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

Transcription

1 Lectures on n Introduction to Grothendieck s Theory of the Fundamental Group By J.P. Murre Notes by. nantharaman 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

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3 Preface These lectures contain the material presented in a course given at the Tata Institute during the period December February 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. ll of the material (and much more) can be found in the éminaire de géométrie algébrique of Grothendieck, Exposé V, IX and X. I thank Mr.. nantharaman for the careful preparation of the notes. J.P. Murre iii

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5 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, and of the EG ( Éléments de Géometrie lgébrique 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 EG, 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. v

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7 Contents Preface Prerequisites iii v 1 ffine chemes The heaf associated to pec ffine chemes Preschemes Product of Preschemes Fibres ubschemes ome formal properties of morphisms ffine morphisms The finiteness theorem Étale Morphisms and Étale Coverings Examples and Comments Étale coverings vii

8 viii Contents 4 The Fundamental Group Properties of the category C Construction of the Fundamental group Galois Categories and Morphisms of Profinite Groups pplication of the Comparison Theorem The tein-factorisation The first homotopy exact sequence The Technique of Descents and pplications n pplication of the Existence Theorem The second homotopy exact sequence The Homomorphism of pecialisation ppendix to Chapter IX 127

9 Chapter 1 ffine chemes 1.1 Let be a commutative ring with 1 and a multiplicatively closed set in, containing 1. We then form fractions a, a, s ; two s fractions a 1, a 2 are considered equal if there is an s 3 such that s 1 s 2 s 3 (a 1 s 2 a 2 s 1 )=0. When addition and multiplication are defined in the obvious way, these fractions form a ring, denoted by 1 and called the ring of fractions of with respect to the multiplicatively closed set. There is a natural ring-homomorphism 1 given by a a/1. This induces a(1 1) correspondence between prime ideals of not intersecting and prime ideals of 1, which is lattice-preserving. If f and f is the multiplicatively closed set{1, f, f 2,...}, the ring of quotients 1 f is denoted by f. If p is a prime ideal of and = p, the ring of quotients 1 is denoted by p ; p is a local ring Let be a commutative ring with 1 and X the set of prime ideals of. For any E, we define V(E) as the subset{p : p a prime ideal E} of X. Then the following properties are easily verified: (i) α V(E α )=V( α E α ) 2 1

10 2 1. ffine chemes (ii) V(E 1 ) V(E 2 )=V(E 1 E 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 pec. Note. pec is a generalisation of the classical notion of an affine algebraic variety. uppose k is an algebraically closed field and let k[x 1,..., X n ]= k[x] be the polynomial ring in n variables over k. Letabe an ideal of k[x] and V be the set in k n defined by V={(α 1,...,α n ) : f (α 1,...,α 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) correspondence with the maximal ideals of k[x]/a. Remarks 1.3. (a) Ifa(E) is the ideal generated by E in, then we have: V(E) = V(a(E)) (b) For f, define X f = X V( f ); then the X f form a basis for the Zariski topology on X. In fact, X V(E)= X f by (i). f E 3 (c) X is not in general Hausdorff; however, it is T 0. (d) X f X fα an n Z + such that f n is in the ideal generated α by the f αs. For any idealaof, define a ={a :a n a for some n Z + }. a is an ideal of and we assert that a = {p : p a prime ideal a}. To prove this, we assume, (as we may, by passing to /a) thata=(0). Clearly, (0) p. On the other hand, if a is such that a n 0 n Z +, then a 1= a is a non-zero ring and so contains a proper prime ideal; the lift p of this prime ideal in is such that a p.

11 1.4. The heaf associated to pec 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( f α )=V( { f 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 X gs; thus, if X f X fα, then by (d), we have f n = r a i f αi (say); again by α i=1 (d), we obtain X f = X f n r X fαi. i=1 (f) There is a (1 1) correspondence between closed sets of X and 4 roots of ideals of ; in this correspondence, closed irreducible sets of X go to prime ideals of 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 is noetherian, pec is a noetherian space (i.e. satisfies the minimum condition for closed sets). α α 1.4 The heaf associated to pec We shall define a presheaf of rings on pec. It is enough to define the presheaf on a basis for the topology on X, namely, on the X f s; we set F (X f )= f. If X g X f, V(g) V( f ) and so g n = a 0 f for some n Z + and a 0. The homomorphism f g given by a f q a aq 0 g qn is independent of the way g is expressed in terms of f and thus defines a natural mapρg f : F (X f ) F (X g ) for X g X f. The transitivity conditions are readily verified and we have a presheaf of rings on X. This defines a sheaf Ã=O X of rings on X. It is easy to check that the stalk O p,x of O X at a point p of X is the local ring p.

12 4 1. ffine chemes If M is an -module and if we define the presheaf X f M f = M f, we get a sheaf M of Ã-modules (in short, an Ã-Module M), whose stalk at p X is M p = M P. 5 Remark. The presheaf X f M f is a sheaf, i.e. it satisfies the axioms (F 1 ) and (F 2 ) of Godement, Théorie des faisceaux, p In this section, we briefly recall certain sheaf-theoretic notions Let f : X Y be a continuous map of topological spaces. uppose 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, open in Y, the restriction maps of this presheaf will be the restriction homomorphismsγ( 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 Γ( f 1 (U), F ) F x given a homomorphismγ(u, f (F )) F x ; by passing to the inductive limit as U shrinks down to f (x), we obtain a natural homomorphism: f x : f (F ) f (x) F x. If O X is a sheaf of rings on X, f (O X ) has a natural structure of a sheaf of rings on Y. If F is an O X -Module, f (F ) has a natural structure of an f (O X )-Module. The direct image f (F ) is a covariant functor on F Let f : X Y be a continuous map of topological spaces andg 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=ρ g :g f (F )

13 and (b) for any sheaf H of abelian groups on X, the homomorphism Hom X (F, H ) Hom Y (g, f (H )) given byρ f (ϕ) ρ g is an isomorphism. The unique sheaf F of abelian groups on X with these properties is called the inverse image f 1 (g) ofgunder f. It can be shown that canonical homomorphism (*) f x ρ f (x) :g f (x) f 1 (g) x is an isomorphism, for every x X. The inverse image f 1 (g) is a covariant functor ongand the isomorphism ( ) shows that it is an exact functor. If O Y is a sheaf of rings on Y, f 1 (O Y ) has a natural structure of a sheaf of rings on X. Ifgis an O Y -Module, f 1 (g) has a natural structure of an f 1 (O Y )-Module ringed space is a pair (X, O X ) where X is a topological space 7 and O X is a sheaf of rings on X, called the structure sheaf of (X, O X ). morphismφ : (X, O X ) (Y, O Y ) of ringed spaces is a pair ( f,ϕ) such that (i) f : X Y is a continuous map of topological spaces, and (ii) ϕ : O Y f (O X ) is a morphism of sheaves of rings on Y. Ringed spaces, with morphisms so defined, form a category. Observe that condition (ii) is equivalent to giving a morphism f 1 (O Y ) O X of sheaves of rings on X (see (1.5.2)). If F is a sheaf of O X -modules, we denote byφ (F ) the sheaf f (F ), considered as an O Y -Module throughϕ. Ifgis an O Y -Module, f 1 (g) is an f 1 (O Y )-Module and the morphism f 1 (O Y ) O X, defined byϕ, gives an O X -Module f 1 (g) f 1 (OY ) O X ; the stalks of this O X -Module are isomorphic tog f (x) O f (x) O x, under the identification f 1 (g) x g f (x). We denote this O X -Module byφ (g). In general,φ is not an exact functor ong.

14 6 1. ffine chemes 1.6 ffine chemes ringed space of the form (pec, Ã), a ring, defined in (1.4) is called an affine scheme Letϕ: B be a ring-homomorphism;ϕ defines a map f= a ϕ : X= pec pec B=Y p ϕ 1 (p). ince a ϕ 1 (V(E))=V(ϕ(E)) for any E B, a ϕ is a continuous map. Let s B; ϕ defines, in a natural way, a homomorphism ϕ s : B s ϕ(s). In view of the remark at the end of (1.4), this gives us a homomorphism: B s =Γ(Y s, B) ϕ(s)=γ(x ϕ(s), Ã)=Γ(Y s, f (Ã)) and hence a homomorphism ϕ : B f (Ã). If x X the stalk map defined by ϕ, namely ϕ x : O f (x) B f (x) O x x is a local homomorphism (i.e. the image of the maximal ideal in B f (x) is contained in the maximal ideal of x ). 9 Definition morphismφ : (pec, Ã) (pec B, B) of two affine schemes, is a morphism of ringed spaces with the additional property thatφis of the form ( a ϕ, ϕ) for a homomorphismϕ : B of rings. It can be shown that a morphismφ = ( f,ϕ) of ringed spaces is a morphism of affine schemes (spec, Ã)(spec B, B) if and only if the stalk-maps are local homomorphisms. O f (x) O x defined by Φ(rather, byϕ)

15 1.6. ffine chemes 7 Remarks (a) If M is an -module, M is an exact covariant functor on M. (b) For any -modules M, N, HomÃ( M, Ñ) is canonically isomorphic to Hom (M, N). (c) If (X, O X )=(spec, Ã), (Y, O Y )=(spec B, B) are affine, there is a natural bijection from the set Hom(X, Y) of morphisms of affine schemes X Y onto the set Hom(B, ) of ring-homomorphisms B. (d) Let (X, O X ) = (pec, Ã) be an affine scheme and F an O X - module. Then one can show that F is quasi-coherent (i.e. for every x X, an open neighbourhood U of x and an exact sequence (O X U) (I) (O X U) (J) F U 0) F M for an -module M. If we assume that is noetherian, one sees that F is coherent F is to M for a finite type -module M. (e) Let X Φ Y be a morphism of affine schemes and F (resp. g) 10 be a quasicoherent O X -Module (resp. O Y -Module). Then one can define a quasi-coherent O Y -Module (resp. O X -Module) denoted byφ (F ) (resp. Φ (g)) just as in (1.5.3); if X = pec, Y = pec B andφ=( a ϕ, ϕ) for aϕ : B and if F= M, M an - module (resp.g=ñ, N a B-module) thenφ (F ) (resp.φ (g)) is canonically identified with [ϕ] M, where [ϕ] M is the abelian group M considered as a B-module through ϕ (resp. M B ). (For proofs see EG Ch. I)

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17 Chapter 2 Preschemes Definition 2.1. ringed space (X, O X ) is called a prescheme if every 11 point x X has an open neighbourhood U such that (U, O X U) is an affine scheme. n open set U such that (U, O X U) is an affine scheme is called an affine open set of X; such sets form a basis for the topology on X. Definition morphismφ : (X, O X ) (Y, O Y ) of preschemes is a morphism ( f,ϕ) of ringed spaces such that for every x X, the stalk-mapϕ x : O f (x) O x defined byφis a local homomorphism. Preschemes then form a category (ch). In referring to a prescheme, we will often suppress the structure sheaf from notation and denote (X, O X ) simply by X uppose C is any category and ObC. We consider the pairs (T, f ) where T Ob C and f Hom C (T, ). If (T 1, f 1 ), (T 2, f 2 ) are two such pairs, we define Hom((T 1, f 1 ), (T 2, f 2 )) to be the set of C -morphismsϕ : T 1 T 2, making the diagram ϕ T 1 f 1 9 f 2 T 2

18 10 2. Preschemes 12 commutative. This way we obtain a category, denoted by C. In the special case C= (ch), the category (ch/ )=(ch) is called the category of - preschemes; its morphisms are called -morphisms. itself is known as the base prescheme of the category. Remark Let pec be an affine scheme and Y any prescheme. Then Hom(,Γ(Y, O Y )) is naturally isomorphic to Hom(Y, pec ). In fact, let (U i ) be an affine open covering of Y andϕ Hom (,Γ(Y, O Y )). The composite maps ϕ i : ϕ Γ(Y, O Y ) restriction Γ(U i, O Y ) give morphisms a ϕ i : U i pec, for every i, since the U i are affine. It is easily checked that a ϕ i = a ϕ j on U i U j, i, j. We then get a morphism a ϕ : Y pec ; the mapϕ a ϕ is a bijection from Hom(,Γ(Y, O Y )) onto Hom(Y, pec ) (cf. (1.6.4)(c)). It follows that every prescheme X can be considered as a peczprescheme in a natural way: (ch)=(ch/pecz)=(ch/z). 13 Remark Let (X, O X ) be a prescheme and F an O X -module. Then 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 U, for aγ(u, O X ) - module M U. We may take this as our definition of a quasi-coherent O X -module. 2.2 Product of Preschemes uppose (X, f ), (Y, g) are -preschemes. We say that a triple (Z, p, q) is a product of X and Y over if: (i) Z is an -prescheme (ii) p : Z X, q : Z Y are -morphisms and

19 2.2. Product of Preschemes 11 (iii) for any T (ch/ ), the natural map: is a bijection. Hom (T, Z) Hom (T, X) Hom (T, Y) f (p. f, q. f ) 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 and Y over ; p, q are called projection morphisms. Theorem If X, Y (ch/ ), the fibre-product of X and Y over always exists. We shall not prove the theorem here. However, we observe that if 14 X= pec, Y= pec B and = pec C are all affine, then pec( C B) 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. (For details see EG, Ch. I, Theorem (3.2.6)). Remarks. (1) The underlying set of X Y is not the fibre-product of the underlying sets of X and Y over that of. However if x X, y Y lie over the same s, then there is a z X Y lying over x and y. (For a proof see Lemma (2.3.1)). (2) n open subset U of a prescheme X can be considered as a prescheme in a natural way. uppose, U X, V Y are open sets such that f (U), g(v) ; we may consider U, V as -preschemes. When this is done, the fibre-product U V is isomorphic to the open set p 1 (U) q 1 (V) in Z=X Y, considered as a prescheme. This follows easily from the universal property of the fibre-product.

20 12 2. Preschemes Change of base: Let X, be -preschemes. Then the fibeproduct X can be considered as an -prescheme in a natural way: X X 15 When this is done, we say that X is obtained from X by the base- change and denote it by X ( ). Note that, in the affine case, this corresponds to the extension of scalars. If X is any prescheme, by its reduction mod p, p Z +, (resp. mod p 2 and so on) we mean the base-change corresponding toz Z /(p) (resp.z Z /(p 2 ) and so on). If f : X X., g : Y Y are -morphisms f and g define, in a natural way an -morphism: X Y X Y, which we denote by f g or by ( f, g). When g = I :, we get a morphism: f I = f ( ) : X ( ) X ( ). 2.3 Fibres Let (X, f ) be an -prescheme and s be any point. Let U be an affine open neighbourhood of s and =Γ(U, O ). If p s is the prime ideal of corresponding to s, O s, is identified with ps. Denote by k(s) the residue field of O s, = ps. The composite ps k(s) defines a morphism pec k(s) pec =U ; i.e. to say, pec k(s) is an - prescheme in a natural way. Consider now the base-change pec k(s) : p X X = X pec k(s) f q pec k(s) 16 The first projection p clearly maps X into the set f 1 (s). We

21 2.3. Fibres 13 claim that p(x )= f 1 (s) and further that, when we provide f 1 (s) with the topology induced from X, p is a homomorphism between X and f 1 (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, are affine say X= pec, = pec C. That p(x )= f 1 (s) will follow as a corollary to the following more general result. Lemma Let X = pec, Y = pec B be affine schemes over = pec C. uppose that x X, y Y lie over the same element s. Then the set E of elements z Z=X Y lying over x, y is isomorphic to pec(k(x) k( ) k(y)) (as a set). Proof. One has a homomorphismψ: B k(x) k(y) which gives C k(s) a morphism a ψ : pec k(x) k(y) Z; clearly the image of a ψ is k(s) contained in the set E=p 1 (x) q 1 (y). That a ψ is injective is seen by factoringψas follows: B x B y k(x) k(y). In order C Cs k(s) to see that a ψ is surjective one remarks that for z E the homomorphism B k(z) factors through k(x), similarly for B, therefore C we have for B k(z) a factorisation 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 pec k(s)=pec( k(s)) C p q X= pec Y= pec B f g s = pec C

22 14 2. Preschemes the map p : X f 1 (s) is a bijection Returning to the assertion that X is homeomorphic to the fibre f 1 (s) (with the induced topology from X) we note that ifϕ:c k(s) is the natural map, then p : X X is the morphism corresponding to 1 ϕ : C k(s). To show that p carries the topology over, it is 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. Now, any element of k(s) can be written in the form ( ) ci a i = ( ) ( C i t (a i c i ) 1 1 ) ( with a i, c i, t C. ince 1 1 ) is a unit i t t of k(s), we can take for E, the set of elements a i c i where C i a i ( c i /t ) is an element of E. Q.E.D. i Note. The fibre f 1 (s) can be given a prescheme structure through this homeomorphism p : X f 1 (s). If, in the above proof, we had taken O s /Ms n+1, instead of k(s)=o s /M s we would still have obtained homeomorphisms p n : X pec ( ) O s /Ms n+1 f 1 (s). The prescheme struc- ture on f 1 (s) defined by means of p n, is known as the n th -infinitesimal neighbourhood of the fibre. 2.4 ubschemes Let X be a prescheme and J a quasi-coherent sheaf of ideals of O X. Then the support Y of the O X -Module O X /J is closed in X and (Y, O X/J Y) has a natural structure of a prescheme. In fact, the question is purely local and we may assume X=pec. Then J is defined by an ideal I of and Y corresponds to V(I) which is surely closed. The ringed space (Y, O X/J Y) has then a natural structure of an affine scheme, namely, that of pec(/i). uch a prescheme is called a closed subscheme of X. n open subscheme is, by definition, the prescheme induced by X on an open subset

23 2.4. ubschemes 15 in a natural way. subscheme of X is a closed subscheme of an open subscheme of X 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 O X such that N x = nil-radical of O x. N defines a closed subscheme Y of X, which we denote by X red and which is reduced, in the sense that the stalks O y,y of Y have no nilpotent elements. X and Y have the same base space ( and red have the same prime ideals). Consider a morphism f : X Y of preschemes. upposeϕ X : X red X,ϕ Y : Y red Y are the natural morphisms. Then amorphism f red : X red Y red making the diagram X f Y ϕ X ϕ Y X red f red Y red commutative. This corresponds to the fact that a homomorphism ϕ : B of rings defines a homomorphismϕ red : red B red such that ϕ B η η B red ϕ red B red morphism f : Z X is called an immersion if it admits a 20 factorization Z f Y j 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.

