Sheaf Theory (London Mathematical Society Lecture Note Series)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES

Editor: PROFESSOR G. C. SHEPHARD, University of East Anglia

This series publishes the records of lectures and seminars on advanced topics in mathematics held at universities throughout the world. For the most part, these are at postgraduate level either presenting new material or describing older material in a new way. Exceptionally, topics at the undergraduate level may be published if the treatment is sufficiently original. Prospective authors should contact the editor in the first instance. Already published in this series 1. 2. 3.

4. 5. 6. 7. 8.

General cohomology theory and K-theory, PETER HILTON. Numerical ranges of operators on normed spaces and of elements of normed algebras, F. F. BONSALL and J. DUNCAN. Convex polytopes and the upper bound conjecture, P. McMULLEN and G. C. SHEPHARD. Algebraic topology: A student's guide, J. F. ADAMS. Commutative algebra, J. T. KNIGHT. Finite groups of automorphisms, NORMAN BIGGS. Introduction to combinatory logic, J. R. HINDLEY, B. LERCHER and J. P. SELDIN. Integration and harmonic analysis on compact groups, R. E. EDWARDS.

9. 10. 11. 12.

13. 14.

Elliptic functions and elliptic curves, PATRICK DU VAL. Numerical ranges II, F. F. BONSALL and J. DUNCAN. New developments in topology, G. SEGAL (ed.). Proceedings of the Symposium in complex analysis, Canterbury 1973, J. Cl UNIE and W. K. HAYMAN (eds.). Combinatorics, Proceedings of the British combinatorial conference, 1973, T. P. McDONOUGH and V. C. MAVRON (eds.). Analytic theory of abelian varieties, H. P. F. SWINNERTONDYER.

15. 16. 17. 18. 19.

Introduction to topological groups, P. J. HIGGINS. Topics in finite groups, TERENCE M. GAGEN. Differentiable germs and catastrophes, THEODOR BROCKER and L. C. LANDER. A geometric approach to homology theory, S. BUONCRISTIANO, C. P. ROURKE and B. J. SANDERSON. Graph theory, designs and coding theory, P. J. CAMERON and J. H. VAN LINT.

London Mathematical Society Lecture Note Series. 20

Sheaf Theory B.R.TENNISON

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE

LONDON NEW YORK MELBOURNE

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521207843

© Cambridge University Press 1975

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1975 Re-issued in this digitally printed version 2007

A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 74-31804 ISBN 978-0-521-20784-3 paperback

Dedicated to my family

Contents

Page Introduction

v

Conventions and notation

vii

Chapter 1: Presheaves and their stalks 1. 2

Definition of presheaves Examples of presheaves

1. 3

Interlude: direct limits

1. 1

Stalks of presheaves 1. 5 Morphisms of presheaves Exercises on Chapter 1 1. 4

Chapter 2: Sheaves and sheaf spaces 2. 1 The sheaf axiom 2. 2 Examples of sheaves Sheaf spaces 2. 4 The sheafification of a presheaf 2. 5 Sheaf spaces of abelian groups Exercises on Chapter 2 2. 3

Chapter 3: Morphisms of sheaves and presheaves 3. 1 Categories and functors 3. 2 The categories of sheaves and presheaves 3. 3 Kernels and monomorphisms 3. 4 Cokernels and epimorphisms 3. 5 Biproducts and the abelianness of Presh and Shv 3. 6 Exact sequences 3. 7 Change of base space 3.8 Restriction and extension Exercises on Chapter 3

14

17 17 22

25

27

31

35

37 41

47 49

53 62

68

iii

Chapter 4: Ringed spaces

The category of ringed spaces over a ring R 4. 2 The prime spectrum of a ring 4. 3 Geometric spaces and manifolds 4. 4 Modules over ringed spaces 4. 5 Locally free Modules Exercises on Chapter 4 4. 1

73 81

87 94

102

107

Chapter 5: Cohomology 5.1

Injective objects

115

5. 2

Derived functors

119

5. 3

Sheaf cohomology Cech cohomology

131

Exercises on Chapter 5

152

5. 4

140

The way ahead: further reading

154

References

156

Hints and answers to some exercises

157

Index of terminology

159

Index of notation

163

iv

Introduction

Sheaf theory provides a language for the discussion of geometric objects of many different kinds. At present it finds its main applications in topology and (more especially) in modern algebraic geometry, where it has been used with great success as a tool in the solution of several longstanding problems. In this course we build enough of the foundations of sheaf theory to give a broad definition of manifold, covering as special cases the algebraic geometer's schemes as well as the topological, differentiable and analytic kinds; and to define sheaf cohomology for application to such objects. Chapters 1 and 2 cover the groundwork of presheaves and sheaves, and show that any presheaf gives rise to a sheaf in a universal way. Chapter 3 defines the categorical viewpoint, shows that the categories of sheaves and presheaves of abelian groups on a fixed topological space are abelian, and investigates the relations between them. It also covers the processes of change of base space of a sheaf, both for the inclusion of a subspace and for a general continuous map. Chapter 4 defines the notions of ringed space and geometric space, and gives as an example the spectrum of a commutative ring. This is proved to be a construction with a universal property among all geometric spaces. The latter are shown to be the prototypical geometric objects, by exhibiting various kinds of manifolds as special cases. The chapter also includes a discussion of Modules over ringed spaces, and the consideration of locally free Modules leads to the definition of the picard group of a ringed space. Chapter 5 gives an introduction to sheaf cohomology, at first in the general context of being the right derived functor of a suitable left exact functor between abelian categories. It is shown how other definitions fit into this picture, particularly those defined by flasque sheaves and by

the tech method. The picard group of a ringed space is interpreted as a v

cohomology group.

