an A-algebra ⊗A 26 3. Etale´ Morphisms and Etale´ Coverings

in a natural way and

p1 : B B B given by b b 1 → ⊗A 7→ ⊗ p2 : B B B given by b 1 b → ⊗A 7→ ⊗

and µ : B B B given by b1 b2 b1b2 are all A-algebra homo- ⊗A → ⊗ 7→ morphisms. We may make B B a B-algebra through p1 i.e. by defining ⊗A b(b b ) = bb b and the kernel I of µ is then a B-module. Since we 1 ⊗ 2 1 ⊗ 2 have µ p = Id . and the equality b b = (b b 1) b (b 1 1 b ) ◦ 1 1 ⊗ 2 1 2 ⊗ − 1 2 ⊗ − ⊗ 2 is follows that B B = (B 1) I, as a B-module. ⊗A ⊗A ⊕ 32 We define the space ΩB/A of A-differentials of B as the B-module 2 2 I/I . We note that the B-module structure on ΩB/A = I/I is natural, in the sense that it does not depend on whether we make B B, a B-algebra ⊗A through p1 or through p2.

Some properties of ΩB/A

2 (1) The map d : b (1 b b 1) (mod I ) from B to Ω / is 7→ ⊗ − ⊗ B A A-linear. Also, since

(1 b b b b 1) = b (1 b b 1) + b (1 b b 1) ⊗ 1 2 − 1 2 ⊗ 1 ⊗ 2 − 2 ⊗ 2 ⊗ 1 − 1 ⊗ =+(1 b b 1)(1 b b 1), ⊗ 1 − 1 ⊗ ⊗ 2 − 2 ⊗ we have: d(b b ) = b db + b db b , b B. 1 2 1 2 2 1 ∀ 1 2 ∈ (2) Since I is generated as a B-module by elements of the form (1 ⊗ b b 1), it follows that Ω / is generated by the elements db. 1 − 1 ⊗ B A (3) Let W be the B-submodule of B B, generated by elements of the ⊗A form (1 bb′ b b′ b′ b); ΩB/A is then isomorphic to (B B)/W. ⊗ − ⊗ − ⊗ ⊗A In fact,

2 2 ΩB/A = I/I (B 1 I)/(B 1 I ) ⊗A ⊕ ⊗A ⊕ 3.3. 27

= (B B)/(B 1 I2) ⊗A ⊗A ⊕ 33 and clearly W B 1; so W = B 1 (W I). ⊃ ⊗A ⊗ ⊕ ∩ On the other hand, we have

(1 bb′ b b′ b′ b) = bb′ 1+(1 b b 1)(1 b′ b′ 1) ⊗ − ⊗ − ⊗ − ⊗ ⊗ − ⊗ ⊗ − ⊗ and hence: W B 1 I2. Also clearly, I2 W. It follows that ⊂ ×A ⊕ ⊂ 2 2 W I I , W = B 1 I and ΩB/A (B B)/W. ∩ − ⊗A ⊕ ⊗A (4) If V is any B-module and D : B V any A-derivation of B in V, → then there is a B-linear map H : Ω / V such that D B A → D B / V -- H -- -- -- -- d -- HD -- $ -- -- - ΩB/A

In fact, D defines an A-linear map B B V given by b1 b2 ⊗A → ⊗ 7→ b1Db2. This is certainly B-linear and is trivial on W. We then obtain a B-linear map H : Ω / V such that H (db) = Db b D B A → D ∀ ∈ B. Also the correspondence D H is a B-isomorphism from 7→ D the B-module of A-derivations B V, onto the B-module Hom → B (ΩB/A, V). (This is the universal property of ΩB/A). In particular, the B-module of A-derivations of B is isomorphic to HomB (ΩB/A, B); and even more in particular, if A = k, B = K are fields, 34 we see that the space of k-differentials ΩK/k is the dual of the space of k-derivations of K and therefore is trivial if K/k is separably algebraic. (5) Consider a base-change A A and the extension of scalars B = → ′ ′ B A′. From the universal property of the space of differentials, it ⊗A 28 3. Etale´ Morphisms and Etale´ Coverings

follows that ΩB /A ΩB/A B′ΩB/A A′. ′ ′ ≃ ⊗B ⊗A

3.3.1 The sheaf of differentials of a morphism. Let f : X S be a morphism. Our aim now is to define a sheaf → ΩX/S on X. Assume, to start with, that X = Spec B, S = Spec A; then ΩB/A is a B-module in a natural way and we define ΩX/S = ΩB/A on X. In the general case, consider the diagonal ∆ of f in X X. Then ∆ ×S e is a closed subscheme of an open subscheme U of X X and X ∆ is ×S → an isomorphism. Let IX be the sheaf of ideals of OU defining ∆. Then I 2 is a sheaf of OU -modules whose support is contained in ∆. By X/IX means of the isomorphism X ∆, we can lift this sheaf to a sheaf of → abelian groups on X. Clearly the lift is independent of the choice of U. It remains to show that this sheaf is an OX-Module. To do this, we can 35 assume again that X S are affine and in this case, I is the sheaf I on × X X X (where I is the ideal of B B defined in 3.3.0) and therefore, our ×S ⊗A e sheaf is ΩB/A considered above. The O -Module thus obtained will be called the sheaf of differen- e X tials of f (or of X over S ) and will be denoted ΩX/S . The stalk Ωx,X/S of this sheaf at a point x X is canonically isomorphic to Ω / . This is ∈ Ox O f (x) seen either from the universal property or by observing that the natural map Ω , / Ω / given by x X S → Ox O f (x) bi bi db′ d(b′ ) (with X = Spec B, bi, b′, si B, si < p ) s i 7→ s i/1 i ∈ x Xi i Xi i is an isomorphism. Since, by assumptions, S is locally noetherian and f is of finite type it is cler from the definition that ΩX/S is coherent. Proposition 3.3.2. For a morphism f : X S and a point x X the → ∈ following are equivalent: (i) f is unramified at x

(ii) Ωx,X/S = (0) 3.3. 29

(iii) ∆ : X X X is an open immersion in a neighbourhood of x. → ×S Proof. We may assume X = Spec B, S = Spec A. 36 (i) (ii): Consider the base change: ⇒ = x X o X′ XS Spec k(s) ∈ ×S

s = f (x) S o Spec k(s) = S ∈ ′ Ω = Ω If x′ X′ is above x, then we have: x,X/S Ox′ x′,X′/S ′ ; this ∈ O⊗x = = follows from property (5) of 3.3.0; furthermore,Ox′ Ox/Ms Ox k(x) Ω = Ω since f is unramified at x. Therefore x,X/S k(x) x′,X′/S ′ and by O⊗x Nakayama it suffices to show that the latter is (0). By the remark made Ω = Ω above, x′,X′/S ′ k(x)/k(s) and this is zero as k(x)/k(s) is separably algebraic. (ii) (iii) ⇒ Let z be the image of x in X X under the diagonal ∆ : X X X ×S → ×S · ∆(X) is a closed subscheme of an open subscheme U of X X, and is ×S defined, therefore, by a sheaf I of OU -ideals. If Ωx,X/S = (0), then, we = 2 = = will have Iz Iz ...; hence Iz (0) (Krull’s intersection theorem). 37 However, I is a sheaf of ideals of finite type and so I vanishes in a neighbourhood of z i.e. to say ∆ : X X X is an open immersion in a → ×S neighbourhood of x X. ∈ (iii) (i) ⇒ The question being local, we may assume that ∆ : X X X is → ×S an open immersion everywhere. An open immersion remains an open immersion under base-change, and since we only have to look at the fibre over f (x), for non-ramification at x, we may take X = Spec A, S = Spec k, k a field and A a k-algebra of finite type. We are to show that A = Ki, where Ki/k are finite separable field finiteL extensions; for this, we have to show that A is artinian and, if k is the 30 3. Etale´ Morphisms and Etale´ Coverings

algebraic closure of k, A k is radical-free. It is thus enough to show that ⊗k A k = k. By making the base-change k k, we may assume k is ⊗k → ﬁniteL algebraically closed. Let a X be any closed point of X. Since k is algebraically closed, ∈ k(a) k and we have then ≃ X ∼ X Spec k(a) = X (a) (say). ←−i Spec× k × X (a) is canonically imbedded in X X × ×S and let ϕ be the composite of the morphisms:

i 1 .X − X (a) ֒ X X −−→ × → ×S 1 38 Then ϕ− (∆) = (a) is open in X, since, by assumption, ∆ is open in X X; i.e. any closed point of X is also open. But X = Spec A is quasi- ×S compact and this implies that A has only ﬁnitely many maximal ideals. But A is a k-algebra of ﬁnite type and so the set of closed points of X n˙ is dense in X. It follows that A is artinian and we may write A = Ai Li=1 where the Ai are artinian local rings. We may then assume A = Ai; the open immersion ∆ : X X X then gives an isomorphism A A A. → ×S ⊗k → This however means A = k. Q.E.D.

3.3.3 Some properties of etale´ morphisms. (1) An open immersion is ´etale.

(2) f , g étale g f étale. ⇒ ◦ (3) f étale f étale for any base change S S ⇒ (S ′) ′ → (not necessarily of finite type), S ′, S locally noetherian. This follows from condition (iii) of Proposition 3.3.2, the facts that an open immersion remains an open immersion and that a flat morphism remains flat under base change. 3.3. 31

(4) f1, f2 ´etale f1 f2 ´etale. ⇒ ×S Follows from (2), (3) and the equality

f1 f2 = ( f1 IY ) (IX f2). ×S ×S 2 ◦ 1 ×S

(5) g f étale, g unramified f étale. 39 ◦ ⇒ Let f : X Y and g : Y Z. The question being local, we may → → assume from (iii) Proposition 3.3.2 that ∆ = ∆(Y) : Y Y Y is → ×Z f an open immersion, hence étale. By the base-change X Y we −→ get ∆(X) : X X Y which again is étale by (3). The diagram: → ×Z X Y X o ×Z

g f p2 ◦

Z o g Y

shows that the second projection p2 : X Y Y is given by ×Z → (g f )(Y); so, again by (3), p2 is also étale and from (2) we obtain ◦ ∆(X) that f = p2 ∆(X) : X X Y Y is étale. ◦ −−−→ ×Z −−→p2 (6) g f étale, f étale g étale. ◦ ⇒ The composite: O O O is flat and O O g f (x) ⇒ f (x) → x f (x) → x is faithfully flat, and thus O O is flat. It is clear that g f (x) → f (x) k( f (x))/k(g f (x)) is a finite separable extension. Finally M f (x)Ox = 40 · Mx and Mg f (x)Ox = Mx = (Mg f (x)O f (x))Ox; since O f (x) Ox · · → is faithfully flat, it follows that Mg f (x)O f (x) = M f (x). · (7) An étale morphism is an open map. In fact, we prove more generally: Proposition 3.3.4. A flat morphism is an open map. 32 3. Etale´ Morphisms and Etale´ Coverings

The proposition will follow as a consequence from the lemmas below. We ﬁrst make the

Deﬁnition. A subset E of a noetherian topological space X is said to be constructible if E is a ﬁnite union of locally closed sets in X.

Lemma 3.3.5. Let X be a noetherian topological space and E any set in X. Then, E is constructible for every irreducible closed set Y of X, ⇔ E Y is either non-dense in Y or contains an open set of Y. ∩ n Proof. : Suppose E = (Oi Fi), Oi open, Fi closed in X. Then ⇒ i=1 ∩ n S E Y Y (Fi Y) for any closed Y X; if Y is irreducible with ∩ ∩ ⊂ i=1 ∩ ⊂ E Y dense inSY, this means Y F Y for some i, i.e., F Y; then ∩ ⊂ i ∩ i ⊃ Y E (O Y) open in Y. ∩ ⊃ i ∩ 41 : We shall prove this by noetherian induction. Let E be the set of ⇐ all closed sets F in X such that E F is not constructible. If E , ∩ ∅ choose a minimal F E . By replacing X by F , we may assume that 0 ∈ 0 for every closed set F properly contained in X, E F is constructible. ∩ If X is reducible, say X = X X , X , X both proper subsets of X 1 ∪ 2 1 2 and closed, then E X , E X are both constructible and hence so is ∩ 1 ∩ 2 E = (E X ) (E X ). If X is irreducible, either ∩ 1 ∪ ∩ 2 (i) E , X and so E = E E is constructible or ∩ (ii) E U , , U open, so that E = U (E (X U)) is still ⊃ ∅ ∪ ∩ − constructible. This contradiction shows that E = . Q.E.D. ∅

Lemma 3.3.6. Let S be a noetherian prescheme and f : X S a → morphism. Then the image, under f , of any constructible set is constructible.

Proof. Using the preceding lemma, the fact that S is noetherian and by passing to subschemes, irreducible components and so on, we are readily reduced to proving the following assertion: if X, S are both aﬃne, 3.3. 33

reduced, irreducible and noetherian and if f : X S is a morphism of → ﬁnite type such that f (X) = S , then f (X) contains an open subset of S . 42 If X = Spec B, S = Spec A, our assumptions mean that A, B are noetherian integral domains and the ring-homomorphism A B deﬁn- → ing f is then easily checked to be an injection. The lemma will then follow from the following purely algebraic result.

Lemma 3.3.7. Let A be an integral domain and B = A[x1,..., xn] an integral A-algebra containing A. Then there exists a g A such that for ∈ every prime ideal p of A with g < p, there is a prime ideal P of B such that P A = p. ∩

Proof. Choose a transcendence base X1,..., Xk of B/A. Then the extension B = A[x1,..., xn] is algebraic over A′ = A[X1,..., Xk]. By writing down the minimal polynomials of the xi over A′ and by dividing out these polynomials by a suitably chosen g A, g , 0, we can make the 1 ∈ xi integral over A′ g . Consider now the tower of extensions. h i 1 B g h i

1 A′ g h i

1 A g h i

If p is a prime ideal of A such that g < p then there is a prime p′ of 1 + 1 A g lying over p. The prime ideal p′ (X1,..., Xk) of A′ g lies over p.h Nowi 1 is a ﬁnite extension of A 1 and by Cohen-Seidenberg,h i g ′ g ∃ h i 1 h i a prime ideal P′ of B g lying over p. The restriction P of P′ to B then sits over p.h i Q.E.D. 34 3. Etale´ Morphisms and Etale´ Coverings

Lemma 3.3.8. Let f : X S be a morphism (of ﬁnite type) and T X 43 → ⊂ be an open neighbourhood of a point x X. Assume that for every ∈ s S such that f (x) (s ), there is a t T such that f (t ) = s . Then 1 ∈ ∈ 1 1 ∈ 1 1 f (T) is a neighbourhood of f (x). Proof. Since the question is local, we may assume S noetherian and then by lemma 3.3.6 it follows that f (T) is constructible. We may then write f (T) = (Oi Fi); and by choosing only those i, for which i=1 ∩ f (x) O , we mayS say that f (T) is closed in a neighbourhood of f (x). ∈ i The hypotheses show that f (T) is also dense in a neighbourhood of f (x). It follows that f (T) is itself a neighbourhood of f (x). Q.E.D.

Proof of Proposition 3.3.4. In view of the lemma 3.3.8, it suﬃces to show that if U X is ⊂ an open neighbourhood of x X, f (U) contains all generisations of ∈ f (x). The generisations of f (x) are the points of Spec O f (x) and those of x are points of Spec Ox. But Ox is O f (x)-faithfully ﬂat and then it is well-known for any prime ideal p of O f (x), there is a prime-ideal P of O contracting to p. U being open we have Spec O U. Hence x x ⊂ Spec O f (U) (cf. Bourbaki, Alg. Comm. Ch. II, 2, n 5, Cor. 4 f (x) ⊂ § ◦ to Proposition 11). Q.E.D.

3.4

44 A morphism f : S S is said to be an eﬀective epimorphism if the ′ → sequence p1 f S ′′ = S ′ S ′ / / S ′ / S is exact ×S p2

p1∗ / i.e., if the sequence Hom(S, Y) / Hom(S ′, Y) / Hom(S ′′, Y) is p2∗ exact, as a sequence of sets, Y. ∀ h2 h1 / (We say that a sequence of sets E1 / E2 / E3 is exact if h1 h2′ is an injection and h (E ) = x E : h (x) = h (x) ). 1 1 { ∈ 2 2 2′ } 3.4. 35

A morphism f : S S is faithfully flat if it is flat and surjec- ′ → tive. Our aim now is to show that any such morphism is an effective epimorphism.

3.4.1 Some algebraic preliminaries; the Amitsur complex. A homomorphism of rings f : A A deﬁnes a sequence: → ′

p21 / f p1 p32 / / A′ A′ = A′′ A′ A′ A′ = A′′′ / A / A′ / / / p2 ⊗A / ⊗A ⊗A / p31 where p (a ) = a 1; p (a ) = 1 a , p (a b ) = a b 1; 1 ′ ′ ⊗ 2 ′ ⊗ ′ 21 ′ ⊗ ′ ′ ⊗ ′ ⊗ p (a b ) = a 1 b ; p (a b ) = 1 a b and so on. 31 ′ ⊗ ′ ′ ⊗ ⊗ ′ 32 ′ ⊗ ′ ⊗ ′ ⊗ ′ We may then deﬁne homomorphisms of A-modules: 45

∂ = p p 0 1 − 2 ∂ = p p + p 1 21 − 31 32 ∂ = p p + p p ... 2 321 − 421 431 − 432 and so on. One then checks that ∂ + ∂ = 0 i; we thus get an augmented i 1 i ∀ cochain complex:

∂0 ∂1 ∂2 ∂3 A A′ A′′ A′′′ A′′′′ .... −→f −−→ −−→ −−→ −−→

This complex is called the Amitsur complex A′/A.

f Lemma 3.4.1.1. If A A is faithfully ﬂat, then −→ ′ (i) A ∼ H0(A /A) −→ ′ (ii) Hq(A /A) = (0) q > 0. ′ ∀ Proof. Suppose B is any faithfully ﬂat A-algebra. Consider then the complex

1 ∂0 B′/B B (A′/A) B B′ = B A′ ⊗ B′′ = B′ B′ = B A′′ ≡ ⊗A ≡ −−−→1 f ⊗A −−−−→ ⊗B ⊗A → ⊗ 36 3. Etale´ Morphisms and Etale´ Coverings

0 Since B is A-faithfully ﬂat, it is enough to prove that H (B′/B) ∼ B 46 q ←− and H (B′/B) = (0) q > 0. As a particular choice we may take B = A′. ∀ f ′=1 f Then the homomorphism A′ ⊗ A′ A′ = A′′ admits a section, i.e., −−−−−−→ ⊗A σ : A A such that σ f = 1 , namely, the homomorphism: ∃ ′′ → ′ ◦ ′ A′ a b a b . ′ ⊗ ′ 7→ ′ ′ We may thus assume, without any loss of generality that the homo- f morphism A A admits a section σ : A A such that σ f = 1 : −→ ′ ′ → ◦ A f ∂0 ∂1 ∂2 . A / A′ / A′′ / A′′′ ... / ] σ

We construct now a homotopy operator in this complex, as follows:

p21 f p1 / / / A′ A′ = A′′ / A′ A′ A′ = A′′′ / A / A′ / p32 / 9 p2 ⊗A ⊗A ⊗A / 9 p / 99 31 99 σ 1 σ 1 1 σ 1A 9 ⊗ ⊗ ⊗ 99 9 f p1 ∼ ∼ / A / A′ A A′ // A′ A′ A A′′ / ⊗A −→ p2 ⊗A ⊗A −→ /

(i) H0(A /A) = ker ∂ ∼ A. ′ 0 ←− 47 Let a A be such that p (a ) = p (a ). Applying 1 σ we get ′ ∈ ′ 1 ′ 2 ′ ⊗ (1 σ)(a′ 1) = (1 σ)(1 a′) i.e. a′ 1 = 1 σ(a′) in A′ A. ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗A Under the canonical identiﬁcation A′ A ∼ A′, this means that ⊗A −→

a′ = f (σ(a′)) i.e. a′ f (A). ∈ On the other hand, f (A) ker ∂ and f being faithfully ﬂat, is ⊂ 0 injective. It follows that ker ∂ ∼ A. 0 ←− q (ii) By using the homotopy operator, one can show that H (A′/A) = (0) q > 0; the proof is omitted. (We do not need it). ∀ 3.4. 37

3.4.2

Proposition 3.4.2.1. A faithfully ﬂat morphism is also an eﬀective epimorphism.

Proof. Case (a). S′ = Spec A′, S = Spec A are affine. ϕ From local algebra, it follows that the ring homomorphism A A −→ ′ defining f : S S , is also faithfully flat. We have to show that, for ′ → any Y, the sequence

(*) Hom(S, Y) Hom(S ′, Y) ⇉ Hom(S ′ S ′, Y) is exact in ns . → ×S E

(i) Suppose Y = Spec B is aﬃne.

Then the sequence (*) is equivalent to a sequence: 48

( )′ Hom(B, Z) Hom(B, A′) ⇉ Hom(B, A′ A′). ∗ → ⊗A

The exactness of this sequence ( ) now follows from assertion (i) ∗ ′ of lemma 3.4.1.1.

(ii) Let Y be arbitrary. Let ϕ : S Y, ϕ : S Y be two morphisms such that ϕ f = 1 → 2 → 1 ◦ ϕ f . Since f is a surjection, it is clear that ϕ (s) = ϕ (s) s S . 2 ◦ 1 2 ∀ ∈ Choose a point s S and a point s S with f (s ) = s; let ∈ ′ ∈ ′ ′ y = ϕ (s) = ϕ (s) Y. Choose an aﬃne open neighbourhood 1 2 ∈ Spec B of y Y and an element θ A such that s Spec Aθ and ∈ ∈ ∈ ϕ (Spec Aθ) Spec B, ϕ (Spec Aθ) Spec B. 1 ⊂ 2 ⊂ 1 Set θ = ϕ(θ) A′. Then s′ Spec A′ , and f (Spec A′ ) ∈ ∈ θ′ θ′ ⊆ Spec Aθ; and by (i) it follows that ϕ1 and ϕ2, when restricted to Spec Aθ, deﬁne the same morphism of preschemes. Since s S ∈ was arbitrary, this proves that Hom(S, Y) Hom(S , Y) is injec- → ′ tive.