24 16 2. Preschemes Example. Let X f be an -prescheme. Then, there is a natural -morphism : X X X such that the diagram X I X =id X X p 1 X X f p 2 X f is commutative. is called the diagonal of f. It is an immersion. Definition morphism f : X is said to be separated (or X is said to be an -scheme) if the diagonal :X X X of f is a closed immersion. prescheme X is called a scheme if the natural map X pec Z is separated. I X 21 Remark. Let Y be an affine scheme, X any prescheme and (U α ) α an 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) U α U β is also affine (ii) Γ(U α U β,o X ) is generated as a ring by the canonical images of Γ(U α, O X ) andγ(u β, O X ). (For a proof see EG, Ch. I, Proposition (5.5.6)) Example of a prescheme which is not a scheme. Let B=k[X], C= k[y] be polynomial rings over a field k. Then pec B X and pec C Y are affine open sets of pec B and pec C respectively; the isomorphism

25 2.5. ome formal properties of morphisms 17 f (X) X m f (Y) Y m of B X onto C Y defines an isomorphism pec C Y pec B X. By recollement of pec B and pec C through this isomorphism, one gets a prescheme, which is not a scheme; in fact, condition (ii) of the proceding remark does not hold: for,γ(pec B, O ) B=k[X] andγ(pec C, O ) C= k[y]; the canonical maps from these intoγ(pec B pec C, O ) k[u, u 1 ] are given by X u, Y u and the image in each case is precisely=k[u]. 2.5 ome 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 -morphism f ( ) : X ( ) Y ( ) is separated for every base-change. (iv) f : X Y, f : X Y are separated -morphisms f f : 22 X X Y Y is separated. (v) g f separated f is separated (vi) f separated f red separated. 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 -morphism having P f ( ) : X ( ) Y ( ) has P for any base-change. (iv) f : X Y has P, f : X Y has P that f f : X X Y Y has P

26 18 2. Preschemes (v) g f has P, g separated f has P. (vi) f has P f red has P. If we suppose that (i) and (ii) hold then (iii) (iv). lso, (v), (vi) are consequences of (i), (ii) and (iii) (or (iv)). Proof. ssume (ii) and (iii). The morphism f f admits a factorization: f f X X Y Y f I X I Y f Y X 23 By (iii), the morphisms f T X and I Y f have P and so by (ii) f f also has P. On the other hand, assume (i) and (iv). I being a closed immersion, has P by (i) and so f ( )= f I has P by (iv). Now assume (i), (ii) and (iii). If g : Y Z is separated, Y Y Z Y is a closed immersion and has P by (i); by making a base-change X f Y we get a morphism X Y X Y I X Y Y X X Y which, by (iii) has Y Z Y Z property p. The projection p 2 : X Z Y Y satisfies the diagram: X ϕ X Z Y g f p 2 Z g Y i.e. to say, p 2 is obtained from g f by the base-change Y Z and so, by (iii) has P.

27 2.6. ffine morphisms 19 Finally, f : X Y is the composite of X Y I X X Y and p 2 : Z X Z Y Y and so by (ii) has P. To prove (vi) from (i), (ii), (iii) use the diagram 24 f red X red Y red ϕ X X f Y ϕ Y and the facts that the canonical morphismsϕ X,ϕ Y are closed immersions and so have P, thatϕ Y f red = f g has P and that a closed immersion 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 ffine morphisms Definition morphism f : X of preschemes is said to be affine (or X affine over ) if, for every affine open U, f 1 (U) is affine in X. It is enough to check that for an affine open cover (U α ) of, the f 1 (U α ) are affine uppose that B is a quasi-coherent O -lgebra. Let (U α ) be an affine open cover of ; set α =Γ(U α, O ), B α =Γ(U α, B) and X α = pec B α. The homomorphism α B α defines a morphism 25 f α : X α U α ; the X α s then patch up together to give an -prescheme X f ; this prescheme X is affine over, is such that f α (O X ) B, and is determined, by this property, uniquely upto an isomorphism. We denote it by pec B. conversely, every affine -prescheme is obtained

28 20 2. Preschemes as pec B, for some quasi-coherent O -lgebra B. (For details, see EG Ch. II, Proposition (1.4.3)). Remarks. (a) ny affine morphism is separated. (Recall the remark at the end of (2.4.2).) (b) If is an affine scheme, a morphism f : X is affine X is an affine scheme. (c) The formal properties (i) to (vi) of (2.5) hold, when P is the property of being affine. (d) uppose X h Y is an -morphism. If f, g are the structural morphisms of X, Y resply, the homomorphism O Y h (O X ) defined by h, given an O -morphism (h) : g (O Y ) g (h (O X ))= f (O X ); then we have a natural map: Hom (X, Y) Hom O (g (O Y ) f (O X )) (the latter in the sense of O -lgebras) defined by h (h). If Y is affine over, it can be shown that this natural map is a bijection. (EG Ch II, Proposition (1.2.7)). (lso, compare with remark (1.6.4) (c) for affine schemes, and with remark (2.1.3)) The finiteness theorem Definition morphism f : X Y of preschemes is said to be of finite type, if, for every affine open set U of Y, f 1 (U) can be written as f 1 (U)= n V α, with each V α affine open in X and eachγ(v α, O X ) a α=1 finite typeγ(u, O Y )-algebra. It is again enough to check this for an affine open cover of Y. n affine morphism f : X Y is of finite type the quasi-coherent O Y -lgebra f α (O X ) is an O Y -lgebra of finite type. In particular, a morphism f : pec B pec is of finite type B is a finite type -algebra (i.e. is finitely generated as -algebra).

29 2.7. The finiteness theorem 21 Definition morphism f : X is universally closed if, for every base-change, the morphism f ( ) : X is a closed map in the topological sense. Definition 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 prescheme Y is locally noetherian, if every y Y has an affine open neighbourhood pec B, with B noetherian. It is said to be noetherian, if it can be written as Y= n Y i where the Y i are affine 27 open sets such that theγ(y i, O Y ) are noetherian rings. If f : X Y is a morphism of finite type and Y is locally noetherian, then X is also locally noetherian Let (X, O X ) and (Y, O Y ) be ringed spaces and f a morphism from X to Y. Let F be an O X -Module. We then define, for every q Z +, a presheaf of modules on Y by defining: U H q ( f 1 (U), F ) for every open U Y (ee EG Ch 0, III 12). The sheaf that this presheaf defines on Y is called the q th -direct image of F and is denoted by R q f (F ). Theorem Let X, Y be preschemes, Y locally noetherian, and f a proper morphism from X to Y. Then, if F is any coherent O X -Module, the direct images R q f (F ) are all coherent O Y -Modules. (For a proof see EG Ch. III, Theorem (3.2.1)). This is the theorem of finiteness for proper morphisms. i=1

30

31 Chapter 3 Étale Morphisms and Étale 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., peco s,, pec pec Â, a noetherian local ring). Definition morphism f : X is said to be unramified at a point x Xif (i) M f (x) O x = M x, (ii) k(x)/k( f (x)) is a finite separable extension. Definition morphism f : X is said to be flat at a point x X if the local homomorphism O f (x) O x is flat (i.e., O x, considered as an O f (x) -module is flat; note that since the homomorphism O f (x) O x is local, O x will be faithfully O f (x) -flat). Definition morphism f : X is said to be étale at a point x X if it is both unramified and flat at x. We say that f : X is unramified (resp. flat, étale) if it is unramified (resp. flat, étale) at every x X. Remarks (1) n unramified morphism X f is étale at x X Ô f (x) Ôx is flat. 23

32 24 3. Étale Morphisms and Étale Coverings 29 (2) morphism f : X is unramified at x X f pec k( f (x)) : X pec k( f (x)) pec k( f (x)) is unramified at the corresponding point; in other words, it is enough to look at the fibre for nonramification. (3) If f : X is unramified at x X and k( f (x)) =k(x) then Ô f (x) Ôx is surjective; if f is étale in addition, then Ô f (x) Ô x. 3.2 Examples and Comments (1) morphism f : pec pec k (k a field) is étale it is r unramified = K i, K i/k finite separable extensions. i=1 (2) Let X and be irreducible algebraic varieties, normal. Then it can be shown that a dominant morphism X is étale= it is unramified. (3) If is non-normal, an unramified dominant morphism X 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. 30 (i) p is unramified. One has to prove this only at the points a, b c sitting over x. Now M a is generated in O a by a function with a simple zero at a, but such a function we can find already in O x, for instance a function induced by a straight line through x not tangent to c at x. (ii) p is not étale: otherwise, Ô x Ô a but we know that Ôx is not a domain, while Ôa is. (4) connected étale variety over an irreducible algebraic variety need not be irreducible.

33 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 way that the points a, b on c are identified with the points b, a on c. We then get a connected but reducible variety X and the morphism p : X c defined in the obvious manner is surely étale. (5) Let X and be irreducible algebraic varieties and suppose that 31 is normal and f : X a dominant morphism. ssume, in addition, that the function field R(X) of X is a finite extension of degree n of the function field R( ) of. If x Xis such that the number of points in the fibre f 1 ( 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 ome algebraic preliminaries. Let ϕ B be a homomorphism of rings defining an -algebra structure on B. Then B B is an -algebra

34 26 3. Étale Morphisms and Étale Coverings in a natural way and p 1 : B B B given by b b 1 p 2 : B B B given by b 1 b 32 andµ : B B B given by b 1 b 2 b 1 b 2 are all -algebra homomorphisms. We may make B B a B-algebra through p 1 i.e. by defining b(b 1 b 2 )=bb 1 b 2 and the kernel I ofµis then a B-module. ince we haveµ p 1 = Id. and the equality b 1 b 2 = (b 1 b 2 1) b 1 (b b 2 ) is follows that B B=(B 1) I, as a B-module. We define the spaceω B/ of -differentials of B as the B-module I/I 2. We note that the B-module structure onω B/ = I/I 2 is natural, in the sense that it does not depend on whether we make B B, a B-algebra through p 1 or through p 2. ome properties ofω B/ (1) The map d : b (1 b b 1) (mod I 2 ) from B toω B/ is -linear. lso, since (1 b 1 b 2 b 1 b 2 1)=b 1 (1 b 2 b 2 1)+b 2 (1 b 1 b 1 1) =+(1 b 1 b 1 1)(1 b 2 b 2 1), we have: d(b 1 b 2 )=b 1 db 2 + b 2 db 1 b 1, b 2 B. (2) ince I is generated as a B-module by elements of the form (1 b 1 b 1 1), it follows thatω B/ is generated by the elements db. (3) Let W be the B-submodule of B B, generated by elements of the form (1 bb b b b b);ω B/ is then isomorphic to (B B)/W. In fact, Ω B/ = I/I 2 (B 1 I)/(B 1 I 2 )

35 = (B B)/(B 1 I 2 ) 33 and clearly W B 1; so W= B 1 (W I). 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 I 2. lso clearly, I 2 W. It follows that W I I 2, W=B 1 I 2 andω B/ (B B)/W. (4) If V is any B-module and D : B V any -derivation of B in V, then there is a B-linear map H D :Ω B/ V such that D B V d H D Ω B/ In fact, D defines an -linear map B B V given by b 1 b 2 b 1 Db 2. This is certainly B-linear and is trivial on W. We then obtain a B-linear map H D :Ω B/ V such that H D (db)=db b B. lso the correspondence D H D is a B-isomorphism from the B-module of -derivations B V, onto the B-module Hom B (Ω B/, V). (This is the universal property ofω B/ ). In particular, the B-module of -derivations of B is isomorphic to Hom B (Ω B/, B); and even more in particular, if =k, B=Kare fields, 34 we see that the space of k-differentialsω K/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 and the extension of scalars B = B. From the universal property of the space of differentials, it

36 28 3. Étale Morphisms and Étale Coverings follows that Ω B / Ω B/ B B Ω B/ The sheaf of differentials of a morphism. Let f : X be a morphism. Our aim now is to define a sheaf Ω X/ on X. ssume, to start with, that X=pec B, = pec ; then Ω B/ is a B-module in a natural way and we defineω X/ = Ω B/ on X. In the general case, consider the diagonal of f in X X. Then is a closed subscheme of an open subscheme U of X X and X is an isomorphism. Let I X be the sheaf of ideals of O U defining. Then I X/I 2 is a sheaf of O U -modules whose support is contained in. By X 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 O X -Module. To do this, we can assume again that X are affine and in this case, I X is the sheaf Ĩ on X X (where I is the ideal of B B defined in 3.3.0) and therefore, our sheaf is Ω B/ considered above. The O X -Module thus obtained will be called the sheaf of differentials of f (or of X over ) and will be denotedω X/. The stalkω x,x/ of this sheaf at a point x X is canonically isomorphic toω Ox /O f (x). This is seen either from the universal property or by observing that the natural mapω x,x/ Ω Ox /O f (x) given by i b i db i s b i d(b i/1 i s ) (with X= pec B, b i, b i, s i B, s i p ) i x i is an isomorphism. ince, by assumptions, is locally noetherian and f is of finite type it is cler from the definition thatω X/ is coherent. Proposition For a morphism f : X and a point x X the following are equivalent: (i) f is unramified at x (ii) Ω x,x/ = (0)

37 (iii) : X X X is an open immersion in a neighbourhood of x. Proof. We may assume X= pec B, = pec. 36 (i) (ii): Consider the base change: x X X = X pec k(s) s= f (x) pec k(s)= If x X is above x, then we have:ω x,x/ O x =Ω x,x / ; this Ox follows from property (5) of 3.3.0; furthermore,o x = O x/ms O x = k(x) since f is unramified at x. ThereforeΩ x,x/ k(x)=ω x,x / and by Ox Nakayama it suffices to show that the latter is (0). By the remark made above,ω x,x / =Ω 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 (X) is a closed subscheme of an open subscheme U of X X, and is defined, therefore, by a sheaf I of O U -ideals. IfΩ x,x/ = (0), then, we will have I z = I 2 z =...; hence I z = (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 neighbourhood of x X. (iii) (i) The question being local, we may assume that : X X X is an open immersion everywhere. n 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 = pec, = pec k, k a field and a k-algebra of finite type. We are to show that = finite K i, where K i/k are finite separable field extensions; for this, we have to show that is artinian and, if k is the

38 30 3. Étale Morphisms and Étale Coverings algebraic closure of k, k k is radical-free. It is thus enough to show that k k= finite k. By making the base-change k k, we may assume k is algebraically closed. Let a Xbe any closed point of X. ince k is algebraically closed, k(a) kand we have then X i X pec k pec k(a)=x (a) (say). X (a) is canonically imbedded in X X and letϕbe the composite of the morphisms: X i 1 X (a) X X. 38 Thenϕ 1 ( )=(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= pec is quasi- compact and this implies that has only finitely many maximal ideals. But is a k-algebra of finite type and so the set of closed points of X ṅ is dense in X. It follows that is artinian and we may write = i i=1 where the i are artinian local rings. We may then assume = i ; the open immersion : X X X then gives an isomorphism k. This however means = k. Q.E.D ome properties of étale morphisms. (1) n open immersion is étale. (2) f, g étale g f étale. (3) f étale f ( ) étale for any base change (not necessarily of finite type),, 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.

39 (4) f 1, f 2 étale f 1 f 2 étale. Follows from (2), (3) and the equality f 1 f 2 = ( f 1 I Y2 ) (I X1 f 2 ). (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 that = (Y) : Y Y Z Y is an open immersion, hence étale. By the base-change X f Y we get (X) : X X Z Y which again is étale by (3). The diagram: X X Z Y g f p 2 Z g Y shows that the second projection p 2 : X Y Y is given by Z (g f ) (Y) ; so, again by (3), p 2 is also étale and from (2) we obtain that f= p 2 (X) : X (X) X Z Y p2 (6) g f étale, f étale g étale. Y is étale. The composite: O g f (x) O f (x) O x is flat and O f (x) O x is faithfully flat, and thus O g f (x) O f (x) is flat. It is clear that k( f (x))/ k(g f (x)) is a finite separable extension. Finally M f (x) O x = 40 M x and M g f (x) O x = M x = (M g f (x) O f (x) )O x ; since O f (x) O x is faithfully flat, it follows that M g f (x) O f (x) = M f (x). (7) n étale morphism is an open map. In fact, we prove more generally: Proposition flat morphism is an open map.

40 32 3. Étale Morphisms and Étale Coverings The proposition will follow as a consequence from the lemmas below. We first make the Definition. 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 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. 41 Proof. : uppose E= n (O i F i ), O i open, F i closed in X. Then i=1 E Y Y n (F i Y) for any closed Y X; if Y is irreducible with i=1 E Y dense in Y, this means Y F i Y for some i, i.e., F i Y; then Y E (O i Y) open in Y. : 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 0 E. By replacing X by F 0, we may assume that for every closed set F properly contained in X, E F is constructible. If X is reducible, say X = X 1 X 2, X 1, X 2 both proper subsets of X and closed, then E X 1, E X 2 are both constructible and hence so is E= (E X 1 ) (E X 2 ). If X is irreducible, either (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 Let be a noetherian prescheme and f : X a morphism. Then the image, under f, of any constructible set is constructible. Proof. Using the preceding lemma, the fact that is noetherian and by passing to subschemes, irreducible components and so on, we are readily reduced to proving the following assertion: if X, are both affine,

41 reduced, irreducible and noetherian and if f : X is a morphism of finite type such that f (X) =, then f (X) contains an open subset of. If X = pec B, = pec, our assumptions mean that, B are noetherian integral domains and the ring-homomorphism B defining f is then easily checked to be an injection. The lemma will then follow from the following purely algebraic result. Lemma Let be an integral domain and B=[x 1,..., x n ] an integral -algebra containing. Then there exists a g such that for every prime ideal p of with g p, there is a prime ideal P of B such that P = p. Proof. Choose a transcendence base X 1,..., X k of B/. Then the extension B= [x 1,..., x n ] is algebraic over = [X 1,..., X k ]. By writing down the minimal polynomials of the x i over and by dividing out these polynomials by a suitably chosen g, g0, we can make the x i integral over [ 1 g]. Consider now the tower of extensions. B [ ] 1 g [ ] 1 g [ ] 1 g If p is a prime ideal of such that g p then there is a prime p of [ ] 1 g lying over p. The prime ideal p + (X 1,..., X k ) of [ 1 g] lies over p. Now [ ] [ 1 g is a finite extension of 1 g] and by Cohen-eidenberg, a prime ideal P of B [ 1 g] lying over p. The restriction P of P to B then sits over p. Q.E.D.