The essence of the usefulness of sheaves is that they express the connexions between the local and global properties of a geometric object. This should become clear from the results of Chapters 4 and 5. The approach to the subject taken here is rather categorical, and the course may be used (and indeed has been, in Part III of the Mathematical Tripos at Cambridge) as an introduction to the usefulness of categories and functors. It presupposes only a knowledge of elementary general topology (topological spaces and open sets) and elementary algebra (abelian groups, rings), although reference is made to other sources for further elucidation of some points. There are exercises scattered throughout the text and at the end of each chapter, and they vary considerably in difficulty. There is a section at the end of the book containing hints and solutions to some of them.

vi

Conventions and notation

We use Bourbaki notation for the sets N, Z, Q, R, C of natural numbers (0, 1, 2, ... ), integers, rationals, reals and complexes. We use the barred arrow I- to indicate where an element of a set is sent under a map with that set as domain: hence for example f

: Z -Z :

defines f as the squaring map. = and = each denote an isomorphism. IL denotes disjoint union of sets, definable for instance as LL ;EAXx=UREA{x} XXX;

the important thing is that IL AEAAXA is the disjoint union of a copy of X f or each A. References within the book are explained by example as follows. 4. 3. 12 is the full reference to part 12 of §3 of Chapter 4; within Chapter 4 this is abbreviated to 3. 12. 4. Ex. 3 refers to Exercise 3 at the end of Chapter 4.

References to other sources are either quoted in full or given as [X], where X is one of a short list of acronyms detailed in the list of references (page 156).

vii

1 Presheaves and their stalks

In this chapter, we give definitions and examples of presheaves of sets and of abelian groups, and of morphisms between them. We study the notion of direct limit of a directed system of sets (or abelian groups), and apply it to construct the stalks of a presheaf, which summarise the nature of the presheaf locally in the neighbourhood of some point. 1. 1

Definition of presheaves 1. 1

Definition.

Let X be a topological space. A presheaf F

of sets on X is given by two pieces of information:

for each open set U of X, a set F(U) (called the set of sections of F over U) (b) for each pair of open sets V S U of X, a restriction map pU : F(U) - F(V) such that V (a)

(bl) (b2)

for all U p U = idU U . U V 0 PV i. e. whenever W S V S U (all open) pW=pW

F(U) -F(W) commutes. F(V)

A presheaf of abelian groups over X is a presheaf F of sets such that

each F(U) has a given abelian group structure every restriction map pU is a group homomorphism with respect to these structures. (a') (b')

Remark. Sets and abelian groups are the two main types of structure with which we shall be concerned for the moment; it should be clear how to phrase the definitions of presheaves of groups, rings, commutative rings, topological spaces,... ; all the sets of sections have the 1. 2

appropriate structure, and all the restriction maps are morphisms of the

1

appropriate kind (homomorphisms, continuous maps, ...). Examples of presheaves

1. 2

Let A be any given set (or abelian group). Then the constant presheaf AX on X is given by Example A.

AX(U) = A U

p V = idA : AX(U) - AX(V)

for U open in X for V C U open in X.

Example B. Let Y be another topological space. CY of continuous Y-valued functions on X is defined by: CY(U)

The presheaf

= set of continuous maps : U - Y (for U open in X)

pv CY(U) - CY(V) : f I- f I V :

(for U

V open in X)

[whence the name 'restriction' for the maps pVU] If in addition Y has the structure of an abelian group, so has each CY(U) by pointwise addition of functions. In this case CY is a presheaf of abelian groups. For example, giving Z the indiscrete topology, CZ is a presheaf of abelian groups on X [it is even a presheaf of rings].

Similarly for CR for R with the usual topology. Suppose that X is an open subset of some Rn. Let r E N (= 10, 1, 2, ... 1). The presheaf Cr of r-times differentiable R-valued functions on X has Example C.

Cr(U) = set of r-times continuously differentiable functions : U-'R

(for open U)

and restrictions as in Example B.

Suppose that X is an open subset of some Cn. The of analytic C-valued functions on X has

Example D.

presheaf C

W

C°(U) = set of analytic functions : U -C (for U open in X) (analytic = regular = holomorphic).

2

Example E. Two more pathological examples. Let X be any topological space with more than one point e.g. X = {0, 1 } or

[0, 11 --R. Define the presheaf P1 by

P1(X)=Z P1 (U) _ {0 } (trivial group) for open U * X

all restrictions except pX being constant maps. Pick x0 E X. Define the presheaf P2 by

P2(U)=Z for U open in X such that U 3 xo P2 (U) = {0 } for U open in X such that U A xo U restrictions pV =

1idZ if xo E V S U

trivial map if not.

Then P1, P2 are both presheaves of abelian groups on X. 1. 3

Interlude: direct limits

A directed set A is a set with a pre-order (that is, a reflexive and transitive relation: a -5 a, and a p s y==> a _ y) which also satisfies: Va, /3 E A 3y c A such that a <_ y and p y. (a) 3. 1

Definition.

We often write Al = { (a, 0) E A X A; a <_ p }. A direct system of sets indexed by a directed set A is a family (Ua)aEA of sets together with, for each (a, p) E Al, a map of sets

pap : Ua - Up, satisfying (b)

ya E A p a a= idU

(c)

ya, p, y E A if a < p <_ y then the triangle U--

a

Up

commutes, i. e. pay =

ppy

cp

ap.

Given a topological space X, the set T of its open sets is directed by the relation 3. 2

Example.

3

U:5 V UDV (condition (a) holds since U n V is open if U, V are).

Given also a presheaf F on X, let pUV = pV be the restriction map when U V. Then the (F(U))UET with the pUV form a direct system of sets.

Picture.

3. 3

U6-

U

We wish to find a 'limit' for the system, i. e. an object which can go 'to the right of everything'. 3. 4 Definition. Given a direct system in the notation of 3. 1, a target for the system is a set V and a collection of maps

(aa : Ua

V)aEA satisfying the compatibility condition:

Va

Ua

(3

a

pal Uis

a

commutes; that is V

va=o1 0 pad

UP

A direct limit for the system is a target U, (Ta:Ua~U)aEA satisfying the universal property:

(*34)

for any target V (with maps as as above) there is a unique map f : U - V such that

Va E A U

3. 5

4

Remark.

a

a

V

commutes.

So a direct limit is a 'best' target.

Proposition. Any two direct limits for a direct system are naturally isomorphic (that is, there is a bijection between them compatible with all the Ta). 3. 6

Proof.

(Archetypal of a large number of proofs of similar statements.) Let them be (U, (T&)ath) and (U', (T'' )ath). Since U is universal and U' a target, we obtain an f : U - U'; since U' is uni-

versal, we obtain a g : U' - U and

But now U is a target, and the universality of U implies that idU : U-U T

is the unique map making all the

Ua

a,, U

commute.