Now, suppose that ϕ : S Y is a morphism such that in 49 ′ ′ → 38 3. Etale´ Morphisms and Etale´ Coverings

p1 f S ′′ = S ′ S ′ / 1 S / S / S × p2 > >> >> ϕ′ >> > Y. We have ϕ p = ϕ p . We want to find a morphism ψ : S Y ′ ◦ 2 ′ ◦ 1 → such that ϕ = ψ f . ′ ◦ In view of the injectivity established above, it is enough to define ψ locally. Let s S and s′ S ′ with f (s′) = s. Choose an affine open ∈ ∈ 1 neighbourhood V of ϕ′(s′) in Y. Then the open neighbourhood ϕ′− (V) 1 of s in S is saturated under f : in fact, let x ϕ − (V) and x S ′ ′ 1 ∈ ′ 2 ∈ ′ such that f (x1) = f (x2). Then, there is an s′′ S ′′ = S ′ S ′ such that ∈ ×S p1(s′′) = x1 and p2(s′′) = x2; so

1 ϕ′(x ) = ϕ′ p (s′′) = ϕ′ p (s′′) = ϕ′(x ) and x ϕ′− (V). 1 ◦ 1 ◦ 2 2 2 ∈ Now, f is ﬂat and hence an open map (Prop. (3.3.4)) and so 1 f (ϕ′− (V)) is an open neighbourhood of s S . Choose an element 1 ∈ 1 θ A such that s Spec Aθ f (ϕ′− (V)); if θ = ϕ(θ) A′ then ∈ = ∈ ⊂ = 1 1∈ f (Spec Aθ′ ) Spec Aθ, and Spec Aθ′ f − (Spec Aθ) ϕ′− (V) since 1 ′ ′ ⊂ ϕ′− (V) is saturated under f . 50 We then have a diagram

Spec(Aθ′ Aθ′ ) ′ A×θ ′

p1 p2

Spec A′ θ′ vv vv vv vv f vv ϕ′ vv vv {vv V Spec Aθ = f (Spec A′ ) θ′

The problem of deﬁning ψ : Spec Aθ V is now a purely aﬃne → problem and we are back to (i). 3.5. Etale´ coverings 39

Case (b). The general case. Without loss of generality, we may assume S affine. Since f is a ffi n morphism of finite type, there is a finite a ne open covering (S α′ )α=1 = n ffi of S ′. Consider the disjoint union S ∗ α=1 S α′ . S ∗ is a ne and the morphism S ∗ S defined in the obvious` way is again faithfuly flat. → Let Y be an arbitrary prescheme and F be the (contravariant) functor X Hom(X, Y). We have a commutative diagram: 7→

F(S ) / F(S ′) // F(S ′′) O

identity

F(S ) / F(S ∗ S ∗) / F(S ∗) / ×S

The lower sequence is exact by case (a); the ﬁrst vertical map is the 51 identity and the second vertical map is clearly injective. Usual diagram - chasing shows that the upper sequence is also exact. Q.E.D.

3.5 Etale´ coverings

Definition . A morphism of preschemes, f : X S , is said to be fi- 1 → nite if, for every affine open U S , f − (U) is also affine and the ring 1 ⊂ Γ( f − (U), OX) is a Γ(U, OS )-module of finite type.

It is again enough to check the conditions for an aﬃne open cover of S .

(1) If S is locally noetherian and f : X S is a finite morphism, → f (OX) is a coherent OS -Module. ∗ (2) A finite morphism remains finite under a base-change. In particular, if f : X S is finite and s S any point, the → ∈ morphism fSpec k(s) : X k(s) k(s) is finite and this means that ×S → 1 the fibre f − (s) is finite, discrete. 40 3. Etale´ Morphisms and Etale´ Coverings

(3) A finite morphism is proper. In face, a finite morphism is affine and is hence separated; it remains finite under any base-change and so it is enough to show that a finite morphism is closed.

52 By obvious reductions, we may take X, S aﬃne reduced and f (X) = S . If X = Spec B and S = Spec A, the corresponding homomorphism A B is an injection and B is a ﬁnite A-module. → Cohen-Seidenberg then shows that f (X) = S .

(4) The following result holds:

Lemma 3.5.1 (Chevalley). Let S be (as always) locally noetherian and f : X S a morphism of preschemes. Then the following conditions → are equivalent:

(a) f is ﬁnite

(b) f is proper and aﬃne

Definition. A morphism f : X S is said to be an étale covering if it → is both étale and finite.

f Let X S be an étale covering. Then f (OX) is a locally free OS - −→ ∗ Algebra of finite rank. For any s S , the fibre f (OX) k(s) is a finite ∈ ∗ ⊗S ns direct sum Ki of finite separable field extensions Ki of k(s). The rank i=1 P ns of f (OX) at s S is then given by [Ki : k(s)] which equals the ∗ ∈ i=1 P 1 number of geometric points in the fibre f − (s). This is constant in each 53 connected component of S ; if S is connected, this constant rank is called the degree or the rank of the covering f . In this case, if this rank equals 1, then f is an isomorphism. 3.5. Etale´ coverings 41

Note . For ´etale coverings we have properties similar to (2), (3) (with S S not necessarily of ﬁnite type) (4), (5) and (6) from 3.3.3; this ′ → follows immediately from 3.3.3 and properties of coverings.

A word of caution: This concept of an “étale covering” (French: revêtementetale) ´ should not be confused with the concept of a “covering in the étale topology” (French: famille couvrante). The latter concept is not treated in this course. We also note that an étale covering, as defined here, is not necessarily surjective if S is not connected.

Chapter 4 The Fundamental Group

Throughout this chapter, we shall denote by S a locally noetherian con- 54 nected prescheme and by C = (E t/S ) the category of ´etale coverings of S . We note that the morphisms of C will all be ´etale coverings. (See 3.3.3 and the note at the end of Ch. 3).

4.1 Properties of the category C

(C ) C has an initial object (the empty prescheme) and a ﬁnal object 0 ∅ S .

(C ) Finite ﬁbre-products exist in C , i.e., if X Z and Y Z are 1 → → morphisms in C , then X Y exists in C (see (3.3.3)) ×Z (C ) If X, Y C , then the disjoint union X Y C (obvious). 2 ∈ ∈ ` (C ) Any morphism u : X Y in C admits a factorisation of the form 3 → u X / Y @@ ? @@ @@ ui @@ j @ @ / Y1

where u1 is an eﬀective epimorphism, j is a monomorphism and Y = Y Y , Y C . 1 2 2 ∈ ` 43 44 4. The Fundamental Group

55 In fact, u is an ´etale covering so u is both open and closed; if we write u(X) = Y we have Y = Y Y and u = u : X Y is then an 1 1 2 1 → 1 eﬀective epimorphism ` (see (3.4.2.1)). Further, this factorisation of u into an epimorphism and a monomorphism is essentially unique in the sense that if

u X / Y @ ? @@ @@ @@ u′ @ j′ 1 @ @ / Y1

is another such factorisation, then there exists an isomorphism ω : Y 1 → Y such that u = ω u and j = j ω. 1′ 1′ ◦ 1 ′ ◦ (Because a factorisation into a product of an eﬀective epimorphism and a monomorphism is unique).

(C ) If X C and g is a ﬁnite group of automorphisms of X acting, 4 ∈ say, to the right on X, then the quotient X/g of X by g exists in C η and the natural morphism X X/g is an eﬀective epimorphism. −→

The quotient, if it exists, is evidently unique upto a canonical isomorphism; also X is affine over S and therefore the existence of X/g has only to be proved in the case X, S affine, say X = Spec A, S = Spec B and G = the group of B-automorphisms of A corresponding to g. Then Spec AG (AG is the ring of G-invariants of A) is the quotient we are looking for. (Compare with Serre, Groupes algébriques et corps de classes, p. 57). Our aim now is to show that X/g is actually in C . The ques- 56 tion is again local and we may assume X = Spec A, S = Spec B, with B noetherian, and X/g = Spec AG as above. X S is finite and so → X/g S is also finite. It remains to show that this morphism is étale. → In order to do this, we first make some simplifications. Suppose that S S is a flat affine base-change. We have a com- ′ → 4.1. Properties of the category C 45 mutative diagram:

= = X′ X S ′ Y′ Y S ′ Spec B = S ×S / ×S / ′ ′

Spec A = X / X/g = Spec AG = Y / Spec B = S. g acts on X′ in the obvious way, as a group of S ′-automorphisms of X′. We assert that Y′ = Y S ′ is the quotient of X′ with respect to this action. ×S Indeed, we have an exact sequence of B-algebras: 0 AG A → σ → → A, where A A is the map given by A (a a ). Since B′ σ G → σ G 7→ σ G − L∈ L∈ G P∈ is B-flat, we get an exact sequence: 0 A B′ A B′ (A B′) → ⊗B → ⊗B → σ G ⊗B G L∈ and this proves that the subring of invariants of A B′ is A B′; hence ⊗B ⊗B our assertion. Let y Y = X/g and s S be its image. Take for B the local ring ∈ ∈ ′ Os,S . Then there is a unique point y′ Y′ = Y S ′ over y and one has ∈ ×S O , = O , ; hence Y S is étale at Y Y S is étale at y . We 57 y′ Y′ y Y → ⇔ ′ → ′ ′ may thus assume that S = Spec B, B a noetherian, local ring. In view of the following lemma, we may assume B complete.

f ϕ Lemma 4.1.1. Let X S be a morphism and S S a faithfully flat −→ ′ −→ base-change. Then f is étale f is étale. ⇔ (S ′) Proof. : is clear. ⇒ : flatness of f is straightforward. To prove non-ramification one ⇐ Ω = Ω observes that in view of (5) (3.3.0), one has X′/S ′ ϕ∗( X/S ); but ϕ being faithfully flat, Ω / = 0 Ω / = 0; one now applies proposi- X′ S ′ ⇔ X S tion 3.3.2. Q.E.D.

Let x1,..., xn be the points of X over s. By hypothesis each k(xi)/ k(s) is a finite separable extension. We choose a sufficiently large finite galois extension K of k(s) such that each k(xi) is imbedded in K. We now need the 46 4. The Fundamental Group

Lemma 4.1.2. Let B be a noetherian local ring with maximal ideal M and residue field k. Let K be an extension field of k. Then a noetherian ∃ local ring C and a local homomorphism ϕ : B C such that (i) ϕ is → B-flat and (ii) C/M C K. (see EGA. Ch. 0III, Prop. (10.3.1)).

58 In addition if [K : k] < , we can choose C to be a finite B-algebra. ∞ (EGA. Ch. 0III, Cor. (10.3.2)). ϕ By making such a base-change B C we may assume that each −→ r k(xi) is trivial over k(s). Under these assumptions we get A = B,afi- Li=1 nite direct sum of copies of B. Under the action of g, the set x ,..., x { 1 n} splits into disjoint subsets x ,..., x , x + ,..., x ,..., on each of { 1 l} { l 1 m} which g acts transitively. The corresponding decomposition of A will l m then be given by A = ( B) ( B) .... The action of G on each i=1 i=l+1 L L L Ll block, for instance, on a (b1,..., bl) B, will then be just a permu- ∈ i=1 G L tation. The subring A will then be the direct sum ∆1 ... ∆α ... l ⊕ ⊕ ⊕ where ∆1 is the diagonal of the block B and so on. Each ∆ is ev- Li=1 idently isomorphic to B. Our assertion that X/g S is étale is now → clear. η Thus X/g C ; the natural morphism X X/g is also then an étale ∈ −→ covering. Therefore η will be an open map and thus if η is not surjective one could replace Y by the image of η; this is clearly impossible. Hence η is surjective and thus an effective epimorphism 3.4.2.1.

4.2

59 We shall now define a covariant functor F from C (E t/S ) to the category of finite sets. We shall fix once and for all a point s S and an ∈ algebraically closed field Ω k(s). ⊃ For any X C , F(X), by definition, will be the set of geomet- ∈ ric points of X over s S , with values in Ω, i.e., is the set of all S - ∈ 4.2. 47 morphisms Spec Ω X for which the diagram is commutative. → Spec Ω / X

Spec k(s) / S

We observe that if x X sits above s S , then giving an S - ∈ ∈ morphism Spec Ω X whose image is x X is equivalent to giving a → ∈ k(s)-monomorphism of k(x) into Ω. Also note that for any X C , F(X) ∈ is a ﬁnite set whose cardinality equals the rank of X over S . Properties of the functor F.

(F ) F(X) = X = . 0 ∅⇔ ∅

(F1) F(S ) = a set with one element; F(X Y) = F(X) F(Y), X, Y, Z C . ×Z F×(Z) ∀ ∈

(F2) F(X1 X2) = F(X1) F(X2).

`u ` (F ) If X Y is an eﬀective epimorphism in C , the map F(u) : 60 3 −→ F(X) F(Y) is onto. → In fact, if y Y is a point above s S and y = u(x), x ∈ ∈ ∈ X, then any k(s)-monomorphism of k(y) to Ω extends to a k(s)- monomorphism of k(x) to Ω.

(F ) Let X C and g a ﬁnite group of S -automorphisms of X (acting 4 ∈ to the right on X). Then g acts in a natural way (again to the right) on F(X) as expressed by

σ g Spec Ω X ∈ X. → −−−→

The natural map η : X X/g (see (C )) deﬁnes a surjection → 3 F(η) : F(X) F(X/g) (see (F )). In view of the commutativity → 3 48 4. The Fundamental Group

of the diagram

σ g Spec Ω / X ∈ / X A AA η }} AA }} η AA }} A ~}} X/g

it follows that F(η) descends to a surjection η : F(X)/g F(X/g). → We claim that η is actually a bijection. This follows immediately from the e e Lemma 4.2.1. (i) g acts transitively on the ﬁbres of η.

61 (ii) Suppose y Y = X/g and x η 1(y). Let g (x) be the subgroup ∈ ∈ − d σ g : σ(x) = x (called the decomposition group of x). Then { ∈ } we have:

(a) k(x)/k(y) is a galois extension. (b) the natural map g (x) the galois group G(k(x)/k(y)) is d → onto.

Proof. (i) We may assume X = Spec A, Y = X/g = Spec AG as before (both noetherian); also one knows that A is ﬁnite on AG. Let P, P1 be prime ideals of A (i.e., points of X) such that P1 , σP σ G, while P AG = P AG = p. We may assume ∀ ∈ ∩ 1 ∩ P and P maximal (otherwise apply the ﬂat base change Y 1 ← Spec O , , where y Y corresponds to p). Then there is an a y Y ∈ ∈ P1 such that a < σP σ (Chinese Remainder Theorem). Thus ∀ G G G b = σ(a) < P; but b A , so b A P1 = A P- σ ∈ ∈ ∩ ∩ contradiction.Q

(ii) We have a diagram:

A / A/P k(x) O ≃O

G / AG/p k(y), A ≃ 4.3. 49

62 and we know that k(x)/k(y) is a ﬁnite, separable extension. Let θ A be such that k(x) = k(y)(θ). The polynomial f = (T σθ) ∈ σ − is in AG[T], has θ as a root and splits completely in AQ[T]. The reduction f of f mod p is in k(y)[T], has θ as a root and splits completely in k(x)[T]. It follows that k(x)/k(y) is normal, hence galois.

Consider now the subgroup Gd(P) of G corresponding to gd(x). We have P , σ 1P σ < G (P) and by the Chinese Remain- − ∀ d der Theorem we can choose a θ A such that θ θ(modP) 1 ∈ 1 ≡ and θ 0(modσ 1P), σ < G (P). 1 ≡ − ∀ d We have k(x) = k(y)(θ1). Consider now the polynomial g = (T σθ1) k(y)[T]. As θ1 is a root of g, for every ϕ σ − ∈ ∈ Q k(x) G( /k(y)), ϕ(θ1) is also a root of g; hence ϕ(θ1) = σ(θ1) for some σ G. But ϕ(θ ) , 0 and, by the choice of θ , σ(θ ) = 0 ∈ 1 1 1 if σ < G (P); hence ϕ(θ ) = σ(θ ) for some σ G (P), i.e., d 1 1 ∈ d ϕ = σ for some σ Gd(P). Q.E.D. ∈

(F ) If u : X Y is a morphism in C such that F(u) : F(X) F(Y) 5 → → is a bijection, then u is an isomorphism

From the fact that F(u) is a bijection it follows (see the remark at 63 the end of Ch. 3) that the rank of u : X Y is 1 at every y Y, hence → ∈ (again by the same remark) u is an isomorphism. A category which has the properties (C0),..., (C4) of 4.1 and from which there is given a functor F into ﬁnite sets with the above properties (F0),..., (F5) is called a galois category; the functor F itself is known as a fundamental functor.

4.3

Before we start our construction of the fundamental group of a galois category we motivate our procedure by two examples. 50 4. The Fundamental Group

Example 1. Let S be a connected, locally arcwise connected, locally simply connected topological space and C the category of connected p coverings of S ; the morphisms of C are covering maps. Let X S be −→ such a covering. Fix a point s S ; we deﬁne F(X) = p 1(s). Then ∈ − (X, p) p 1(s) is a covariant functor F : C ns. 7→ − →E Each member of C determines (upto conjugacy) a sub-group of the fundamental group Π1(S, s); and to each subgroup H of Π1(S, s) there corresponds a member of C determining H. To the subgroup e corre- { } sponds, what is known as, the universal covering S of S ; and Π1(S, s) is isomorphic to the group of S -automorphisms of S , i.e., to the group of e covering transformations of S over S . Further we have the isomorphism e (in ns): E e Hom (S , X) ∼ F(X), X C . C −→ ∀ ∈ 64 The functor F : C ns is thus representable in the following →e E sense.

Definition 4.3.1. A covariant functor G from a category C to ns is E representable if an object Y C such that: ∃ ∈ Hom (Y, X) ∼ G (X) X C . C −→ ∀ ∈ Example 2. Let k be a field and Ω a fixed algebraically closed field extension of k. Set S = Spec k and C = the category of connected étale coverings of S ; any member of C is of the form Spec K, where K/k is a finite separable field extension. For any X C we define F(X) = the set ∈ of geometric points of X with values in Ω. Then, F(X) Hom (K, Ω ) ≃ k s is X = Spec K, where Ωs is the separable closure of k in Ω. If Ωs is finite over k, we can further write F(X) Hom (Spec Ω , X) and the functor ≃ C s F : C (finite sets), defined above, will be representable. However → this is not the case in general; out we can find an indexed, filtered family (Ni)i I of finite galois extensions of k, (namely, the set of finite galois ∈ extensions contained in Ω ) such that for any X C , we can find an s ∈ i = i (X) such that F(X) Hom (Spec N , X), i i (X). In other 0 0 ≃ C i ∀ ≥ 0 words, we may write F(X) lim Hom (Spec N , X), X C ≃ C i ∀ ∈ i I −−→∈ 4.3. 51

and the family (Spec Ni)i I is in fact a projective family of objects in C . 65 ∈ Suppose now that C is any category and G : C ns is a covariant →E functor. If X C and ξ G (X), we write, as a matter of notation, ξ ∈ξ ∈η u G X. If G X, and G Y and X Y is a C -morphism, we say −→ −→ −→ −→ that the diagram ξ G / X ?? η ?? ?? u ? Y is commutative if G (u)(ξ) = η. ξ If G X, then for any Z C , we have a natural map Hom (X, Z) −→ ∈ C G (Z) deﬁned by u G (u)(ξ). → 7→ Deﬁnition 4.3.2. We say that G is pro-representable if a projective ∃ system (S i,ϕi j)i I of objects of C and elements τi G (S i) (called the ∈ ∈ canonical elements of G (S i)) such that (i) the diagrams τi G / S i ? ? ?? ~~ ?? ~~ τ j ? ~ ϕij( j i) ? ~~ ≥ S j are commutative.

(ii) for any Z C , the (natural) map 66 ∈ lim Hom (S , Z) F(Z) C i → i I −−→∈ is bijective.

In addition, if the ϕi j are epimorphisms of C , we say that G is strictly pro-representable. Thus, our functor F in Example 2 is pro-representable. Example 1 and 2 show that representable and pro-representable functors arise naturally in the consideration of the fundamental group. 52 4. The Fundamental Group

4.4 Construction of the Fundamental group

4.4.1 Main theorem

(1) Let C be a galois category with a fundamental functor F. Then there exists a pro-finite group π (i.e., a group π which is a projective limit of finite discrete groups provided with the limit topology) such that F is an equivalence between C and the category C (π) of finite sets on which π acts continuously.

F (2) If C ′ C (π ) is another such equivalence, then π is continu- −−→ ′ ′ ously isomorphic to π and this isomorphism between π and π′ is canonically determined upto an inner automorphism of π.

67 The proﬁnite group π, whose existence is envisaged in assertion (1) above will be called the fundamental group of the galois category C . The theorem is a consequence of the following series of lemmas.

Deﬁnition 4.4.1.1. A category C is artinian if any “decreasing” sequence ... ֓ T ֓ T ֓ T 1 ← j1 2 ← j2 3 ← j3 of monomorphisms in C is stationary, i.e., the jr are isomorphisms for large r.

A (covariant) functor F : C ns is left-exact if it commutes with →E ﬁnite products i.e., if F(X Y) = F(X) F(Y) and if, for every exact × × u u1 sequence X / Y / Z in C , the sequence u2

F(u) F(u1) F(X) / F(Y) // F(Z) F(u2)

is exact as a sequence of sets. u1 If Y // Z are morphisms in C , a kernel for u1, u2 in C is a pair u2 u u1 (X, u) with X C and u : X Y in C such that X / Y / Z ∈ → u2 4.4. Construction of the Fundamental group 53 is exact in C . Clearly a kernal is determined uniquely upto an isomorphism in C .