42 34 3. Étale Morphisms and Étale Coverings Lemma Let f : X be a morphism (of finite type) and T X 43 be an open neighbourhood of a point x X. ssume that for every s 1 such that f (x) (s 1 ), there is a t 1 T such that f (t 1 )= s 1. Then f (T) is a neighbourhood of f (x). Proof. ince the question is local, we may assume noetherian and then by lemma it follows that f (T) is constructible. We may then write f (T) = i=1(o i F i ); and by choosing only those i, for which f (x) O i, we may say that f (T) is closed in a neighbourhood of f (x). 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 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 pec O f (x) and those of x are points of peco x. But O x 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 x contracting to p. U being open we have pec O x U. Hence pec O f (x) f (U) (cf. Bourbaki, lg. Comm. Ch. II, 2, n 5, Cor. 4 to Proposition 11). Q.E.D morphism f : is said to be an effective epimorphism if the sequence = p 1 f is exact p 2 i.e., if the sequence Hom(, Y) Hom(, Y) exact, as a sequence of sets, Y. p 1 Hom(, Y) is h 1 h 2 (We say that a sequence of sets E 1 E 2 E 3 is exact if h 1 is an injection and h 1 (E 1 )={x E 2 : h 2 (x)=h 2 (x)}). h 2 p 2

43 morphism f : is faithfully flat if it is flat and surjective. Our aim now is to show that any such morphism is an effective epimorphism ome algebraic preliminaries; the mitsur complex. homomorphism of rings f : defines a sequence: f p 1 p 2 = p 21 p 32 p 31 = where p 1 (a )=a 1; p 2 (a )=1 a, p 21 (a b )=a b 1; p 31 (a b )=a 1 b ; p 32 (a b )=1 a b and so on. We may then define homomorphisms of -modules: 45 0 = p 1 p 2 1 = p 21 p 31 + p 32 2 = p 321 p p 431 p and so on. One then checks that i+1 i = 0 i; we thus get an augmented cochain complex: f This complex is called the mitsur complex /. Lemma If f is faithfully flat, then (i) H 0 ( /) (ii) H q ( /)=(0) q>0. Proof. uppose B is any faithfully flat -algebra. Consider then the complex B /B B ( /) B B = B 1 0 B = B B = B 1 f B

44 36 3. Étale Morphisms and Étale Coverings ince B is -faithfully flat, it is enough to prove that H 0 (B /B) B 46 and H q (B /B)=(0) q>0. s a particular choice we may take B=. Then the homomorphism f =1 f = admits a section, i.e., σ : such thatσ f = 1, namely, the homomorphism: a b a b. We may thus assume, without any loss of generality that the homomorphism f admits a sectionσ: such thatσ f= 1 : f σ We construct now a homotopy operator in this complex, as follows: f p 1 = p 2 σ 1 σ 1 f p 21 p 32 = p σ p 1 p 2 47 (i) H 0 ( /)=ker 0. Let a be such that p 1 (a )= p 2 (a ). pplying 1 σ we get (1 σ)(a 1)=(1 σ)(1 a ) i.e. a 1=1 σ(a ) in. Under the canonical identification, this means that a = f (σ(a )) i.e. a f (). On the other hand, f () ker 0 and f being faithfully flat, is injective. It follows that ker 0. (ii) By using the homotopy operator, one can show that H q ( /)= (0) q>0; the proof is omitted. (We do not need it).

45 Proposition faithfully flat morphism is also an effective epimorphism. Proof. Case (a). = pec, =pec are affine. From local algebra, it follows that the ring homomorphism ϕ defining f :, is also faithfully flat. We have to show that, for any Y, the sequence (*) Hom(, Y) Hom(, Y) Hom(, Y) is exact inens. (i) uppose Y = pec B is affine. Then the sequence (*) is equivalent to a sequence: 48 ( ) Hom(B, Z) Hom(B, ) Hom(B, ). The exactness of this sequence ( ) now follows from assertion (i) of lemma (ii) Let Y be arbitrary. Letϕ 1 : Y,ϕ 2 : Y be two morphisms such thatϕ 1 f= ϕ 2 f. ince f is a surjection, it is clear thatϕ 1 (s)=ϕ 2 (s) s. Choose a point s and a point s with f (s )=s; let y=ϕ 1 (s)=ϕ 2 (s) Y. Choose an affine open neighbourhood pec B of y Yand an elementθ such that s pec θ and ϕ 1 (pec θ ) pec B,ϕ 2 (pec θ ) pec B. etθ 1 = ϕ(θ). Then s pec θ, and f (pec θ ) pec θ ; and by (i) it follows thatϕ 1 andϕ 2, when restricted to pec θ, define the same morphism of preschemes. ince s was arbitrary, this proves that Hom(, Y) Hom(, Y) is injective. Now, suppose thatϕ : Y is a morphism such that in 49

46 38 3. Étale Morphisms and Étale Coverings = p 1 p 2 1 f ϕ Y. We haveϕ p 2 =ϕ p 1. We want to find a morphismψ: Y such thatϕ =ψ f. In view of the injectivity established above, it is enough to define ψ locally. Let s and s with f (s )= s. Choose an affine open neighbourhood V ofϕ (s ) in Y. Then the open neighbourhoodϕ 1 (V) of s in is saturated under f : in fact, let x 1 ϕ 1 (V) and x 2 such that f (x 1 )= f (x 2 ). Then, there is an s = such that p 1 (s )= x 1 and p 2 (s )= x 2 ; so ϕ (x 1 )=ϕ p 1 (s )=ϕ p 2 (s )=ϕ (x 2 ) and x 2 ϕ 1 (V). 50 Now, f is flat and hence an open map (Prop. (3.3.4)) and so f (ϕ 1 (V)) is an open neighbourhood of s. Choose an element θ such that s pec θ f (ϕ 1 (V)); ifθ 1 =ϕ(θ) then f (pec θ )=pec θ, and pec θ = f 1 (pec θ ) ϕ 1 (V) since ϕ 1 (V) is saturated under f. We then have a diagram pec( θ θ θ ) p 1 p 2 pec θ ϕ f V pec θ = f (pec θ ) The problem of definingψ : pec θ V is now a purely affine problem and we are back to (i).

47 3.5. Étale coverings 39 Case (b). The general case. Without loss of generality, we may assume affine. ince f is a morphism of finite type, there is a finite affine open covering ( α )n α=1 of. Consider the disjoint union = n α=1 α. is affine and the morphism 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: F( ) F( ) F( ) identity F( ) F( ) F( ) 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 Étale coverings Definition. morphism of preschemes, f : X, is said to be finite if, for every affine open U, f 1 (U) is also affine and the ring Γ( f 1 (U), O X ) is aγ(u, O )-module of finite type. It is again enough to check the conditions for an affine open cover of. (1) If is locally noetherian and f : X is a finite morphism, f (O X ) is a coherent O -Module. (2) finite morphism remains finite under a base-change. In particular, if f : X is finite and s any point, the morphism f pec k(s) : X k(s) k(s) is finite and this means that the fibre f 1 (s) is finite, discrete.

48 40 3. Étale Morphisms and Étale Coverings 52 (3) 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. By obvious reductions, we may take X, affine reduced and f (X)=. If X = pec B and = pec, the corresponding homomorphism B is an injection and B is a finite -module. Cohen-eidenberg then shows that f (X) =. (4) The following result holds: Lemma (Chevalley). Let be (as always) locally noetherian and f : X a morphism of preschemes. Then the following conditions are equivalent: (a) f is finite (b) f is proper and affine (c) f is proper and f 1 (s) is finite s. (For a proof see EG Ch.III (a) Proposition (4.4.2)). Definition. morphism f : X is said to be an étale covering if it is both étale and finite. 53 Let X f be an étale covering. Then f (O X ) is a locally free O - lgebra of finite rank. For any s, the fibre f (O X ) k(s) is a finite direct sum n s i=1 K i of finite separable field extensions K i of k(s). The rank of f (O X ) at s is then given by n s [K i : k(s)] which equals the number of geometric points in the fibre f 1 (s). This is constant in each connected component of ; if 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. i=1

49 3.5. Étale coverings 41 Note. For étale coverings we have properties similar to (2), (3) (with not necessarily of finite type) (4), (5) and (6) from 3.3.3; this follows immediately from and properties of coverings. word of caution: This concept of an étale covering (French: revêtement étale) 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 is not connected.

50

51 Chapter 4 The Fundamental Group Throughout this chapter, we shall denote by a locally noetherian con- 54 nected prescheme and by C= (E t/ ) the category of étale coverings of. We note that the morphisms of C will all be étale coverings. (ee and the note at the end of Ch. 3). 4.1 Properties of the category C (C 0 ) C has an initial object (the empty prescheme) and a final object. (C 1 ) Finite fibre-products exist in C, i.e., if X Z and Y Z are morphisms in C, then X Z Y exists in C (see (3.3.3)) (C 2 ) If X, Y C, then the disjoint union X Y C (obvious). (C 3 ) ny morphism u : X Y in C admits a factorisation of the form X u Y u i j Y 1 where u 1 is an effective epimorphism, j is a monomorphism and Y= Y 1 Y2, Y 2 C. 43

52 44 4. The Fundamental Group 55 In fact, u is an étale covering so u is both open and closed; if we write u(x)=y 1 we have Y= Y 1 Y2 and u=u 1 : X Y 1 is then an effective epimorphism (see ( )). Further, this factorisation of u into an epimorphism and a monomorphism is essentially unique in the sense that if X u 1 u Y j Y 1 is another such factorisation, then there exists an isomorphismω : Y 1 Y 1 such that u 1 =ω u 1 and j= j ω. (Because a factorisation into a product of an effective epimorphism and a monomorphism is unique). (C 4 ) If X C andgis a finite group of automorphisms of X acting, say, to the right on X, then the quotient X/g of X bygexists in C and the natural morphism X η X/g is an effective epimorphism. 56 The quotient, if it exists, is evidently unique upto a canonical isomorphism; also X is affine over and therefore the existence of X/g has only to be proved in the case X, affine, say X=pec, = pec B and G= the group of B-automorphisms of corresponding tog. Then pec G ( G is the ring of G-invariants of ) is the quotient we are looking for. (Compare with erre, Groupes algébriques et corps de classes, p. 57). Our aim now is to show that X/g is actually in C. The question is again local and we may assume X= pec, = pec B, with B noetherian, and X/g=pec G as above. X is finite and so X/g is also finite. It remains to show that this morphism is étale. In order to do this, we first make some simplifications. uppose that is a flat affine base-change. We have a com-

53 4.1. Properties of the category C 45 mutative diagram: X = X Y = Y pec B = pec =X X/g=pec G = Y pec B=. g acts on X in the obvious way, as a group of -automorphisms of X. We assert that Y = Y is the quotient of X with respect to this action. Indeed, we have an exact sequence of B-algebras: 0 G, where is the map given by (a a σ ). ince B σ G σ G σ G is B-flat, we get an exact sequence: 0 G B B B B B σ G( ) B and this proves that the subring of invariants of B B is G B B ; hence our assertion. Let y Y= X/g and s be its image. Take for B the local ring O s,. Then there is a unique point y Y = Y over y and one has O y,y = O y,y; hence Y is étale at Y Y is étale at y. We 57 may thus assume that = pec B, B a noetherian, local ring. In view of the following lemma, we may assume B complete. Lemma Let X f be a morphism and ϕ a faithfully flat base-change. Then f is étale f ( ) is étale. Proof. : is clear. : flatness of f is straightforward. To prove non-ramification one observes that in view of (5) (3.3.0), one hasω X / =ϕ (Ω X/ ); butϕ being faithfully flat,ω X / = 0 Ω X/= 0; one now applies proposition Q.E.D. Let x 1,..., x n be the points of X over s. By hypothesis each k(x i )/ k(s) is a finite separable extension. We choose a sufficiently large finite galois extension K of k(s) such that each k(x i ) is imbedded in K. We now need the

54 46 4. The Fundamental Group Lemma Let B be a noetherian local ring with maximal ideal M and residue field k. Let K be an extension field of k. Then anoetherian local ring C and a local homomorphismϕ : B Csuch that (i)ϕis B-flat and (ii) C/M CK. (see EG. Ch. 0 III, Prop. (10.3.1)). 58 In addition if [K : k] <, we can choose C to be a finite B-algebra. (EG. Ch. 0 III, Cor. (10.3.2)). By making such a base-change B ϕ C we may assume that each r k(x i ) is trivial over k(s). Under these assumptions we get = B, a finite direct sum of copies of B. Under the action ofg, the set{x 1,..., x n } splits into disjoint subsets{x 1,..., x l },{x l+1,..., x m },..., on each of which g acts transitively. The corresponding decomposition of will l then be given by =( B) m ( B)... The action of G on each i=1 i=l+1 block, for instance, on a (b 1,...,b l ) i=1 l B, will then be just a permutation. The subring G will then be the direct sum 1... α... l where 1 is the diagonal of the block B and so on. Each is evidently isomorphic to B. Our assertion that X/g is étale is now clear. Thus X/g C ; the natural morphism X η X/g is also then an étale covering. Thereforeηwill be an open map and thus ifηis not surjective one could replace Y by the image ofη; this is clearly impossible. Hence η is surjective and thus an effective epimorphism i=1 i= We shall now define a covariant functor F from C (E t/ ) to the category of finite sets. We shall fix once and for all a point s and an algebraically closed field Ω k(s). For any X C, F(X), by definition, will be the set of geometric points of X over s, with values inω, i.e., is the set of all -

55 morphisms pecω Xfor which the diagram is commutative. pecω X pec k(s) We observe that if x X sits above s, then giving an - morphism pecω Xwhose image is x Xis equivalent to giving a k(s)-monomorphism of k(x) into Ω. lso note that for any X C, F(X) is a finite set whose cardinality equals the rank of X over. Properties of the functor F. (F 0 ) F(X)= X=. (F 1 ) F( )= a set with one element; F(X Z Y)=F(X) F(Z) F(Y), X, Y, Z C. (F 2 ) F(X 1 X2 )= F(X 1 ) F(X 2 ). (F 3 ) If X u Y is an effective epimorphism in C, the map F(u) : 60 F(X) F(Y) is onto. In fact, if y Y is a point above s and y = u(x), x X, then any k(s)-monomorphism of k(y) toωextends to a k(s)- monomorphism of k(x) toω. (F 4 ) Let X C andgafinite group of -automorphisms of X (acting to the right on X). Then g acts in a natural way (again to the right) on F(X) as expressed by pecω X σ g X. The natural mapη : X X/g (see (C 3 )) defines a surjection F(η) : F(X) F(X/g) (see (F 3 )). In view of the commutativity

56 48 4. The Fundamental Group of the diagram pecω X σ g η η 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 X Lemma (i) g acts transitively on the fibres ofη. 61 (ii) uppose y Y= X/g and x η 1 (y). Letg d (x) be the subgroup {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 mapg d (x) the galois group G( k(x) /k(y)) is onto. Proof. (i) We may assume X = pec, Y = X/g = pec G as before (both noetherian); also one knows that is finite on G. Let P, P 1 be prime ideals of (i.e., points of X) such that P 1 σp σ G, while P G = P 1 G = p. We may assume P and P 1 maximal (otherwise apply the flat base change Y pec O y,y, where y Ycorresponds to p). Then there is an a P 1 such that aσp σ (Chinese Remainder Theorem). Thus b= σ(a)p; but b G, so b G P 1 = G P- σ contradiction. (ii) We have a diagram: /P k(x) G G /p k(y),

57 and we know that k(x)/k(y) is a finite, separable extension. Let θ be such that k(x)=k(y)(θ). The polynomial f= (T σθ) σ is in G [T], hasθas a root and splits completely in [T]. The reduction f of f mod p is in k(y)[t], hasθas a root and splits completely in k(x)[t]. It follows that k(x)/k(y) is normal, hence galois. Consider now the subgroup G d (P) of G corresponding tog d (x). We have Pσ 1 P σg d (P) and by the Chinese Remainder Theorem we can choose aθ 1 such thatθ 1 θ(modp) andθ 1 0(modσ 1 P), σ G d (P). We have k(x) = k(y)(θ 1 ). Consider now the polynomial g = (T σθ 1 ) k(y)[t]. sθ 1 is a root of g, for everyϕ σ G( k(x) /k(y)),ϕ(θ 1 ) is also a root of g; henceϕ(θ 1 )=σ(θ 1 ) for someσ G. Butϕ(θ 1 )0 and, by the choice ofθ 1,σ(θ 1 )=0 ifσg d (P); henceϕ(θ 1 )=σ(θ 1 ) for someσ G d (P), i.e., ϕ=σ for someσ G d (P). Q.E.D. (F 5 ) If u : X Yis a morphism in C such that F(u) : F(X) F(Y) 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. category which has the properties (C 0 ),...,(C 4 ) of 4.1 and from which there is given a functor F into finite sets with the above properties (F 0 ),...,(F 5 ) 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.

58 50 4. The Fundamental Group Example 1. Let be a connected, locally arcwise connected, locally simply connected topological space and C the category of connected coverings of ; the morphisms of C are covering maps. Let X p be such a covering. Fix a point s ; we define F(X)= p 1 (s). Then (X, p) p 1 (s) is a covariant functor F : C Ens. 64 Each member of C determines (upto conjugacy) a sub-group of the fundamental groupπ 1 (, s); and to each subgroup H ofπ 1 (, s) there corresponds a member of C determining H. To the subgroup{e} corresponds, what is known as, the universal covering of ; andπ 1 (, s) is isomorphic to the group of -automorphisms of, i.e., to the group of covering transformations of over. Further we have the isomorphism (inens): Hom C (, X) F(X), X C. The functor F : C Ens is thus representable in the following sense. Definition covariant functor G from a category C toens is representable if an object Y C such that: Hom C (Y, X) G (X) X C. Example 2. Let k be a field andωafixed algebraically closed field extension of k. et = pec k and C = the category of connected étale coverings of ; any member of C is of the form pec 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 inω. Then, F(X) Hom k (K,Ω s ) is X= pec K, whereω s is the separable closure of k inω. IfΩ s is finite over k, we can further write F(X) Hom C (pecω s, X) and the functor 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 (N i ) i I of finite galois extensions of k, (namely, the set of finite galois extensions contained inω s ) such that for any X C, we can find an i 0 = i 0 (X) such that F(X) Hom C (pec N i, X), i i 0 (X). In other words, we may write F(X) lim i I Hom C (pec N i, X), X C

59 and the family (pec N i ) i I is in fact a projective family of objects in C. 65 uppose now that C is any category and G : C Ens is a covariant functor. If X C andξ G (X), we write, as a matter of notation, G ξ X. If G ξ X, and G η Y and X u Y is a C -morphism, we say that the diagram ξ G X η u Y is commutative if G (u)(ξ) = η. If G ξ X, then for any Z C, we have a natural map Hom C (X, Z) G (Z) defined by u G (u)(ξ). Definition We say that G is pro-representable if a projective system ( i,ϕ i j ) i I of objects of C and elementsτ i G ( i ) (called the canonical elements of G ( i )) such that (i) the diagrams are commutative. τ i G i τ j ϕ i j( j i) j (ii) for any Z C, the (natural) map 66 is bijective. lim i I Hom C ( i, Z) F(Z) In addition, if theϕ i 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.