Hence

J

Ta'U

g o f = idU and similarly f o g = idU,. // Thus there is some justification in speaking of the direct limit and denoting it by lim Ua. 3. 7

Notation.

aEA

Suppose U, (Ta : U a - U)aeA is a target for the system (Ua)aEA' (Pa(3)(a, /3)EAl' such that: 3. 8

Theorem.

(i)

Vu E U Ha E A such that u E Im(Ta)

(ii)

if a, /3 E A and ua E Ua and u9 E U0 then

Ta(ua) = TT(Ua) = Hy E A

such that a

/3 .

Then U is a direct limit of the system. [Remark: We can paraphrase (ii): if ua, uR get identified in U, they must have been identified along the way. ]

5

Proof. Suppose V, (aa)aEA is another target. If f : U - V is to satisfy the compatibility condition (*34) then it must be obtained as follows:

for u E U, pick a E A such that u E Im(Ta), say u = Ta(ua);

(*38)

then f(u) = aq(ua). Hence, if f exists, it is unique. If we choose /3 E A such that u E Im(T too, say u = then by condition (ii) ay E A with pay (u a) = pay(uhhence aq(ua) = ay(pay(ua)) U

(uU

= ay(P y(ua)) = a9

U

T0(uh),

->V

So f is well-defined by (*38), and so U satisfies the condition (*34). // Construction. Given a direct system (Ua)aEA' (pa(3)(a,0)EA1 of sets, we can now construct a direct limit. Let 3. 9

W=

-aEAUa

be the disjoint union of all the sets Ua. On W define the relation - by

u-v4

if u c U01 and v E U13, then

ay with any, a<_y and

Ua

such that pay(u) = pRy(v).

UA,

Then ~ is an equivalence relation (clearly reflexive and symmetric; if

u-v and v-w then we have

u E Ua Ub v E U0"U

WE

U

U--'V Y

so that 6 < C, E: C to see that u - w). Let U = W/- and Ta : Ua -. U be the composite maps

and we can pick

6

U with the (Ta)aEA is a direct limit for the Hence every direct system of sets has a direct limit.

Theorem.

3.10

system (U a) a EA. Proof.

U, (Ta)aEA satisfy the conditions of Theorem 3. 8. //

A direct system of abelian groups is a direct system of sets (Ga)aEAI (pag)(a,(3)EA such that each Ga has an Definition.

3. 11

1

abelian group structure, and all the are homomorphisms with respal3 pect to these structures. 3.12 Example. sheaf of abelian groups. Definition.

3. 14

As in Example 3. 2 except we start with a pre-

A target for a direct system of abelian groups

is a target G, (va : Ga - G) aEA for the underlying direct system of sets, together with an abelian group structure on G such that all the as are abelian group homomorphisms. A direct limit is defined as in the set case, all maps involved being required to be abelian group homomorphisms. 3.15

Clearly we could define the same concepts

Remark.

relative to any structure e. g. direct limits of rings, groups, topological spaces, modules etc. Proposition. Given a direct system of abelian groups, any two direct limits for it are naturally isomorphic (as abelian groups). 3.16

Proof.

Exactly as in Proposition 3. 6. //

3.18 Theorem.

Suppose (Ga)0EA, (10

a direct

system of abelian groups and G, (Ta)aEA is a target satisfying (i) Vg E G as c A such that g E Im(Ta) (ii)

Va, for ga E Ga we have

Ta(ga) = 0

ai3 such that a

/3

and pap(ga) = 0.

Then G is a direct limit for the system.

7

Given a direct system (Ga)aEA' (Pao) (a, (3)EA1 of abelian groups, let H = 4) aEA Ga be the direct sum of the Ga, with is : Ga -0 H the natural injections. [Recall: H is the subgroup of UaEAGa (with pointwise operations) generated by the images of all the ia, where 3. 19

Construction.

go

(ia(ga))/3 = 0

a#a

for ga EGa.]

Let H1 be the subgroup of H generated by all the ia(ga) - i/3(pao(ga))

as (a, A) runs through A 1 and (then) ga runs through Ga. Let G = H/H1 and Ta : Ga - G be the natural maps. G, (Ta : Ga - G)aEA is a direct limit for the system. Hence any direct system of abelian groups has a direct limit. 3.20 Theorem.

Proofs of 3. 18, 3. 20 are very similar to those of 3. 8 and 3. 10,

and are left as an exercise. // Remark. In fact we could do without Construction 3. 19 by constructing instead the set lim and imposing an appropriate abelian group structure on it. But 3. 19 works in more general circumstances (see 1. Ex. 9). 3. 21

Stalks of presheaves

1. 4

Let F be a presheaf (of sets or abelian groups) over a topological space X. Fix X E X. The F(U), as U runs through all open sets such that U 3 x, form a direct system with maps 4. 1

pV: F(U) - F(V) whenever U ? V (3x). Definition. equipped with maps

The stalk Fx of F at x is 1m F(U). This comes

F(U) - Fx : s'- sx 8

IT-91,x

whenever an open U a x. The members of Fx are sometimes called germs (of sections of F). 4. 2 Proposition. (a) Each germ t E Fx arises as t = sx for some s c F(U) for some open neighbourhood U of x. (b) Two germs sx, tx E Fx (with s E F(U), t E F(V) say) are

equal

sx = tx

a open W S U n V such that pW(s) = P Wt).

Proof. This is just a restatement of Theorems 3. 8 and 3. 18, taking account of Proposition 3. 6. // Examples. A. For a constant presheaf AX over X we have AX x = A for each x EX. B. For a presheaf of functions such as CY, Cr or C w each 4. 3

germ at x c X extends to a function on some neighbourhood of x, and two germs are equal iff corresponding functions agree on some neighbourhood of x. Hence a germ summarises the 'local' behaviour of a function at a point. C.

For the pathological example P1 of 2. Ewe have

Vx EX

P1 x= {0) r

and yet it is not the constant presheaf 1. 5

{0 } X.

Morphisms of presheaves

5. 1 Definition. If F. G are presheaves of sets over X, a morphism f : F - G is given by maps

f(U) : F(U) -+ G(U)

for each open set U of X, such that whenever U? V are open in X, the diagram f(U)

F(U)

G(U)

p,U

pVVl F(V)

V f(V)

G(V) 9

commutes, i. e. PT U

(U) = f (V)P v.