Lemma 4.4.1.2. Let C be a category in which finite products exists. 68 Then finite fibre-products exist in C kernels exist in C . ⇔ u1 Proof. : Let Y // Z be morphisms in C . We have a commutative ⇒ u2 diagram: Z pp8 dII u1 pp II u2 pp II ppp II ppp II Y p Y eLLL x; LL xx LLL xx p1 LL xxp2 LL xx Y Y Z : × uu CC uu C (p1,p2) uu CC uu C uu CC (Y Y) Y C! ×Z (Y×Y) Y Y × x< × KKK xx KK xx KKK xxdiagonal KKK xx % Y x and it easily follows that (Y Y Y is a solution for the kernal of u1 ×Z (Y×Y) × and u2. f g : Suppose X Z and Y Z are morphisms in . If p and q are ⇐ −→ −→p q the canonical projections X Y X, X Y Y, we have an exact × −→ × −→ sequence: fp ker( f p, gq) / X Y // Z. × gq It follows that ker( f p, gq) is a solution for the fibre-product X Y. 69 ×Z Q.E.D.

In fact, we have shown that finite fibre products and kernels can be expressed in terms of each other. Hence, F commutes with finite fibre- products it is left-exact. ⇔ 54 4. The Fundamental Group

Corollary . A fundamental functor is left-exact (see (F1))

Lemma 4.4.1.3. A galois category is artinian.

Proof. Let

... T1 ֓ j1 T2 ֓ j2 ... ֓ jr 1 Tr ֓ jr Tr+1 ← ← ← − ← be a decreasing sequence of monomorphisms in C . We have then:

jr T + T is a monomorphism r 1 −→ r Tr+1 ∼ Tr+1 Tr+1 ⇔ −→∆ T×r

F(Tr+1) ∼ F(Tr+1) F(Tr+1) (by (F1), (F5)) ⇒ −→∆ F(×Tr )

F( jr ) F(T + ) F(T ) is a monomorphism. ⇔ r 1 −−−−→ r

Since the F(Tr) are ﬁnite, this implies that the F( jr) are isomorphisms for large r; we are through by (F5). Q.E.D.

70 Lemma 4.4.1.4. Let C be a galois category with a fundamental functor F. Then F is strictly pro-representable.

Proof. With the notations of 4.3.2, consider the set E of pairs (X,ξ) with ξ F X. We order E as follows: −→

(X,ξ) (X′,ξ′) a commutative diagram: ≥ ⇔ ∃

ξ F / X @ @@ ~~ @@ ~~ ξ @@ ~~ ′ @ ~ ~ X′

We claim that E is ﬁltered for this ordering; in fact, if (X,ξ), (X′,ξ′) 4.4. Construction of the Fundamental group 55

E , in view of (F ) we get a commutative diagram: ∈ 1

m6 X mmmz= mmm zz ξ mmm zz mmm zz mmm zzp mmm zz mmm zz mmm(ξ,ξ′) z F m / X X QQQ D′ QQQ × DD QQQ DD QQQ ξ′ D p′ QQQ DD QQQ DD QQQ DD QQQ DD QQ(! X′ where p and p′ are the natural projections. We say that a pair (X,ξ) E is minimal in E if for any commutative 71 ∈ diagram ξ F / X ?? ? ?? ?? η ?? j ? ? / Y with a monomorphism j, one necessarily has that j is an isomorphism.

(*) Every pair in E is dominated, in this ordering, by a minimal pair in E . Observe that C is artinian (Lemma 4.4.1.3).

(**) If (X,ξ) E is minimal and (Y, η) E then a u Hom (X, Y) in ∈ ∈ ∈ C a commutative diagram

ξ F / X ? ?? ?? η ?? u ?? ? Y

is uniquely determined. 56 4. The Fundamental Group

In fact, if u , u Hom (X, Y) such that the diagrams 72 1 2 ∈ C η η F / Y F / Y ?? ? ?? ? ?? and ?? ξ ?? u1 ξ ?? u2 ? ? X X

are commutative then by (C1) and Lemma 4.4.1.2 ker(u1, u2) exists; since F is left exact we get a commutative diagram

η F G / Y GG O O GG GG ξ GG u1 u2 ξ GG GG GG G# ker(u , u ) / X 1 2 j

with a monomorphism j. As(X,ξ) is minimal j must be an isomorphism, i.e., u1 = u2. From (*), (**) it follows that the system I of minimal pairs of E is directed. If (X,ξ) I, (Y, η) E and u Hom (Y, X) appears in a commuta- ∈ ∈ ∈ C tive diagram ξ F / X ? ? ?? ?? η ?? u ?? ? Y

73 then u must be an eﬀective epimorphism. In fact, be (C3) we get a factorisation

u Y / X1 X2 = X ?? t: ?? tt` ? tt ?? tt u1 ?? tt j ?? tt , tt X1 4.4. Construction of the Fundamental group 57

with an eﬀective epimorphism u1 and a monomorphism j. By(F2) and (F3) we then obtain a commutative diagram:

ξ F / X = X1 X2 . OO 9 . OOO ξ r9 . OO rrr ` .. OOO rr . OO rrr j . OO' + rr η . X .. 1 . ~? .. ~~ . ~~u1 . ~~ Y

By minimality of (X,ξ), it follows that j is an isomorphism; thus u is an eﬀective epimorphism. In particular:

*** The structure morphisms occurring in the projective family I are eﬀective epimorphisms.

Consider now the natural map

lim Hom (S , X) F(X), X C . C i → ∈ i I −−→∈ By (*) this is onto; by (**) it is injective. From (***) it thus follows 74 that F is strictly pro-representable. Q.E.D.

Deﬁnition 4.4.1.5. Let C be a category with zero ( ) in which disjoint ∅ unions exist in C . An X C is connected in C X , X X in C ∈ ⇔ 1 2 with X1, X2 , . ` ∅ Note. In (E t/S ), a prescheme is connected it is connected as a topo- ⇔ logical space.

With the notations of the preceding lemma, we have:

Lemma 4.4.1.6. (i) (X,ξ) E is minimal X is connected in C . ∈ ⇔ (ii) If X is connected in C , then any u Hom (X, X) is an automor- ∈ C phism. 58 4. The Fundamental Group

(iii) For any X C , Aut X acts on F(X) as follows: ∈ ξ σ Aut X F X ∈ X. It X C is connected, then for any ξ F(X) −→ −−−−−−→ ∈ ∈ the map Aut X F(X) deﬁned by u F(u)(ξ) = u ξ, is an → 7→ ◦ injection.

Proof. (i) Suppose X = X1 X2 in C , X1, X2 , and that (X,ξ) = ∅ ∈ E ; then ξ F(X) F(X1`) F(X2) say, ξ F(X1). ∈ ∈ We then have a commutative` diagram

ξ F / X @@ ~? @@ ~~ @@ ~~ ξ @@ ~~ j @ ~ @ / ~~ X1

75 with a monomorphism j which is not an isomorphism. Thus (X,ξ) is not minimal. On the other hand, let X C be connected and (X,ξ) E . Sup- ∈ ∈ pose we have a commutative diagram:

ξ F / X ?? ? ?? ?? η ?? j ? ? / Y

with a monomorphism j. By(C3) we get a factorisation:

j Y / X = X1 X2 ?? t: ?? tt ` ? tt ?? tt j1 ?? tt j2 ?? tt , tt X1

with an eﬀective epimorphism j1 and a monomorphism j2. As j is a monomorphism, so is j1 and thus j1 is an isomorphism; since X is connected, X = and one gets that j = j j is an 2 ∅ 2 ◦ 1 isomorphism. 4.4. Construction of the Fundamental group 59

(ii) As X is connected, it follows by (C3) that u is an eﬀective epimorphism; by (F ), F(u) : F(X) F(X) is onto and thus is a 3 → bijection (F(X) ﬁnite). By (F ) it follows that u Aut X. 5 ∈ (iii) Let u , u Aut X such that F(u )(ξ) = F(u )(ξ), i.e., ξ 76 1 2 ∈ 1 2 ∈ ker(F(u1), F(u2)) = F(ker(u1, u2)) (F is left-exact). We thus have a commutative diagram

ξ u1 F / X / X D = u / DD zz 2 DD zz DD zz D z j ξ′ DD zz D! . zz ker(u1, u2) with a monomorphism j; as (X,ξ) is minimal by (i), j is an isomorphism, in other words, u1 = u2. Q.E.D.

We briefly recall now the example 2 of 4.3. Assertion (iii) of the above lemma simply says in this case that if K/k is a finite separable extension field and if ξ Hom (K, Ω), then the map Aut K Hom (K, Ω) ∈ k → k given by u ξ u is injective. We know that K/k is galois this map 7→ ◦ ⇔ is also onto. Following this, we now make the

Definition 4.4.1.7. A connected-object X C is galois if for any ξ ∈ ∈ F(X), the map Aut X F(X) defined by u u ξ is a bijection. → 7→ ◦ Note that this is equivalent to saying that the action of Aut X on F(X) is transitive. Also observe that this definition is independent of F because the cardinality of F(X) is the degree of the covering X over S ; the action is already effective since X is connected (by (iii), Lemma 4.4.1.6). 77

η Lemma 4.4.1.8. If F Y, then there is a galois object X C , a −→ ∈ ξ F(X) and a u Hom (X, Y) such that the diagram ∈ ∈ C ξ F / X ? ?? ?? η ?? u ?? ? Y 60 4. The Fundamental Group

is commutative. In other words, the system I1 of galois pairs of E is coﬁnal in E .

Proof. Let (S i)i I be a projective system of minimal objects of C such that ∈ F ∼ lim Hom (S , ). ←− C i ∗ i I ←−−∈ Let η1, . . . , ηr be the elements of F(Y). We can choose i large enough such that as u varies over Hom (S , Y), the u τ give all the η s (τ is C i ◦ i ′ i the canonical element of F(S i)). We then get:

p τi α r j F S i Y = Y ... Y Y −→ −→ × × −−→ r times | {z } where p is the jth canonical projection Yr Y; the elements p α τ , j → j ◦ ◦ i 78 1 j r, are precisely the elements η , . . . , η of F(Y). By (C ) we get ≤ ≤ 1 r 3 a factorisation: α r S i / Y ?? ~? ?? ~~ ?? ~~ α1 ?? ~~ β ? ~ ? / ~~ X

with a monomorphism β and an eﬀective epimorphism α1. We claim that X is galois. (i) X is connected. Suppose X = X X , X , X in C , , ; the element α τ 1 2 1 2 ∅ 1 ◦ i ∈ F(X1), say. We can` then choose a j large enough for us to get a commutative diagram:

α1 S i / X = X1 X2 ? O r9 τi rrr ` rr rr α′ + rrr F ϕij X1 ?? > ?? ~~ ? ~~ τ j ?? ~ α ? ~~ ′ S j 4.4. Construction of the Fundamental group 61

Hence β α = α ϕ is an epimorphism, which is absurd. ′ ◦ ′ 1 ◦ i j (ii) Set ξ = α τ F(X). We shall prove that the map Aut X F(X) 79 1◦ i ∈ → deﬁned by u u ξ is onto. 7→ ◦ Let ξ F(X); we may assume i is so large that we get a commu- ′ ∈ tative diagram:

X pp7@ ppp ÐÐ ξ ppp ÐÐ ppp ÐÐ pp Ð α1 ppp ÐÐ ppp ÐÐ ppp τi Ð F / S i NNN > NNN >> NN > α NNN >> 1′ NNN >> ξ′ NN > NNN >> NNN> ' X

= Our aim is to find a σ Aut X such that α1′ σ α1. Since ∈ ff ◦ X is connected, α1′ is also an e ective epimorphism. Since the manner in which a morphism in C is expressed as the composite of an effective epimorphism and a monomorphism is essentially unique, we will be through if we find a ρ Aut Yr such that the ∈ diagram

β X / Yr pp7@ ppp ÐÐ ξ ppp ÐÐ ppp ÐÐ pp Ð α1 ppp ÐÐ ppp ÐÐ ppp τi Ð F / S i ρ NNN > NNN >> NN > α NNN >> 1′ NNN >> ξ′ NN > NNN >> NNN> β ' X / Yr

is commutative. By assumption the elements p β ξ, 1 j r, 80 1 ◦ ◦ ≤ ≤ 62 4. The Fundamental Group

are all the distinct elements of F(Y); so the morphisms p β α j ◦ ◦ 1 are all distinct. This means that the p β are all distinct; since j ◦ α is an eﬀective epimorphism the p β α are all distinct; and 1′ j ◦ ◦ 1′ as X is connected and β is a monomorphism, it follows that the p β ξ are all distinct and therefore form the set F(Y). j ◦ ◦ ′

If we set p β ξ = η , 1 j r, and p β ξ = ηρ , 1 j r, we j◦ ◦ j ≤ ≤ j ◦ ◦ ′ ( j) ≤ ≤ get a permutation ρ of the set 1, 2,..., r ; this permutation determines { } an automorphism of Yr with the required property. Q.E.D.

By this lemma we may clearly assume now that F is strictly pro- represented by a projective system (S i) of galois objects of C . Let g = Aut S and θ be the bijection g F(S ) deﬁned by u i i i i → i 7→ u C where C is the canonical element of F(S ). For j i, we deﬁne ◦ i i i ≥ ψ : g g as the composite i j j → i

1 θ j F(ϕij) θ− g F(S ) F(S ) i g . j −→ j −−−−→ i −−→ i For any u g , ψ (u) is the uniquely determined automorphism of ∈ j i j 81 S i which makes either one (and hence also the other) of the diagrams

ψ (u) ψij(u) ij S S 8 S i / S i i / i τi qqq O O O qqq qq ϕ ϕ ϕ F L ij ij ij LL LLL τ j LL & S S j u / j S j u / S j

commutative. It follows from this easily that the ψi j are group homomorphisms. We thus obtain a projective system gi, ψi j i I of ﬁnite groups with { } ∈ each ψi j surjective. Denote by πi, ψi j i I the projective system of the { } ∈ opposite groups. The group π = lim πi with the limit topology is pro- i I ﬁnite and we shall prove that it is←−− the∈ fundamental group of the galois category C ; we denote it by π1(S, s) when C and F are as in 4.1 and 4.2. 4.4. Construction of the Fundamental group 63

πi acts on HomC (S i, X) to the left and hence π acts continuously on the set lim Hom (S i, X) ∼ F(X), to the left. Since F(X) is ﬁnite, the i C −→ action of−−→π on F(X) comes from the action of some πi on F(X).

4.4.1.9 We shall ﬁnd it convenient now to introduce informally the no- 82 tion of the procategory Pro C of C . An object of Pro C = (called a pro-object of C ) will be a projective system p = (Pi)i I in C . If P, ∈ P′ = (P′ ) j J are pro-objects of C , we deﬁne Hom(P, P′) as the double j ∈ e e limit lim lim Hom (Pi, P′ ). An object of C will be considered an e j J i I C j e e object←−− of Pro∈ −−→C ∈in a natural way. We may look at a pro-representable functor on C , as a functor “represented” in a sense by a pro-object of C . For instance, in the case of 4.4.1.4, we have:

F(X) ∼ lim Hom (S , X), X C ←− C i ∀ ∈ i I −−→∈ Hom (S , X) ≃ Pro C e where S is the pro-object (S i)i I of C . Also for any i I, Hom ∈ (S , S ) Hom (S , S ) = g and we e ∈ Pro C i ≃ C i i i may then write: e = = g lim gi lim HomC (S i, S i) ←−−i ←−−i = = lim HomPro C (S , S i) HomPro C (S , S ) ←−−i e e e = and hence AutPro C S . If we call S a pro-representative of F (in the case C = (E t/S ) we 83 e call it a universal covering of S ) then π is the opposite of the group of e automorphisms of S . Lemma 4.4.1.10. eLet E C (π); then an object G(E) C , and a ∈ ∃ ∈ C (π)-isomorphism γ : E FG(E) such that the map Hom (G(E), X) E → C Hom π (E, F(X)) given by u F(u) γ is a bijection for all X C . → C ( ) 7→ ◦ E ∈ The assignment E G(E) can be extended to a functor G : C (π) C 7→ → such that F and G establish an equivalence of C and C (π). 64 4. The Fundamental Group

Proof. If E = Ei is a decomposition of E into connected sets in C (π) = and if G(Ei) are` defined we may define G(E) G(Ei). We may thus assume that π acts transitively on E. Fix an element` E E and consider ∈ the surjection π E defined by σ σ E . As E is finite, there is an i → 7→ · such that the diagram

π / E ?? ? ?? ~~ ?? ~~ ?? ~~ ?? ~~ ? ~~ πi

(π π is the natural projection and π E is the map σ σ E ) is → i i → 7→ · commutative. Let H π be the isotropy group of E in π . It is easily i ⊂ i i proved that the set πi/Hi of left-cosets of πimod Hi is C (π)-isomorphic = = ·0 84 to E. We then deﬁne: G(E) G(πi/Hi) S i/Hi , the quotient of S i by the opposite H0 of H (remark: H0 g Aut S ). By (F ) we have: i i i ⊂ i ⊂ i 4 F(G(E)) = F(S /H0) ∼ F(S )/H0 π /H E, and hence a C (π)- i i ←− i i ≃ i i ≃ isomorphism γ : E FG(E). If j i and H π is the isotropy E → ≥ j ⊂ j group of E in π , then we have a C -morphism S /H0 S /H0; since j j j → i i F(S /H0) F(S /H0) is a C (π)-isomorphism, it follows from (F ) j j → i i 5 that S /H0 ∼ S /H0 and that G(E) is independent of the choice of i j j −→ i i (upto a C -isomorphism). Let X C . Consider the map ∈

ω : Hom (G(E), X) Hom π (E, F(X)) C → C ( ) u F(u) γ 7→ ◦ E (i) ω is an injection: Let u , u Hom (G(E), X) be such that F(u ) γ = F(u ) γ . 1 2 ∈ C 1 ◦ E 2 ◦ E But γ : E FG(E) is an isomorphism and so ker(F(u ), F(u )) E → 1 2 -FG(E), i.e., F(ker(u1, u2))֒ FG(E) (F is left-exact). It fol ֒ → →≃ .lows from (F5) that ker(u1, u2)֒ G(E), in other words, u1 = u2 →≃ 85 (ii) ω is a surjection. 4.4. Construction of the Fundamental group 65

α Let E F(X) be any C (π)-morphism; put δ = α(E ) F(X). −→ ∈ The Pro C (π)-morphism δ : S X can be factored through some → S i: e δ S / X >> Ð? >> ÐÐ e > Ð τ >> ÐÐ i >> ÐÐ δi > ÐÐ S i Let H be the isotropy group of δ in π then H H where H i′ i i i′ ⊃ i i is as before (take i large enough). By the construction of G(E) = S S /H0, we have a morphism i X making the diagram i i 0 Hi →

τi δi S / S i / X CC |= CC || e CC || CC || CC || C! || 0 S i/Hi commutative; one easily checks that this morphism goes to α under ω. It only remains to show that the assignment E G(E) 7→ can be extended to a functor. Let E, E C (π) and θ Hom π ′ ∈ ∈ C ( ) (E, E ). To the composite γ θ : E FG(E ) there corresponds 86 ′ E′ ◦ → ′ a unique θ Hom (G(E), G(E )) such that the diagram ∈ C ′ θ E / E′

γE γ E′ FG(E) / FG(E′) F(θ)

is commutative. We set G(θ) = θ. It follows easily that G is a covariant functor from C (π) to C . One now checks that there are functorial isomorphisms Φ : I G F C → ◦ 66 4. The Fundamental Group

and Ψ : I π F G, C ( ) → ◦ such that, for any X C and E C (π), ∈ ∈ F(Φ(X)) Ψ 1(F(X)) F(X) FGF(X) − F(X) −−−−−−→ −−−−−−−−→ G(Ψ(E)) Φ 1(G(E)) and G(E) GFG(E) − G(E) −−−−−−→ −−−−−−−−→ and the identity maps. This completes the proof of assertion (1) of Theorem (4.4.1).

87 Lemma 4.4.1.11. Let C be a galois category and F, F′ be two fundamental functors C (finite sets). Suppose π, π are the profinite groups → ′ defined by F, F′ respectively as above; then π and π′ are continuously isomorphic and this isomorphism is canonically determined upto an inner automorphism of π.