60 52 4. The Fundamental Group 4.4 Construction of the Fundamental group 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 whichπacts continuously. (2) If C F C (π ) is another such equivalence, thenπ is continuously isomorphic toπand this isomorphism betweenπandπ is canonically determined upto an inner automorphism ofπ. 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 category C is artinian if any decreasing sequence T 1 j1 T 2 j2 T 3 j3... of monomorphisms in C is stationary, i.e., the j r are isomorphisms for large r. (covariant) functor F : C Ens is left-exact if it commutes with finite products i.e., if F(X Y)=F(X) F(Y) and if, for every exact u sequence X Y u 1 Z in C, the sequence u 2 F(X) F(u) F(Y) F(u1) F(Z) F(u 2 ) is exact as a sequence of sets. If Y u 1 u 2 Z are morphisms in C, a kernel for u 1, u 2 in C is a pair (X, u) with X C and u : X Y in C such that X u Y u 1 u 2 Z

61 4.4. Construction of the Fundamental group 53 is exact in C. Clearly a kernal is determined uniquely upto an isomorphism in C. Lemma Let C be a category in which finite products exists. 68 Then finite fibre-products exist in C kernels exist in C. Proof. : Let Y u 1 Z be morphisms in C. We have a commutative u 2 diagram: Z u 1 u 2 Y Y p 1 p2 Y Y Z (Y Y) Y Z (Y Y) Y (p 1,p 2 ) Y Y diagonal and it easily follows that (Y Y Y is a solution for the kernal of u 1 Z (Y Y) and u 2. : uppose X f Z and Y g Z are morphisms in. If p and q are the canonical projections X Y p X, X Y q Y, we have an exact sequence: f p ker( f p, gq) X Y Z. Q.E.D. It follows that ker( f p, gq) is a solution for the fibre-product X Z Y. 69 In fact, we have shown that finite fibre products and kernels can be expressed in terms of each other. Hence, F commutes with finite fibreproducts it is left-exact. gq

62 54 4. The Fundamental Group Corollary. fundamental functor is left-exact (see (F 1 )) Lemma galois category is artinian. Proof. Let T 1 j1 T 2 j2... jr 1 T r jr T r+1... be a decreasing sequence of monomorphisms in C. We have then: T r+1 j r T r is a monomorphism T r+1 T r+1 Tr T r+1 F(T r+1 ) F(T r+1 ) F(Tr ) F(T r+1) (by (F 1 ), (F 5 )) F(T r+1 ) F( j r) F(T r ) is a monomorphism. ince the F(T r ) are finite, this implies that the F( j r ) are isomorphisms for large r; we are through by (F 5 ). Q.E.D. 70 Lemma 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 F ξ X. We order E as follows: (X,ξ) (X,ξ ) a commutative diagram: F ξ ξ X We claim that E is filtered for this ordering; in fact, if (X,ξ), (X,ξ ) X

63 4.4. Construction of the Fundamental group 55 E, in view of (F 1 ) we get a commutative diagram: X ξ p (ξ,ξ ) F X X ξ p 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 F η ξ 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 ). (**) If (X,ξ) E is minimal and (Y,η) E then a u Hom C (X, Y) in a commutative diagram F is uniquely determined. X X ξ X η u Y

64 56 4. The Fundamental Group In fact, if u 1, u 2 Hom C (X, Y) such that the diagrams 72 F η ξ X u 1 Y and F η ξ X u 2 Y are commutative then by (C 1 ) and Lemma ker(u 1, u 2 ) exists; since F is left exact we get a commutative diagram F η Y ξ ker(u 1, u 2 ) ξ j u 1 X u 2 73 with a monomorphism j. s (X, ξ) is minimal j must be an isomorphism, i.e., u 1 = u 2. From (*), (**) it follows that the system I of minimal pairs of E is directed. If (X,ξ) I, (Y,η) E and u Hom C (Y, X) appears in a commutative diagram F η ξ Y u then u must be an effective epimorphism. In fact, be (C 3 ) we get a factorisation X u Y X 1 X2 = X u 1 j X 1

65 4.4. Construction of the Fundamental group 57 with an effective epimorphism u 1 and a monomorphism j. By (F 2 ) and (F 3 ) we then obtain a commutative diagram: ξ F X=X 1 X2 ξ j η X 1 u 1 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 i I Hom C ( i, X) F(X), X C. By (*) this is onto; by (**) it is injective. From (***) it thus follows 74 that F is strictly pro-representable. Q.E.D. Definition Let C be a category with zero ( ) in which disjoint unions exist in C. n X C is connected in C XX 1 X2 in C with X 1, X 2. Note. In (E t/ ), a prescheme is connected it is connected as a topological space. With the notations of the preceding lemma, we have: Lemma (i) (X, ξ) E is minimal X is connected in C. (ii) If X is connected in C, then any u Hom C (X, X) is an automorphism.

66 58 4. The Fundamental Group (iii) For any X C, ut X acts on F(X) as follows: F ξ X σ ut X X. It X C is connected, then for anyξ F(X) the map ut X F(X) defined by u F(u)(ξ)=u ξ, is an injection. Proof. (i) uppose X=X 1 X2 in C, X 1, X 2 and that (X,ξ) E ; thenξ F(X)=F(X 1 ) F(X 2 ) say,ξ F(X 1 ). We then have a commutative diagram F ξ ξ j X 1 X 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. uppose we have a commutative diagram: F η ξ j Y with a monomorphism j. By (C 3 ) we get a factorisation: j Y X=X 1 X2 j 1 j 2 X 1 with an effective epimorphism j 1 and a monomorphism j 2. s j is a monomorphism, so is j 1 and thus j 1 is an isomorphism; since X is connected, X 2 = and one gets that j= j 2 j 1 is an isomorphism. X

67 4.4. Construction of the Fundamental group 59 (ii) s X is connected, it follows by (C 3 ) that u is an effective epimorphism; by (F 3 ), F(u) : F(X) F(X) is onto and thus is a bijection (F(X) finite). By (F 5 ) it follows that u ut X. (iii) Let u 1, u 2 ut X such that F(u 1 )(ξ) = F(u 2 )(ξ), i.e.,ξ 76 ker(f(u 1 ), F(u 2 ))=F(ker(u 1, u 2 )) (F is left-exact). We thus have a commutative diagram ξ F X u 1 X u 2 ξ j ker(u 1, u 2 ) with a monomorphism j; as (X,ξ) is minimal by (i), j is an isomorphism, in other words, u 1 = u 2. Q.E.D. We briefly recall now the example 2 of 4.3. ssertion (iii) of the above lemma simply says in this case that if K/k is a finite separable extension field and ifξ Hom k (K,Ω), then the map ut K Hom k (K,Ω) given by u ξ uis injective. We know that K/k is galois this map is also onto. Following this, we now make the Definition connected-object X C is galois if for anyξ F(X), the map ut X F(X) defined by u u ξ is a bijection. Note that this is equivalent to saying that the action of ut X on F(X) is transitive. lso observe that this definition is independent of F because the cardinality of F(X) is the degree of the covering X over ; the action is already effective since X is connected (by (iii), Lemma ). 77 Lemma If F η Y, then there is a galois object X C, a ξ F(X) and a u Hom C (X, Y) such that the diagram F ξ X η u Y

68 60 4. The Fundamental Group is commutative. In other words, the system I 1 of galois pairs of E is cofinal in E. Proof. Let ( i ) i I be a projective system of minimal objects of C such that F lim Hom C ( i, ). i I Letη 1,...,η r be the elements of F(Y). We can choose i large enough such that as u varies over Hom C ( i, Y), the u τ i give all theη s (τ i is the canonical element of F( i )). We then get: F τ i α i Y r = }{{} Y... Y r times p j Y 78 where p j is the j th canonical projection Y r Y; the elements p j α τ i, 1 j r, are precisely the elementsη 1,...,η r of F(Y). By (C 3 ) we get a factorisation: α i Y r α 1 β X with a monomorphismβ and an effective epimorphismα 1. We claim that X is galois. (i) X is connected. uppose X=X 1 X2, X 1, X 2 in C, ; the elementα 1 τ i F(X 1 ), say. We can then choose a j large enough for us to get a commutative diagram: α 1 i X=X 1 X2 τ i α F ϕ i j X 1 τ j α j

69 4.4. Construction of the Fundamental group 61 Henceβ α =α 1 ϕ i j is an epimorphism, which is absurd. (ii) etξ=α 1 τ i F(X). We shall prove that the map ut X F(X) 79 defined by u u ξ is onto. Letξ F(X); we may assume i is so large that we get a commutative diagram: X ξ α 1 τ i F i α 1 ξ X Our aim is to find aσ ut X such thatα 1 =σ α 1. ince X is connected,α 1 is also an effective epimorphism. ince 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 aρ ut Y r such that the diagram X ξ α 1 τ i F i β Y r ρ α 1 ξ X β Y r is commutative. By assumption the elements p 1 β ξ, 1 j r, 80

70 62 4. The Fundamental Group are all the distinct elements of F(Y); so the morphisms p j β α 1 are all distinct. This means that the p j β are all distinct; since α 1 is an effective epimorphism the p j β α 1 are all distinct; and as X is connected andβis a monomorphism, it follows that the p j β ξ are all distinct and therefore form the set F(Y). If we set p j β ξ=η j, 1 j r, and p j β ξ =η ρ( j), 1 j r, we get a permutationρof the set{1, 2,...,r}; this permutation determines an automorphism of Y r with the required property. Q.E.D. By this lemma we may clearly assume now that F is strictly prorepresented by a projective system ( i ) of galois objects of C. Letg i = ut i andθ i be the bijectiong i F( i ) defined by u u C i where C i is the canonical element of F( i ). For j i, we define ψ i j :g j g i as the composite θ j g j F( j ) F(ϕ i j) F( i ) θ 1 i g i. 81 For any u g j,ψ i j (u) is the uniquely determined automorphism of i which makes either one (and hence also the other) of the diagrams F i τ i ψ i j (u) i ϕ i j i ϕ i j ψ i j (u) i ϕ i j τ j j u j j u j commutative. It follows from this easily that theψ i j are group homomorphisms. We thus obtain a projective system{g i,ψ i j } i I of finite groups with eachψ i j surjective. Denote by{π i,ψ i j } i I the projective system of the opposite groups. The group π = lim π i with the limit topology is profinite and we shall prove that it is the fundamental group of the galois i I category C ; we denote it byπ 1 (, s) when C and F are as in 4.1 and 4.2.

71 4.4. Construction of the Fundamental group 63 π i acts on Hom C ( i, X) to the left and henceπacts continuously on the set lim Hom C ( i, X) F(X), to the left. ince F(X) is finite, the i action ofπon F(X) comes from the action of someπ i on F(X) We shall find it convenient now to introduce informally the no- 82 tion of the procategory Pro C of C. n object of Pro C = (called a pro-object of C ) will be a projective system p=(p i ) i I in C. If P, P = (P j ) j J are pro-objects of C, we define Hom( P, P ) as the double limit lim lim Hom C (P i, P j J i I j ). n object of C will be considered an 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 , we have: F(X) lim i I Hom C ( i, X), X C Hom ProC (, X) where is the pro-object ( i ) i I of C. lso for any i I, Hom ProC (, i ) Hom C ( i, i )=g i and we may then write: g=limg i = lim Hom C ( i, i ) i i = lim i Hom ProC (, i )=Hom Pro C (, ) and hence=ut ProC. If we call a pro-representative of F (in the case C = (E t/ ) we 83 call it a universal covering of ) thenπis the opposite of the group of automorphisms of. Lemma Let E C (π); then an object G(E) C, and a C (π)-isomorphismγ E : E FG(E) such that the map Hom C (G(E), X) Hom C (π) (E, F(X)) given by u F(u) γ E is a bijection for all X C. The assignment E G(E) can be extended to a functor G : C (π) C such that F and G establish an equivalence of C and C (π).

72 64 4. The Fundamental Group Proof. If E= E i is a decomposition of E into connected sets in C (π) and if G(E i ) are defined we may define G(E)= G(E i ). We may thus assume that π acts transitively on E. Fix an element E E and consider the surjectionπ E defined byσ σ E. s E is finite, there is an i such that the diagram π π i E 84 (π π i is the natural projection andπ i E is the mapσ σ E ) is commutative. Let H i π i be the isotropy group of E inπ i. It is easily proved that the setπ i /H i of left-cosets ofπ i mod H i is C (π)-isomorphic to E. We then define: G(E)=G(π i /H i )= i /Hi 0, the quotient of i by the opposite Hi 0 of H i (remark: Hi 0 g i ut i ). By (F 4 ) we have: F(G(E))=F( i /Hi 0) F( i )/Hi 0 π i /H i E, and hence a C (π)- isomorphismγ E : E FG(E). If j i and H j π j is the isotropy group of E inπ j, then we have a C -morphism j /H 0 j i /Hi 0; since F( j /H 0 j ) F( i/hi 0) is a C (π)-isomorphism, it follows from (F 5) that j /H 0 j i /Hi 0 and that G(E) is independent of the choice of i (upto a C -isomorphism). Let X C. Consider the map (i) ω is an injection: ω : Hom C (G(E), X) Hom C (π) (E, F(X)) u F(u) γ E Let u 1, u 2 Hom C (G(E), X) be such that F(u 1 ) γ E = F(u 2 ) γ E. Butγ E : E FG(E) is an isomorphism and so ker(f(u 1 ), F(u 2 )) FG(E), i.e., F(ker(u 1, u 2 )) FG(E) (F is left-exact). It follows from (F 5 ) that ker(u 1, u 2 ) G(E), in other words, u 1 = u (ii) ω is a surjection.

73 4.4. Construction of the Fundamental group 65 Let E α F(X) be any C (π)-morphism; putδ=α(e ) F(X). The ProC(π)-morphismδ: X can be factored through some i : τ i δ X δ i i Let H i be the isotropy group ofδ i inπ i then H i H i where H i is as before (take i large enough). By the construction of G(E)= i /Hi 0, we have a morphism i Hi 0 X making the diagram τ i δ i i i /Hi 0 X commutative; one easily checks that this morphism goes to α underω. It only remains to show that the assignment E G(E) can be extended to a functor. Let E, E C (π) andθ Hom C (π) (E, E ). To the compositeγ E θ : E FG(E ) there corresponds 86 a uniqueθ Hom C (G(E), G(E )) such that the diagram E θ E γ E FG(E) F(θ) γ E FG(E ) 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 C G F

74 66 4. The Fundamental Group and Ψ : I C (π) F G, such that, for any X C and E C (π), and F(X) F(Φ(X)) FGF(X) Ψ 1 (F(X)) F(X) G(E) G(Ψ(E)) GFG(E) Φ 1 (G(E)) G(E) and the identity maps. This completes the proof of assertion (1) of Theorem (4.4.1). 87 Lemma Let C be a galois category and F, F be two fundamental functors C (finite sets). upposeπ,π are the profinite groups defined by F, F respectively as above; thenπandπ are continuously isomorphic and this isomorphism is canonically determined upto an inner automorphism ofπ. 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 thatπ T in ProC. In order to do this, let (T j ) j J be a projective family of galois objects (with respect to F ) of C such that T= lim T j j; we denote the canonical maps T j T i by q i j and T T j by q j. Let t j T j be a coherent system of points and consider the continuous mapsα j :π T j determined by α j (e)=t j ; there exists a continuousα:π T such that π α j α T q j T j 88 is commutative. We haveα(e)= t=(t j ) T. The T j are connected, henceα j transitive and asπis compact it follows thatαis onto. Let H be the isotropy group of t, then we haveπ/h T. (π/h is the set

75 4.4. Construction of the Fundamental group 67 of left-cosets ofπ mod.h) and this is an isomorphism of topological spaces (it is a continuous map of compact Hausdorff spaces). However F = Hom ProC ( T, ) and then clearly F (X) = X H (the points of X invariant underh). We know that F (X)= X= ; from this it follows thath=(e) (take for X theπ i 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 ProC. For this it is enough to show the following: ifβ : T π isα 1 and if p i :π π i are the canonical maps then everyβ i = p i β : T π i must factor through some T j. In other words, we must find some morphism T j π i making the diagram T β π β q i j T j π i commutative. For this we must show that given i, j J such that 89 for every t T j, an s π i such that q 1 j (t) β 1 i (s). ince the setsβ 1 i (s) are open we can find for every point x β 1 i (s) an open neighbourhood U x of the form q 1 j (t x ) for some j x J and t x T jx, such that U x β 1 i (s). ince T is compact, afinite covering U x1,...,u xn of T of this type and j>max( j x1,..., j xn ) satisfies our requirements. This shows thatπ T in Pro C, and hence the map π 0 = ut ProC T ut ProC π=π 0 ρ β ρ α is a group isomorphism; it remains to be shown that this map is continuous and hence a homeomorphism. Take a fixedγ 0 =β ρ 0 α inπ, some i I and consider the commutative diagram p i π γ 0 π p i π i γ 0 i π i p i

76 68 4. The Fundamental Group 90 Let U be the neighbourhood ofγ 0 consisting of allγ π such that p i γ=γi 0 p i. We want to find a neighbourhood V ofρ 0 such that β ρ α U ρ V. There is an index j J and l Iand morphisms making all the following diagrams π α T ρ 0 T β π p 1 q j q j π l T j ρ 0 j T j p i p l π i γ 0 i π i commutative. Consider allρ : T T such that q j ρ=ρ 0 j q j; theseρform a neighbourhood V ofρ 0 andβ ρ α U ρ V. Hence we are through. Finally we observe that the isomorphismπ π is fixed as soon asα :π T is fixed and this is in turn fixed by the choice of t = (t j ) T. By a different choice of t we obtain an isomorphismπ π which differs from the first one by an inner automorphism ofπ(orπ ). Q.E.D. 91 Remark One can, in fact, show that ProC(π) is precisely the category of compact, totally disconnected (Hausdorff) topological spaces on whichπacts continuously.