If F, G are presheaves of abelian groups, for f to be a morphism

of presheaves of abelian groups we require each f(U) to be a homomorphism of abelian groups. Composition of such morphisms is defined in the obvious way: (g ° f)(U) = g(U) ° f(U) if F f -b- G g H. As usual, we say that f : F -' G is an isomorphism of presheaves (of sets or abelian groups) iff there is a

morphism g : G - F such that f ° g = idG and g ° f = idF (where id : F - F is defined by idF(U) = idF(U) for each open U in X). F Proposition. f : F - G is an isomorphism of presheaves (of sets or abelian groups) 5. 2

iff V open U of X f(U) is an isomorphism iff V open U of X f(U) is bijective. Proof.

f isomorphism

a g such that f o g = idG and g ° f=idF, H g VU f(U) ° g(U) = idG(U) and g(U) ° f(U) = idF(U) 4

VU f(U) isomorphism.

For if f : F - G is a morphism with all f(U) isomorphisms, the inverses f(U) i : G(U) - F(U) satisfy the conditions of compatibility with restriction. [This point needs checking. ] // 5. 3 Remark.

We shall investigate later the extent to which this Proposition has meaning and is true with 'isomorphism' replaced by 'monomorphism' and 'epimorphism'. 5. 4 Construction. Given a morphism of presheaves f : F - G on X, for each point x e X we can produce a morphism of stalks f

X

10

: Fx -Gx

f

in such a way that whenever F - G

9 H we have

(g°f)x=gx0fx. Given x E X we define fx as follows: any e E Fx is of the form e =sx for some open U 3 x and some s E F(U) (by 4. 2); set fx(e) = (f (U) (s))x(i. e. take the germ of the image of s). If also e = sx = tx with t E F(V), then by 4. 2 21W S U n V with x E W and PW(s)

= PW(t); so PW(f(U)(s)) = f(W)PW(s) = f(W)PW(t) = PW(f(V)(t))

so that (f(U)(s))x = (f(V)(t))x and fx is well-defined. The functoriality (g ° Ox = gx ° fx is easy to check; note that (idF)x = idF is obvious too. X

Remark. 5. 4 is in fact a special case of a generality concerning 'maps of direct systems'. 5. 5

Exercises on Chapter 1

1. Prove that the lim of the direct systems of the examples (3. 2 and 3. 12) is F(c in each case. Generalise.

2. Prove directly from the definitions (i. e. without Theorem 3. 8 or the Construction 3. 9) that if U is a direct limit of a direct system (Ua)aEA of sets, then

U= 3.

(i)

UaEA

Image(Ua - U).

Interpret and prove: a set is the direct limit of its finite

subsets.

Interpret and prove: an abelian group is the direct limit of its finitely generated subgroups. (iii) Can you obtain Z as a direct limit of finite abelian groups? (ii)

Characterise direct systems of sets with lim = 0. (ii) Produce an interesting direct system of abelian groups with lim = { 0 ), the trivial group. Characterise such systems. 4.

(i)

11

What can you say about the direct limit of a direct system all of whose maps are injective? Surjective? 5.

For n E N*, let Cn(x) denote a cyclic group of order n with generator x. Let P E N be a prime number. Let G be the direct 6.

limit of the following direct system of abelian groups:

{0 } =Cp0(x0)-'Cp(x1)-'Cp2(x2)-'... -'Cpn(xn)-'C n+1(xn+1)"... P

(where Cpn(xn) -. Cpn+l(xn+l) takes xn - pxn+1)' Preferably without resorting to the explicit construction prove: (i) G is infinite, but torsion (i. e. every element has finite order). (ii)

Every finitely-generated subgroup of G is finite. Find all

of them.

Deduce that G has no proper infinite subgroup, and no maximal proper subgroup. Can either of these situations arise for subspaces of a vector space (using dimension instead of order) ? Identify a realisation of G inside the unit circle C C (under x.).

Consider the following direct system of abelian groups: fix r E Z; for all n E N let Un = Z and for n ? m let pmn : Um - Unbe multiplication by rn-m. Identify the li_m as a subring of Q. 7.

8. Interpret and prove: the direct limit of a system of exact sequences is exact.

9. The notions of target and direct limit can be formulated without the restriction (a) of Definitions 3. 1 and 3. 11. What difference does this make to the Constructions? Find a system of abelian groups (in this generalised sense) with direct limit A ® B without having this abelian group appear in the system. Justify Remark 3. 21.

Formulate the dual notions of inverse system and inverse limit lim (reverse the arrows). Find inverse systems: (a) of finite sets whose lim is infinite of finite abelian groups whose lim is infinite (b) 10.

r

(c)

12

r

of abelian groups whose lim is Z (without Z in the system).

Verify that if (Ra) is a direct system of abelian groups such that each Ra is a ring and all the are ring morphisms, then 1m Ra 11.

paR

has a natural ring structure such that all the maps R -1m Ra are ring morphisms. 12.

What are the stalks of the presheaf P 2 of 2. E ?

Construct a topological space X and presheaf F of abelian groups on X with the properties: (a) for any open USX F(U) 101 (b) for all x e X the stalk Fx = { 0 } . (If you cannot, prove that it is impossible.) (Compare Q4(ii). ) 13.

13

2 Sheaves and sheaf spaces

We now study presheaves which satisfy additional axioms concerning the existence and uniqueness of sections with prescribed local nature (in the form of a given set of restrictions). In particular, we find that the geometric examples of presheaves of functions are in fact sheaves. We show that sheaves may be viewed as local homeomorphisms over the

base space, and deduce that every presheaf gives rise to a sheaf in a universal way. The sheaf axiom

2. 1

Let X be a topological space and F a presheaf of sets over X. F is called a monopresheaf (or separated pre1. 1

Definition.

sheaf) iff it satisfies the condition (M): Suppose that U is an open set of X and U = uX EAU

(M)

is an open

covering of U (i.e. each U. open in X), and s, s' E F(U) are two sections of F such that VA E A

PUU(s)=PUU(s') Ix

then s = s'. The presheaves of 1. 2. A-1. 2. D are monopresheaves, but P1 of 1. 2. E is not. 1. 2

Examples.