Proof. We know that F : C C (π) is an equivalence. Replacing F → ′ by F G (with G as before) we can assume that C = C (π), that F is the ′ ◦ trivial functor identifying an object of C (π) with its underlying set and π itself is the pro-object pro-representing this functor. Let T Pro C (π) ∈ pro-represent F ; ﬁrst we show that π ∼ T in Pro C . ′ −→ e In order to do this, let (T j) j J be a projective family of galois objects ∈ e (with respect to F′) of C such that T = lim T j; we denote the canonical j ←−− maps T j Ti by qi j and T T j byeq j. Let t j T j be a coherent system → → ∈ of points and consider the continuous maps α : π T determined by e j ∈ j α (e) = t ; there exists a continuous α : π T such that j j → α e π / > T >> Ð > ÐÐ >> Ð e α >> ÐÐq j >> ÐÐ j > ÐÐ T j

is commutative. We have α(e) = t = (t ) T. The T are connected, j ∈ j 88 hence α j transitive and as π is compact it follows that α is onto. Let e e H be the isotropy group of t, then we have π/H ∼ T. (π/H is the set −→ e e 4.4. Construction of the Fundamental group 67 of left-cosets of π mod. H) and this is an isomorphism of topological spaces (it is a continuous map of compact Hausdorff spaces). However F = Hom (T, ) and then clearly F (X) = XH (the points of X ′ Pro C ∗ ′ invariant under H). We know that F (X) = X = ; from this it e ′ ∅ ⇔ ∅ follows that H = (e) (take for X the πi for larger and larger i). Hence α : π ∼ T is an isomorphism of topological spaces. Our aim is to show −→ that this is an isomorphism in Pro C . e For this it is enough to show the following: if β : T π is α 1 and if → − p : π π are the canonical maps then every β = p β : T π must i → i i ei ◦ → i factor through some T . In other words, we must find some morphism j e T π making the diagram j → i β / π T > >> >> βi q je > pi >> >> > T j / πi commutative. For this we must show that given i, j J such that 89 ∃ ∈ for every t T , an s π such that q 1(t) β 1(s). Since the ∈ j ∃ ∈ i −j ⊂ i− sets β 1(s) are open we can find for every point x β 1(s) an open i− ∈ i− neighbourhood U of the form q 1(t ) for some j J and t T , such x −j x x ∈ x ∈ jx that U β 1(s). Since T is compact, a finite covering U ,..., U x ⊂ i− ∃ x1 xN of T of this type and j > max( j ,..., j ) satisfies our requirements. e x1 xN This shows that π ∼ T in Pro C , and hence the map e −→ 0 = = 0 π′ Aute Pro C T AutPro C π π → ρ β ρ α 7→e ◦ ◦ is a group isomorphism; it remains to be shown that this map is continuous and hence a homeomorphism. Take a fixed γ0 = β ρ0 α in π, ◦ ◦ some i I and consider the commutative diagram ∈ γ0 π / π

pi pi γ0 i πi / πi 68 4. The Fundamental Group

90 Let U be the neighbourhood of γ0 consisting of all γ π such that ∈ p γ = γ0 p . We want to ﬁnd a neighbourhood V of ρ0 such that i ◦ i ◦ i β ρ α U ρ V. There is an index j J and l I and morphisms ◦ ◦ ∈ ∀ ∈ ∈ ∈ making all the following diagrams

α ρ0 β π / T / T / π

p1 eq j eq j

ρ0 j πl / T j / T j pi @@ @@ pl @@ @@ γ0 @@ i πi / πi

commutative. Consider all ρ : T T such that q ρ = ρ0 q ; these ρ form a → j ◦ j ◦ j neighbourhood V of ρ0 and β ρ α U ρ V. Hence we are through. e e ◦ ◦ ∈ ∀ ∈ Finally we observe that the isomorphism π π is fixed as soon → ′ as α : π T is fixed and this is in turn fixed by the choice of t = → (t ) T. By a different choice of t we obtain an isomorphism π π j ∈ e → ′ which differs from the first one by an inner automorphism of π (or eπ ). e ′ Q.E.D. e

91 Remark 4.4.1.12. One can, in fact, show that Pro C (π) is precisely the category of compact, totally disconnected (Hausdorﬀ) topological spaces on which π acts continuously. Chapter 5 Galois Categories and Morphisms of Proﬁnite Groups

5.1 92 Suppose π and π are proﬁnite groups and u : π π a continuous ho- ′ ′ → momorphism. Then u deﬁnes, in a natural way, a functor H : C (π) u → C (π′); and Hu being the identity functor on the underlying sets is fundamental.

On the other hand, let C , C ′ be galois categories, π′ a proﬁnite group and H : C C , F : C C (π ) be functors such that F = F H → ′ ′ ′ → ′ ′ ◦ is fundamental. Then we can choose a pro-object S = S i,ϕi j i I of C { } ∈ such that, X C , F(X) ∼ Hom (S , X). Moreover we may assume ∀ ∈ ←− Pro C e that the S are galois objects of C , therefore the F(S ) are principal ho- i e i mogeneous spaces under the action of the πi (on the right) (notations from Ch. 4) and if we identify F(S i), by means of the canonical element τ , with π then the maps F(ϕ ) = ψ : F(S ) F(S ) are i i i j i j j → i group homomorphisms (see Ch. 4). However, in the present situation the F(S i) are not merely sets but are objects of C (π′); as such, the group π′ acts continuously upon the sets to the left, and this action commutes with the right-action of the πi. This gives a continuous homomorphism

69 70 5. Galois Categories and Morphisms of Proﬁnite Groups

u : π π determined by the condition that for σ π the u (σ ) is i ′ → i ′ ∈ ′ i ′ 93 the unique element of π such that σ τ = τ u (σ ) (τ is the canonical i ′ · i i · i ′ i element of F(S )); for j i we clearly have: ψ u = u ; thus, we i ≥ i j ◦ j i obtain a continuous homomorphism thus, we obtain a continuous homomorphism u : π′ lim πi = π and u corresponds to H if we identify → i C with C (π). ←−−

Example. We shall apply the above to the particular case C = (E t/S ), C ′ = (E t/S ′) where S , S ′ are, as usual locally noetherian, connected preschemes. Suppose ϕ : S S is a morphism of finite type and ′ → s S , s = ϕ(s ) S . Let Ω be an algebraically closed field contain- ′ ∈ ′ ′ ∈ ing k(s ). We define functors F : C Finite sets and F : C ′ → { } ′ ′ → Finite sets by defining: { } F(X) = Hom (Spec Ω, X), X C , S ∈ F′(X′) = Hom (Spec Ω, X′), X′ C ′. S ′ ∈

Denote by π and π′ the fundamental groups π1(S, s) and π1(S ′, s′). Corresponding to the morphism ϕ : S S , we obtain a functor ′ → Φ : C C ′ given by X X S ′. We then have: → 7→ ×S

F′(X S ′) = Homs (Spec Ω, X S ′) ×S ′ ×S Hom (Spec Ω, X) = F(X). ≃ s

94 Thus F = F′ Φ; in view of the fact that F is fundamental and the F ◦ F equivalences C C (π), C C (π ), we obtain a continuous homomor- ∼ ′∼ ′ phism π π. ′ → 5.2

In this section, we shall correlate the properties of a homomorphism u : π π and those of the corresponding functor H : C (π) C (π ). ′ → u → ′

5.2.1 Suppose u : π π is onto; for any connected object X of C (π) ′ → (i.e., π acts transitively on X) any π-morphism π X defined by, say, → 5.2. 71 e x is onto and therefore so is the map π X defined by e x; in 7→ ′ → ′ 7→ other words, Hu(X) is a connected object of C (π′). Conversely, suppose that for any connected X C (π), the object ∈ Hu(X) is again connected in C (π′). Write π = lim πi where the πi are i finite groups and the structure-homomorphisms←−−π π , j i, are all j → i ≥ onto; this implies that the π π are all onto and by our assumption → i then all the π π are onto. Since π and π are both profinite it follows ′ → i ′ that u : π π is onto. Thus: u : π π is onto for any connected ′ → ′ → ⇔ X C (π), the object H (X) is connected in C (π ). ∈ u ′

5.2.2 A pointed object of C (π) is, by definition, a pair (X, x) with X 95 ∈ C (π) and x X. By the definition of the topology on π, it is clear that ∈ giving a pointed, connected object of C (π) is equivalent to giving an open subgroup H of π; the object is π/H = set of left-cosets of π mod H and the point is the class H. A final object of C (π) is a point ec on which π acts trivially. We say that an X C (π) has a section if there is a C (π)- ∈ morphism from a final object ec to X; giving a section of X is equivalent to giving a point of X, invariant under the action of π. A pointed object (X, x) of C (π) admits a pointed section (i.e., e is mapped onto x) x c ⇔ is invariant under π. Suppose u : π π is a homomorphism and H is an open subgroup ′ → of π such that u(π ) H. Let (X, x) be the pointed, connected object of ′ ⊂ C (π) determined by H. Then, in the action of π′ on Hu(X), x remains invariant, i.e., to say, the pointed object (Hu(X), x) of C (π′) admits a pointed section. The converse situation is clear. Thus: For an open subgroup H of π, one has u(π ) H H (π/H) admits ′ ⊂ ⇔ u a pointed section in C (π′).

5.2.3 We say that an X C (π) is completely decomposed if X is a ∈ finite disjoint sum of final objects of C (π), i.e., if the action of π on X is trivial. Suppose u : π π is trivial, then for any X C (π), H (X) is 96 ′ → ∈ u completely decomposed in C (π ). Conversely, assume that for any X ′ ∈ C (π)Hu(X) is completely decomposed in C (π′). Write π = lim πi as i ←−− 72 5. Galois Categories and Morphisms of Profinite Groups

usual; by assumption, each composite π π π is trivial. Hence ′ → → i u : π π is also trivial. Thus: ′ → u : π π is trivial for any X C (π),H (X) is completely ′ → ⇔ ∈ u decomposed in C (π′).

5.2.4 Let H be an open subgroup of π and X C (π ) the con- ′ ′ ′ ∈ ′ nected, pointed object defined by H . Assume that ker u H ; then ′ ⊂ ′ u(π )/u(H ) π /H in C (π ). This means that u(H ) is a subgroup ′ ′ ≃ ′ ′ ′ ′ of finite index of the pro-finite group u(π′) and hence is open in u(π′). Since π′ is compact and π Hausdorff we can find an open subgroup H of π such that H u(π ) u(H ). ∩ ′ ⊂ ′ Consider now the connected, pointed object X = π/H of C (π). De- note by Hu(X)0 the C (π′)-component of the pointed object Hu(X) of C (π′), containing the distinguished point of Hu(X). There exists then an open subgroup H of π such that H (X) π /H in C (π ). We claim 1′ ′ u 0 ≃ ′ 1′ ′ that H H ; in fact, u(H ) H and so u(H ) H u(π ) u(H ), 1′ ⊂ ′ 1′ ⊂ 1′ ⊂ ∩ ′ ⊂ ′ hence H u 1(u(H )) u 1(u(H )) = H , since, by assumption, H is 1′ ⊂ − 1′ ⊂ − ′ ′ ′ saturated under u. 97 Thus, there is a pointed C (π )-morphism H (X) π /H π /H ′ u 0 ≃ ′ 1′ → ′ ′ X . If, on the other hand, we assume that a pointed, connected ≃ ′ ∃ object X of C (π) such that we have a pointed C (π )-morphism H (X) ′ u 0 ≃ π /H X π /H , then, we must have H H and hence ker u ′ 1′ → ′ ≃ ′ ′ 1′ ⊂ ′ ⊂ H H . If u is surjective, then we can say that H (X) X (see 1′ ⊂ ′ u ≃ ′ 5.2.1). Also ker u H is a relation independent of the choice of the ⊂ ′ distinguished point in X π /H . Thus: ′ ≃ ′ ′ ker u H a connected object X of C (π) and a C (π )-mor- ⊂ ′ ⇔ ∃ ′ phism of a connected C (π′)-component of Hu(X) to X′ = π′/H′. If u is onto then X = π /H H (X) for a connected object X C (π). ′ ′ ′ ≃ u ∈ In particular: u is injective for every connected X C (π ), there is a connected ⇔ ′ ∈ ′ X C (π) and a C (π )-morphism from a C (π )-component of H (X) to ∈ ′ ′ u X′.

u u 5.2.5 Let π ′ π ′′ π be a sequence of morphisms of proﬁnite ′ −→ −−→ ′′ groups. From 5.2.3 and 5.2.4 we obtain the following necessary and 5.2. 73 suﬃcient conditions for the sequence to be exact: (a) u u is trivial for any X C (π ), H H (X ) is com- 98 ′′ ◦ ′ ⇔ ′′ ∈ ′′ u′ ◦ u′′ ′′ pletely decomposed in C (π′). (b) Im u ker u for any open subgroup H of π, with H Im u , ′ ⊃ ′′ ⇔ ⊃ ′ we also have H ker u for any connected pointed object X ⊃ ′′ ⇔ of C (π) such that Hu′ (X) admits a pointed section in C (π′), there is a connected object X C (π ) and a C (π)-morphism of a ′′ ∈ ′′ C (π)-component of Hu′′ (X′′) to X.

5.2.6 Let C be a galois category with a fundamental functor F. Let S = (S i)i I be a pro-object of C with usual properties (in particular, S i ∈ are galois) pro-representing F. Let π be the fundamental group of C e F determined by F. We know then that C C (π). ∼ Let now T C be a connected object and t F(T) be fixed for ∈ ∈ our considerations. We form the category C = C T; it is then readily ′ | checked that C ′ satisfies the axioms (C0),..., (C4) of Ch. IV. We have an exact functor P : C C defined by X X T. We now define → ′ 7→ × a functor F : C Finite sets by setting, for any X C , F (X) = ′ ′ → { } ∈ ′ ′ inverse image of t under the map F(X) F(T). Again it is easily → checked that C ′, equipped with F′, is galois. A cofinal subsystem S ′ of S is defined by the condition: S S (S , τ ) dominates (T, t) in i ∈ ′ ⇔ i i e the sense of Ch. 4. An S S can be considered in the obvious way 99 e i ∈ ′ e as an object of C ; it is then easily shown that they are galois in C and ′ e ′ the pro-object S ′ of C ′ pro-represents F′ (To do the checking one may identify C and C (π)). e Let H be the isotropy group of t F(T) in π; let also N be the ∈ i isotropy groups of τ F(S ), S S . We have a diagram of the form i ∈ i i ∈ ′ π e / F(T) t GG 9 ∋ GG rr GG rr GG rrr GG rr GG rrr G# rr τ F(S ) i ∈ i which is commutative and thus Ni H, i. Since the F(S i) and F(T) ⊂ ∀ π are all connected objects of C (π), we have F(S i) and π/H F(T). Ni ≃ ≃ 74 5. Galois Categories and Morphisms of Profinite Groups

The maps F(S ) F(T) are then the natural maps π/N π/H and it i → i → follows that F (S ) H/N in C (π). But as we have already remarked ′ i ≃ i the S i in S ′ form a system of galois objects with respect to F′, pro- representing F and one thus obtains: e ′

π′ lim F′(S ) = lim H/N H. ≃ i i ≈ N H S←−−i S ←−−i ∈ ′ ⊂ e Finally we remark that the composite functor F P is isomorphic ′ ◦ with F and therefore fundamental; following the procedure of 5.1 we see that the corresponding continuous homomorphism u : π π is ′ → .nothing but the canonical inclusion H ֒ π → Chapter 6 Application of the Comparison Theorem an Exact Sequence for Fundamental Groups

6.1 100 As usual we make the convention that the preschemes considered are locally noetherian and the morphisms are of ﬁnite type (with the same remark as in the beginning of Ch. 3).

Definition 6.1.1. (a) A morphism X Spec k, k a field, is said to be → separable if, for any extension field K/k, the prescheme X K is ⊗k reduced.

f (b) A morphism X Y is separable if f is ﬂat and for any y Y, −→ ∈ X Spec k(y) is separable over k(y). ⊗Y We shall now state a few results which we will need for our next main theorem. Proofs can be found in EGA.

Theorem 6.1.2. Let f : X Y be a proper morphism and F a coherent → 75 76 6. Application of the Comparison Theorem....

q O -Module. If Y Y is a ﬂat base-change X 1 −→ q1 X1 = X Y1 X o ×Y

f f1

q Y o Y1 then we have the isomorphisms

n ∼ n R f1 (q1∗(F )) R f (F ) OY1 ∗ ←− ∗ O⊗Y for any n Z+. (Prop. (1.4.15), EGA, Ch. III). ∈

101 6.1.3 Suppose Y is a noetherian prescheme and f : X Y a proper → morphism. Let Y ֒ Y be a closed subscheme of Y, defined by a ′ → coherent Ideal T of OY . The “inverse image” of Y′ by f , namely the fibre-product X Y′ = X′ is then a closed subscheme of X, defined by ×Y the OX-Ideal T = f ∗(T )OX. + Let F be any coherent OX-Module; for n Z , consider Fn = n+1 ∈ F OX/T (this is a coherent OX-Module, concentrated on the pre- O⊗X = n+1 scheme Xn (X′, OX/T ) and may also be considered as an OXn - q Module). Consider R f (Fn); this is a coherent OY -Module (finiteness ∗ n+1 theorem), is concentrated on the prescheme Yn = (Y′, OY /T ) and is

in fact an OYn -Module. From the homomorphism F Fn we obtain q q → a homomorphism R f (F ) R f (Fn) and since the latter is an OYn - ∗ → ∗ Module, we get a natural homomorphism:

q n+1 q (*) R f (F ) OY /T R f (Fn). ∗ O⊗Y → ∗ As n varies, we get a projective system of homomorphisms. With the assumptions we have made about X, Y, f , the comparison theorem (EGA, Ch. III, Theorem (4.1.5)) states that in the limit, this gives an isomorphism:

q n+1 q (**) lim R f (F ) OY /T ∼ lim R f (Fn). ∗ ⊗ −→ ∗ ←−−n OY ←−−n 6.2. The Stein-factorisation 77

102 (Note: Both sides of (**) are concentrated on Y′ and they are also equal q n+1 to R f (lim F OX/T ) where f is the morphism of the ringed ∗ n O⊗X ←−− n+1 n+1 spacesbX = (X′, lim OX/T ) Yb = (Y′, lim OY /T ) obtained n → n from the morphisms←−− f : X Y , induced by ←−−f ). b n n → n b We shall now “specialise” the above comparison theorem to the case Y affine, say Y = Spec A. Then Y′ is defined by an ideal I of A. The first q A member of (**) corresponds to lim H (X, F ) A + which is precisely n ⊗ In 1 the completion of Hq(X, F ) under←−− the I-adic topology, while the second q member of (**) corresponds to lim H (X, Fn) and thus: n ←−− (***) Hq(X, F ) ∼ lim Hq(X, F ). −→ n ←−−n b 6.2 The Stein-factorisation

f Let X Y be a proper morphism. Then the coherent OY -Algebra −→ q f (OX) (finiteness theorem) defines a Y-prescheme Y′ T, finite on Y. ∗ −→ To the identity OY -morphism q (OY ) = f (OX) f (OX) corresponds ∗ ′ ∗ → ∗ a Y-morphism f : X Y , i.e., we have a commutative diagram ′ → ′

f ′ X / Y′ = Spec f (OX) >> r ∗ >> rr > rrr >> rrq f >> rrr > rrr Y yr

The morphism f is again proper. This factorisation f = q f is 103 ′ ◦ ′ known as the Stein-factorisation of f . For details the reader is referred to EGA Ch. III. We shall now prove a theorem, which is of great importance to us.

Theorem 6.2.1. Let f : X Y be a separable, proper morphism. Let f ′ →q X Y′ = Spec f (OX) Y be the Stein-factorisation of f . Then −→q ∗ −→ Y Y is an ´etale covering. ′ −→ 78 6. Application of the Comparison Theorem....

Proof. We have only to show that q is étale; this is a purely local problem and we may thus assume that Y is affine, say, Y = Spec A. We shall make a few simplifications to start with. Suppose X Y 1 → is a flat base-change; then from 6.1.2 one gets:

f(Y1) (OX OY1 ) f (OX) OY1 ∗ O⊗Y ≃ ∗ O⊗Y

This means that in the commutative diagram

X1 = X Y1 X o ×Y

f ′ f ′ (Y1) = = = Y1′ Y′ Y1 Spec f(Y1) (OX OY1 ) Spec f (OX) Y′ o Y ∗ ∗ × O⊗Y

q q(Y1) Y o Y1

the second vertical sequence is the Stein-factorisation of f(Y1). 104 In view of this and the fact that it is enough to look at Spec Oy,Y for étaleness over y Y we may make the base change Y Spec O , and ∈ ← y Y assume that A is a local ring. Again, in view of the same remark and Lemma 4.4.1 we may assume that A is complete. Consider now the functor T on the category of finite type A-modules M, defined by M T(M) = Γ(X, O M). We shall show that the 7→ X ⊗A assumption “T is right-exact” implies the theorem. We remark that the right-exactness of T is equivalent to the assumption that the natural map

(+) Γ(X, OX) M Γ(X, OX M) ⊗A → ⊗A

is an isomorphism. Denote by T ′ the right-exact functor

M Γ(X, OX) M. 7→ ⊗A 6.2. The Stein-factorisation 79

Suppose T is right exact. Then for any exact sequence An Am → → M 0 we have a commutative diagram of exact sequences: → n m T ′(A ) / T ′(A ) / T ′(M) / 0

≀ ≀ T(An) / T(Am) / T(M) / 0 the first two vertical maps are isomorphisms; hence the third is also an 105 isomorphism. Conversely, if (+) is an isomorphism, clearly T is right- exact. Now from the flatness of f and the assumed right exactness of T it follows that Γ(X, OX M) is exact in M and thus [from the above remark] ⊗A Γ(X, O ) is A-flat. But Y = Spec Γ(X, O ) and therefore q : Y Y is X ′ X ′ → flat. It remains to show that q is unramified. Let y Y be the unique closed point of Y; denote by k the residue ∈ = = Γ = Γ field A/M k(y). Then T(k) (X, OX k) (Xy, OXy ) where Xy is ⊗A the fibre X k(y). But as X is Y-proper, T(k) =Γ(X, OX k) is an artinian ⊗Y ⊗A k-algebra which, by the separability of X over Y, is radical-free for any r base-change K/k, K an extension field of k. Hence T(k) = Ki where Li=1 the Ki are finite separable field extensions of k. Also, by our assumption, T(k) = Γ(X, OX) k = Γ(X, OX)/M Γ(X, OX). Let y′ Y′ be a point ⊗A ∈ r above y. From the equality Γ(X, OX)/M Γ(X, OX) = Ki, (Ki/k are Li=1 separable) it follows that the maximal ideal M of Oy. generates the maximal ideal of Oy′ and moreover that Oy′ is unramified over Oy.

The proof of the theorem is thus complete modulo the assumption 106 that T is right-exact. Before proceeding to prove this we may make some = Γ = more simplifications. First we may assume that T(k) (Xy, OXy ) r k (this can be done as in Lemma 4.1.2 by making a faithfully flat Li=1 base-change which “kills” the extensions Ki/k). Then finally we may 80 6. Application of the Comparison Theorem....