77 Chapter 5 Galois Categories and Morphisms of Profinite Groups 5.1 upposeπandπ are profinite groups and u :π π a continuous homomorphism. Then u defines, in a natural way, a functor H u : C (π) C (π ); and H u 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 ={ i,ϕ i j } i I of C such that, X C, F(X) Hom ProC (, X). Moreover we may assume that the i are galois objects of C, therefore the F( i ) are principal homogeneous spaces under the action of theπ i (on the right) (notations from Ch. 4) and if we identify F( i ), by means of the canonical elementτ i, withπ i then the maps F(ϕ i j ) = ψ i j : F( j ) F( i ) are group homomorphisms (see Ch. 4). However, in the present situation the F( 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 theπ i. This gives a continuous homomorphism 92 69

78 70 5. Galois Categories and Morphisms of Profinite Groups 93 u i :π π i determined by the condition that forσ π the u i (σ ) is the unique element ofπ i such thatσ τ i =τ i u i (σ ) (τ i is the canonical element of F( i )); for j i we clearly have:ψ i j u j = u i ; thus, we obtain a continuous homomorphism thus, we obtain a continuous homomorphism u :π lim π i i=π and u corresponds to H if we identify C with C (π). Example. We shall apply the above to the particular case C= (E t/ ), C = (E t/ ) where, are, as usual locally noetherian, connected preschemes. upposeϕ : is a morphism of finite type and s, s=ϕ(s ). LetΩbe an algebraically closed field containing k(s ). We define functors F : C {Finite sets} and F : C {Finite sets} by defining: F(X)=Hom (pecω, X), X C, F (X )=Hom (pecω, X ), X C. Denote byπandπ the fundamental groupsπ 1 (, s) andπ 1 (, s ). Corresponding to the morphismϕ :, we obtain a functor Φ : C C given by X X. We then have: F (X )=Hom s (pecω, X ) Hom s (pecω, X)=F(X). 94 Thus F= F Φ; in view of the fact that F is fundamental and the equivalences C C F (π), C F C (π ), we obtain a continuous homomorphismπ π. 5.2 In this section, we shall correlate the properties of a homomorphism u :π πand those of the corresponding functor H u : C (π) C (π ) uppose u :π πis onto; for any connected object X of C (π) (i.e.,πacts transitively on X) anyπ-morphismπ X defined by, say,

79 e x is onto and therefore so is the mapπ X defined by e x; in other words, H u (X) is a connected object of C (π ). Conversely, suppose that for any connected X C (π), the object H u (X) is again connected in C (π ). Writeπ=lim π i where theπ i are i finite groups and the structure-homomorphismsπ j π i, j i, are all onto; this implies that theπ π i are all onto and by our assumption then all theπ π i are onto. inceπandπ are both profinite it follows that u :π π is onto. Thus: u :π π is onto for any connected X C (π), the object H u (X) is connected in C (π ) 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 onπ, it is clear that giving a pointed, connected object of C (π) is equivalent to giving an open subgroup H ofπ; the object isπ/h= set of left-cosets ofπ mod H and the point is the class H. final object of C (π) is a point e c 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 e c to X; giving a section of X is equivalent to giving a point of X, invariant under the action ofπ. pointed object (X, x) of C (π) admits a pointed section (i.e., e c is mapped onto x) x is invariant under π. uppose u :π π is a homomorphism and H is an open subgroup ofπsuch that u(π ) H. Let (X, x) be the pointed, connected object of C (π) determined by H. Then, in the action ofπ on H u (X), x remains invariant, i.e., to say, the pointed object (H u (X), x) of C (π ) admits a pointed section. The converse situation is clear. Thus: For an open subgroup H ofπ, one has u(π ) H H u (π/h) admits a pointed section in C (π ) 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 ofπon X is trivial. uppose u :π π is trivial, then for any X C (π), H u (X) is 96 completely decomposed in C (π ). Conversely, assume that for any X C (π)h u (X) is completely decomposed in C (π ). Writeπ=lim i π i as

80 72 5. Galois Categories and Morphisms of Profinite Groups usual; by assumption, each compositeπ π π i is trivial. Hence u :π πis also trivial. Thus: u :π π is trivial for any X C (π), H u (X) is completely decomposed in C (π ) Let H be an open subgroup ofπ and X C (π ) the connected, pointed object defined by H. ssume 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 ). inceπ is compact andπhausdorff 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). Denote by H u (X) 0 the C (π )-component of the pointed object H u (X) of C (π ), containing the distinguished point of H u (X). There exists then an open subgroup H 1 ofπ such that H u (X) 0 π /H 1 in C (π ). We claim that H 1 H ; in fact, u(h 1 ) H and so u(h 1 ) H u(π ) u(h ), hence H 1 u 1 (u(h 1 )) u 1 (u(h ))=H, since, by assumption, H is saturated under u. Thus, there is a pointed C (π )-morphism H u (X) 0 π /H 1 π /H X. If, on the other hand, we assume that apointed, connected object X of C (π) such that we have a pointed C (π )-morphism H u (X) 0 π /H 1 X π /H, then, we must have H 1 H and hence ker u H 1 H. If u is surjective, then we can say that H u (X) X (see 5.2.1). lso 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 )-morphism of a connected C (π )-component of H u (X) to X =π /H. If u is onto then X =π /H H u (X) for a connected object X C (π). 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 u (X) to X Letπ u π π u be a sequence of morphisms of profinite groups. From and we obtain the following necessary and

81 sufficient conditions for the sequence to be exact: (a) u u is trivial for any X C (π ), H u H u (X ) is com- 98 pletely decomposed in C (π ). (b) Im u ker u for any open subgroup H ofπ, with H Im u, we also have H ker u for any connected pointed object X of C (π) such that H u (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 H u (X ) to X Let C be a galois category with a fundamental functor F. Let = ( i ) i I be a pro-object of C with usual properties (in particular, i are galois) pro-representing F. Let π be the fundamental group of C determined by F. We know then that C F 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 (C 0 ),...,(C 4 ) of Ch. IV. We have an exact functor P : C C defined by X X T. We now define 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). gain it is easily checked that C, equipped with F, is galois. cofinal subsystem of is defined by the condition: i ( i,τ i ) dominates (T, t) in the sense of Ch. 4. n i can be considered in the obvious way 99 as an object of C ; it is then easily shown that they are galois in C and the pro-object of C pro-represents F (To do the checking one may identify C and C (π)). Let H be the isotropy group of t F(T) inπ; let also N i be the isotropy groups ofτ i F( i ), i. We have a diagram of the form π F(T) t τ i F( i ) which is commutative and thus N i H, i. ince the F( i ) and F(T) are all connected objects of C (π), we have π N i F( i ) andπ/h F(T).

82 74 5. Galois Categories and Morphisms of Profinite Groups The maps F( i ) F(T) are then the natural mapsπ/n i π/h and it follows that F ( i ) H/N i in C (π). But as we have already remarked the i in form a system of galois objects with respect to F, prorepresenting F and one thus obtains: π lim F ( i )= lim i N i H H/N i H. 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.

83 Chapter 6 pplication of the Comparison Theorem an Exact equence for Fundamental Groups 6.1 s 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). 100 Definition (a) morphism X pec k, k a field, is said to be separable if, for any extension field K/k, the prescheme X K is k reduced. (b) morphism X f Y is separable if f is flat and for any y Y, X pec 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 EG. Theorem Let f : X Y be a proper morphism and F a coherent 75

84 76 6. pplication of the Comparison Theorem... O X -Module. If Y 1 q Y is a flat base-change X q 1 X 1 = X Y Y 1 then we have the isomorphisms f f 1 q Y Y 1 R n f 1 (q 1 (F )) R n f (F ) OY O Y1 for any n Z +. (Prop. (1.4.15), EG, Ch. III) uppose Y is a noetherian prescheme and f : X Ya proper morphism. Let Y Y be a closed subscheme of Y, defined by a coherent Ideal T of O Y. 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 O X -Ideal T = f (T )O X. Let F be any coherent O X -Module; for n Z +, consider F n = F OX O X /T n+1 (this is a coherent O X -Module, concentrated on the prescheme X n = (X, O X /T n+1 ) and may also be considered as an O Xn - Module). Consider R q f (F n ); this is a coherent O Y -Module (finiteness theorem), is concentrated on the prescheme Y n = (Y, O Y /T n+1 ) and is in fact an O Yn -Module. From the homomorphism F F n we obtain a homomorphism R q f (F ) R q f (F n ) and since the latter is an O Yn - Module, we get a natural homomorphism: (*) R q f (F ) OY O Y /T n+1 R q f (F n ). s n varies, we get a projective system of homomorphisms. With the assumptions we have made about X, Y, f, the comparison theorem (EG, Ch. III, Theorem (4.1.5)) states that in the limit, this gives an isomorphism: (**) lim R q f (F ) O Y /T n+1 lim R q f OY (F n ). n n

85 6.2. The tein-factorisation (Note: Both sides of (**) are concentrated on Y and they are also equal to R q f (lim F O X /T n+1 ) where f is the morphism of the ringed n O X spaces X=(X, lim O n X/T n+1 ) Ŷ= (Y, lim O n Y/T n+1 ) obtained from the morphisms f n : X n Y n, induced by f ). We shall now specialise the above comparison theorem to the case Y affine, say Y= pec. Then Y is defined by an ideal I of. The first member of (**) corresponds to lim Hq (X, F ) which is precisely n I n+1 the completion of H q (X, F ) under the I-adic topology, while the second member of (**) corresponds to lim Hq (X, F n ) and thus: n (***) H q (X, F ) lim n H q (X, F n ). 6.2 The tein-factorisation Let X f Y be a proper morphism. Then the coherent O Y -lgebra f (O X ) (finiteness theorem) defines a Y-prescheme Y q T, finite on Y. To the identity O Y -morphism q (O Y )= f (O X ) f (O X ) corresponds a Y-morphism f : X Y, i.e., we have a commutative diagram X f Y = pec f (O X ) f q Y The morphism f is again proper. This factorisation f = q f is 103 known as the tein-factorisation of f. For details the reader is referred to EG Ch. III. We shall now prove a theorem, which is of great importance to us. Theorem Let f : X Y be a separable, proper morphism. Let X f Y q = pec f (O X ) Y be the tein-factorisation of f. Then Y q Y is an étale covering.

86 78 6. pplication 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= pec. We shall make a few simplifications to start with. uppose X 1 Y is a flat base-change; then from one gets: f (Y1 ) (O X OY O Y1 ) f (O X ) OY O Y1 This means that in the commutative diagram X f pec f (O X )=Y X 1 = X Y Y 1 f (Y 1 ) Y 1 = Y Y 1 = pec f (Y1 ) (O X O Y1 ) Y OY q Y Y 1 q (Y1 ) 104 the second vertical sequence is the tein-factorisation of f (Y1 ). In view of this and the fact that it is enough to look at pec O y,y for étaleness over y Y we may make the base change Y pec O y,y and assume that is a local ring. gain, in view of the same remark and Lemma we may assume that is complete. Consider now the functor T on the category of finite type -modules M, defined by M T(M)=Γ(X, O X M). We shall show that the 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, O X ) M Γ(X, O X M) is an isomorphism. Denote by T the right-exact functor M Γ(X, O X ) M.

87 6.2. The tein-factorisation 79 uppose T is right exact. Then for any exact sequence n m M 0 we have a commutative diagram of exact sequences: T ( n ) T ( m ) T (M) 0 T( n ) T( m ) 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 rightexact. Now from the flatness of f and the assumed right exactness of T it follows thatγ(x, O X M) is exact in M and thus [from the above remark] Γ(X, O X ) is -flat. But Y = pecγ(x, O X ) and therefore q : Y Y is flat. It remains to show that q is unramified. Let y Ybe the unique closed point of Y; denote by k the residue field /M = k(y). Then T(k)=Γ(X, O X k)=γ(x y, O Xy ) where X y is the fibre X Y k(y). But as X is Y-proper, T(k)=Γ(X, O X k) is an artinian 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) = K i where the K i are finite separable field extensions of k. lso, by our assumption, T(k)=Γ(X, O X ) k=γ(x, O X )/MΓ(X, O X ). Let y Y be a point r above y. From the equalityγ(x, O X )/MΓ(X, O X )= K i, (K i /k are separable) it follows that the maximal ideal M of O y. generates the maximal ideal of O y and moreover that O y is unramified over O y. 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 more simplifications. First we may assume that T(k)=Γ(X y, O Xy )= r k (this can be done as in Lemma by making a faithfully flat i=1 base-change which kills the extensions K i/k ). Then finally we may i=1 i=1

88 80 6. pplication of the Comparison Theorem... assumeγ(x y, O Xy )=k, i.e., X y connected, because the connected components of X over y correspond bijectively with the connected components of X y and it is enough to prove the theorem separately for each component over y. Case (a). ssume artinian. Our aim is to show that for any finite type -module M, T (M) T(M). We shall first show that T (M) T(M) is surjective. For n large, one has M n M= 0 so that T (M n M) T(M n M) is onto. ssume now that T (M i+1 M) T(M i+1 M) is onto, and consider the exact sequence: 0 M i+1 M M i M M i M M i+1 M 0 of -modules. We get a commutative diagram T (M i+1 M) T (M i M) T ( ) M i M M i+1 M 0 T(M i+1 M) T(M i M) T ( ) M i M M i+1 M 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 map is a surjection by assumption, the last is a surjection since M i M M i+1 M is a finite direct sum of copies of k and by our assumptionγ(x y, O 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 p M 0. We then obtain a commutative diagram T (R) T ( p ) T (M) 0 onto T(R) T( p ) T(M) of exact sequences. If follows easily that T (M) T(M) is also injection. Thus, the theorem is completely proved in the case (a).

89 6.2. The tein-factorisation 81 Case (b). The general case Let be a complete noetherian local ring. Denote by Y 1 the closed subscheme of Y defined by the maximal ideal M. s in the comparison theorem we may define the functors T n (M n )=Γ(X n, O X M /M n+1 ) =Γ(X, O X M /M n+1 ) where M n = M /M n+1. If M is an -module of finite type, so 108 isγ(x, O X M); as is complete under the M -adic topology, so is Γ(X, O X M) and from we obtain: Γ(X, O X M) lim n T n (M n ). To show that T is right-exact it is enough to show that for any exact sequence M u N 0 of finite type -modules, T(M) T(u) T(N) 0 is again exact. But as each /M n+1 is a complete artinian local ring, it follows from case (a) that T n (M n ) T n(u) T n (N n ) 0 is exact; also for each n, ker T n (u) is a module of finite length over. Thus, it is enough now to prove the Lemma Let (K n,ϕ nm ), (M n,ψ nm ), (N n,θ nm ), n Z +, be projective systems of abelian groups and u=(u n ), v=(v n ) be morphisms such that, for every n Z + u n v n 0 K n M n N n 0 is exact. ssume, in addition, that for each n, m 0 = m 0 (n) such that ϕ nm (K m )=ϕ nm0 (K m0 ) m m 0 (n)>n (this is the so-called Mittag- Leffler (ML) condition; it is certainly satisfied if the ker u n = K n are of finite length). Then the sequence of projective limits is also exact. Proof. The only difficult point is to show that lim n v n is onto. By hy- 109 pothesis, for each n, m 0 (n)>n withϕ nm (K m )=ϕ nm0 (K m0 ) m m 0 (n). By passing to a cofinal subsystem we may suppose, that, for any n Z + ϕ nm (K m )=ϕ n,n+1 (K n+1 ) m n+1.

90 82 6. pplication of the Comparison Theorem... Let now (y n ) lim N n n. Choose x 0 M 0 with v 0 (x 0 )=y 0. ssume inductively that we have chosen (x 0, x 1,..., x n 1, x n ) with (i)ψ r,r+1(x r+1 ) = x r for 0 r n 2, andψ n 1,n (x n )= x n 1. (ii) v r (x r )=y r for 0 r n 1, and v n (x n)=y n. Our aim now is to find (x 0, x 1,..., x n 1, x n, x n+1 ) for which the above properties (i), (ii) hold when n is replaced by n+1. Choose x n+1 M n+1 such that v n+1 (x n+1 )=y n+1; then, v n (ψ n,n+1 (x n+1 ) x n )=y n y n = 0 i.e. to say,ψ n,n+1 (x n+1 ) x n K n. By assumption, we can find z n+1 K n+1 such that: ψ n 1,n+1 (z n+1 )=ϕ n 1,n+1 (z n+1 ) =ϕ n 1,n (ψ n,n+1 (x n+1 ) x n) =ψ n 1,n+1 (x n+1 ) ψ n 1,n(x n). 110 We now set x n+1 = (x n+1 z n+1) and x n =ψ n,n+1 (x n+1 ). We then have: ψ n 1,n (x n )=ψ n 1,n+1 (x n+1 ) ψ n 1,n+1(z n+1 ) =ψ n 1,n (x n)= x n 1. v n+1 (x n+1 )=v n+1(x n+1 )=y n+1 and finally v n (x n )=v n (ψ n,n+1 (x n+1 ))=y n. Q.E.D. 6.3 The first homotopy exact sequence ome properties of the tein-factorisation of a proper morphism. Let Y be locally noetherian and f : X Y be proper. uppose X f Y q Y is the tein factorisation of f. By using the comparison theorem one can prove the (Zariski s connection theorem). The morphism f : X Y is also proper. nd for any y Y the fibre f 1 (y ) is non-void and geometrically connected (i.e., for any field k k(y ) the prescheme X Y pec k is connected).

91 6.3. The first homotopy exact sequence 83 (For a proof see EG Ch. III (4.3)). One can then easily draw the following corollaries. Corollary For any y Y, the connected components of the fibre f 1 (y) are in a (1 1) correspondence with the set of points of the fibre q 1 (y) (which is finite and discrete). Corollary For any y Y, let k(y) be the algebraic closure of 111 k(y) and X y be the prescheme X Y k(y). The connected components of X y (known as the geometric components of the fibre over y) are in (1 1) correspondence with the geometric points of Y over y Now, in addition, suppose that (i) Y is connected. (ii) f is separable (iii) f (O X ) O Y. ssumption (iii) implies that X f Y is its own tein-factorisation. From??, it follows that the fibres f 1 (y) are (geometrically) connected. f is a closed map and therefore X is also connected. imilarly, for any y Y, X y is also connected. Fix y Y. We have a commutative diagram: a X X y a f y Y f k(y) LetΩbe an algebraically closed field k(y) and let a X y be a geometric point overω; let a Xbe the image of a in X. We have the fundamental groupsπ 1 (X y, a),π 1 (X, a),π 1 (Y, y) and continuous homo- 112 ϕ ψ morphisms: π 1 (X y, a) π 1 (X, a),π 1 (X, a) π 1 (Y, y) (ee 5.1). We now have the following:

92 84 6. pplication of the Comparison Theorem... Theorem The sequence is exact. Proof. π 1 (X y, a) ϕ π 1 (X, a) ψ π 1 (Y, y) 0 (a) ψ is surjective. In view of it is enough to show that, if Y /Y is connected étale covering, then the étale covering X = X Y Y over X is connected. X X Y Y = X f Y f (proper) Y (connected) But O Y = f (O X ) and therefore, we have (from 6.1.2) f (O X )= f (O X OY O Y )= f (O X ) OY O Y = O Y. It follows that the fibres of f are connected, and hence X = X Y Y is also connected. 113 (b)ψ ϕis trivial. In view of it is enough to show that if Y /Y is any étale covering, the étale covering (Y X y )/X y is completely decomposed. Y But Y k(y)= k(y) and hence: Y X y X (Y k(y)) X y. Y Y = γ (c) Imϕ kerψ. finite finite In view of it is enough to prove that: uppose X g X is a connected étale covering of X and X g y X y admits a section σ (over X y ). Then aconnected étale covering Y /Y such that X X Y Y. We need, for proving this, the following

93 6.3. The first homotopy exact sequence 85 Lemma The composite h : X g X f Y is proper and separable. 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 pec K, K a field. For this, it is enough to prove that: X /X étale covering, X reduced X reduced. We may assume X= pec, X = pec ; we have = red and we want to show that, = red. Let (P i ) n i=1 be the minimal prime ideals of. By assumption, the natural map n (/P i ) is an injection. o, n ( /P i ) 114 i=1 is also injective and it is enough then to show that each /P i is reduced. By making the base-change /P i, we may then assume that is an integralldomain. Let a X= pec be the generic point of X; then k(a)=k the field of fractions of. The fibre over a is= pec( K) and since this is non-ramified over k(a)=k, we have K= r K i, K i/k finite separable field extensions. In particular, i=1 K is reduced and hence for each x X, O x Og(x ) K is reduced and hence so is O x O x g(x ) K. It follows that = red. Coming back to the proof of the assertion, let now X h Y Y be the tein-factorisation of h : X Y. From the above lemma and Theorem it follows that Y Y is an étale covering. We have a commutative diagram X X α X = X Y Y p 1 f f Y Y = pec h (O X ) h i=1