1. 3 We have also a 'glueing' condition (G): Suppose that U is open in X and U = uXEAUX is an open covering of U; suppose we are given a family (sA)AEA of sections of F

(G)

with VA E A sA E F(UX), such that

VA, µ E A P x

14

U

µ

(sµ);

(sA) = PU1, µ

then there is s E F(U) such that

VXEA pU(s)=s In other words, if the system (sx) is given on a covering and is consistent on all the overlaps, then it comes from a section over all of U. 1. 4 Definition. A presheaf of sets over X satisfying (M) and (G) is called a sheaf of sets. Similarly, an abelian sheaf is a presheaf of abelian groups which satisfies (M) and (G).

For a presheaf of abelian groups, we can simplify

Remark. (M) by putting s' = 0. 1. 5

There is a neat way to summarise the conditions (M) and (G). Given an open cover of an open set U = UXEAUx we can define maps 1. 6

b (*16)

F(U)

IXEAF(UA)

c

11

(X, µ)EAxA F(UA

AU

by:

a(s) = (PU (s))AEA U

(sx))(A,µ)EAxA

b((sA)XEA) = U

c ((sx)X EA) =

(PUAn

(sA))(A,

µ) EAxA

If A -a+ B -b C is a diagram of sets and maps, we say that a is an c (i. e. iff A equaliser of (b, c) iff f a is injective, and Image(a) = {x E B; b(x) = c(x) }

bijects with the subset of B on which b and c are equal). Then we have:

Proposition. A presheaf F is a sheaf iff whenever U = UxEAUA is an open cover of an open set, the associated diagram (*16) of sets is an equaliser diagram (i. e. a is an equaliser of (b, c)). 1. 7

Proof.

An easy translation of the definition. // 15

If F is a presheaf of abelian groups, then the maps a, b, c are group morphisms and the equaliser condition is that 1. 8

Remark.

the sequence of abelian groups a

0 - F(U) - UAEAF(UA)

b-.c

U(A, P)EAxAF(UX r Uµ)

be exact (that is, kernel = image at each point). 1. 9

Exercise.

Show that if G is an abelian sheaf, then

G(O) _ 10 }, the trivial group.

If F is a presheaf and G a monopresheaf over X, and f, g : F - G are two morphisms such that 1. 10

Proposition.

Vx EX fx = gx

(i. e.

f, g agree on all stalks)

then f = g. Proof. Let U be open in X and s E F(U). Then f(s), g(s) E G(U) (f(s) is shorthand for f(U)(s)) and we wish to prove

that f(s) = g(s). Vx E U

that is

Now fx(sX)

= gx(sx)

(f(s))x = (g(s))X

and so x has an open neighbourhood Ux S U such that PU (f (s)) = PU (g(s)).

x

X

Applying the condition (M) for G to the covering (U X) XEU of U we see

that f(s) = g(s). // 1. 11

Remark.

We shall mainly be interested in Proposition 1. 10

when G is a sheaf (or even both F, G are sheaves). 1. 12 Exercise. where 1. 10 fails.

16

Find an example with G not a monopresheaf

Definition. If F, G are sheaves (of sets or abelian groups) and f : F - G is a presheaf morphism, we also call f a morphism of 1. 13

sheaves. A moment's thought and Proposition 1. 5. 2 show that if F - G

is an isomorphism of presheaves, and G is a sheaf, then F is a sheaf. 2.2

Examples of sheaves 2.A

The presheaf P2

of

1. 2. E is a sheaf.

2. B All the examples of presheaves of functions CY, Cr, C of 1. 2. B-D are in fact sheaves, and all for the same reason: in order for an arbitrary map to Y, R, C to satisfy the appropriate condition (continuity, differentiability, analyticity) it is necessary and sufficient that the condition be satisfied in some small neighbourhood of each point: thus a 'glued' function will also satisfy the condition.

Let E be a topological space and p : E - X a continuous map. We can construct a sheaf F of sections of p: for U open in X 2. C

let

p

F(U)

X commutes, i, e. continuous maps a : U - E; U p o a= idU}

If U D V are open in X we have an easy restriction:

pv: The conditions (M) and (G) are easy to verify.

Note that the condition on a means that a gives for each x E U a (continuous) choice of a point a(x) E p-1(x), the fibre of p over x.

2.3

Sheaf spaces In 2. 3 and 2. 4 we shall be concerned with presheaves of sets.

If X is a topological space and F a sheaf on X, then for any open U and s, s' E F(U) we have 3. 1

Proposition.

S=S'<-- VX EU Sx=s' ; 17

that is, two sections of F agree iff their germs are everywhere equal. is clear. Conversely, given s, s' E F(U) such that Vx e U sx = s', for each x e U we can find an open Ux 3 x such that Proof.

pU (s) = pU (s'). Applying the condition (M) to x

s=s'. //

we see that

x

3. 2 Remark. This is not true for arbitrary presheaves, e. g. the presheaf P1 of 1. 2. E (cf. 1. 4. 3). But we are led to try to represent a sheaf as a collection of functions with values in its stalks.

Let X be a topological space. A sheaf space over X is a pair (E, p) of a topological space E and a continuous map p : E -+ X such that p is a local homeomorphism, that is: Vy E E a open N 3 y, open U 3 p(y) such that p IN : N - U is a homeo3. 3

Definition.

morphism. A morphism of sheaf spaces f : (E, p) - (E', p') is a continuous

map f: E - E' such that Ep

f -

E' commutes i. e. p=p' ° f.

X

3. 4 Construction. For each sheaf space E we can construct a sheaf of sets FE (the sheaf of sections of E) in such a way that a morphism f : E - E' of sheaf spaces gives rise to a morphism Ff : FE -+ FE' of sheaves.

We have seen in 2. 2. C how to construct the sheaf of sections of

the pair (E, p). We let, for U open in X, -eE Q

continuous maps a : U -+ E; p o or = idu } U ' and then the presheaf FE : U I-+ r(U, E) is a sheaf. Given a morphism f : E - E' of sheaf spaces, we obtain

I'(U, E)

f E=E'

r(U, E) - r(U, E')

ar-'faa

U

'X

and this gives a morphism of sheaves Ff : FE : FE'. 18

'

X

Let (E, p) be a sheaf space over X. Then (a) p is an open map (b) if U is open in X and a E I'(U, E), then a[U] is open in E; furthermore sets of this form give a basis for the topology of F. 3. 5

Lemma.

(c)

If Ep

0

-' 3-

p

is a commutative diagram of maps, and p, p, are local homeomorphisms, then

x 0 continuous

0 local homeomorphism.

