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

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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