94 86 6. pplication of the Comparison Theorem Our assertion will follow if we show thatα : X X = X Y Y is an isomorphism. We do this by showing that: (i) α is an étale covering. (ii) X is connected. (iii) rank ofαis 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 soαis an étale covering. (ii) We know that h is onto ( ) 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 y g α X y X p y h 1 f k(y) Y y 116 It is enough to show that rankα=1 at some point of X y. ince Y /Y is an étale covering, Y y= n k i each k i = k(y) and so X y = X y Y y= i=1 k(y) n X yi each X yi = X y. The sectionσ:x y X y is an étale covering and i=1 X y is connected and henceσ(x y ) is a component Z of X y;α(z) must then be some X yi. lso the projection p 1 is an isomorphism from X yi

95 6.3. The first homotopy exact sequence 87 to X y ; it follows that Z are α inverses of one another. lso X yi (σ p 1 ) we know that the number of components of X y 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 X y and X y are the some. α is surjective because it is both open and closed and X is connected, thereforeα is surjective and étale and we get rankα=1 at every pt. of X y.q.e.d. Remark One may drop the assumptions f (O X )=O Y and Y connected, in the above theorem; the assertion of the theorem will then be: Denote byπ 0 (X y, a),π 0 (X, a),π 0 (Y, y) the (pointed) sets of (connected) components of the preschemes X y, X, Y. Then, if f is proper 117 and separable, we have the following exact sequence: π 1 (X y, a) π 1 (X, a) π 1 (Y, y) π 0 (X y, a) π 0 (X, a) π 0 (Y, y) (1) Proof. ssume to start with that X, Y are connected, dropping only the assumption f (O X ) O Y. We have then the tein-factorisation: X f f q Y = pec f (O X ). pplying the above theorem to f we obtain an exact sequence: π 1 (X y, a) π 1 (X, a) π 1 (Y, y ) (e) where y Y is the image of a X. We know then thatπ 1 (Y, y ) π 1 (Y, y) is an injection (5.2.6) and the quotientπ 1 (Y, y)/π 1 (Y, y ) (set of left cosets modπ 1 (Y, y )) is isomorphic to the set of geometric points of Y over y. By corollary this is isomorphic toπ 0 (X y, a). We thus obtain the exact sequence: π(x y, a) π 1 (X, a) π 1 (Y, y) π 0 (X y, a) (1). 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. Y

96

97 Chapter 7 The Technique of Descents and pplications 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, be rings andϕ : be a ring-homomorphism. Let C (resp. C ) be the category of -(resp. -) modules. ϕ defines a covariant functorϕ : C C, viz.,ϕ (M)= M = M. uppose M, N C and u : M N is an -linear map. ϕ (u)=u 1 : M N is an -linear map. Then Problem 1. uppose u : M =ϕ (M) ϕ (N)=N is an -linear map. When can we say that u =ϕ (u) for an -linear map u : M N? Problem 2. uppose M C. When can we say that M = M for an M C? Example 2. Let, be preschemes andϕbe a morphismϕ:. Let C (resp. C ) be the category of quasicoherent O -(resp. C -) Modules. ϕ defines a (covariant) functor C C given by F 89

98 90 7. The Technique of Descents and pplications ϕ (F ). We again have: Problem 1. If u : F =ϕ (F ) g =ϕ (g) is an O -morphism when can we say that u =ϕ (u) for an O -morphism u : F g? 119 Problem 2. If F is any quasi-coherent O -Module, when can we say that F =ϕ (F ) for a quasi-coherent O -Module? Example 3. Let, be preschemes and let C = (ch/ ), C = (ch/ ). upposeϕ : is a morphism;ϕ defines a (covariant) functorϕ : C C given by X ϕ (X)=X = X. 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 C, we are given a category C such that for every morphismϕ : in C, we are given a covariant functorϕ : C C. ssume, in addition that (1) If ψ ϕ is a sequence in C, there is a natural isomorphism:ψ ϕ (ϕ ψ). (2) If ϕ3 ϕ2 ϕ1 is a sequence in C, the diagram, obtained from (1), (ϕ 1 ϕ 2 ϕ 3 ) ϕ 3 (ϕ 1ϕ 2 ) (ϕ 2 ϕ 3 ) ϕ 1 ϕ 3 ϕ 2 ϕ is commutative.

99 (3) (Id) = Id. (4) The isomorphismψ ϕ (ϕ ψ) of (1) has the property that if eitherψorϕis the identity, the isomorphism also becomes the identity. Problem 1. Givenξ,η C and a morphism u :ξ =ϕ (ξ) ϕ (η)=η in C when can we say that u=ϕ (u) for a morphism u :ξ η in C? Problem 2. Givenξ C, when isξ of the formϕ (ξ) for aξ C? We shall now obtain certain conditions necessary for the above two questions to have an answer in the affirmative. For Problem 1. uppose u :ξ η in C such that u =ϕ (u) :ξ η. ssume that = exists in C ; consider the sequence: p 1 ϕ p 2 Letϕ p 1 =ϕ p 2 =ψ and denote byξ,η the elementsψ (ξ), ψ (η) of C. Consider the morphism p 1 (u) : p 1 (ξ ) p 2 (ξ ); according to our assumption p i ϕ ψ. There is then a (dotted) morphism 121 making the diagram p 1 (ξ ) p 1 (u ) p 1 (η ) ξ =ψ (ξ) ψ (η)=η commutative. In the following we also denote this morphism by p 1 (u ), i.e., we identify p 1 (ξ ) withξ and p 1 (η ) withη. We introduce similarly p 2 (u ). It is clear that with these notations we must have p 1 (u )= p 2 (u );

100 92 7. The Technique of Descents and pplications in factα: both equalψ (u). If this necessary condition is also sufficient (as they turn out to be in some cases) we say thatϕ: is a morphism of descent. For Problem (2). uppose ξ C, withϕ (ξ)=ξ. Then, with the above notations, we have: p 1 (ξ ) ψ (ξ) p 2 (ξ ), 122 i.e., an isomorphismα: p 1 (ξ ) p 2 (ξ ) making the diagram p 1 (ξ α ) p 2 (ξ ) ψ (ξ) commutative. We shall now get conditions onα. ssume that = exists in C ; let q i be the i th projection ; also let :. Consider the se- p ji ( j i) be the morphism (q i, q j ) quence: p 1 ϕ p. 2 p 31 Letλ=ϕ p 1 p 21 =ϕ p 1 p 31 = :. We then have commutative diagrams of the type: p 21 p 32 λ (ξ) q 1 (ξ ) q 2 (ξ) p 21 p 1 (ξ ) p 21 (α) p 21 ψ (ξ) p 21 p 2 (ξ ) 123 We also denote by p 21 (α) the (dotted) morphism q 1 (ξ ) q 2 (ξ )

101 which will make the diagram q 1 (ξ ) p 21 p 1 (ξ ) p 21 (α) q 2 (ξ ) p 21 p 2 (ξ ) commutative. Using this diagram and the large diagram above we see that q 1 (ξ ) λ (ξ) q 2 (ξ) p 21 (α) is commutative. We make similar conventions and get similar commutative diagrams for p 32 (α) and p 31 (α). Then we must have p 32 (α)p 21 (α)= p 31 (α) (the so-called cocycle condition). Finally, if : = is the diagonal, by the assumptions made at the beginning we must have 124 (α)=identity. If these necessary conditions are also sufficient and ifϕis also a morphism of descent, we say thatϕis a morphism of effective descent. nαof the above type is then called a descent-datum onξ. Remark. Let C be the category of preschemes; if we agree that we take and locally noetherian but if we don t assume of finite type then it may very well happen that = is not locally noetherian (e.g. = pec, = pec Â, a noetherian local ring). We shall deal with this difficulty in the sections , ,

102 94 7. The Technique of Descents and pplications Example (1). Proposition faithfully flat ring-homomorphismϕ: is a morphism of effective descent. Proof. (a) We shall first show thatϕis a morphism of descent. Let M, N be -modules and u : M N be an -linear map. Consider the commutative diagram M M = M p 1 p 2 M = M u p 1 (u ) p 2 (u ) N N = N p 1 p 2 N = N 125 We have to show that if p 1 (u ) = p 2 (u ) then an -linear map u : M N such that u = u I. This will follow, if we show that (i) for any M C, M M is an injection, (ii) for any N C, N N p 1 p 2 N is exact, and (iii) u (M) ker(n, p 1, p 2 ). (i) We know that is a direct factor of by means of the map a a 1; thus, for any M C, M M makes M a direct summand of M and is in particular an injection. s is faithfully -flat it follows that M M is an injection. (ii) We have to prove that the sequence N N p 1 p 2 N

103 is exact. It suffices to prove this after tensoring the sequence with a ring B faithfully flat over ; note that we get a similar situation with the pair B B = B as with. Take for B the ring itself. But then = admits a sectionλ: given byλ(a 1 a 2 )=a 1 a 2. We may thus assume without loss of generality thatϕ : itself admits a sectionλ :. 126 Consider then the commutative diagram: N ψ=1 ϕ N p 1 p 2 N Id. µ=1 λ N N µ 1 N N Let x = (x i a i ) be an element of N = N such that p 1 (x )= p 2 (x ); that is, (x i a i 1)= (x i 1 a i ). On applyingµ 1 we obtain: (x 1 λ(a i ) 1)= (x i 1 a i ); under the identification N N, this means that (x i a i )= (λ(a i )x i 1), i.e., x =ψ(µ(x )) ψ(n). (iii) By the commutativity of the diagram at the beginning of the proof, we have: p 1 (u ) p 1 = p 1 u, p 2 (u ) p 2 = p 2 u and by assumption p 1 (u )= p 2 (u ) while for m M one has p 1 (m)= p 2 (m)=m 1 1 M. Thus, for m M, p 1 u (m)= p 2 u (m) and (iii) is proved. (b) It remains to show thatϕis effective. uppose N C and we have 127

104 96 7. The Technique of Descents and pplications a commutative diagram: N = p 1 (N ) p 1 N α p 2 N = p 2 (N ) whereαis an -isomorphism satisfying the cocycle condition. We want to find an N C such that N ρ N and such that N p 1 (ρ) N (natural) α N p 2 (ρ) N 128 commutes. et N= ker(α p 1 p 2 ) C (this choice of N is motivated by (ii) of (a)). We always have an -linear map N ρ N. To show thatρ is an -isomorphism, it is enough to show thatρis an -isomorphism; and for this, we may assume, as in (a), that there is a sectionλ: forϕ:. For the sake of clarity and ease, we now go back to the general case and prove: Lemma Ifϕ : admits a sectionσ:, thenϕis a morphism of effective descent. Proof. Letξ C andα : p 1 (ξ ) p 2 (ξ ) be an isomorphism such that p 32 (α)p 21 (α)= p 31 (α) (notations as before). Our aim is to find an

105 η C and an isomorphismη =ϕ (η) ρ ξ such that the diagram η p 1 (η ) p 2 (η ) p 1 (ρ) p 1 (ξ ) α p 2 (ρ) p 2 (ξ ) is commutative. We have the diagram: ϕ p 1 p 2 Id. σ σ 1 P with p 1 (σ 1)=σ ϕ and p 2 (σ 1)= identity. Hence, ifη=σ (ξ ) 129 we haveη =ϕ σ (ξ )=(p 1 (σ 1)) (ξ )=(σ 1) p 1 (ξ ) and ( ) (σ 1) p 2 (ξ )=ξ. We then get aθ :η ξ, namely,θ=(σ 1) (α). We obtain thus aβ: p 1 (η) p 2 (η ) which makes the diagram p 1 (ξ ) α p 2 (ξ ) p 1 (θ) p 2 (θ) p 1 (η ) β p 2 (η ) commutative. However, we do not, in general, have a commutative dia-

106 98 7. The Technique of Descents and pplications gram p 1 (η ) p β 2 (η ) (natural) (natural) η = (p 1 ϕ) (η) 130 Therfore, we want to modifyθ :η ξ to aρ :η ξ by means of an automorphismγ:η η (i.e.,ρ=θ γ) such that the diagram p 1 (η ) p 1 (γ) β p 2 (η ) p 2 (γ) p 1 (η ) p 2 (η ) (natural) η is commutative. This ρ will satisfy our requirements. We first observe thatβ : p 1 (η ) p 2 (η ) also satisfies the cocycle condition. We may considerβ as an element of ut(η ). To find aγin utη such as above, i.e., such thatβ= p 2 (γ) p 1 (γ) 1 we have only to prove the following: Lemma Letη C and consider the complex (ut) utη p 1 utη p 2 p 31 p 21 utη p 32 (notations as before). If there is a sectionσ: then H 1 (ut)=(e).

107 Proof. By definition, the 1-cocycles are elementsβ utη for which the cocycle condition is satisfied and the 1-coboundaries are those β utη which are of the formβ= p 2 (γ) p 1 (γ) 1, for aγ utη. Consider now the corresponding complex in C ; the sectionσ: defines a homotopy operator for this complex in the following manner: ϕ p 1 p p 31 2 p 32 p 21 Id. σ σ 1 σ 1 1 ϕ p 1 p 2 with p 1 (σ 1)=σ ϕ, p 2 (σ 1)=Id., (σ 1) p 1 = p 21 (σ 1 1), (σ 1) p 2 = p 31 (σ 1 1), and p 32 (σ 1 1)=Id. Ifβis a 1-cocycle, setγ=(σ 1) (β). We then have: p 2 (γ) p 1 (γ) 1 = p 2 (σ 1) (β) (p 1 (σ 1) (β)) 1 = (((σ 1) p2 ) (β)) (((σ 1) p1 ) (β)) 1 = (σ 1 1) p 31 (β) ((σ 1 1) p 21 (β)) 1 = (σ 1 1) (p 31 (β) p 21 (β) 1 ) = (σ 1 1) p 32 (β)=β. Q.E.D. Example 2. C= (ch),, (ch); C (resp. C ) is the category of 132 quasi-coherent O -(resp. O -)-Modules. Proposition Ifϕ : is faithfully flat, quasi compact thenϕ is a morphism of effective descent.

108 The Technique of Descents and pplications ( morphism f is quasi-compact if f 1 (U) is quasi-compact for every quasi-compact: U). (Note: We do not assume here thatϕ is a morphism of finite type-our hypothesis is much weaker). Proof. Case (a), both affine. The proposition in this case follows from Proposition Case (b) affine. There is a finite affine open cover ( i ) i I of (ϕ quasi-compact). et = i I i. Then is affine and the morphism ψ (composite of the natural mapθ : andϕ) is also faithfully flat. Now consider the diagram: q 1 q 2 ψ θ (natural) q 31 q 32 q 21 ϕ p p p 1 p 2 p Ifξ,η C and u :ξ η is a C -morphism then u defines a C -morphism ũ :ψ (ξ) ψ (η). nd from the equality p 1 (u )= p 2 (u ) and the commutativity of the above diagram follows q 1 (ũ)=q 2 (ũ). By case (a), au :ξ η, a C -morphism, such that ũ=ψ (u); it is immediate that u =ϕ (u); in fact this holds in every open set i. Ifξ C andαis a descent-datum forξ, α defined in the obvious way is a descent-datum forθ (ξ )= ξ and again by case (a), ξ C such that ξ ψ (ξ). It is easy to see thatξ ϕ (ξ). Case (c)., arbitrary. Let (T j ) be an affine open cover of and T j=ϕ 1 (T j ). Then the T j form an open cover of. Let T = j T j andθ : T the natural map. If F and G are quasi-coherent O - Modules denote byφ( ) the group Hom (F, G ), byφ( ) the group

109 Hom (F, G ) and so on. If we set T = T j andθ : T is the j natural map, we have a commutative diagram: Φ( ) ϕ Φ( ) p 1 Φ( ) p 2 Φ(T) Φ(T ) p 1 Φ(T ) p 2 Φ(T T) Φ(T T ) By case (b), the morphisms T j T j are morphisms of effective 134 descent and therefore clearly so is T = T j T j = T. It follows j j that the second row is exact whileφ(t T) Φ(T T ) is an injection. s it is clear thatθ : T 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, thatϕis a morphism of descent. We show similarly thatϕis 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, then a faithfully flat, quasi-compact morphism is a morphism of effective descent. In fact, such preschemes are defined by quasi-coherent O -and O -lgebras 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 O -lgebra homomorphism (resp. an O -lgebra) provided we start with O -lgebras and homomorphisms of O -lgebras instead of O - Modules (look at the proof of Proposition 7.1.1)]. 7.2 Let be a locally noetherian prescheme and X, Y be étale coverings of. Let 0 be a closed subscheme of defined by a Nil-Ideal F

110 The Technique of Descents and pplications 135 of O (F is a Nil-Ideal of O F s nil-radical of O s, s ; this is equivalent to saying that 0 and have the same base space or ( 0 ) red = red ). We then have an obvious functor (E t/ ) Φ (E t/ 0 ) given by X X 0 = X 0. We assert thatφdefines an equivalence of categories in the sense of the following Main Theorem (a) Hom (X, Y) Hom 0 (X 0, Y 0 ) (b) If X 0 (E t/ 0 ), then X (E t/ ) and an isomorphism X 0 X0. The theorem will follow from the series of lemmas below. Lemma Let be locally noetherian and f : X a separated morphism of finite type. Then f is an open immersion f is étale and universally injective. (Note: Universally injective - radiciel= injective+ radiciel residue field extensions). 136 Proof. clear. : f is an open map and thus by passing to an open sub-scheme of, we may assume that f (X)=. Clearly f is then a homeomorphism onto. ince étale remains étale under base-change, f ( ) is still a homeomorphism onto, for any base-change ; it follows that f is universally closed and thus proper. By Chevalley s lemma we deduce that f is finite. uppose then that X= pec, where is a coherent O -lgebra. s f is étale and universally injective, it follows that is locally free of rank 1 at every point of, hence X. Q.E.D. Lemma Let f : X be a separated étale morphism of finite type ( locally noetherian). uppoe Y is a morphism of finite type and Y 0 = V(F ) is a closed subscheme of Y defined by a Nil-Ideal F of

111 O Y. Letα 0 : Y 0 X be an -morphism. Then aunique -morphism α : Y X making the diagram α 0 X Y 0 f α Y commutative. Proof. By making the base-change Y and considering the morphism Y 0 X Y we are reduced to proving the lemma in the case α 0 = Y. That is, given a Y-morphism Y 0 X, we want to extendα 0 to a sectionαof X Y. Now supposeσ:y X is a section of X f Y; f σ= identity is étale and f is étale, henceσis étale. lso f σ is a closed immersion and f is separated and thusσis 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 underσ; in view of lemma then, the sections of f are in (1 1) correspondence with the components X i of X such that f X i is 137 surjective and universally injective on X i to Y. Now Y 0, Y have the same base-space and hence so have X 0 = X Y 0 Y and X; also, the morphism f (Y0 ) : X 0 Y 0 obtained from f by the base-change Y 0 Y is topologically the same map as f ; hence, f is universally injective and surjective on X i to Y f (Y0 ) is so on (X i ) 0 to Y 0, i.e., the set of sections of X f Y is the same as the set of sections of X 0 f (Y0 ) Y 0 and the (1 1) correspondence is given in the obvious way. The lemma now follows from the fact that Y 0 α 0 X can be considered as a section for X 0 f (Y0 ) Y 0. Q.E.D. This lemma proves part (a) of Theorem To prove part (b) we need a generalisation of the lemma.