0 open

(a) Let W be open in E and X E p[W]. Pick any e c W such that p(e) = x. Then by the definition of a sheaf space, e has an open neighbourhood W' S W mapped by p onto an open set in X; i. e. x has an open neighbourhood p[W'] inside p[W]. (b) Any e E a[U] has an open neighbourhood W S E such that Proof.

p I W is a homeomorphism onto an open set V c X. Then p I W maps W n a[U] bijectively to U n V, which is open in X; hence W n a[U] is an open neighbourhood of e inside a[U]. The last part is easy, using (a).

By definition and part (a), local homeomorphism continuous and open. Let us prove $ continuous 0 local homeomorphism. Given y E E, O(y) E E' and so since p' is a local homeomorphism there are open N', V such that O(y) E N' pIN' V is a homeomorphism. Also 411 (N') is open in E, so we can construct a diagram (c)

yEN

OIN 0. N"

pIN\ AN" U

with N, N", U open and p I N, p'I N" both homeomorphisms. Hence 0 I N is a homeomorphism. 0 local homeomorphism is similar (first pick a suitable 0 open neighbourhood of y). //

is a sheaf space, then the stalk of I'E at x E X is (up to natural bijection) just the fibre p 1(x) of p over 3.6 Proposition.

If (E, p)

19

x, which has the discrete topology as a subspace of E.

For X E U open in X we have the map

Proof.

rE(U) = r(u, E) -

(*36)

p-1(x) : Qi- u(x)

(since or maps U - E); these maps are clearly compatible with re-

strictions. Thus we wish to prove that this target is a direct limit, and we use the criterion of 1. 3. 8: (a) each e E p-1(x) arises as an image under one of these maps; for, since p is a local homeomorphism e has a neighbourhood W in E such that p I W: W - U is a homeomorphism, with U open in of this map has Q E r(u, E) and maps to X. The inverse o = (p e under (*36). IW)-1

(b)

if

s E r(u, E) and t E r(v, E) agree at x, by I emma

3. 5 W = s[U] n t[V] is open in E, and s, t agree on p[W] (which is open by I emma 3. 5) since they are both inverses of p I W. Hence np[W](s) = np[WI(t) E r(p[W], E). Thus

p-1(x) = lim r(u, E). U3x

To see that p-1(x) is a discrete subspace of E, note that for e E p (x) and W as constructed in (a) above, W is open and W n p-1(x) = {e }. // 1

r(f ° g)=rf G rg r(id)=id. Check also that if f : E - E' is a morphism of sheaf spaces over X, then 3. 7

(cf)x

Exercise.

Check the functorial properties

: (rE)x - (rE') x

and fIp-1(x) : p-1(x)

-P,-'(X)

are isomorphic maps.

20

{

For each presheaf F on X we can construct a sheaf space LF in such a way that any morphism f : F - F' of presheaves gives rise to a morphism Lf : LF - I F' of sheaf spaces. 3. 8

Construction.

Set L F = JJ-xEXFx (the disjoint union of the stalks of F) with

p : I F - X the natural projection, so that p 1(x) = F. We topologise IF as follows: let U be open in X and s E F(U); then we can define a map x

EF x

We prescribe that all the sets s[U] = {sx E LF; x E U } be open sets. Then {9[U]; s E F(U) } forms a basis for the topology it generates, for e c s[U] n t [VI

(with s E F(U), t E F(V))

s, t agree in germ at p(e) = x say s, t agree in a neighbourhood W of x (with W C U n V) e has a basic neighbourhood s[W] = t[W] inside s[U] n t[V] (where

we really mean p(s)[W] by s[W]). Furthermore p is continuous with respect to this topology on LF, since for any open U of X p-1(U) = U {s[V]; s E F(V) with V s U open },

and p is a local homeomorphism since on s[U] it has the continuous

inverse s. A presheaf morphism f : F -, F' gives a collection of stalk maps fx : Fx

- F'x and so a map Lf : LF - LF' such that LF -> LF'

commutes; also Lf[s[U]] = f(U(s)[U], so Lf is continuous by 3. 5(c). o g)=Lf o Lg Check the functorial properties rL(f l L(id)=id. Now it is natural to ask what happens when we do L, IF in succession. 3. 9

Exercise.

If E is a sheaf space over X, then LFE is isomorphic to E as sheaf spaces over X (i. e. there is a morphism 3. 10

Theorem.

21

0 : E - LrE of sheaf spaces with a two-sided inverse). Proof. E

Fix nomenclature: I. rE

e-

p

pi

Y

X

For X E X, the fibre p-1(x) bijects with the stalk of rE at x, and so with the fibre p11(x) in LTE. These bijections fit together to give a bijection 0 such that p = p1 0 0. If U is open in X and v E r(U, E), then $[a[Ull = a[UM.

Hence 0 is open, and by 3. 5 it is also continuous; since this means that 0 is a homeomorphism. //

is bijective

The sheafification of a presheaf

2. 4

4. 1 Given a presheaf F over X we can construct the sheaf space LF and then obtain a sheaf rLF called the sheafification of F. Now we have a morphism of presheaves nF : F - TLF defined as follows: given U open in X and s E F(U), s defines the function

s:

x

x

as in 3. 8 and so s E r(U, LF); then nF(U) : s i- s. Now this construction enjoys the following universal property: Theorem. Let F be a presheaf and G a sheaf over X. Then any morphism of presheaves f : F -+ G factors uniquely through 4. 2

F - rLF; that is, given Ff G , there is a unique sheaf morphism rLF y G makingn F

f

G

g

rL F

22

FLF commute.

This theorem gives precise meaning to the notion that I'L F is the 'best' sheaf that can be made out of F. Before the proof we need two lemmas.

If (and only if) G is a sheaf, then (and only then) G - rLG is an isomorphism of sheaves. 4. 3

Lemma.

Proof.

'Only if' is clear, by 1. 13. For 'if' we check that each

G(U) - rl G(U) = r(U, LG) : s i- s (for U open in X) is bijective; this is enough by Proposition 1. 5. 2. The map is injective; for by Proposition 3. 1 (i) V XEU U

SX =S'X

S=S'.