Γ = assume (Xy, OXy ) k, i.e., Xy connected, because the connected components of X over y correspond bijectively with the connected components of Xy and it is enough to prove the theorem separately for each component over y. Case (a). Assume A artinian. Our aim is to show that for any ﬁnite type A-module M, T (M) ∼ ′ −→ T(M). We shall ﬁrst show that T ′(M) T(M) is surjective. n → n n For n large, one has M M = 0 so that T ′(M M) T(M M) is i+1 i+1 → onto. Assume now that T ′(M M) T(M M) is onto, and con- i+1→ i M iM sider the exact sequence: 0 M M M M + 0 of → → → M i 1M → A-modules. We get a commutative diagram

i+1 i M i M / / T + / T ′(M M) T ′(M M) ′ M i 1M 0

i+1 i M iM / / T + T(M M) T(M M) M i 1M 107 of exact sequences (the second row is exact at the spot T(M i M) because T is semi-exact, i.e., for 0 M M M 0 exact, the sequence → ′ → → ′′ → T(M ) T(M) T(M ) is exact at the spot T(M)). The ﬁrst vertical ′ → → ′′ M i M map is a surjection by assumption, the last is a surjection since M i+1 M is a ﬁnite direct sum of copies of k and by our assumption Γ(X , O ) y Xy ≃ k, we have T (k) ∼ T(k). It follows that T (M) T(M) is onto, by ′ −→ ′ → a downward induction. Finally consider an exact sequence 0 R → → Ap M 0. We then obtain a commutative diagram → → p T ′(R) / T ′(A ) / T ′(M) / 0

onto ≃ T(R) / T(Ap) / T(M) of exact sequences. If follows easily that T (M) T(M) is also injec- ′ → tion. Thus, the theorem is completely proved in the case (a). 6.2. The Stein-factorisation 81

Case (b). The general case

Let A be a complete noetherian local ring. Denote by Y1′ the closed subscheme of Y deﬁned by the maximal ideal M . As in the comparison theorem we may deﬁne the functors

n+1 Tn(Mn) =Γ(Xn, OX M A/M ) ⊗A ⊗A n+1 =Γ(X, OX M A/M ) ⊗A ⊗A n+1 where Mn = M A/M . If M is an A-module of ﬁnite type, so 108 ⊗A is Γ(X, OX M); as A is complete under the M -adic topology, so is ⊗A Γ(X, OX M) and from 6.1.3 we obtain: ⊗A

Γ(X, OX M) ∼ lim Tn(Mn). ⊗A −→ ←−−n To show that T is right-exact it is enough to show that for any exact u T(u) sequence M N 0 of finite type A-modules, T(M) T(N) 0 −→ → + −−−→ → is again exact. But as each A/M n 1 is a complete artinian local ring, it T (u) follows from case (a) that T (M ) n T (N ) 0 is exact; also for n n −−−−→ n n → each n, ker Tn(u) is a module of finite length over A. Thus, it is enough now to prove the Lemma 6.2.2. Let (K ,ϕ ), (M , ψ ), (N ,θ ), n Z+, be projec- n nm n nm n nm ∈ tive systems of abelian groups and u = (un), v = (vn) be morphisms u v such that, for every n Z+ 0 K n M n N 0 is ex- ∈ → n −−→ n −→ n → act. Assume, in addition, that for each n, m = m (n) such that ∃ 0 0 ϕ (K ) = ϕ (K ) m m (n) > n (this is the so-called Mittag- nm m nm0 m0 ∀ ≥ 0 Leffler (ML) condition; it is certainly satisfied if the ker un = Kn are of finite length). Then the sequence of projective limits is also exact.

Proof. The only diﬃcult point is to show that lim vn is onto. By hy- 109 n pothesis, for each n, m (n) > n with ϕ (K ←−−) = ϕ (K ) m ∃ 0 nm m nm0 m0 ∀ ≥ m0(n). By passing to a coﬁnal subsystem we may suppose, that, for any n Z+ ∈ ϕ (K ) = ϕ , + (K + ) m n + 1. nm m n n 1 n 1 ∀ ≥ 82 6. Application of the Comparison Theorem....

Let now (yn) lim Nn. Choose x′ M0 with v0(x′ ) = y0. Assume ∈ n 0 ∈ 0 inductively that we have←−− chosen (x0, x1,..., xn 1, xn′ ) with (i) ψr,r+1(xr+1) = = − = xr for 0 r n 2, and ψn 1,n(xn′ ) xn 1. (ii) vr(xr) yr for ≤ ≤ − − − 0 r n 1, and v (x ) = y . ≤ ≤ − n n′ n Our aim now is to ﬁnd (x0, x1,..., xn 1, xn, x′ + ) for which the above − n 1 properties (i), (ii) hold when n is replaced by n + 1. Choose x M + n′′+1 ∈ n 1 such that v + (x ) = y + ; then, v (ψ , + (x ) x ) = y y = 0 i.e. n 1 n′′+1 n 1 n n n 1 n′′+1 − n′ n − n to say, ψ , + (x ) x K . By assumption, we can ﬁnd z + K + n n 1 n′′+1 − n′ ∈ n n 1 ∈ n 1 such that:

ψn 1,n+1(zn+1) = ϕn 1,n+1(zn+1) − − = ϕn 1,n(ψn,n+1(xn′′+1) xn′ ) − − = ψn 1,n+1(xn′′+1) ψn 1,n(xn′ ). − − −

We now set x = (x z + ) and x = ψ , + (x ). n′ +1 n′′+1 − n 1 n n n 1 n′ +1 110 We then have:

= ψn 1,n(xn) ψn 1,n+1(xn′′+1) ψn 1,n+1(zn+1) − − − − = = ψn 1,n(xn′ ) xn 1. − − = = vn+1(xn′ +1) vn+1(xn′′+1) yn+1 = = and ﬁnally vn(xn) vn(ψn,n+1(xn′ +1)) yn. Q.E.D.

6.3 The ﬁrst homotopy exact sequence

6.3.1 Some properties of the Stein-factorisation of a proper morphism. Let Y be locally noetherian and f : X Y be proper. Suppose f q → X ′ Y Y is the Stein factorisation of f . By using the comparison −→ ′ −→ theorem one can prove the

6.3.1.1 (Zariski’s connection theorem). The morphism f ′ : X Y′ is also proper. And for any y′ Y′ the 1 → ∈ fibre f ′− (y′) is non-void and geometrically connected (i.e., for any field k′ k(y′) the prescheme X Spec k′ is connected). ⊃ Y×′ 6.3. The first homotopy exact sequence 83

(For a proof see EGA Ch. III (4.3)). § One can then easily draw the following corollaries.

Corollary 6.3.1.2. For any y Y, the connected components of the fibre 1 ∈ f − (y) are in a (1 1) correspondence with the set of points of the fibre 1 − q− (y) (which is finite and discrete).

Corollary 6.3.1.3. For any y Y, let k(y) be the algebraic closure of 111 ∈ k(y) and Xy be the prescheme X k(y). The connected components of Xy ×Y (known as the geometric components of the ﬁbre over y) are in (1 1) − correspondence with the geometric points of Y′ over y.

6.3.2 Now, in addition, suppose that

(i) Y is connected.

(ii) f is separable

(iii) f (OX) ∼ OY . ∗ ←− f Assumption (iii) implies that X Y is its own Stein-factorisation. −→1 From ??, it follows that the ﬁbres f − (y) are (geometrically) connected. f is a closed map and therefore X is also connected. Similarly, for any y Y, X is also connected. ∈ y Fix y Y. We have a commutative diagram: ∈

a X o Xy a ∈ ∋

f f y Y o k(y) ∈ Let Ω be an algebraically closed ﬁeld k(y) and let a X be a ⊃ ∈ y geometric point over Ω; let a X be the image of a in X. We have the ∈ fundamental groups π1(Xy, a), π1(X, a), π1(Y, y) and continuous homo- 112 ϕ ψ morphisms: π (X , a) π (X, a), π (X, a) π (Y, y) (See 5.1). We 1 y −→ 1 1 −→ 1 now have the following: 84 6. Application of the Comparison Theorem....

Theorem 6.3.2.1. The sequence

ϕ ψ π (X , a) π (X, a) π (Y, y) 0 1 y −→ 1 −→ 1 → is exact.

Proof. (a) ψ is surjective.

In view of 5.2.1 it is enough to show that, if Y′/Y is connected ´etale covering, then the ´etale covering X′ = X Y′ over X is con- ×Y nected. X Y′ = X′ X o ×Y

f f ′ (proper)

Y o Y′ (connected)

But OY = f (OX) and therefore, we have (from 6.1.2) ∗ = = = f ′(OX′ ) f ′(OX OY′ ) f (OX ) OY′ OY′ . ∗ ∗ O⊗Y ∗ O⊗Y

It follows that the ﬁbres of f ′ are connected, and hence X′ = X Y′ ×Y is also connected.

(b) ψ ◦ ϕ is trivial.

In view of 5.2.3 it is enough to show that if Y′/Y is any étale cov- 113 ering, the étale covering (Y′ Xy)/Xy is completely decomposed. ×Y But Y k(y) = k(y) and hence: Y X X (Y k(y)) X . ′ ′ y = ′ y ×Y finite ×Y × ×γ finite ` ` (c) Im ϕ ⊃ ker ψ. g In view of 5.2.4 it is enough to prove that: Suppose X′ X is g −→ a connected étale covering of X and X′ X admits a section y −→ y σ (over X ). Then a connected étale covering Y /Y such that y ∃ ′ X′ ∼ X Y′. We need, for proving this, the following −→ ×Y 6.3. The first homotopy exact sequence 85

g f Lemma 6.3.2.2. The composite h : X X Y is proper and separa- ′ −→ −→ ble. Proof. h is obviously proper and flat. We have only to show that the fibres of h are reduced, and remain reduced after any base-change Y ← Spec K, K a field. For this, it is enough to prove that:

X′/X étale covering, X reduced X′ reduced. ⇒ We may assume X = Spec A, X′ = Spec A′; we have A = Ared and = we want to show that, A′ Ared′ . n Let (Pi)i=1 be the minimal prime ideals of A. By assumption, the n n natural map A (A/Pi) is an injection. So, A′ (A′/PiA′) 114 → i=1 → i=1 Q Q is also injective and it is enough then to show that each A′/PiA′ is reduced. By making the base-change A A/P , we may then as- → i sume that A is an integralldomain. Let a X = Spec A be the generic ∈ point of X; then k(a) = K the field of fractions of A. The fibre over a is = Spec(A′ K) and since this is non-ramified over k(a) = K, we ⊗A r have A′ K = Ki, Ki/K finite separable field extensions. In particular, ⊗A i=1 P A′ K is reduced and hence for each x′ X′, Ox′ K is reduced and ⊗A ∈ O⊗ g(x′) hence so is K. It follows that A = A . Ox′ Ox′ ′ red′ ⊂ g(⊗x′) h Coming back to the proof of the assertion, let now X ′ Y Y ′ −→ ′ → be the Stein-factorisation of h : X Y. From the above lemma 6.3.2.2 ′ → and Theorem 6.2.1 it follows that Y Y is an étale covering. We have ′ → a commutative diagram

X′ sss ss ss α sss ss sss yss = X o X′′ X Y′ h p1 ×Y ′

f f ′

Ð Y o Y′ = Spec h (OX ) ∗ ′ 86 6. Application of the Comparison Theorem....

115 Our assertion will follow if we show that α : X′ X′′ = X Y′ is → ×Y an isomorphism. We do this by showing that:

(i) α is an ´etale covering.

(ii) X′′ is connected.

(iii) rank of α is 1 at some point of X′′. (see the remark at the end of Ch. 3).

(i) Y Y is an étale covering and so X X is an étale covering. ′ → ′′ → The composite X X X is the étale covering g and so α is ′ → ′′ → an étale covering.

(ii) We know that h′ is onto (6.3.1.1) and X′ is connected (hypothesis). Hence Y′ is connected and (ii) follows now from (a). (iii) We make the base-change Y k(y) and obtain: ←

σ - X o ′y oo ooo g ooα ooo ooo ooo ooo wo ′′ Xy o Xy ′ p1 h f × ′ k(y) o Yy

= ′′ 116 It is enough to show that rank α 1 at some point of Xy . Since Y′/Y n ′ = = ′′ = ′ = is an ´etale covering, Yy ki each ki k(y) and so Xy Xy Yy i=1 k×(y) n ` = ′ Xyi each Xyi Xy. The section σ : Xy Xy is an ´etale covering and i=1 → ` ′ Xy is connected and hence σ(Xy) is a component Z of Xy; α(Z) must

then be some Xyi . Also the projection p1 is an isomorphism from Xyi 6.3. The ﬁrst homotopy exact sequence 87

α / to Xy; it follows that Z o Xyi are inverses of one another. Also (σ p ) ◦ 1 we know that the number of components of Xy′ is equal to the number of geometric points of Y′ over y, i.e., is n. It follows that the number of connected components of Xy′ and Xy′′ are the some. α is surjective because it is both open and closed and X′′ is connected, therefore α is = ′′ surjective and ´etale and we get rank α 1 at every pt. of Xy .Q.E.D.

Remark 6.3.2.3. One may drop the assumptions f (OX) = OY and Y connected, in the above theorem; the assertion of the∗ theorem will then be: Denote by π0(Xy, a), π0(X, a), π0(Y, y) the (pointed) sets of (connected) components of the preschemes Xy, X, Y. Then, if f is proper 117 and separable, we have the following exact sequence: π (X , a) π (X, a) π (Y, y) π (X , a) π (X, a) π (Y, y) (1) 1 y → 1 → 1 → 0 y → 0 → 0 → Proof. Assume to start with that X, Y are connected, dropping only the assumption f (OX) ∼ OY . We have then the Stein-factorisation: ∗ ←− f X / Y LLL rr8 LLL rrr LLL rrr f LL rrq ′ LLL rrr L& rr Y′ = Spec f (OX). ∗ Applying the above theorem to f ′ we obtain an exact sequence: π (X , a) π (X, a) π (Y , y ) (e) where y Y is the image 1 y → 1 → 1 ′ ′ → ′ ∈ ′ of a X. We know then that π (Y , y ) π (Y, y) is an injection (5.2.6) ∈ 1 ′ ′ → 1 and the quotient π1(Y, y)/π1(Y′, y′) (set of left cosets mod π1(Y′, y′)) is isomorphic to the set of geometric points of Y′ over y. By corollary 6.3.1.3 this is isomorphic to π0(Xy, a). We thus obtain the exact sequence: π(X , a) π (X, a) π (Y, y) π (X , a) (1). y → 1 → 1 → 0 y → Now the assumptions about the connectivity of X and Y are dropped in turn to get the general exact sequence. The procedure is obvious and the proof is omitted.

Chapter 7 The Technique of Descents and Applications

7.1

Before stating the problem with which we shall be concerned in this 118 section, in its most general form, we shall look at it in three particular cases of interest.

Example 1. Let A, A be rings and ϕ : A A be a ring-homomorphism. ′ → ′ Let CA (resp. CA′ ) be the category of A-(resp. A′-) modules. ϕ de- ﬁnes a covariant functor ϕ∗ : CA CA , viz., ϕ∗(M) = M A′ = M′. → ′ ⊗A Suppose M, N C and u : M N is an A-linear map. Then ∈ A → ϕ∗(u) = u 1A : M′ N′ is an A′-linear map. ⊗A ′ → Problem 1. Suppose u : M = ϕ (M) ϕ (N) = N is an A -linear ′ ′ ∗ → ∗ ′ ′ map. When can we say that u = ϕ (u) for an A-linear map u : M N? ′ ∗ →

Problem 2. Suppose M′ CA . When can we say that M′ = M A′ for ∈ ′ ⊗A an M C ? ∈ A Example 2. Let S , S be preschemes and ϕ be a morphism ϕ : S S . ′ ′ → Let CS (resp. CS ′ ) be the category of quasicoherent OS -(resp. CS ′ -) Modules. ϕ deﬁnes a (covariant) functor C C given by F S → S ′ 7→ 89 90 7. The Technique of Descents and Applications

ϕ∗(F ). We again have:

Problem 1. If u : F = ϕ (F ) g = ϕ (g) is an O -morphism when ′ ′ ∗ → ′ ∗ S ′ can we say that u = ϕ (u) for an O -morphism u : F g? ′ ∗ S → 119 Problem 2. If F ′ is any quasi-coherent OS ′ -Module, when can we say that F ′ = ϕ∗(F ) for a quasi-coherent OS -Module? = = Example 3. Let S , S ′ be preschemes and let CS (Sch /S ), CS ′ (Sch /S ). Suppose ϕ : S S is a morphism; ϕ deﬁnes a (covariant) ′ ′ → functor ϕ : C C given by ∗ S → S ′

X ϕ∗(X) = X′ = X S ′. 7→ ×S We may again pose the two problems as in the previous examples. It is clear that the two problems posed are of the same nature in the diﬀerent examples and admit a generalisation in the following manner: Let C be a category and suppose that for every S C , we are given ∈ a category C such that for every morphism ϕ : S S in C , we are S ′ → given a covariant functor ϕ : C C . Assume, in addition that ∗ S → S ′ (1) If S ′′ S ′ S is a sequence in C , there is a natural isomor- −→ψ −→ϕ phism: ψ ϕ (ϕ ψ) . ∗ ◦ ∗ ≃ ◦ ∗ (2) If S ′′′ S ′′ S ′ S is a sequence in C , the diagram, −−→ϕ3 −−→ϕ2 −−→ϕ1 obtained from (1),

(ϕ1ϕ2ϕ3)∗ 8 fN ppp NNN ppp NNN ∼ppp N∼NN ppp NNN ppp NN ϕ3∗ (ϕ1ϕ2)∗ (ϕ2ϕ3)∗ ϕ1∗ ◦ fM 8 ◦ MMM qqq MMM qqq M∼MM q∼qq MMM qqq MM qqq ϕ ϕ ϕ 3∗ ◦ 2∗ ◦ 1∗ 120 is commutative. 7.1. 91

(3) (Id)∗ = Id.

(4) The isomorphism ψ ϕ ∼ (ϕ ψ) of (1) has the property that ∗ ◦ ∗ −→ ◦ ∗ if either ψ or ϕ is the identity, the isomorphism also becomes the identity.

Problem 1. Given ξ, η C and a morphism ∈ S u′ : ξ′ = ϕ∗(ξ) ϕ∗(η) = η′ in C → S ′ when can we say that u = ϕ (u) for a morphism u : ξ η in C ? ∗ → S Problem 2. Given ξ C , when is ξ of the form ϕ (ξ) for a ξ C ? ′ ∈ S ′ ′ ∗ ∈ S We shall now obtain certain conditions necessary for the above two questions to have an answer in the aﬃrmative. For Problem 1. Suppose u : ξ η in C such that u = ϕ (u) : ξ ∃ → S ′ ∗ ′ → η′. Assume that S ′ S ′ = S ′′ exists in C ; consider the sequence: ×S

p1 o S o ϕ S ′ o S ′′ p2

Let ϕ p = ϕ p = ψ and denote by ξ , η the elements ψ (ξ), ◦ 1 ◦ 2 ′′ ′′ ∗ ψ (η) of C . Consider the morphism p (u) : p (ξ ) p (ξ ); accord- ∗ S ′′ 1∗ 1∗ ′ → 2∗ ′ ing to our assumption p ϕ ∼ ψ . There is then a (dotted) morphism 121 i∗ ◦ ∗ −→ ∗ making the diagram

p1∗ (u′) p1∗(ξ′) / p1∗(η′)

≀ ≀ ξ′′ = ψ∗(ξ) ____ / ψ∗(η) = η′′ commutative. In the following we also denote this morphism by p1∗(u′), i.e., we identify p1∗(ξ′) with ξ′′ and p1∗(η′) with η′′. We introduce similarly p2∗(u′). It is clear that with these notations we must have = p1∗(u′) p2∗(u′); 92 7. The Technique of Descents and Applications

in fact α: both equal ψ∗(u). If this necessary condition is also suﬃcient (as they turn out to be in some cases) we say that ϕ : S S is a morphism of descent. ′ → For Problem (2). Suppose ξ C , with ϕ (ξ) = ξ . Then, with the ∃ ∈ S ∗ ′ above notations, we have:

p∗(ξ′) ∼ ψ∗(ξ) ∼ p∗(ξ′), 1 −→ ←− 2 122 i.e., an isomorphism α : p (ξ ) p (ξ ) making the diagram 1∗ ′ → 2∗ ′ α p1∗(ξ′) / p2∗(ξ′) F FF xx FF xx FF∼ ∼ xx FF xx FF xx F# |xx ψ∗(ξ)

commutative. We shall now get conditions on α. Assume that S ′′′ = th S ′ S ′ S ′ exists in C ; let qi be the i projection S ′′′ S ′; also let ×S ×S → p ( j i) be the morphism (q , q ) : S S . Consider the se- ji ≥ i j S ′′′ → ′′ quence: p p1 21 o o S o ϕ S ′ o S ′′ o p S ′′′. p2 o 31 p32 Let λ = ϕ p p = ϕ p p = : S S . We then have ◦ 1 ◦ 21 ◦ 1 ◦ 31 ··· ′′′ → commutative diagrams of the type:

q1∗(ξ′) o ∼ p21∗ p1∗(ξ′) x LL xx LL ∼xx LL∼L xx LLL {xx p21∗ (α) % λ∗(ξ) l p21∗ ψ∗(ξ) cFF ∼ 9 FF rrr FF∼ ∼rr FF rrr F rr q2∗(ξ) o ∼ p21∗ p2∗(ξ′)

123 We also denote by p (α) the (dotted) morphism q (ξ ) q (ξ ) 21∗ 1∗ ′ → 2∗ ′ 7.1. 93 which will make the diagram

q1∗(ξ′) o ∼ p21∗ p1∗(ξ′) p (α) 21∗ q2∗(ξ′) o ∼ p21∗ p2∗(ξ′) commutative. Using this diagram and the “large” diagram above we see that q1∗(ξ′) q qqq q∼qq xqqq λ∗(ξ) p21∗ (α) fM MMM MM∼ MMM MM q2∗(ξ) is commutative. We make similar conventions and get similar commutative diagrams for p32∗ (α) and p31∗ (α). Then we must have = p32∗ (α)p21∗ (α) p31∗ (α)

(the so-called “cocycle” condition). Finally, if ∆ : S ′ S ′′ = S ′ S ′ is → ×S the diagonal, by the assumptions made at the beginning we must have 124

∆∗(α) = identity.

If these necessary conditions are also suﬃcient and if ϕ is also a morphism of descent, we say that ϕ is a morphism of eﬀective descent. An α of the above type is then called a descent-datum on ξ′.

Remark. Let C be the category of preschemes; if we agree that we take S and S locally noetherian but if we don’t assume S S of ﬁnite type ′ ′ → then it may very well happen that S ′′ = S ′ S ′ is not locally noetherian ×S (e.g. S = Spec A, S ′ = Spec A, A a noetherian local ring). We shall deal with this diﬃculty in the sections 7.2.1.3, 7.2.1.4, 7.2.1.5. b 94 7. The Technique of Descents and Applications

Example (1).

Proposition 7.1.1. A faithfully ﬂat ring-homomorphism ϕ : A A is → ′ a morphism of eﬀective descent.