112 The Technique of Descents and pplications Warning. In the following sections 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 morphism f : X Yof preschemes is of finite presentation if (i) f is quasi-compact (ii) : X X Y X is quasi-compact, 138 (iii) for every x X, anbd. U x of x Xand a nbd. V f (x) of f (x) in Y such that f (U x ) V f (x) andγ(u x, O X ) is aγ(v f (x), O Y )-algebra of finite presentation; i.e.,γ(u x, O X )Γ(V f (x), O Y )[T 1,...,T s ]/a where a is a finitely generated ideal of a polynomial algebra Γ(V f (x), O Y )[T 1,...,T s ]. n example of a morphism of finite presentation is a finite type separated morphism X f Y with Y locally noetherian. Lemma Let U be a noetherian prescheme and ( α ) α I an inductive family of quasi-coherent O U -lgebras. Let V α be the U-affine prescheme pec α defined by α, α I, and V= pec where = lim α α. (a) uppose we have a diagram: U X α Y α X β X β X V V α V β where X α, Y α are finitely presented preschemes over V α α I such that β α, X β = X α Vα V β Y β = Y α Vα V β ; let X= X α Vα V, Y= Y α Vα V Y

113 Then, lim α Hom V α (X α, Y α ) Hom V (X, Y). (b) uppose is a finitely presented prescheme over V. Then, for all 139 largeα I, X α /V α, finitely presented, such that (i) β α, X β X α Vα V β and (ii) X X α Vα V. (c) With assumptions as in (a), if the Y α s and Y are locally noetherian, X/Y étale covering X α /Y α étale covering already, for someα I. Proof. (a) We shall be content with merely observing that it is clear how to prove this in case everything is affine. (b) ince U is noetherian, in view of (a) we may assume that U= pec C, V α = pec α, V= pec where =lim α α. Case (1). ssume X= pec B. X/V finitely presented B/ finitely presented; let B [T 1,...,T s ]/a,a finitely generated, say, by P 1,..., P n [T 1,...,T s ]. Chooseαso large that the coefficients of the P i come from α. Consider the ideal a α = (P 1,..., P n ) of α [T 1,...,T s ] and set B α = α [T 1,...,T s ]/a α and X α = pec B α ; forβ α, set X β = X α Vα V β. Case (2). X arbitrary. Let (X i ) r i=1 be a finite affine open cover of X. In view of case (1) and (a), it is enough now to show that each X i X j is quasi-compact. But the underlying space of X i X j is that of X (X (X i X j ) as is seen from X) V V the commutative diagram: 140 X (X V X) (X i V X j ) X i V X j X X V X (natural)

114 The Technique of Descents and pplications s is a quasi-compact morphism the top -morphism is also quasi-compact; since X i V X j is quasi-compact, our result follows. (c) Here again we give only some indications and leave the details to the reader. To start with we may assume everything affine. ay U= pec C, V α = pec α, V = pec, X α = pec B α 1, Y α = pec B α 2, X=pec B 1, Y= pec B 2. We also note that we can assume V α = Y α, and V= Y. One checks thatω B1/B2 lim Ω B α, say. Now if X/Y is α 1 /Bα 2 unramified thenω B1 /B 2 = 0. But sinceω α B 0 1 /Bα 0 is a finite B α module one hasω B α 1 /B α= 0 for largeα. lso, if X/Y is finite and flat then B 1 is 2 a locally free B 2 -module of finite rank but then the same is true for B α 1 with respect to B α 2 for largeα. ssertion (c) follows. Q.E.D. 141 Lemma uppose we have a commutative diagram: X X (T) = X T α 0 Y 0 = V(F ) f Y T with: noetherian, f étale and separated, T affine, Y T finitely presented and Y 0 = V( F ) where F is a Nil-Ideal of O Y. Then a uniqueα:y X (T) keeping the diagram still commutative. Proof. ince T is affine, T= pec B, where B is a quasi-coherent O -lgebra. Write B= lim B λ where the B λ λ s are O -sub-lgebras, of finite type, of B. et T λ = pec B λ. By (a) and (b) of Lemma

115 we have, for largeλ, a situation of the following type: X T λ X α 0,λ α λ? X T pec B λ = T λ pec B=T Y 0,λ Y 0 = V(F )=Y 0,λ Y Yλ Y λ Y= Y λ Tλ T For largeλ, inductive families (Y λ ), (Y 0,λ ) and morphismsα 0,π : 142 Y 0,λ X T λ such that lim λ α 0,λ =α 0 (see diagram). lso it is not very difficult to see (using arguments similar to those in Lemma ) that the Y 0,λ are subschemes defined by Nil-Ideals F 0,λ of O Yλ. Now T λ is of finite type over and is therefore noetherian; also, since Y T is of finite presentation so is Y λ T λ and is in particular of finite type. Hence Lemma applies and we get a (dooted) morphism α λ : Y λ X T λ keeping the diagram commutative. Passing to the limit with respect toλwe get anα : T X T making the diagram commutative. The uniqueness ofαfollows from Lemma and (a) of Lemma Q.E.D Proof of (b) of Theorem Case 1. ssume = pec, a noetherian, complete local ring; 0 is f 0 then given by pec(/j)=pec 0, J a nil-ideal of. Let X 0 0 be the given étale covering; assume that if s 0 0 is the unique closed point, the residual extensions at the points of f0 1(s 0) are all trivial. Then X 0 is given by pec B 0 where B 0 is a finite direct-product of copies of 0, say B 0 = r i=1 0; X= pec B, with B= r i=1 is then a solution for our problem. Case 2. From among the assumptions in case 1, drop completeness of 143

116 The Technique of Descents and pplications and the triviality of residual extensions along the fibre f 1 0 (s 0). faithfully flat = pec pec = quasi-compact X 0 X 0 = X f 0 pec 0 = 0 f 0 0 = pec( J ) 144 In this case we can choose a complete, noetherian, local overring of such that the following situation holds: and such that if s 0 0 is the unique closed point, then the residual extensions along the fibre f 1 0 (s 0 ) are all trivial. By case 1, an étale covering X / such that X 0. X 0 We have then the following situation:

117 X p 1 (X ) p 2 (X ) p 1 p 1 = p = 2 p 31 X 0 p X 0 1 (X 0 ) p 2 (X 0 ) f 0 f 0 p p 1 0 = 0 0 p p p 32 p 2 0 =

118 The Technique of Descents and pplications 145 ince X 0 / 0 comes from below, it clearly has a descent-datum α 0 : p 1 (X 0 ) p 2 (X 0 ) satisfying the cocycle condition. We want to lift this isomorphismα 0 to anα : p 1 (X ) p 2 (X ). For this one may be tempted to use part (a) of Theorem which we already have proved. But observe that we are now in a type of situation we anticipated while making the remark preceding Proposition we cannot make sure that, 0 are locally noetherian and hence cannot apply part (a). It is here that Lemma comes to our rescue. By using this lemma in the obvious way we get an isomorphismα : p 1 (X ) p 2 (X ); the assertion of uniqueness in this lemma proves that the α we obtained satisfies the cocycle condition. s is faithfully flat and quasicompact, it is a morphism of effective descent for étale coverings (an easy corollary to Proposition 7.1.4) and thus an étale covering X/ such that X X. Therefore X 0 X 0 X 0 (X 0) 0 0. lso, by construction, the descent-datum for X / goes down to that for X 0 / 0. It follows that X 0 X Case 3. arbitrary. By part (a), which we have already proved, it is enough to prove the existence of an étale covering X/ (with the required property) locally. Let s and U= pec be an affine open neighbourhood of s. We may assume = pec, noetherian. The local ring s is given by s = lim f ; we then have pec f pec s and the given f f (s)0 étale covering X 0 over 0 = pec(/j), J a nil-ideal, defines an étale covering X s,0 over pec( s /J s )=pec(/j) s and then, by case 2, an étale covering X s over pec s. By (b) and (c) of lemma , f, f (s)0 and an étale covering X f / pec f such that X s X f s. The f covering X f is a solution for our problem in the neighbourhood pec f of s. Q.E.D. Remark. The Theorem shows that if 0 is such that ( 0 ) red = red then the natural functor (E t/ ) Φ (E t/ 0 ) is an equivalence; in

119 particular, it proves thatπ 1 ( red, s) π 1 (, s). Proposition Let f be a faithfully flat, quasi-compact, radiciel morphism. Then the natural functor (E t/ ) Φ (E t/ ) is an equivalence. Proof. Consider the diagonal =. ince f is radiciel ( )= (Cor. (3.5.10) EG, Ch.I), i.e. is a closed subscheme of, having the same base-space. (a) Given X, Y (E t/ ), set X = X, Y = Y. To prove Hom (X, Y) Hom (X, Y ). ince f : is faithfully flat, quasi-compact it is a morphism of descent for étale coverings. We have thus only to show that if u Hom (X, Y ), and if are = p 2 the canonical projections, then, considered as morphisms from X = X to p 1 Y = Y, p 1 (u) and p 2 (u) are equal. We have the diagram: 147 = X u p Y X 1 (u ) Y X u p 2 (u ) In view of our remarks about, and Theorem 7.2.1, it is enough to show that (p 1 (u ))= (p 2 (u )). But each of (p 1 (u )), (p 2 (u )) is equal to u considered as morphisms form X to Y. Y (b) Given X (E t/ ), to show that X (E t/ ) such that X X.

120 The Technique of Descents and pplications For this, again it is enough to find a descent-datumα: p 1 (X ) p 2 (X ). We have the diagram: p p 1 32 p p diagonal p 1 (X ) p diagonal 2 (X ) p 31 X X = 1 p 1 (X )= 1 p 2 (X ) 148 in view of our remarks about 1 : and Theorem 7.2.1, the identity morphism i : X = 1 p 1 (X ) 1 p 2 (X )=X, lifts to an isomorphismα: p 1 (X ) p 2 (X ). The other diagonal morphism 2 : = also imbeds as a closed subscheme of having the same base-space. Then one checks easily that α satisfies the cocycle condition again by using Theorem Q.E.D. Proposition simply says that if is faithfully flat, quasicompact and radiciel, s and s its image, thenπ 1 (, s ) π 1 (, s). 7.3 Let k be an algebraically closed field and X, Y be connected k-preschemes. uppose 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 k Y over a and b. We claim first that X k Y is connected. ince Y is connected and

121 X k Y Y is proper, it is enough to show that the fibres of X k Y Y are connected and for this one has only to show that for any field k k (*) X k k is connected. The question is purely topological and we may assume X=X red. Looking at the tein-factorisation X pecγ(x, O X ) k it follows 149 (by making use of and the finiteness theorem) thatγ(x, O X )=k, since X is connected and k algebraically closed. On the other hand, if X = X k k, one hasγ(x, O X )=Γ(X, O X ) k k (flat base-change) hence = k ; again looking at the tein-factorisation we see that X is connected We can then formπ 1 (X k Y, c) and form the product of the natural mapsπ 1 (X k Y, c) π 1 (X, a), andπ 1 (X k Y, c) π 1 (Y, b). We then have the Proposition With the assumptions made aboveπ 1 (X k Y, c) π 1 (X, a) π 1 (Y, b). Proof. We may assume X=X red. Case (1). ssume K= k. The morphism X k Y Y is proper and separable and the fibre over b Y is to X k k=x. We then get the exact sequence: π 1 (X, a) π 1 (X k Y, c) π 1 (Y, b) e (Theorem ). But the fibre over b, namely, X k k = X is imbedded in X k Y and the composite X=X k k X k Y p 1 X is identity. This means that continuous sections for the homomorphismπ 1 (X, a) π 1 (X k Y, c). Thus, we get: (i) e π 1 (X, a) π 1 (X k Y, c) π 1 (Y, b) e is exact,

122 The Technique of Descents and pplications (ii) and the sequence splits (also topologically). 150 Case (2). K arbitrary. The reasoning as in case (1) gives an isomorphism π 1 (X k Y k K, c ) π 1 (X k K, a ) π 1 (Y k K, b ) where c, a, b are points of (X k Y) k K, X k K, Y k K respectively, above c, a, b. The theorem then is a consequence of the following: Proposition 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 k be any geometric point. If a X be the image of a, then π 1 (X k k, a ) π 1 (X, a). 151 Proof. That X k k is connected follows from assertion (*) of 7.3; the same assertion also proves that if Z is a connected étale covering of X then Z = Z k k is a connected étale covering of X = X k k. In other words,π 1 (X k k, a ) π 1 (X, a) is surjective (cf ). We shall prove that it is injective by showing that every connected étale covering Z of X = X k k is of the form Z k k for some Z (E t/x). By Lemma ak-algebra of finite type, k, and Z (E t/(x k )) such that Z Z k. If Y= pec, y Ysuch that k(y)=k[since is of finite type over (the algebraically closed) k]. One can apply case (1) of Theorem to the k-rational point (a, y) X k Y to obtain π 1 (X k Y, (a, y)) π 1 (X, a) π 1 (Y, y). If Z is defined by an open subgroup H ofπ 1 = π 1 (X k Y, (a, y)) this means that open (normal) subgroups G π 1 (X) and G π 1 (Y) (defining galois coverings X/X, Ỹ/Y respectively) such that H G G (i.e. such that Z is obtained as a quotient of the galois covering X k Ỹ

123 of X Y). Let Z be the lift of the covering Z /(X k Y) to X k Ỹ; we than have the commutative diagram: X k Ỹ X Ỹ k Z Z X Y k We claim that Z is connected. 152 In fact, let y Y= pec be the generic point of Y; then k(y)= field of fractions of k. For any ỹ Ỹlying above y Ywe thus have k(ỹ) k since k is algebraically closed. If we then apply the basechange X k Ỹ Ỹ k X k Ỹ to the étale covering Z X k Ỹ, we obtain the étale covering Z Y k = Z X which is connected. It follows that Z is connected. Thus the morphism X Ỹ Z is surjective and Z is sandwiched between X Ỹ k k and X k Ỹ; this implies that Z /(X k Ỹ) is defined by an open subgroup ofπ 1 (X k Ỹ) = π 1 (X) G which containsπ 1 ( X k Ỹ) = G G ; i.e. Z = X 1 k Ỹ for an X 1 (E t/x). We now obtain (Z )ỹ= the fibre of Z over ỹ = X 1 k Ỹ Ỹ k(ỹ)= X 1 k k(ỹ) and Z = Z Y k = (Z ) y k(y) k = (Z )ỹ k(ỹ) k= X 1 k k Q.E.D.

124

125 Chapter 8 n pplication of the Existence Theorem 8.1 The second homotopy exact sequence Let be a complete noetherian local ring and = pec ; let s 0 be the closed point of. Let X be a proper -scheme and set X 0 = X k(s 0 ). Let a 0 X 0 be a geometric point of X 0 (over some fixed algebraically closed fieldω k(s 0 )) and a 0 X be its image in X. ssume that X 0 is connected. 153 a 0 X X 0 a 0 proper f s 0 pec k(s 0 ) Theorem The sequence e π 1 (X 0, a 0 ) π 1 (X, a 0 ) π 1 (, s 0 ) e is exact; and we have the isomorphism π 1 (, s 0 )G(k(s 0 )/k(s 0 )). 117

126 n pplication of the Existence Theorem (Note: Compare with Theorem ). The proof of the theorem is a consequence of the following results. 154 Proposition In the above Theorem assume is an artinian local ring. With the same notation, the same assertions hold. Proof. ince the introduction of nilpotent elements does not affect the fundamental groups (Theorem 7.2.1). We may assume that =k(s 0 ). In this case, the assertionπ 1 (, s 0 ) G(k(s 0 )/k(s 0 )) is clear. Next, if characteristic k(s 0 )= p and if k = (k(s 0 )) p, then the morphism pec k(s 0 ) pec k is faithfully flat, quasi-compact and radiciel and hence by Proposition 7.2.2, we may replace k(s 0 ) by (k(s 0 )) p, i.e., we may assume k(s 0 ) is perfect. In this case, k(s 0 ) is the inductive limit lim k i of finite galois extensions k i of k(s 0 ); set X i = X k i and a i = i I k(s0 ) image of a 0 in X i. a 0 X X i a i X 0 a 0 k(s 0 ) k(s k i 0 ) 155 By Lemma an étale covering of X 0 is determined by an étale covering of some X i and the latter is uniquely determined modulo passage to X j, j i. One thus gets the isomorphism: π 1 (X 0, a 0 ) lim i π 1 (X i, a i ) [The injectivity follows from the fact that for any open subgroup H of π=π 1 (X 0, a 0 ) there exists, by Lemma an index i and an open subgroup H (i) ofπ (i) =π 1 (X i, a i ) such thatπ/h π (i) /H (i). The surjectivity follows because otherwise there would exist a set E C (π (i) ) with two points a and b in Ė such that a and b are in the same connected component of E with respect to the action of allπ ( j) ( j i) but a and b lie in different components with respect to the action ofπ; again by

127 8.1. The second homotopy exact sequence 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 X i /X is galois and G(k i /k(s 0 )) ut(x i /X). In fact, suppose we have a situation of the following type: a 0 X ϕ X = X k k a k k with X universally connected over k. Then X /X is a connected étale 156 covering. lso, we have: deg(x /X)=rankϕ= number of geometric points in the fibreϕ 1 (a 0 )=deg(k /k)= number of automorphisms of k /k number of automorphisms of X /X number of geometric points in the fibreϕ 1 (a 0 ) (because X is connected - see the proof of Lemma ). Hence ut(k /k) ut(x /X) and X /X is galois. Now for every i I we have an exact sequence: (e) π 1 (X i, a i ) π 1 (X, a 0 ) ut(x i /X) (e) (see 5.2.6). ince eachπ 1 is pro-finite, by passing to the projective limit we obtain an exact sequence: (e) π 1 (X 0, a 0 ) π 1 (X, a 0 ) G(k(s 0 )/k(s 0 ))=π 1 (, s 0 ) (e) Proposition Let be a complete, noetherian local ring and = pec. Let X be a proper -scheme such that if s 0 is the closed point of the fibre X 0 = X k(s 0 ) is universally connected. Let a 0 be a geometric point of X 0 with values in some algebraically closed Ω k(s 0 ). If a 0 X is the image of a 0 then canonicallyπ 1(X, a 0 ) π 1 (X 0, a 0 ). Proof. This will come from the fact that the natural functor (E t/x) Φ (E t/x 0 ) is an equivalence.