The map is surjective; for given t E r(U, LG), t[U] is open in LG by Lemma 3. 5. For each x E U, t(x) E Gx has a basic neighbourhood inside t[U], of the form sx[Ux] for some open Ux S U and some sX E G(UX) (by 3. 8). This means that the sxU G(Ux) satisfy the glueing condition (G) of 1. 3; for in V = Ux n Uy, pV (sX) and pv (sy) have the same germ everywhere (viz. t(z) at z E V) and so are equal by Proposition 3. 1. Hence since G is a sheaf as E G(U) such that Vx E U sx = (X) x = t(x) i. e. s = t. // (ii)

4.4 Remark. The morphism nF : F - rLF (and so the isomorphism of 4. 3 when F is a sheaf) is natural in the sense that if f : F - F' is a morphism of presheaves, then the following diagram commutes:

F F -rLF n

f

Inif F'

4. 5

nF' Lemma.

rLF' For any presheaf F on X, all the maps

Fx - (rLF) x induced on stalks by nF are isomorphisms. 23

Clear, since the stalk (rLF) x is naturally the fibre of LF over x, which is Fx (by 3. 6 and 3. 8), and the map Fx - (rLF)x is induced by the maps (*36) of Proposition 3. 6 . // Proof.

F

If g : rLF - G exists making

Proof of Theorem 4. 2. f

nF

,

G

r LF

commute, then its stalk maps gx for x E X are determined as the composites f

(rLF)X

Fx

xo Gx

(nFl x exists by Lemma 4. 5); '

n F, x

hence g is determined uniquely, by 1. 10. f To show that g exists, we take F - G and apply 3. 8 to get Lf LF - LG, and then apply 3. 4 to get

rLF

r-Lf

rLG - G,

by Lemma 4. 3. It is easy to check that the resulting triangle commutes (use 4. 4). //

If A is a given set, what should we call the constant sheaf A over X? If we take the constant presheaf AX of 1. 2. A and apply the sheafification procedure FL, we first obtain a sheaf space LAX p X such that 4. 6

dx E. X

p-1 (x) = AX

=A

by 1. 4. 3A so that set-theoretically LAX = A x X and p = 7r2 (projection on second factor). By 3. 8 the topology on A X X has as basis sets of the form la l x U for a e A and U open in X; hence it is the product topology, with A given the discrete topology. The sections of F = FLAX are given by

24

a

AXX

F(U) = r(u, LAX) _ set of continuous functions a making commute

1 IT2

U

X

set of continuous functions s : U -A (with A discrete) set of locally constant functions s : U - A.

Note that for U disconnected (and A with > 1 element) this has F(U) * A, so that A x was not a sheaf originally by 4. 3. The constant sheaf over X modelled on A is the sheaf whose sheaf space is A X X -2 X (A given discrete topology). It Definition.

is sometimes denoted (confusingly) by AX. 2. 5

Sheaf spaces of abelian groups 5. 1

We now wish to define a sheaf space of abelian groups and

constructions r, L in such a way as to make Theorem 4. 2 (in particular) true for sheaves and presheaves of abelian groups. Given a presheaf of abelian groups F over X, each stalk is an abelian group and so the corresponding sheaf space (LF, p) has the property: (a) For each x E X, the fibre p 1(x) is an abelian group. However, the abelian group structures also 'vary continuously' as x varies; more precisely: 5. 2 Proposition. For a sheaf space (E, p) satisfying (a) of 5. 1 the following two conditions are equivalent: (b)

For any U open in X the set r(U, E) is an abelian group

under pointwise addition of functions; (b') Let E7rE = {(e, e') E E X E; p(e) = p(e')) ; then the map m : ErrE - E : (e, e') l-' e - e' is continuous (where - denotes subtraction in p-1(p(e)) ).

Proof.

/

(b')

(b): We have the diagram

E \, E m X 25

where p' : (e, e') 1- p(e) = p(e'). If f, g E r(u, E) then f-g : U '' E can be written as the composite

U(f,-g) x-

m

E 7iE - E

(f W, g(x))

and so is continuous i. e. f - g E r(U, E). The result follows easily. (b) (b') By Lemma 3. 5(b) we need to check that for any open

U S X and f c r(U, E) we have m-1(f[U])

open in ErE.

for x = p(e) = p(e'), e - e' = f(x). Pick by Proposition 3. 6 an open W in X and g, g' E r(W, E) such that g(x) = e, g'(x) = e'. Then g - g' and f agree in germ at x and so for some open VSWnU But (e, e') E m-1(f[U])

PV(g

- g') = PV(f )

by Proposition 3. 6 again. Then (e, e') has the basic neighbourhood (g[V] x g'[V]) n EvvE

inside m-1(f[U]). //

(Aliter use 3. 5(c). )

Remark. (b') is often summarised by saying that subtraction is continuous on E . 5. 3

A sheaf space of abelian groups over X is a sheaf space (E, p) satisfying condition (a) of 5. 1 and conditions (b) and (b') of 5. 2. A morphism (F, p) - (E', p') of such is a sheaf space morphism such that Vx E X the map p-1(x) - p'-1(x) is a homomorphism of abelian groups. 5. 4

Definition.

5. 5

With this definition and the results of 5. 2 we see that the

constructions r, L of 3. 4 and 3. 8 take us from sheaf spaces of abelian groups to abelian sheaves, and from presheaves of abelian groups to sheaf spaces of abelian groups.

26

The following results remain true in the context of sheaves, presheaves and sheaf spaces of abelian groups, and their appropriate mor-

phisms: 3. 5, 3. 6, 3. 7, 3. 9, 3. 10, 4. 2, 4. 3, 4. 4, 4. 5.

If F, G are abelian sheaves over X, and f : F - G is a morphism as sheaves of sets, then f is a morphism of Corollary.

5. 6

abelian sheaves iff dx E X fx : Fx '- Gx is a homomorphism of abelian groups.

Proof. 'Only if' is clear. If f has the above property, then Lf : LF - LG is a morphism of sheaf spaces of abelian groups; hence

rLf : rLF - rLG is a morphism of abelian sheaves; but by 4. 3, 4. 4 and 5. 5 we see that this implies that f is too. // Some authors define a sheaf over a topological space to be what we have called a sheaf space over X. The existence of Terminology.