Proof. A (a) We shall ﬁrst show that ϕ is a morphism of descent. Let M, N be A-modules and u′ : M A′ N A′ be an A′-linear map. Consider ⊗A → ⊗A the commutative diagram

p M A = M 1 M A A = M M / ′ ′ / ′ ′ ′′ ⊗A p2 ⊗A ⊗A

u′ p1∗ (u′) p2∗(u′)

p N A = N 1 N A A = N N / ′ ′ // ′ ′ ′′ ⊗A p2 ⊗A ⊗A

125 We have to show that if p (u ) = p (u ) then an A-linear map 1∗ ′ 2∗ ′ ∃ u : M N such that u′ = u IA . This will follow, if we show that → ⊗A ′

(i) for any M CA, M M A′ is an injection, ∈ → ⊗A

p1 / (ii) for any N CA, N / N′ / N′′ is exact, and ∈ p2 (iii) u (M) ker(N , p , p ). ′ ⊂ ′ 1 2

(i) We know that A′ is a direct factor of A′ A′ by means of the map ⊗A a′ a′ 1; thus, for any M CA, M A′ M A′ A′ makes 7→ ⊗ ∈ ⊗A → ⊗A ⊗A M A′ a direct summand of M A′ A′ and is in particular an in- ⊗A ⊗A ⊗A jection. As A′ is faithfully A-ﬂat it follows that M M A′ is an → ⊗A injection.

(ii) We have to prove that the sequence

p1 / N / N′ / N′′ p2 7.1. 95

is exact. It suﬃces to prove this after tensoring the sequence with a ring B faithfully ﬂat over A; note that we get a similar situation with the pair B B′ = B A′ as with A A′. Take for B the ring A′ → ⊗A → itself. But then A′ A′ A′ = A′′ admits a section λ : A′′ A′ → ⊗A → given by λ(a a ) = a a . We may thus assume without loss of 1′ ⊗ 2′ 1′ 2′ generality that ϕ : A A itself admits a section λ : A A. 126 → ′ ′ → Consider then the commutative diagram:

p ψ=1 ϕ N A 1 N A A N ⊗ / ′ / ′ ′ Eb ⊗A p2 ⊗A ⊗A EE EE EE EE µ=1 λ µ 1 Id . EE ⊗ ⊗ EE E" ∼ N A ∼ N / N A A′ N A′ ⊗ −→ ⊗A ⊗A −→ ⊗A

Let x′ = (xi a′) be an element of N′ = N A′ such that p1(x′) = ⊗ i ⊗A p (x ); thatP is, (x a 1) = (x 1 a ). On applying µ 1 2 ′ i ⊗ i′ ⊗ i ⊗ ⊗ i′ ⊗ we obtain: P P = (x1 λ(ai′) 1) (xi 1 ai′); X ⊗ ⊗ X ⊗ ⊗

under the identiﬁcation N A A′ ∼ N A′, this means that ⊗A ⊗A −→ ⊗A

= = (xi ai′) (λ(ai′)xi 1), i.e., x′ ψ(µ(x′)) ψ(N). X ⊗ X ⊗ ∈ (iii) By the commutativity of the diagram at the beginning of the proof, we have: p (u ) p = p u , p (u ) p = p u and by 1∗ ′ ◦ 1 1 ◦ ′ 2∗ ′ ◦ 2 2 ◦ ′ assumption p (u ) = p (u ) while for m M one has p (m) = 1∗ ′ 2∗ ′ ∈ 1 p (m) = m 1 1 M . Thus, for m M, p u (m) = p u (m) 2 ⊗ ⊗ ∈ ′′ ∈ 1 ◦ ′ 2 ◦ ′ and (iii) is proved.

(b) It remains to show that ϕ is eﬀective. Suppose N C and we have 127 ∈ A′ 96 7. The Technique of Descents and Applications

a commutative diagram:

N′ A′ = p∗(N′) ⊗A 1 7 p1 ppp ppp ppp ppp N α ′ NN ≀ NNN NNN p NN 2 NN' A′ N′ = p∗(N′) ⊗A 2

where α is an A′-isomorphism satisfying the “cocycle” condition. We want to ﬁnd an N CA such that N A′ ∼ N′ and such that ∈ ⊗A −→ρ

p∗(ρ) N A′ A′ 1 N′ A′ ⊗A ⊗A / ⊗A

(natural) α ≀ ≀

p∗(ρ) A′ N A′ 2 A′ N′ ⊗A ⊗A / ⊗A

commutes. Set N = ker(α p1 p2) CA (this choice of N is motivated by (ii) ◦ − ∈ ρ of (a)). We always have an A′-linear map N A′ N′. To show that ρ ⊗A −→ is an A′-isomorphism, it is enough to show that ρ is an A-isomorphism; and for this, we may assume, as in (a), that there is a section λ : A A ′ → for ϕ : A A . → ′ 128 For the sake of clarity and ease, we now go back to the general case and prove:

Lemma 7.1.2. If ϕ : S S admits a section σ : S S , then ϕ is a ′ → → ′ morphism of eﬀective descent.

ξ α ξ ξ Proof. Let ′ CS ′ and : p1∗( ′) p2∗( ′) be an isomorphism such ∈ = → that p32∗ (α)p21∗ (α) p31∗ (α) (notations as before). Our aim is to ﬁnd an 7.1. 97

ρ η C and an isomorphism η = ϕ (η) ξ such that the diagram ∈ S ′ ∗ −→ ′

p1∗(η′) ∼ / p1∗(ξ′) p1∗(ρ) rrr ∼rrr rrr yrr η′′ α eLL ≀ LLL LL∼ LLL L p2∗(η′) ∼ / p2∗(ξ′) p2∗(ρ) is commutative. We have the diagram:

ϕ p1 o S ′ S ′ S o S ′ o ×S ];; O p2 O ;; ;; ;; σ σ 1 Id. ;; ×S ;; ;; P S o S S ′ ∼ S ′ ×S −→ with p (σ 1) = σ ϕ and p (σ 1) = identity. Hence, if η = σ (ξ ) 129 1 ◦ × ◦ 2 ◦ × ∗ ′ we have η = ϕ σ (ξ ) = (p (σ 1)) (ξ ) = (σ 1) p (ξ ) and ( ) ′ ∗ ∗ ′ 1◦ × ∗ ′ × ∗ 1∗ ′ ∗∗∗∗∗ ∗ (σ 1) p (ξ ) = ξ . We then get a θ : η ∼ ξ , namely, θ = (σ 1) (α). × ∗ 2∗ ′ ′ ′ −→ ′ × ∗ We obtain thus a β : p (η) p (η ) which makes the diagram 1∗ → 2∗ ′

p1∗(ξ′) ∼α / p2∗(ξ′) O O

p∗(θ) p∗(θ) ≀ 1 2 ≀

p∗(η′) ∼ / p∗(η′) 1 β 2 commutative. However, we do not, in general, have a commutative dia- 98 7. The Technique of Descents and Applications

gram p∗(η′) ∼ / p∗(η′) 1 β 2 cHH v; HH vv HH vv HH vv ∼HH ∼vv (natural) HH vv (natural) HH vv H vv η = (p ϕ) (η) ′′ 1 ◦ ∗ 130 Therfore, we want to modify θ : η ξ to a ρ : η ∼ ξ by means ′ → ′ ′ −→ ′ of an automorphism γ : η η (i.e., ρ = θ γ) such that the diagram ′ → ′ ◦

p1∗(η′) ∼ / p2∗(η′) O β O

p1∗(γ) p2∗(γ)

p1∗(η′) p2∗(η′) ]< A << ÒÒ << ÒÒ << ÒÒ << ÒÒ <∼< (natural) ∼ÒÒ << ÒÒ << ÒÒ << ÒÒ < ÒÒ η′′

is commutative. This ρ will satisfy our requirements. We first observe that β : p (η ) p (η ) also satisfies the cocycle condition. We may 1∗ ′ → 2∗ ′ consider β as an element of Aut(η′′). To find a γ in Aut η′ such as above, = 1 i.e., such that β p2∗(γ) p1∗(γ)− we have only to prove the following: ◦

Lemma 7.1.3. Let η C and consider the “complex” ∈ S

p∗ p 31 1∗ p / (Aut) Aut η / Aut η 21∗ Aut η ′ / ′′ / ′′′ p2∗ / p32∗

(notations as before). If there is a section σ : S S then → ′ H1(Aut) = (e). 7.1. 99

131 Proof. By definition, the 1-cocycles are elements β Aut η for which ∈ ′′ the cocycle condition is satisfied and the 1-coboundaries are those β ∈ Aut η which are of the form β = p (γ) p (γ) 1, for a γ Aut η . ′′ 2∗ ◦ 1∗ − ∈ ′ Consider now the corresponding complex in C ; the section σ : S S → ′ defines a homotopy operator for this complex in the following manner:

p21 ϕ p1 o o p31 S o S ′ o S ′′ o S ′′′ \: O p2 O o ······ : p32 ······O :: :: :: σ σ 1 σ 1 1 Id. :: × × × :: : ϕ p1 ∼ ∼ S o S S ′ S ′ oo S S ′ S ′ S ′′ ×S −→ p2 ×S ×S −→

with p (σ 1) = σ ϕ, p (σ 1) = Id., 1 ◦ × ◦ 2 ◦ × (σ 1) p = p (σ 1 1), (σ 1) p = p (σ 1 1), × ◦ 1 21 ◦ × × × ◦ 2 31 ◦ × × and p (σ 1 1) = Id . 32 ◦ × × If β is a 1-cocycle, set γ = (σ 1) (β). We then have: × ∗ 1 1 p∗(γ) p∗(γ)− = p∗(σ 1)∗(β) (p∗(σ 1)∗(β))− 2 ◦ 1 2 × ◦ 1 × 1 = (((σ 1) )∗(β)) (((σ 1) )∗(β))− × p2 ◦ × p1 1 = (σ 1 1)∗ p∗ (β) ((σ 1 1)∗ p∗ (β))− × × 31 ◦ × × 21 1 = (σ 1 1)∗(p∗ (β) p∗ (β)− ) × × 31 · 21 = (σ 1 1)∗ p∗ (β) = β. × × 32 Q.E.D.

Example 2. C = (Sch), S , S (Sch); C (resp. C ) is the category of 132 ′ ∈ S S ′ quasi-coherent OS -(resp. OS ′ -)-Modules. Proposition 7.1.4. If ϕ : S S is faithfully ﬂat, quasi compact then ϕ ′ → is a morphism of eﬀective descent. 100 7. The Technique of Descents and Applications

1 (A morphism f is quasi-compact if f − (U) is quasi-compact for every quasi-compact: U). (Note: We do not assume here that ϕ is a morphism of ﬁnite type-our hypothesis is much weaker).

Proof. Case (a) S , S ′ both affine. The proposition in this case follows from Proposition 7.1.1. ffi Case (b) S affine. There is a finite a ne open cover (S i′)i I of S ′ = ffi ∈ (ϕ quasi-compact). Set S S i′. Then S is a ne and the morphism i I ψ `∈ S S (composite ofe the natural map eθ : S S and ϕ) is also −→ → ′ faithfully flat. Now consider the diagram: e e

q31 q1 o q32 S oo S S o S S S q2 ×S o ×S ×S {{ q21 ψ {{ e e e e e {{ e {{ }{{ S θ (natural) aD DD DD ϕ D DD p31 D p1 o o S ′ S ′ p32 S ′ S ′ S ′ S ′ o S o S S p2 × o × × p21

133 If ξ, η C and u : ξ η is a C -morphism then u defines a ∈ S ′ → ′ S ′ ′ C -morphism u : ψ (ξ) ψ (η). And from the equality p (u ) = p (u ) S ∗ → ∗ 1∗ ′ 2∗ ′ and the commutativity of the above diagram follows q∗(u) = q∗(u). By e e 1 2 case (a), a u : ξ η, a CS -morphism, such that u = ψ∗(u); it is ∃ = → e e immediate that u′ ϕ∗(u); in fact this holds in every open set S i′. If ξ C and α is a descent-datum for ξ , α definede in the obvious ′ ∈ S ′ ′ way is a descent-datum for θ (ξ ) = ξ and again by case (a), ξ C ∗ ′ ∃ ∈ S such that ξ ∼ ψ (ξ). It is easy to see that ξ ∼ eϕ (ξ). ←− ∗ e ′ ←− ∗ Casee (c). S , S ′ arbitrary. Let (T j) be an affine open cover of S and = 1 = T ′j ϕ− (T j). Then the T ′j form an open cover of S ′. Let T T j j and θ : T S the natural map. If F and G are quasi-coherentÒ - → S Modules denote by Φ(S ) the group HomS (F , G ), by Φ(S ′) the group 7.2. 101

= HomS ′ (F ′, G ′) and so on. If we set T ′ T ′j and θ : T ′ S ′ is the j → natural map, we have a commutative diagram:`

p∗ Φ ϕ∗ Φ 1 Φ (S ) / (S ′) / (S ′′) p2∗

p∗ Φ Φ 1 Φ (T) / (T ′) / (T ′′) p2∗

Φ Φ (T T) / (T ′ T ′) ×S S×′ ff By case (b), the morphisms T ′j T j are morphisms of e ective 134 →= = descent and therefore clearly so is T ′ T ′j T j T. It follows j → j ` ` that the second row is exact while Φ(T T) Φ(T ′ T ′) is an injection. ×S → S×′ As it is clear that θ : T S is a morphism of (effective) descent, the → first column is also exact. Usual diagram-chasing shows that the first row is also exact, in other words, that ϕ is a morphism of descent. We show similarly that ϕ is also effective. Q.E.D.

Example 3. We shall not discuss the problems (1) and (2) in their general form, in this case. However, if we restrict ourselves to the case of preschemes affine over S , S ′ then a faithfully flat, quasi-compact morphism S S is a morphism of effective descent. In fact, such ′ → preschemes are defined by quasi-coherent OS -and OS ′ -Algebras and we are essentially back to example (2). [It is not difficult to see that u (resp. F ) in example (2), problem (1) (resp. problem (2)) is an OS -Algebra homomorphism (resp. an OS -Algebra) provided we start with OS -Algebras and homomorphisms of OS -Algebras instead of OS - Modules (look at the proof of Proposition 7.1.1)].

7.2

Let S be a locally noetherian prescheme and X, Y be ´etale coverings of S . Let S ֒ S be a closed subscheme of S deﬁned by a Nil-Ideal F 0 → 102 7. The Technique of Descents and Applications

of O (F is a Nil-Ideal of O F nil-radical of O , s S ; S S ⇔ s ⊂ sS ∀ ∈ 135 this is equivalent to saying that S 0 and S have the same base space or Φ (S ) = S ). We then have an obvious functor (E t/S ) (E t/S ) 0 red red −→ 0 given by X X S 0 = X0. We assert that Φ deﬁnes an equivalence of 7→ ×S categories in the sense of the following

7.2.1 Main Theorem

(a) Hom (X, Y) ∼ Hom (X , Y ) S −→ S 0 0 0 (b) If X (E t/S ), then X (E t/S ) 0 ∈ 0 ∃ ∈

and an isomorphism X S 0 ∼ X0. ×S −→ The theorem will follow from the series of lemmas below.

Lemma 7.2.1.1. Let S be locally noetherian and f : X S a separated → morphism of ﬁnite type. Then f is an open immersion f is ´etale and ⇔ universally injective.

(Note: Universally injective - radiciel = injective + radiciel residue ﬁeld extensions).

Proof. clear. ⇒ : f is an open map and thus by passing to an open sub-scheme ⇐ of S , we may assume that f (X) = S . Clearly f is then a homeomor-

phism onto. Since étale remains étale under base-change, f(S ′) is still a homeomorphism onto, for any base-change S S ; it follows that f ′ → is universally closed and thus proper. By Chevalley’s lemma we deduce that f is finite. Suppose then that X = Spec A , where A is a coherent 136 OS -Algebra. As f is étale and universally injective, it follows that A is locally free of rank 1 at every point of S , hence X ∼ S . Q.E.D. −→ Lemma 7.2.1.2. Let f : X S be a separated étale morphism of finite → type (S locally noetherian). Suppoe Y S is a morphism of finite type → and Y0 = V(F ) is a closed subscheme of Y defined by a Nil-Ideal F of 7.2. 103

O . Let α : Y X be an S -morphism. Then a unique S -morphism Y 0 0 → ∃ α : Y X making the diagram →

α0 X o Y _@ _0 @@ @ α f @@ @@ @@ @ SYo commutative.

Proof. By making the base-change Y S and considering the mor- → phism Y0 X Y we are reduced to proving the lemma in the case → ×S α S = Y. That is, given a Y-morphism Y 0 X, we want to extend α to 0 −−→ 0 a section α of X Y. → f Now suppose σ : Y X is a section of X Y; f σ = identity is → −→ · étale and f is étale, hence σ is étale. Also f σ is a closed immersion · and f is separated and thus σ is a closed immersion. Y being locally noetherian its connected components are open and we may then assume Y connected. σ(Y) will then be a connected component of X, isomorphic to Y under σ; in view of lemma 7.2.1.1 then, the sections of f are in (1 1) correspondence with the components X of X such that f X is 137 − i | i surjective and universally injective on Xi to Y. Now Y0, Y have the same base-space and hence so have X0 = X Y0 ×Y and X; also, the morphism f : X Y obtained from f by the (Y0) 0 → 0 base-change Y Y is topologically the same map as f ; hence, f is 0 → universally injective and surjective on Xi to Y f(Y0) is so on (Xi)0 to f ⇔ Y , i.e., the set of sections of X Y is the same as the set of sections of 0 −→ f(Y0) X0 Y0 and the (1 1) correspondence is given in the obvious way. −−−→ − α0 The lemma now follows from the fact that Y0 X can be considered f(Y ) −−→ as a section for X 0 Y . Q.E.D. 0 −−−→ 0 This lemma proves part (a) of Theorem 7.2.1. To prove part (b) we need a generalisation of the lemma. 104 7. The Technique of Descents and Applications

Warning. In the following sections 7.2.1.3–7.2.1.5 until the proof of (b) of Theorem 7.2.1, we drop the assumptions made in 6.1 that the preschemes are locally noetherian and the morphisms are of ﬁnite type. We begin by making the

Deﬁnition 7.2.1.3. A morphism f : X Y of preschemes is of ﬁnite → presentation if (i) f is quasi-compact

(ii) ∆ : X X X is quasi-compact, → ×Y (iii) for every x X, a nbd. U of x X and a nbd. V of f (x) in ∈ ∃ x ∈ f (x) 138 Y such that f (U ) V and Γ(U , O ) is a Γ(V , O )-algebra x ⊂ f (x) x X f (x) Y of ﬁnite presentation; i.e., Γ(Ux, OX) Γ(V f (x), OY )[T1,..., Ts]/a where a is a ﬁnitely generated ideal of a polynomial algebra Γ(V f (x), OY )[T1,..., Ts].

An example of a morphism of finite presentation is a finite type sep- f arated morphism X Y with Y locally noetherian. −→ Lemma 7.2.1.4. Let U be a noetherian prescheme and (Aα)α I an in- ∈ ductive family of quasi-coherent OU -Algebras. Let Vα be the U-affine prescheme Spec Aα defined by Aα, α I, and V = Spec A where A = ∀ ∈ lim Aα. α −−→ (a) Suppose we have a diagram:

Xα Yα Xβ Xβ X Y 0 / .. 00 / . 0 // . 00 / .. 0 // . 00 / . × × . Ø U o Vα o Vβ o V′

where Xα,Yα are ﬁnitely presented preschemes over Vα α I ∀ ∈ such that β α,Xβ = Xα Vβ ∀ ≥ V×α

Yβ = Yα Vβ; let X = Xα V, Y = Yα V V×α V×α V×α 7.2. 105

Then, lim HomV (Xα, Yα) ∼ HomV (X, Y). α α −→ ←−− (b) Suppose is a ﬁnitely presented prescheme over V. Then, for all 139 large α I, Xα/Vα, ﬁnitely presented, such that (i) β α, ∈ ∃ ∀ ≥ Xβ Xα Vβ and (ii) X Xα V. ≃ V×α ≃ V×α

(c) With assumptions as in (a), if the Yα′ s and Y are locally noetherian, X/Y étale covering Xα/Yα étale covering already, for ⇒ some α I. ∈ Proof. (a) We shall be content with merely observing that it is clear how to prove this in case everything is affine. (b) Since U is noetherian, in view of (a) we may assume that U = Spec C′, Vα = Spec Aα, V = Spec A where A = lim Aα. α Case (1). Assume X = Spec B. ←−− X/V finitely presented B/A finitely presented; let ⇒

B A[T1,..., Ts]/a, a

finitely generated, say, by P ,..., P A[T ,..., T ]. Choose α so 1 n ∈ 1 s large that the coefficients of the Pi come from Aα. Consider the ideal aα = (P1,..., Pn) of Aα[T1,..., Ts] and set Bα = Aα[T1,..., Ts]/aα and Xα = Spec Bα; for β α, set Xβ = Xα Vβ. ≥ V×α Case (2). X arbitrary. r ffi Let (Xi)i=1 be a finite a ne open cover of X. In view of case (1) and (a), it is enough now to show that each X X is quasi-compact. But i ∩ j the underlying space of Xi X j is that of X (Xi X j) as is seen from ∩ (X×X) ×V ×V the commutative diagram: 140

X (Xi X j) (X×X) ×V / Xi X j ×V ×V

(natural)

∆ X X X / ×V 106 7. The Technique of Descents and Applications

As ∆ is a quasi-compact morphism the “top”-morphism is also quasi-compact; since Xi X j is quasi-compact, our result follows. ×V

(c) Here again we give only some indications and leave the details to the reader. To start with we may assume everything affine. Say U = = = = α = α Spec C, Vα Spec Aα, V Spec A, Xα Spec B1 , Yα Spec B2 , X = Spec B1, Y = Spec B2. We also note that we can assume Vα = Yα, and V = Y. One checks that ΩB ∼ lim ΩBα/Bα , say. Now if X/Y is 1/B2 α 1 2 ←− α Ω = Ω α α 0 unramified then B1/B2 0. But since←−− 0 0 is a finite B -module B1 /B2 1 one has Ω α α = 0 for large α. Also, if X/Y is finite and flat then B is B1 /B2 1 α a locally free B2-module of finite rank but then the same is true for B1 α with respect to B2 for large α. Assertion (c) follows. Q.E.D.