128 n pplication of the Existence Theorem (a) If Z, Z (E t/x) then Hom X (Z, Z ) Hom X0 (Z X X 0, Z X X 0 ) In fact, let (Z), (Z ) be the coherent, locally free O X -lgebras defining Z, Z over X. If M is the maximal ideal of, set n = /M n+1, X n = X /M n+1, Z n = Z X n and so on, n Z +. We then X have: Hom X (Z, Z ) Hom OX lg.( (Z ), (Z)). Now for the O X -Modules, we have: Hom OX ( (Z ), (Z))=Γ(X, H om OX ( (Z ), (Z))) limγ(x n, H om OX ( (Z ), (Z)) /M n+1 ) n by the Comparison theorem, (where H om MX is the sheaf of germs of O X -homomorphisms); since the (Z ), (Z) are coherent and locally free, we have: H om OX ( (Z ), (Z)) /M n+1 = H om OXn ( (Z n ), (Z n)); therefore, Hom OX ( (Z ), (Z)) lim Hom OXn ( (Z n ), (Z n)). n However, this holds also for homomorphisms of O X -lgebras, because the condition for an O X -Module homomorphism to be an O X -lgebra homomorphism can be expressed by means of commutativity in diagrams of O X -Modules. Therefore, Hom X (Z, Z ) lim Hom X n n (Z n, Z n). But by Theorem 7.2.1, the natural map Hom Xn (Z n, Z n) Hom X0 (Z 0, Z 0 ) is an isomorphism; therefore one obtains: lim n Hom Xn (Z n, Z n ) Hom X0 (Z 0, Z 0 ). Q.E.D. (b) If Z 0 is an étale covering over X 0, then there exists an étale covering Z/X such that Z 0 Z X X 0. For proving this we need another powerfull theorem from the EG.

129 8.1. The second homotopy exact sequence The Existence Theorem for proper morphisms Let be a noetherian ring and I and ideal of such that is complete for the I-adic topology. Let Y= pec and f : X Y be a proper morphism. et n = /I n+1, n Z +, and Y n = pec n, X n = X Y Y n. uppose, for every n, F n is a coherent O Xn -Module such that F n 1 F n OXn O Xn 1. Then a coherent O X -Module F such that, for each n, F n F OX O Xn. (For a proof see EG, Ch. III, (5.1.4).) Coming back to the proof of (b), the étale covering Z 0 X 0 is defined by a coherent locally free O X0 -lgebra B 0. By Theorem 7.2.1, for each n, we get a coherent locally free O Xn -lgebra such that B n 1 B n O Xn 1. By the existence theorem, a coherent O X -lgebra B OXn such that B n B OX O Xn. et Z= pec B. We claim that Z/X is the 159 étale covering we are looking for. It is clear that Z X X 0 Z0. It remains to show that Z/X is étale. We first observe that since is a local ring and f is closed any (open) neighbourhood of the fibre f 1 (s 0 ) is the whole of X. lso, if Z/X is étale over the points of the fibre X 0 = f 1 (s 0 ), then Z is étale over X at points of an open neighbourhood of X 0 ; for, if x 0 X 0 then B x0 is free over O x0,x and hence B is a free O X -Module in a neighbourhood of x 0 ; and similarly for non-ramification (use Ch. 3, Proposition 3.3.2). Therefore it is enough to prove that Z Xis étale at points of Z lying over the fibre X 0 = f 1 (s 0 ). Let then x 0 X 0 ; choose an affine open neighbourhood U of x 0 in X such that B 0 is O X0 -free over U 0 = U X 0. etγ(u, O X )=Cand J= M C, the ideal generated by M ; then, for n Z +, C n = C/J n+1 Γ(U, O Xn ). imilarly, if M=Γ(U, B) then M n = M/J n+1 M Γ(U, B n ). We know that M 0 = Γ(U, B 0 ) Γ(U 0, B 0 ) is free as a C 0 = Γ(U 0, O X0 )-module. Choose a basis for M 0 over C 0. Lift this basis to M and let L be the free C-module on these elements andϕ : L M be the

130 n pplication of the Existence Theorem natural C-linear map; then we have an exact sequence of C-modules: 0 kerϕ L M cokerϕ For n Z +, set L n = L/J n+1 L and letϕ n be the C n -linear map ϕ n L n M n defined byϕ. We claim that eachϕ n is an isomorphism. Thatϕ 0 is an isomorphism is clear from the construction. ssume thatϕ 0,...,ϕ n 1 are all isomorphisms and consider the exact sequence 0 J n /J n+1 C n C n 1 0 of C n -modules. ince L n, M n are C n -flat (recall that B n is O Xn -flat) we get the following commutative diagram of exact sequences of C n - modules: 0 J n /J n+1 Cn M n M n M n 1 0 ϕ n ϕ n 1 0 J n /J n+1 L n Cn L n L n 1 0. But and J n /J n+1 M n r M n+1 C Γ(U, O X0 ), J n /J n+1 M n Cn J n /J n+1 L n Cn i=1 r r Γ(U, B 0 )= M 0, i=1 i=1 r L 0. i=1 161 It follows that the first vertical map is an isomorphism; so isϕ n 1 by inductive assumption. Henceϕ n is an isomorphism. If Ĉ, L, M are the J-adic completions of C, L, M and if ϕ is the Ĉ-linear map defined by ϕ, it follows that ϕ is an isomorphism. Thus,

131 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 0 X 0, i.e., Z X is flat at points over X 0. It is also unramified at points above X 0 since Z 0 X 0 is unramified and for non-ramification it suffices to look at the fibre. The proof of Proposition is complete. Proof of Theorem The isomorphismπ 1 (, s 0 ) G(k(s 0 )/k(s 0 )) follows from Proposition when X =. Consider now the diagram a 0 X X 0 a 0 X 0 a 0 s 0 k(s 0 ) 0 ) By Proposition we get the exact sequence: (e) π 1 (X 0, a 0 ) π 1 (X 0, a 0 ) G(k(s 0)/k(s 0 )) (e). One now uses the isomorphismsπ 1 (X 0, a 0 ) π1 (X, a 0 ) (Proposition 8.1.3) and G(k(s 0 )/k(s 0 )) π 1 (, s 0 ) to get he required exact sequence. Q.E.D.

132

133 Chapter 9 The Homomorphism of pecialisation of the Fundamental Group 9.1 With the notations and assumptions of Chapter 8, let s 1 be an arbitrary point of = pec and X 1 = X k(s 1 ). If a 1 is a geometric point of X 1 and a 1 its image in X and if X 1 is connected we get a sequence 162 π 1 (X 1, a 1 ) π 1 (X, a 1 ) π 1 (, s 1 ) (e) such that the composites are trivial. We have continuous isomorphisms β :π 1 (X, a 1 ) π 1 (X, a 0 ) andα :π 1 (, s 1 ) π 1 (, s 0 ). Consider now the diagram: π 1 (X 1, a 1 ) π 1 (X, a 1 ) π 1 (, s 1 ) (e) (e) π 1 (X 0, a 0 ) π 1 (X, a 0 ) π 1 (, s 0 ) (e). This is not necessarily commutative; but it is commutative upto an inner automorphism ofπ 1 (, s 0 ). (It is readily seen in the example of β 125 α

134 The Homomorphism of pecialisation section 5.1 that, with the notation therein, the continuous homomorphism there, is determined upto an inner automorphism ofπ 1 (, s) if we take for s a point different fromϕ(s )-see Ch. 4.) Now the lower row is exact. This implies that there is a continuous homomorphismπ 1 (X 1, a 1 ) π 1 (X 0, a 0 ) which is clearly determined upto an inner automorphism ofπ 1 (X). This is called the homomorphism of specialisation of the fundamental group. 9.2 With the same assumptions as above further suppose X/ separable. ince we have assumed X 0 connected, the condition f (O X )=O 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. uppose y 0, y 1 Y with y 0 (y 1 ). Let a 0, a 1 be geometric points of X 0 = X Y k(y 0 ) and X 1 = X Y k(y 1 ) respectively. Then there is a (natural) homomorphism of specialisation π 1 (X 1, a 1 ) π 1 (X 0, a 0 ) which is surjective. In fact, in view of Proposition we may make the base-change Y pec Ôy 0,Y and apply the above considerations to points above y 0 and y 1. This is the so-called semi-continuity of the fundamental group.

135 ppendix to Chapter IX The Fundamental Group of an lgebraic Curve Over an lgebraically 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 computeπ 1 (X) in this case by assuming results of transcendental geometry. In view of Lemma and Proposition 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 GG - J.P. erre); one can then show that the natural functor { } Finite topological coverings of the (E t/x) analytic space X h defined by X is an equivalence [cf. GG and Espaces fibrés algébriques - Śeminair Chevally 1958, (nneaux de Chow) - ee especially Prop. 19, Cor. to Prop. 20 and Cor. to Theorem 3]. One thus obtains: π 1 (X, a) π top 1 (Xh, a), the completion of the topological fundamental groupπ top 1 (Xh, a) with respect to subgroups of finite index. This groupπ top 1 (Xh, a) is known if X is a nonsingular, proper, connected curve of genus g. It is a group with 2g generators u i, v i (i=1,...,g) subject to one relation, namely (u 1 v 1 u 1 1 v 1 1 )(u 2v 2 u 1 2 v 1 2 )...(u gv g u 1 g v 1 g )=1. Before going to the case charac. k= p0, we shall briefly recall 165 some evaluation-theoretic results. (Ref.. Lang-lgebraic Numbers or J.P. erre-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

136 The Homomorphism of pecialisation... closure of V in K. Choose any maximal ideal M of V and consider the decomposition group of M, i.e., the subgroupg d (M ) of G(K /K) of elements leaving M stable. Then there is a natural homomorphism g d (M ) G(k(M )/k(m )); the kernelg i (M ) of this homomorphism is called the inertial group of M. The maximal ideals of V are transformed 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 writeg i, or sometimesg i (K /K) instead ofg i (M ). We say that K is tamely ramified over V if the order of g i is prime to the characteristic of k(m ) and that K is unramified over V ifg i is trivial. One has then the following: (1) If K /V is tamely ramified then the inertial groupg i (K /K) is cyclic. 166 (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. In addition, one has an exact sequence: (e) g i (K /K ) g i (K /K) g i (K /K) (e) (cf. Corps Locaux, Ch. I, Propostion 22). (3) Letτ M be a uniformising parameter and n Z + be prime to charac. k(m ). If K contains the n th roots of unity, then K = K[X] (X n is a finite galois extension, tamely ramified, with galois τ) group=g i (K /K) Z/nZ. Lemma (bhyankar). Let L, K be finite galois extensions of K, tamely ramified over V with orderg i (L/K) (= n) dividing orderg i (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.

137 Proof. One checks that amonomorphismg i (L /K) g i (K /K) g i (L/K) such that the projectionsg i (L /K) g i (K /K) andg i (L /K) g i (L/K) are onto (see 2)). ince the orders ofg i (L/K) andg i (K /K) are prime to p= charac. k(m ), so is the order ofg i (L /K), i.e., L /V is tamely ramified; henceg i (L /K) is cyclic (1)). s n m, it follows that each element ofg i (L /K) has order dividing m. Butg i (L /K) g i (K /K) is onto. Thus,g i (L /K) g i (K /K). From (2) the kernel of this map isg i (L /K ) which is therefore trivial. Q.E.D. Case 2. Charac. k= p Definition. We say that a Y-prescheme X is smooth at a point x X (or X Yis simple at x) if an open neighborhood U of x such that the natural morphism U Y admits a factorisation of the form (the T i are indeterminates). U étale pec O Y [T 1,...,T n ] Y Note. moothness 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-ModuleΩ X Y ; it is denoted byg X Y. We shall assume the following Theorem 1. (G, 1960, III, Theorem (7.3)). Let be a complete noetherian local ring with residue field k. If X 0 /k is a projective, smooth scheme such that H 2 (X 0,g X0 k)=(0) and H 2 (X 0, O X0 )=(0); then aprojective, smooth -scheme X such that X k X 0. Let k be an algebraically closed field of charac. p0 and X 0 a 168 nonsingular (= smooth in this case) connected curve, proper (hence, as

138 The Homomorphism of pecialisation... is well-known projective) over k. Consider the ring =W(k) of Wittvectors; this is a complete, discrete valuation ring with residue field k; if K is the field of fractions of, then charac. K= 0, and K is complete for the valuation defined by. The conditions of Theorem 1 are satisfied by X 0. Therefore aprojective smooth -scheme X such that X 0 X k. The X 0 is universally connected; but then it follows that the generic fibre is also universally connected (Use tein-factorisation; Note that pec has only two points). Furthermore, as X is smooth over, 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 1 (X) π 1 (X 0 ). We may thus replace X 0 by X. Let a 0 be the closed point of pec and a 1 the generic point; set X K = X K, X K = X K, where as usual K is the algebraic closure of K. If a 0 X k = X k is a geometric point over a 0 and a 1 X K, a geometric point over a 1 then we have the homomorphism of specialisation π 1 (X K, a 1 ) π 1 (X k, a 0 ) π 1 (X) 169 which is surjective (9.2). From case 1 one already knows aboutπ 1 (X K, a 1 ) (charac. K= 0). Thus it only remains to study the kernel of the above epimorphism. This amounts by to studying the following question: Given a connected étale covering Z of X K which is galois, when does there exist a Z (E t/x) such that Z Z K. [Remember: a connected étale cov- ering is galois if the degree of the covering equals the number of automorphisms. lso 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(X K ) of Z and X K. Clearly if Z is galois over X K then R(Z) is a galois extension of R(X K ).]. By Lemma , afinite subextension K 1 of K and a Z K1 (E t/x K1 = X K 1 ) such that Z Z K1 K1 K. Then the above question takes the form: Given a connected Z (E t/x K ), which is galois when does

139 there exist a Z (E t/x) and a finite extension K 2 of K 1 such that Z K2 = Z K1 K1 K 2 Z K 2? Now, for any finite extension K of K, let be the integral closure of in K ; then by a corollary to Hensel s lemma it follows that is again a discrete valuation ring whose residue field is again k (recall that k is algebraically closed). Thusπ(X) π 1 (X k ) π 1 (X ) where X = X, again by Proposition Thus the question becomes: Given Z K1 (E t/x K1 ), universally connected and galois, when does 170 there exist a finite extension K 2 of K 1 such that Z K2 = Z K1 K 2 comes K1 from a Z 2 in (E t /X2 = X 2 ) (where 2 is the integral closure of in K 2 )? Consider any finite extension K of K 1 and let be the integral closure of in K. We are given a situation of the form: Z K X K + p - generic point of the fibre over the closed point. generic point pec K 0 }{{} pec -closed point (residue field k). The Z K1 (resp. Z K = Z K1 K1 K ) are connected and hence, as we have already remarked, integral. Let R(Z K1 ), R(Z K ), R(X K1 ) and R(X K ) be the function fields of Z K1, Z K, X K1 and X K. Then R(Z K1 ) is a finite galois extension of R(X K1 ) and R(Z K )=R(Z K1 ) K = R(Z K1 ) R(X K ). K1 R(X K1 ) Our aim now is to choose, if possible, the field K K 1 such that R(Z K ) is unramified over X. By this we mean the following: consider the normalisation Z of X = X in R(Z K ); we want Z to be unramified 171 over X. (Note: ince Z K is regular, so certainly normal, we have that Z K Z K, because the process of normalisation is unique. ee EG, II, (6.3) and also. Lang, Introduction to lgebraic Geometry, Ch. V).

140 The Homomorphism of pecialisation... 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 X K = X K. Consider the integral closure 1 of in K 1. Let p be the generic point of the fibre in X 1 = X 1 over the closed point (as a space, this is nothing but X 0 ). Then p is a point of codimension 1 in X 1 ; furthermore X 1 / 1 is smooth and therefore the local ring O 1 of p (as a point in X 1 ) is regular. Therefore O 1 is a discrete valuation ring in R(X 1 )=R(X K1 ). imilarly define O in R(X K )=R(X ) for any K K 1. It is easily checked that O is the integral closure of O 1 in R(X K ). Now, any open set in X, containing X K and the generic point p of the fibre over the closed point of pec is such that its complement is of codimension 2 in X. We have the following theorem depending on the so-called purity of the branch locus: Theorem 2. (G, expose X, Cor (3.3)) 172 Let P be a locally noetherian regular prescheme and U the complement of a closed set of codimension 2 in P. Then the natural functor (E t/p) Φ (E t/u) is an equivalence. (For a proof see G, X, 1962). [For the special case of the theorem which we need, there is a direct proof in G, X, 1961, p. 16. However even for that proof one needs the following result (which we have not proved in these lectures). n unramified covering of a normal prescheme is étale. (G, I, , Theorem (9.5)).] Thus, it is enough to prove that K K 1 such that R(Z ) is unramified over X at the point p because Z is clearly flat over X = X at the point p (the local ring O of p in X is a discrete valuation ring). Ifτis a uniformising parameter of 1, it is also a parameter of O 1. Let then n Z + be such that (n, p)=1 and set K = K 1 [T]/(T n τ). Then K /K 1 is a finite galois extension and R(X K ) R(X K1 )[T]/(T n e). Hence R(X K ) is tamely ramified over O 1, and has an inertial group of order n. ssume now that the degree of the galois covering Z over X K is prime to p= charac.k. Then one may take n equal to this degree. By bhyankar s lemma one then has that R(Z ) is unramified over X above the point p. Thus one has proved the

141 Proposition 3. The kernel of the (surjective) homomorphism of specialisationπ 1 (X K, a 1 ) π 1 (X 0, a 0 ) is contained in the inter-section of the kernels of continuous homomorphisms fromπ 1 (X K ) 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 G i where the limit is taken over all quotients G i of G which are finite of order prime to p, then we have: π (p) 1 (X K, a 1 ) π (p) 1 (X 0, a 0 ). (One can also say: G (p) is the quotient of G by the closed normal subgroup generated by the p-ylow subgroups of G ). ince we know, by topological methods, the groupπ 1 (X K, a 1 ) we obtain: Theorem 4. If X is a nonsingular, connected, proper curve of genus g over an algebraically closed field k of charac. p0, thenπ (p) 1 (X) G (p) where G is the completion with respect to subgroups of finite index of a group with 2g generators u, u i, v i (i=1, 2,...,g) with one relation: (u 1 v 1 u 1 v 1 1 )(u 2v 2 u 1 2 v 1 2 )...(u gv g u 1 g v 1 g )=1.

142

143 Bibliography [1] Bourbaki, N. lgébre Commutatif (Hermann, Paris) [2] Grothendieck,. Éléments de Géométrie lgébrique (EG) - Volumes I, II, III, IV. (I.H.E.. Publication) [3] éminaire de Géométrie lgébrique (G) - Volumes I, II, ( ) I, II (1962) [4] Lang,. Introduction to lgebraic Geometry (Interscience Publication) [5] erre, J.P. Corps locaus (Hermann, Paris) [6] Groupes algébriques et corps de classes (Hermann, Paris) [7] Espaces Fibrés algébriques (Exposé 1, éminaire Chevalley (1958) nneaux de Chow et applications) [8] Géométrie lgébrique et Géométrie nalytique (nnales de l Institut Fourier, t.6, , pp.1-42) 135