141 Lemma 7.2.1.5. Suppose we have a commutative diagram:

= α0 X o X(T) X T o Y = V(F ) ×S 0 _

S o T o Y

with: S noetherian, f étale and separated, T S affine, Y T → → finitely presented and Y = V(F ) where F is a Nil-Ideal of O . Then 0 Y ∃ a unique α : Y X keeping the diagram still commutative. → (T) f

Proof. Since T S is aﬃne, T = Spec B, where B is a quasi-coherent → OS -Algebra. Write B = lim Bλ where the B′ s are OS -sub-Algebras, λ λ of ﬁnite type, of B. Set Tλ←−−= Spec Bλ. By (a) and (b) of Lemma 7.2.1.4 7.2. 107 we have, for large λ, a situation of the following type:

α0,λ X Tλ Y ×S ko 0,λ T P fLL vv L LLL vv H αλ? LL vv C LL vv LL zvv = = SX o X T Y0 V(F ) Y0,λ Y ×S Y×λ ; _ H R Spec Bλ = Tλ o V Yλ fM fMM uu MM MM uu MMM MM uu MM MMM uu MMM MM uu M = z Spec = T Y Yλ T S S o B o T×λ

For large λ, inductive families (Yλ), (Y ,λ) and morphisms α ,π : 142 ∃ 0 0 Y0,λ X Tλ such that lim α0,λ = α0 (see diagram). Also it is not → ×S λ very difficult to see (using−−→ arguments similar to those in Lemma 7.2.1.4) that the Y0,λ are subschemes defined by Nil-Ideals F0,λ of OYλ . Now Tλ is of finite type over S and is therefore noetherian; also, since Y → T is of finite presentation so is Yλ Tλ and is in particular of finite → type. Hence Lemma 7.2.1.2 applies and we get a (dooted) morphism αλ : Yλ X Tλ keeping the diagram commutative. Passing to the → ×S limit with respect to λ we get an α : T X T making the diagram → ×S commutative. The uniqueness of α follows from Lemma 7.2.1.2 and (a) ofLemma7.2.1.4. Q.E.D.

7.2.1.6 Proof of (b) of Theorem 7.2.1

Case 1. Assume S = Spec A, A a noetherian, complete local ring; S 0 is f then given by Spec(A/J) = Spec A , J a nil-ideal of A. Let X 0 S 0 0 −→ 0 be the given ´etale covering; assume that if s0 S 0 is the unique closed ∈ 1 point, the residual extensions at the points of f0− (s0) are all trivial. Then X0 is given by Spec B0 where B0 is a ﬁnite direct-product of = r = = r copies of A0, say B0 i=1 A0; X Spec B, with B i=1 A is then a solution for our problem.L L Case 2. From among the assumptions in case 1, drop completeness of A 143 108 7. The Technique of Descents and Applications

1 and the triviality of residual extensions along the ﬁbre f0− (s0).

faithfully flat S = Spec A o Spec A = S quasi-compact ′ ′ Y33 33 33 33 33 33 33 33 33 X = X S 3 X0 o 0′ 0 0′ 3 S×0 33 33 3 f0 f0′ 33 3 + K A′ Spec A0 = S 0 o S ′ = Spec 0 JA′ In this case we can choose a complete, noetherian, local overring A′ of A such that the following situation holds: and such that if s S 0′ ∈ 0′ is the unique closed point, then the residual extensions along the fibre 1 f − (s ) are all trivial. By case 1, an étale covering X /S such that ′0 0′ ∃ ′ ′ ∼ X0′ X′ S 0′ . ←− S×′ 144 We have then the following situation: 7.2. 109 ′′ 0 S 0 × S ′ 0 S 0 × S ′′′ 0 S = ′′′ 0 S ′ S S × ′ ) S ′ 0 21 31 32 S × p p p X ′ ( S ∗ 2 p = r r r ′′′ r r S r r r r r o o o r ′ 0 r x r S 0 ) × S ′ 1 2 ′ 0 31 p p X p S ( ∗ 2 = p t % ′′ t 0 J t J S t J t J J t J t J t o o o J t J t ′ J t J y t S J ) S × 1 2 ′ 0 ′ p p X S ( ∗ 1 = " p F ′′ F F S F F F F F F F 1 2 F ) o o ′ ′ p p 0 f ′ ′ 0 0 X ( X S ∗ 1 p o o o o 0 f ′ ′ 0 0 & E & X & S X S & % & & & & & & & & & & & & & & & & & o & & & & S & S 110 7. The Technique of Descents and Applications

Since X0′ /S 0′ “comes from below”, it clearly has a descent-datum α : p (X ) ∼ p (X ) satisfying the cocycle condition. We want to 0 1∗ 0′ −→ 2∗ 0′ “lift” this isomorphism α to an α : p (X ) p (X ). For this one 0 1∗ ′ → 2∗ ′ may be tempted to use part (a) of Theorem 7.2.1 which we already have proved. But observe that we are now in a type of situation we anticipated while making the remark preceding Proposition 7.1.1 – we cannot make

sure that S ′′, S 0′′ are locally noetherian and hence cannot apply part (a). It is here that Lemma 7.2.1.5 comes to our rescue. By using this lemma 145 in the obvious way we get an isomorphism α : p (X ) ∼ p (X ); the 1∗ ′ −→ 2∗ ′ assertion of uniqueness in this lemma proves that the α we obtained satisfies the cocycle condition. As S S is faithfully flat and quasi- ′ → compact, it is a morphism of effective descent for étale coverings (an easy corollary to Proposition 7.1.4) and thus an étale covering X/S ∃ such that X′ ∼ X S ′. Therefore ←− ×S

∼ X0′ X′ S 0′ X S ′ S 0′ (X S 0) S 0′ . ←− S×′ ≃ ×S S×′ ≃ ×S S×0

Also, by construction, the descent-datum for X′/S ′ goes down to that for X′ /S ′ . It follows that X S 0 ∼ X0. 0 0 ×S −→ Case 3. S arbitrary. By part (a), which we have already proved, it is enough to prove the existence of an étale covering X/S (with the required property) locally. Let s S and U = Spec A be an affine open neighbourhood of s. We ∈ may assume S = Spec A, A noetherian. The local ring As is given by As = lim f A A f ; we then have S Spec A f Spec As and the given ∈ ← ← −−→ f (s),0 étale covering X0 over S 0 = Spec(A/J), J a nil-ideal, defines an étale covering Xs,0 over Spec(As/JAs) = Spec(A/J)s and then, by case 2, an étale covering X over Spec A . By (b) and (c) of lemma 7.2.1.4, f A, s s ∃ ∈ 146 f (s) , 0 and an étale covering X f / Spec A f such that Xs ∼ X f As. The ←− A⊗f covering X f is a solution for our problem in the neighbourhood Spec A f of s. Q.E.D.

Remark. The Theorem 7.2.1 shows that if S 0 ֒ S is such that (S 0)red Φ → = S then the natural functor (E t/S ) (E t/S ) is an equivalence; in red −→ 0 7.2. 111 particular, it proves that π (S , s) ∼ π (S, s). 1 red −→ 1 f Proposition 7.2.2. Let S ′ S be a faithfully ﬂat, quasi-compact, radi- −→ Φ ciel morphism. Then the natural functor (E t/S ) (E t/S ) is an equiv- −→ ′ alence.

∆ Proof. Consider the diagonal S ′ S ′′ = S ′ S ′. −→ ×S Since f is radiciel ∆(S ′) = S ′′ (Cor. (3.5.10) – EGA, Ch.I), i.e. S ′ is a closed subscheme of S ′′, having the same base-space.

(a) Given X, Y (E t/S ), set X′ = X S ′, Y′ = Y S ′. ∈ ×S ×S

To prove Hom (X, Y) ∼ Hom (X , Y ). S −→ S ′ ′ ′ Since f : S S is faithfully ﬂat, quasi-compact it is a mor- ′ → phism of descent for ´etale coverings. We have thus only to show that p1 = , o S ′′ S ′ S ′ if u′ HomS ′ (X′ Y′), and if S ′ o S are the canoni- ∈ p2 × cal projections, then, considered as morphisms from X′′ = X′ S ′′ to S× = ′ Y′′ Y′ S ′′, p1∗(u) and p2∗(u) are equal. We have the diagram: 147 S×′

= ∆ S o S o S ′′ S ′ S ′ o S G ′W. o ×S G ′W. .. ÓA ];; .. . ÓÓ ;; . .. ÓÓ ;; .. . ÓÓ ;; . .. Ó ; .. . Ó p (u ) ; . ÓÓ 1∗ ′ ; u′ / u′ X′ / Y′ X′′ / Y′′ X′ / Y′ p2∗(u′)

In view of our remarks about ∆, and Theorem 7.2.1, it is enough to ∆ = ∆ ∆ ∆ show that ∗(p1∗(u′)) ∗(p2∗(u′)). But each of ∗(p1∗(u′)), ∗(p2∗(u′)) is equal to u′ considered as morphisms form X′ to Y′.

(b) Given X (E t/S ), to show that X (E t/S ) such that X ∼ ′ ∈ ′ ∃ ∈ ′ ←− X S ′. ×S 112 7. The Technique of Descents and Applications

For this, again it is enough to ﬁnd a descent-datum α : p (X ) ∼ 1∗ ′ −→ p2∗(X′). We have the diagram:

p31 p1 o p o oo 32 S o S ′ o S ′′ jU p21 S ′′′ p2 UU L ∆ O ÙD Z55 UUU e LL 2 Ù 5 UUUU LL diagonal ÙÙ 55 UUUU LL Ù 5 ∆ UUU LL Ù 5 1 UUU LL Ù 5 diagonal UUU LL Ù UUUUL p∗(X′) p∗(X′) S 1 2 O ′

X′

= ∆ = ∆ X′ 1∗ p1∗(X′) 1∗ p2∗(X′)

148 in view of our remarks about ∆ : S S and Theorem 7.2.1, the 1 ′ → ′′ identity morphism i : X = ∆ p (X ) ∆ p (X ) = X , lifts to an ′ 1∗ 1∗ ′ → 1∗ 2∗ ′ ′ isomorphism α : p (X ) p (X ). The other diagonal morphism ∆ : 1∗ ′ → 2∗ ′ 2 S ′ S ′′′ = S ′ S ′ S ′ also imbeds S ′ as a closed subscheme of S ′′′ → ×S ×S having the same base-space. Then one checks easily that α satisﬁes the cocycle condition again by using Theorem 7.2.1. Q.E.D.

Proposition 7.2.2 simply says that if S S is faithfully ﬂat, quasi- ′ → compact and radiciel, s S and s S its image, then π (S , s ) ∼ ′ ∈ ′ ∈ 1 ′ ′ −→ π1(S, s).

7.3

Let k be an algebraically closed field and X, Y be connected k-preschemes. Suppose X is k-proper and Y locally noetherian. Let a X, b Y ∈ ∈ be geometric points with values in an algebraically closed field extension K of k. Consider a geometric point c = (a, b) X Y over a and ∈ ×k b. We claim first that X Y is connected. Since Y is connected and ×k 7.3. 113

X Y Y is proper, it is enough to show that the ﬁbres of X Y Y ×k → ×k → are connected and for this one has only to show that for any ﬁeld k k ′ ⊃

(*) X k′ is connected. ×k

The question is purely topological and we may assume X = Xred. Looking at the Stein-factorisation X Spec Γ(X, O ) k it follows 149 → X → (by making use of 6.3.1.1 and the ﬁniteness theorem) that Γ(X, OX) = k, since X is connected and k algebraically closed. On the other hand, if X′ = X k′, one has Γ(X′, OX ) = Γ(X, OX) k′ (ﬂat base-change) hence ×k ′ ⊗k = k′; again looking at the Stein-factorisation we see that X′ is connected 6.3.1.1. We can then form π1(X Y, c) and form the product of the natural ×k maps π1(X Y, c) π1(X, a), and π1(X Y, c) π1(Y, b). We then have ×k → ×k → the

Proposition 7.3.1. With the assumptions made above π1(X Y, c) ∼ ×k −→ π (X, a) π (Y, b). 1 × 1 Proof. We may assume X = Xred. Case (1). Assume K = k. The morphism X Y Y is proper and separable and the ﬁbre over ×k → b Y is to X k = X. We then get the exact sequence: ∈ ≃ ×k

π1(X, a) π1(X Y, c) π1(Y, b) e (Theorem 6.3.2.1). → ×k → → But the ﬁbre over b, namely, X k = X is imbedded in X Y and ×k ×k p the composite X = X k ֒ X Y 1 X is identity. This means that ×k → ×k −−→ continuous sections for the homomorphism π1(X, a) π1(X Y, c). ∃ → ×k Thus, we get:

(i) e π1(X, a) π1(X Y, c) π1(Y, b) e → → ×k → → is exact, 114 7. The Technique of Descents and Applications

(ii) and the sequence splits (also topologically). 150

Case (2). K arbitrary. The reasoning as in case (1) gives an isomorphism

π1(X Y K, c′) ∼ π1(X K, a′) π1(Y K, b′) ×k ×k −→ ×k × ×k

where c′, a′, b′ are points of (X Y) K, X K, Y K respectively, above ×k ×k ×k ×k c, a, b. The theorem then is a consequence of the following:

Proposition 7.3.2. Let k be an algebraically closed ﬁeld and X k → a proper connected k-scheme. Let k′ be an algebraically closed ﬁeld extension of k and let a′ X k′ be any geometric point. If a X be the ∈ ×k ∈ image of a′, then π1(X k, a′) ∼ π1(X, a). ×k −→

Proof. That X k′ is connected follows from assertion (*) of 7.3; the ×k same assertion also proves that if Z is a connected étale covering of X then Z′ = Z k′ is a connected étale covering of X′ = X k′. In other ×k ×k words, π1(X k′, a′) π1(X, a) is surjective (cf. 5.2.1). We shall prove ×k → that it is injective by showing that every connected étale covering Z′ of X′ = X k′ is of the form Z k′ for some Z (E t/X). ×k ×k ∈ By Lemma 7.2.1.4 a k-algebra A of finite type, A k , and Z ∃ ⊂ ′ A ∈ 151 (E t/(X A)) such that Z′ ∼ ZA k. If Y = Spec A, y Y such that ×k ←− ×A ∃ ∈ k(y) = k [since A is of finite type over (the algebraically closed) k]. One can apply case (1) of Theorem 7.3.1 to the k-rational point (a, y) X Y ∈ ×k to obtain π1(X Y, (a, y)) ∼ π1(X, a) π1(Y, y). ×k −→ ×

If ZA is deﬁned by an open subgroup H of π1 = π1(X Y, (a, y)) ×k this means that open (normal) subgroups G π (X) and G π (Y) ∃ ⊂ 1 ′ ⊂ 1 (deﬁning galois coverings X/X, Y/Y respectively) such that H G G ⊃ × ′ (i.e. such that ZA is obtained as a quotient of the galois covering X Y e e ×k e e 7.3. 115

of X Y). Let ZA be the lift of the covering ZA/(X Y) to X Y; we than × ×k ×k have the commutative diagram: e X Y ×k = }} e e == }} == }} = ~ = X Y o / Z ×k ZA A B e BB BB B X Y ×k

We claim that ZA is connected. 152 In fact, let y Y = Spec A be the generic point of Y; then k(y) = ∈ field of fractions of A k. For any y Y lying above y Y we thus ⊂ ∈ ∈ have k(y) k′ since k′ is algebraically closed. If we then apply the base- ⊂ e e change X Y k′ X Y to the étale covering ZA X Y, we obtain e ×k ×Y → ×k → ×k the étale covering ee e e ZA k′ = Z′ X′ ×Y → which is connected. It follows that ZA is connected. Thus the morphism X Y ZA is surjective and ZA is sandwiched between X Y ×k → ×k and X eY;e this implies that ZA/(X Y) is defined by an open subgroupe e ×k ×k of π1(XeY) = π1(X) G′ whiche contains π1(X Y) = G G′; i.e. ×k × ×k × ZA = X1 eY for an X1 (E t/X). We now obtain e e ×k ∈ e e e (ZA)y = the fibre of ZA over y

e = X1 Y k(y) = X1 k(ey) ×k ×Y ×k e ee e e e and Z′ = ZA k′ = (ZA)y k′ ×Y k×(y)

= (ZA)y k = X1 k′ k×(y) ×k e Q.E.D.e e

Chapter 8 An Application of the Existence Theorem

8.1 The second homotopy exact sequence 153 Let A be a complete noetherian local ring and S = Spec A; let s S be 0 ∈ the closed point of S . Let X be a proper S -scheme and set X0 = X k(s0). ×S Let a X be a geometric point of X (over some ﬁxed alge- 0 ∈ 0 0 braically closed ﬁeld Ω k(s )) and a X be its image in X. Assume ⊃ 0 0 ∈ that X0 is connected.

a0 X o X a ∈ 0 ∋ 0 proper f

s0 Spec A o k(s ) ∈ 0 Theorem 8.1.1. The sequence

e π (X , a ) π (X, a ) π (S, s ) e → 1 0 0 → 1 0 → 1 0 → is exact; and we have the isomorphism

π1(S, s0) G(k(s0)/k(s0)).

117 118 8. An Application of the Existence Theorem

(Note: Compare with Theorem 6.3.2.1). The proof of the theorem is a consequence of the following results. 154

Proposition 8.1.2. In the above Theorem 8.1.1 assume A is an artinian local ring. With the same notation, the same assertions hold.

Proof. Since the introduction of nilpotent elements does not aﬀect the fundamental groups (Theorem 7.2.1). We may assume that A = k(s0).

In this case, the assertion π1(S, s0) G(k(s0)/k(s0)) is clear. Next, p if characteristic k(s0) = p and if k′ = (k(s0)) −∞ , then the morphism Spec k(s0) Spec k′ is faithfully ﬂat, quasi-compact and radiciel and ← p hence by Proposition 7.2.2, we may replace k(s0) by (k(s0)) −∞ , i.e., we may assume k(s0) is perfect. In this case, k(s0) is the inductive limit lim ki of ﬁnite galois extensions ki of k(s0); set Xi = X ki and ai = i I k(×s0) −−→ ∈ image of a0 in Xi.

a0 X o Xi ai o X0 a ∈ ∋ ∋ 0

k(s0) o ki o k(s0)

155 By Lemma 7.2.1.4 an ´etale covering of X0 is determined by an ´etale covering of some Xi and the latter is uniquely determined modulo pas- sage to X , j i. j ≥ One thus gets the isomorphism:

π (X , a ) ∼ lim π (X , a ) 1 0 0 −→ 1 i i −−→i [The injectivity follows from the fact that for any open subgroup H of π = π1(X0, a0) there exists, by Lemma 7.2.1.4 an index i and an open subgroup H(i) of π(i) = π (X , a ) such that π/H π(i)/H(i). The sur- 1 i i ≃ jectivity follows because otherwise there would exist a set E C (π(i)) ∈ with two points a and b in E˙ such that a and b are in the same connected component of E with respect to the action of all π( j)( j i) but a and ≥ b lie in diﬀerent components with respect to the action of π; again by 8.1. The second homotopy exact sequence 119

7.2.1.4 this is impossible because a connected component of E in C (π) can be realised in some C (π( j))]. On the other hand, we assert that each Xi/X is galois and G(ki/k(s0)) ∼ Aut(X /X). −→ i In fact, suppose we have a situation of the following type: ϕ = a X o X′ X k′ a′ 0 ∈ ×k ∋

k o k′ with X universally connected over k. Then X′/X is a connected étale 156 covering. Also, we have: deg(X′/X) = rank ϕ = number of geometric 1 points in the fibre ϕ− (a0) = deg(k′/k) = number of automorphisms of k′/k number of automorphisms of X′/X number of geometric points ≤ 1 ≤ in the fibre ϕ− (a0) (because X is connected - see the proof of Lemma 4.4.1.6). Hence Aut(k /k) ∼ Aut(X /X) and X /X is galois. Now for ′ −→ ′ ′ every i I we have an exact sequence: ∈ (e) π (X , a ) π (X, a ) Aut(X /X) (e) → 1 i i → 1 0 → i → (see 5.2.6). Since each π1 is pro-finite, by passing to the projective limit we obtain an exact sequence:

(e) π (X , a ) π (X, a ) G(k(s )/k(s )) = π (S, s ) (e) → 1 0 0 → 1 0 → 0 0 1 0 →

Proposition 8.1.3. Let A be a complete, noetherian local ring and S = Spec A. Let X be a proper S -scheme such that if s S is the closed 0 ∈ point of S the ﬁbre X0 = X k(s0) is universally connected. Let a′ ×S 0 be a geometric point of X0 with values in some algebraically closed Ω k(s ). If a X is the image of a then canonically π (X, a ) ∼ ⊃ 0 0 ∈ 0′ 1 0 ←− π1(X0, a0′ ). Φ Proof. This will come from the fact that the natural functor (E t/X) −→ (E t/X0) is an equivalence. 120 8. An Application of the Existence Theorem

(a) If Z, Z (E t/X) then ′ ∈ HomX(Z, Z′) ∼ HomX (Z X0, Z′ X0). −→ 0 ×X ×X

157 In fact, let A (Z), A (Z′) be the coherent, locally free OX-Algebras deﬁning Z, Z′ over X. If M is the maximal ideal of A, set An = n+1 n+1 + A/M , Xn = X A/M , Zn = Z Xn and so on, n Z . We then ×A ×X ∈ have: ∼ HomX(Z, Z′) HomOX Alg.(A (Z′), A (Z)). −→ − Now for the OX-Modules, we have: =Γ HomOX (A (Z′), A (Z)) (X, H omOX (A (Z′), A (Z))) + ∼ Γ n 1 lim (Xn, H omOX (A (Z′), A (Z)) A/M ) −→ ⊗A ←−−n

by the Comparison theorem, (where H omMX is the sheaf of germs of OX-homomorphisms); since the A (Z′), A (Z) are coherent and locally free, we have: n+1 = H omOX (A (Z′), A (Z)) A/M H omOX (A (Zn′ ), A (Zn)); ⊗A n therefore,

Hom (A (Z′), A (Z)) ∼ lim Hom (A (Z′ ), A (Z )). OX −→ OXn n n ←−−n However, this holds also for homomorphisms of OX-Algebras, because the condition for an OX-Module homomorphism to be an OX-Algebra homomorphism can be expressed by means of commutativity in diagrams of OX-Modules. Therefore, HomX(Z, Z′) ∼ lim HomX (Zn, Z′ ). −→ n n n But by Theorem 7.2.1, the natural map Hom (Z , Z←−−) Hom (Z , Z ) Xn n n′ → X0 0 0′ 158 is an isomorphism; therefore one obtains:

lim Hom (Z , Z′ ) ∼ Hom (Z , Z′ ). Xn n n −→ X0 0 0 ←−−n Q.E.D.

(b) If Z0 is an ´etale covering over X0, then there exists an ´etale covering Z/X such that Z0 ∼ Z X0. ←− ×X For proving this we need another powerfull theorem from the EGA. 8.1. The second homotopy exact sequence 121

8.1.4 The Existence Theorem for proper morphisms Let A be a noetherian ring and I and ideal of A such that A is complete for the I-adic topology. Let Y = Spec A and f : X Y be a proper morphism. Set n+1 + → An = A/I , n Z , and Yn = Spec An, Xn = X Yn. Suppose, for every ∈ ×Y n, Fn is a coherent OXn -Module such that Fn 1 Fn OXn 1 . Then − ≃ O⊗Xn − ∼ a coherent OX-Module F such that, for each n, Fn F OXn . ∃ ←− O⊗X (For a proof see EGA, Ch. III, (5.1.4).) Coming back to the proof of (b), the étale covering Z X is 0 → 0 defined by a coherent locally free OX0 -Algebra B0. By Theorem 7.2.1, ∼ for each n, we get a coherent locally free OXn -Algebra such that Bn 1 − ←− Bn OXn 1 . By the existence theorem, a coherent OX-Algebra B O⊗Xn − ∃ ∼ = such that Bn B OXn . Set Z Spec B. We claim that Z/X is the 159 ←− O⊗X étale covering we are looking for. It is clear that Z X0 ∼ Z0. It remains ×X −→ to show that Z/X is étale. We first observe that since A is a local ring and f is closed any (open) 1 neighbourhood of the fibre f − (s0) is the whole of X. Also, if Z/X is 1 étale over the points of the fibre X0 = f − (s0), then Z is étale over X at points of an open neighbourhood of X ; for, if x X then B is 0 0 ∈ 0 x0 free over Ox0,X and hence B is a free OX-Module in a neighbourhood of x0; and similarly for non-ramification (use Ch. 3, Proposition 3.3.2). Therefore it is enough to prove that Z X is étale at points of Z lying 1 → over the fibre X0 = f − (s0). Let then x0 X0; choose an affine open neighbourhood U of x0 in ∈ = Γ = X such that B0 is OX0 -free over U0 U X0. Set (U, OX) C and ∩ + n+1 J = M C, the ideal generated by M ; then, for n Z , Cn = C/J ∼ Γ =Γ ∈ −→ (U, OXn ). Similarly, if M (U, B) then

+ M = M/Jn 1 M ∼ Γ(U, B ). n −→ n

We know that M0 = Γ(U, B0) ∼ Γ(U0, B0) is free as a C0 = Γ −→ (U0, OX0 )-module. Choose a basis for M0 over C0. Lift this basis to M and let L be the free C-module on these elements and ϕ : L M be the → 122 8. An Application of the Existence Theorem

natural C-linear map; then we have an exact sequence of C-modules:

0 ker ϕ L M coker ϕ 0. → → → → → + n+1 160 For n Z , set Ln = L/J L and let ϕn be the Cn-linear map ϕ ∈ L n M deﬁned by ϕ. We claim that each ϕ is an isomorphism. n −−→ n n That ϕ0 is an isomorphism is clear from the construction. Assume that ϕ0,...,ϕn 1 are all isomorphisms and consider the exact sequence − n n+1 0 J /J Cn Cn 1 0 → → → − →

of Cn-modules. Since Ln, Mn are Cn-ﬂat (recall that Bn is OXn -ﬂat) we get the following commutative diagram of exact sequences of Cn- modules:

Jn/Jn+1 M 0 / n / Mn / Mn 1 / 0 C⊗n O O − O

ϕn ϕn 1 −

Jn/Jn+1 L 0 / n / Ln / Ln 1 / 0. C⊗n −

But

n r n n+1 M Γ J /J + C (U, OX0 ), ≃ n 1 ⊗A M Mi=1 r r n n+1 J /J Mn Γ(U, B0) = M0, C⊗n ≃ Mi=1 Mi=1 r n n+1 and J /J Ln L0. C⊗n ≃ Mi=1

161 It follows that the first vertical map is an isomorphism; so is ϕn 1 by − inductive assumption. Hence ϕn is an isomorphism. If C, L, M are the J-adic completions of C, L, M and if ϕ is the C-linear map defined by ϕ, it follows that ϕ is an isomorphism. Thus, b b b b b b 8.1. The second homotopy exact sequence 123 one obtains: (coker ϕ) = 0 and ker ϕ ker(L L). It follows then that ⊂ → B is locally free at x X , i.e., Z X is flat at points over X . It is 0 ∈ 0 → b 0 also unramified at pointsb above X since Z X is unramified and for 0 0 → 0 non-ramification it suffices to look at the fibre. The proof of Proposition 8.1.3 is complete. Proof of Theorem 8.1.1. The isomorphism π (S, s ) ∼ G(k(s )/k(s )) 1 0 ←− 0 0 follows from Proposition 8.1.3 when X = S . Consider now the diagram

a0 X o X0 a0′ o X a ∈ ∋ 0 ∋ 0

s0 So k(s0) o k(s ) ∈ 0 By Proposition 8.1.2 we get the exact sequence:

(e) π (X , a ) π (X , a′ ) G(k(s )/k(s )) (e). → 1 0 0 → 1 0 0 → 0 0 → One now uses the isomorphisms π (X , a ) ∼ π (X, a ) (Propo- 1 0 0′ −→ 1 0 sition 8.1.3) and G(k(s )/k(s )) ∼ π (S, s ) to get he required exact 0 0 −→ 1 0 sequence. Q.E.D.

Chapter 9 The Homomorphism of Specialisation of the Fundamental Group

9.1 162 With the notations and assumptions of Chapter 8, let s1 be an arbitrary point of S = Spec A and X1 = X k(s1). If a1 is a geometric point of X1 ×S and a1 its image in X and if X1 is connected we get a sequence

π (X , a ) π (X, a ) π (S, s ) (e) 1 1 1 → 1 1 → 1 1 → such that the composites are trivial. We have continuous isomorphisms β : π (X, a ) π (X, a ) and α : π (S, s ) π (S, s ). Consider now 1 1 → 1 0 1 1 → 1 0 the diagram:

π1(X1, a1) / π1(X, a1) / π1(S, s1) / (e)

β α (e) / π1(X0, a0) / π1(X, a0) / π1(S, s0) / (e). This is not necessarily commutative; but it is commutative upto an inner automorphism of π1(S, s0). (It is readily seen in the example of

125 126 9. The Homomorphism of Specialisation.....

section 5.1 that, with the notation therein, the continuous homomorphism there, is determined upto an inner automorphism of π1(S, s) if we take for s a point diﬀerent from ϕ(s′)-see Ch. 4.) Now the lower row is exact. This implies that there is a continuous 163 homomorphism π (X , a ) π (X , a ) which is clearly determined 1 1 1 → 1 0 0 upto an inner automorphism of π1(X). This is called the homomorphism of specialisation of the fundamental group.

9.2

With the same assumptions as above further suppose X/S separable. Since we have assumed X0 connected, the condition f (OX) = OS is automatically satisﬁed and then the upper sequence in∗ the diagram of 9.1 is also exact. In this case the homomorphism of specialisation is surjective.

9.3

Let Y be locally noetherian and X Y be a separable, proper morphism → with ﬁbres universally connected. Suppose y , y Y with y (y ). 0 1 ∈ 0 ∈ 1 Let a0, a1 be geometric points of X0 = X k(y0) and X1 = X k(y1) ×Y ×Y respectively. Then there is a (natural) homomorphism of specialisation π (X , a ) π (X , a ) which is surjective. 1 1 1 → 1 0 0 In fact, in view of Proposition 7.3.2 we may make the base-change Y Spec O , and apply the above considerations to points above y ← y0 Y 0 and y . 1 b This is the so-called semi-continuity of the fundamental group. 9.3. 127

Appendix to Chapter IX The Fundamental Group of an Algebraic Curve Over an Algebraically Closed Field

Let k be an algebraically closed ﬁeld and X a proper, nonsingular, 164 connected algebraic curve over k (i.e., a k-scheme of dimension 1). Case 1. Charac. k = 0.

We compute π1(X) in this case by assuming results of transcendental geometry. In view of Lemma 7.2.1.4 and Proposition 7.3.2 one can assume that k = C, the field of complex numbers. Then one has an analytic structure on X and on every X (E t/X) (see GAGA - J.P. Serre); one can then ′ ∈ show that the natural functor Finite topological coverings of the (E t/X) → ( analytic space Xh defined by X ) is an equivalence [cf. GAGA and Espaces fibrés algébriques - Seminair´ Chevally 1958, (Anneaux de Chow) - See especially Prop. 19, Cor. to Prop. 20 and Cor. to Theorem 3]. One thus obtains: top h π1(X, a) π1 (X , a), the completion of the topological fundamen- top≃ ch tal group π1 (X , a) with respect to subgroups of finite index. This top h group π1 (X , a) is “known” if X is a nonsingular, proper, connected curve of genus g. It is a group with 2g generators ui, vi(i = 1,..., g) subject to one relation, namely 1 1 1 1 1 1 = (u1v1u1− v1− )(u2v2u2− v2− ) ... (ugvgug− vg− ) 1.

Before going to the case charac. k = p , 0, we shall briefly recall 165 some evaluation-theoretic results. (Ref. S. Lang-Algebraic Numbers or J.P. Serre-Corps Locaux). Let V be a discrete valuation ring with maximal ideal M and field of fractions K. Let K′/K be a finite galois extension and V′ the integral 128 9. The Homomorphism of Specialisation.....

closure of V in K′. Choose any maximal ideal M ′ of V′ and consider the decomposition group of M ′, i.e., the subgroup gd(M ′) of G(K′/K) of elements leaving M ′ stable. Then there is a natural homomorphism g (M ) G(k(M )/k(M )); the kernel g (M ) of this homomorphism d ′ → ′ i ′ is called the inertial group of M ′. The maximal ideals of V′ are trans- formed into each other by the action of the galois group G(K′/K) and the various inertial groups are conjugates to each other (Compare with Lemma 4.2.1). For the following definition it is irrelevant which of the M ′ we take and therefore we shall write gi, or sometimes gi(K′/K) instead of gi(M ′). We say that K′ is tamely ramified over V if the order of gi is prime to the characteristic of k(M ) and that K′ is unramified over V if gi is trivial. One has then the following:

(1) If K′/V is tamely ramiﬁed then the inertial group gi(K′/K) is cyclic.

(2) If K K K is a tower of finite galois extensions and if V is a ′′ ⊃ ′ ⊃ ′ localisation of the integral closure of V in K′ and if K′/V, K′′/V′ are tamely ramified then K′′/V is tamely ramified. 166 In addition, one has an exact sequence:

(e) g (K′′/K′) g (K′′/K) g (K′/K) (e) → i → i → i → (cf. Corps Locaux, Ch. I, Propostion 22).

+ (3) Let τ M be a uniformising parameter and n Z be prime ∈ th ∈ to charac. k(M ). If K contains the n roots of unity, then K′ = K[X] is a ﬁnite galois extension, tamely ramiﬁed, with galois (Xn τ) group− = g (K /K) Z/nZ. i ′ ≃

Lemma (Abhyankar). Let L, K′ be finite galois extensions of K, tamely ramified over V with order gi(L/K) (= n) dividing order gi(K′/K)(= m). Let L′ be the composite extension of L and K′. Then L′ is unramified over the localisations of the integral closure V′ of V in K′. 9.3. 129

Proof. One checks that a monomorphism g (L /K) g (K /K) ∃ i ′ → i ′ × g (L/K) such that the projections g (L /K) g (K /K) and g (L /K) i i ′ → i ′ i ′ → gi(L/K) are onto (see 2)). Since the orders of gi(L/K) and gi(K′/K) are prime to p = charac. k(M ), so is the order of gi(L′/K), i.e., L′/V is tamely ramiﬁed; hence g (L /K) is cyclic (1)). As n m, it follows i ′ | that each element of g (L /K) has order dividing m. But g (L /K) i ′ i ′ → g (K /K) is onto. Thus, g (L /K) ∼ g (K /K). From (2) the kernel of i ′ i ′ −→ i ′ this map is gi(L′/K′) which is therefore trivial. Q.E.D.

Case 2. Charac. k = p , 0. 167

Deﬁnition. We say that a Y-prescheme X is smooth at a point x X (or ∈ X Y is simple at x) if an open neighborhood U of x such that the → ∃ natural morphism U Y admits a factorisation of the form → ´etale U Spec O [T ,..., T ] Y −−−→ Y 1 n →

(the Ti are indeterminates).

Note. Smoothness is stable under base-change.

Deﬁnition. For a Y-prescheme X, the sheaf of derivations of X over Y is the dual of the O-Module ΩX∤Y; it is denoted by gX∤Y.

We shall assume the following

Theorem 1. (SGA, 1960, III, Theorem (7.3)).

Let A be a complete noetherian local ring with residue ﬁeld k. If X0/k is a projective, smooth scheme such that

2 = H (X0, gX0 k) (0) | 2 = and H (X0, OX0 ) (0); then a projective, smooth A-scheme X such that X k X0. ∃ ⊗A ≃ Let k be an algebraically closed ﬁeld of charac. p , 0 and X0 a 168 nonsingular (= smooth in this case) connected curve, proper (hence, as 130 9. The Homomorphism of Specialisation.....

is well-known projective) over k. Consider the ring A = W(k) of Witt- vectors; this is a complete, discrete valuation ring with residue field k; if K is the field of fractions of A, then charac. K = 0, and K is complete for the valuation defined by A. The conditions of Theorem 1 are satisfied by X0. Therefore a projective smooth A-scheme X such that X0 ∼ X k. ∃ ←− ⊗A The X0 is universally connected; but then it follows that the generic fibre is also universally connected (Use Stein-factorisation; Note that Spec A has only two points). Furthermore, as X is smooth over A, the local rings of X are regular (one has to show that if X Y is étale and Y is → regular then X is regular). Finally we mention the important fact that by Proposition 8.1.3, we have: π (X) ∼ π (X ). 1 ←− 1 0 We may thus replace X0 by X. Let a0 be the closed point of Spec A and a1 the generic point; set XK = X K, XK = X K, where as usual ⊗A ⊗A K is the algebraic closure of K. If a0 Xk = X k is a geometric ∈ ⊗A point over a and a X , a geometric point over a then we have the 0 1 ∈ K 1 homomorphism of specialisation

π (X , a ) π (X , a ) π (X) 1 K 1 → 1 k 0 ≃ 1 which is surjective (9.2). 169 From case 1 one already “knows” about π1(XK, a1) (charac. K = 0). Thus it only remains to study the kernel of the above epimorphism. This amounts by 5.2.4 to studying the following question: Given a connected étale covering Z of XK which is galois, when does there exist a Z (E t/X) such that Z ∼ Z K. [Remember: a connected étale cov- ∈ ←− ⊗A ering is galois if the degree of the covering equals the number of automorphisms. Also note: we have integral schemes because, for instance, our schemes are connected and regular. Therefore we may consider the function fields R(Z) and R(XK) of Z and XK. Clearly if Z is galois over XK then R(Z) is a galois extension of R(XK).]. By Lemma 7.2.1.4, a finite subextension K of K and a Z ∃ 1 K1 ∈ (E t/XK1 = ∼ X K1) such that Z ZK1 K. Then the above question takes the ⊗A ←− K⊗1 form: Given a connected Z (E t/X ), which is galois when does ∈ K 9.3. 131

there exist a Z (E t/X) and a finite extension K2 of K1 such that = ∼∈ ZK2 ZK1 K2 Z K2? K⊗1 ←− ⊗A Now, for any finite extension K′ of K, let A′ be the integral closure of A in K′; then by a corollary to Hensel’s lemma it follows that A′ is again a discrete valuation ring whose residue field is again k (recall that k is algebraically closed). Thus π(X) π (X ) π (X ) where ≃ 1 k ≃ 1 A′ XA = X A′, again by Proposition 8.1.3. Thus the question becomes: ′ ⊗A Given ZK1 (E t/XK1 ), universally connected and galois, when does 170 ∈ = there exist a finite extension K2 of K1 such that ZK2 ZK1 K2 comes K⊗1 = from a ZA2 in (E t/XA X A2) (where A2 is the integral closure of A in 2 ⊗A K2)? Consider any finite extension K′ of K1 and let A′ be the integral closure of A in K′. We are given a situation of the form: Z K′ + p′- generic point of the fibre over the closed point. X K′

generic point Spec K′ 0 -closed point (residue field k). ∼ − · Spec A′ = The ZK1 (resp.|{z}ZK′ ZK1 K′) are connected and hence, as we have K⊗1 already remarked, integral. Let R(ZK1 ), R(ZK′ ), R(XK1 ) and R(XK′ ) be the function fields of ZK1 , ZK′ , XK1 and XK′ . Then R(ZK1 ) is a finite galois extension of R(XK1 ) and

= = R(ZK′ ) R(ZK1 ) K′ R(ZK1 ) R(XK′ ). K⊗1 R(X⊗K1 )

Our aim now is to choose, if possible, the field K K such that R(Z ) ′ ⊃ 1 K′ is unramified over XA′ . By this we mean the following: consider the = normalisation Z′ of X′ XA′ in R(ZK′ ); we want Z′ to be unramified 171 over X′.

(Note: Since ZK′ is regular, so certainly normal, we have that Z′ K′ A⊗′ ∼ Z , because the process of normalisation is unique. – See EGA, II, −→ K′ (6.3) and also S. Lang, Introduction to Algebraic Geometry, Ch. V). § 132 9. The Homomorphism of Specialisation.....

We can also express this as follows: we want Z′ to be unramiﬁed over = the whole of X′ and not merely on the open subscheme XK′ X′ K′. A⊗′ Consider the integral closure A1 of A in K1. Let p be the generic = point of the ﬁbre in XA1 X A1 over the closed point (as a space, this is ⊗A nothing but X0). Then p is a point of codimension 1 in XA1 ; furthermore

XA1 /A1 is smooth and therefore the local ring O1 of p (as a point in XA1 ) = is regular. Therefore O1 is a discrete valuation ring in R(XA1 ) R(XK1 ). Similarly deﬁne O in R(X ) = R(X ) for any K K . It is easily ′ K′ A′ ′ ⊃ 1 checked that O′ is the integral closure of O1 in R(XK′ ). Now, any open

set in XA′ , containing XK′ and the generic point p′ of the fibre over the closed point of Spec A′ is such that its complement is of codimension 2 in X . We have the following theorem depending on the so-called ≥ A′ purity of the branch locus: Theorem 2. (SGA, 1960-1961 expose X, Cor (3.3)) Let P be a locally noetherian regular prescheme and U the comple- 172 ment of a closed set of codimension 2 in P. Then the natural functor Φ ≥ (E t/P) (E t/U) is an equivalence. (For a proof see SGA, X, 1962). −→ [For the special case of the theorem which we need, there is a direct proof in SGA, X, 1961, p. 16. However even for that proof one needs the following result (which we have not proved in these lectures). An unramified covering of a normal prescheme is étale. (SGA, I, 1960- 1961, Theorem (9.5)).] Thus, it is enough to prove that K′ K1 such that R(Z′) is unrami- ∃ ⊃ = fied over X′ at the point p′ because Z′ is clearly flat over X′ XA′ at the point p (the local ring O′ of p′ in X′ is a discrete valuation ring). If τ is a uniformising parameter of A1, it is also a parameter of O1. Let then n Z+ be such that (n, p) = 1 and set K = K [T]/(T n τ). ∈ ′ 1 − Then K /K is a finite galois extension and R(X ) R(X )[T]/(T n e). ′ 1 K′ ≃ K1 − Hence R(XK′ ) is tamely ramified over O1, and has an inertial group of order n. Assume now that the degree of the galois covering Z over XK is prime to p = charac.k. Then one may take n equal to this degree. By Abhyankar’s lemma one then has that R(Z′) is unramified over X′ above the point p′. Thus one has “proved” the 9.3. 133

Proposition 3. The kernel of the (surjective) homomorphism of specialisation π (X , a ) π (X , a ) is contained in the inter-section of the 1 K 1 → 1 0 0 kernels of continuous homomorphisms from π1(XK) to ﬁnite groups of order prime to p.

Therefore if, for a profinite group G , we denote by G (p) the profinite 173 group lim Gi where the limit is taken over all quotients Gi of G which are finite←−− of order prime to p, then we have:

(p) (p) π (X , a ) ∼ π (X , a ). 1 K 1 −→ 1 0 0 (One can also say: G (p) is the quotient of G by the closed normal subgroup generated by the p-Sylow subgroups of G ). Since we “know”, by topological methods, the group π1(XK, a1) we obtain: Theorem 4. If X is a nonsingular, connected, proper curve of genus g (p) over an algebraically closed ﬁeld k of charac. p , 0, then π (X) ∼ 1 −→ G (p) where G is the completion with respect to subgroups of ﬁnite index of a group with 2g generators u, ui, vi(i = 1, 2,..., g) with one relation: 1 1 1 1 1 1 = (u1v1u− v1− )(u2v2u2− v2− ) ... (ugvgug− vg− ) 1.

Bibliography

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[3] – Séminaire de Géométrie Algébrique (SGA) - Volumes I, II, (1960-61) I, II (1962)

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