5. 7

constructions r, L with properties 4. 3 and 4. 4 shows that there is no essential difference between these definitions (a sheaf determines a unique sheaf space, and conversely; and similarly for morphisms). This leads to the following widely used notation: if F is a sheaf

(of sets or abelian groups) over X and U is open in X, we shall from now on write r(U, F) instead of F(U) for the set (or abelian group) of sections of F over U. By 4. 3 r(U, F) = r(U, LF), and this is the origin of the terminology 'sections of F over U'. This approach involves thinking of a sheaf F as the collection of its stalks: hence some authors use the terminology 'sheaf of germs of continuous (or differentiable, or analytic) functions' for the sheaves described in 1. 2. B-D and 2. 2B.

Exercises on Chapter 2

Let I = [0, 1]'*R. Show that there is a unique (up to isomorphism) sheaf F on I with stalks: 1.

F =F =Z 0

1

Fx= {O} if xEI\{0, 1}.

27

What is r(I, F) ?

Let G be the constant sheaf Z on I

(4. 6).

How many morphisms

are there from F to G? From G to F? Show that the following conditions are equivalent for a topological

2.

space X: (a)

X is locally connected (that is, each point has a base of

connected neighbourhoods); (b)

for any set A, the constant sheaf Ax (4. 6) has

r(U, AX)

UteU,A

for U open in A where U' is the set of connected components of U; (b) holds for A = 10, 11, some set with two elements. When these conditions hold, what are the restriction maps in terms of the representation given in (b)? [Hint, if necessary: Bourbaki, Gen. Top. Ch. I, §11. 6, Prop. 11.] (c)

Let F be a presheaf on a space X, and let V be open in X. Then we can define a presheaf F I V on v by the same recipe as F; that is 3.

(F I V)(U) = F(U)

for U open in V. Show that if F is a sheaf, so is F IV. Show also that if F has sheaf space LF p X, then F I V has sheaf space (p-1 V, p I p-1 V). What can you say when V is not open? (Compare Q4 below and U. 8. )

Let F be a sheaf on a space X with sheaf space LF y X, and let A be a subspace of X. We can define the set (or abelian group) of sections of F over A by 4.

r(A, F) = r(A, LF) = set of sections of the continuous map p -lA

+ A.

(Compare 2. C and 3. 4.) Show that we can define T(A, F) in terms of

F alone as

r(A, F) = 1m r(U, F) 28

where the direct limit is taken over the set of open subsets U of X such that U D A. (Colloquially, this says that a section of F over A extends uniquely into a small neighbourhood of A. )

Let F be a sheaf on a space X and let (Mi)fEI be a locally finite covering of X by closed sets (so that for each x E X, 5.

{ i E I; X E Mi } is finite). In the notation of Q4, suppose we are given

a family (si)W with Vi

1

si E r(Mi, F)

and

Vi, j E I si = sj on Mi n M Show that there is a unique s E r(X, F) with Vi E I

s = s.1 on M..1

Let K be any infinite field and L = K(t) a simple transcendental extension (= the field of fractions of the polynomial ring K[t]). Let X be the topological space obtained by giving K the topology whose closed sets are the finite subsets of K. Define a sheaf 0 of commutative-rings-with-a-one on X as follows: for U open in X, U # 0, let 6.

(*)

O (U) = if EL; ag, h E K[t] with f =

and VP E U h(P) # 0 } c L. h

If 0 # V S U then 0 (U) S 0 (V) C L and we take the inclusion as the restriction map pV. Show that 0 is a sheaf of rings on X (it is called the sheaf of rational functions on the affine line X over K). [Hint: first prove that X is compact. ]

Identify the stalk OP of 0 at P E X as a subring of L, and show that it is a local ring (i. e. has a unique maximal ideal mP); what is its residue field (= OP/nP) and its field of fractions? Show that the set of all non-empty open sets of X is directed by

and that lim r(u, 0) = L. When does 0 have non-polynomial global sections? That is, we certainly have K[t] S r(X, 0); when is the inequality strict? (Give a

29

necessary and sufficient condition on K ) We can consider f E r(U, 0) as a function on U; namely,

express f = g/h as in (*) and define for P E U f(P) = g(P)/h(P) e K

Show that this defines a morphism 0 : 0 - F where F is the sheaf of K-valued functions on X (giving K the indiscrete topology), and that putting 0'(U) = Image(O(U)) defines a sheaf 0' with a morphism 0 - 0'. Prove that 0 - 0' is an isomorphism of sheaves. Hence we may regard 0 as a sheaf of K-valued functions on X. For K = C show that 0' is a subsheaf of the sheaf Cw of analytic C-valued functions on X = C (in a suitable sense). [We shall see later (§4. 2) that Q6 is a special case of a very powerful construction (the prime spectrum of a commutative ring) which will yield analogous results for X = Kn instead of K, or even any subset X of Kn defined by polynomial equations. ]

30

3 Morphisms of sheaves and presheaves In this chapter we first give an account of the elementary language of category theory, and show how this gives a unified way of looking at many of the ideas we have been considering. We are led to look for convenient properties of the categories of sheaves and of presheaves over a given topological space, and we find that they each have a list of such properties which are summarised in the definition of abelian category. However, the construction of cokernels differs in the two categories; this expresses what is perhaps the basic question in sheaf theory: to what extent does a sheaf epimorphism (a map of sheaves which is 'locally' surjective) have surjective section maps? This is studied further when we consider cohomology (Chapter 5). Lastly, we consider what happens in a change of base space by a continuous map. We find that there is a covariant (that is, going in the same direction as the map) method of changing the base space of presheaves, and a contraviant (opposite direction) construction which is a generalisation of sheafification. These are connected by an adjointness relation, which may be interpreted as expressing their universal nature. In the case of an inclusion map of a locally closed subspace, we also consider the process of extension by zero. 3. 1

Categories and functors 1. 1 (a) (b)

A category C consists of a class ObC of objects for each A, B E ObC a set HomC(A, B) of morphisms

Definition.

from A to B (c)

for each A, B, D E ObC a function (composition)

HomC (B, D) X HomC (A, B) - HomC (A, D)

31

written (g, f) F g o f

such that (i)

for each A E ObC a an identity 1A E HomC(A, A) such that

VB E ObC Vf E Hom(A, B)

f o 1A = f

Yf E Hom(B, A)

1A o f = f

and (ii)

whenever A, B, D, E E ObC and f E Hom(A, B) then (associativity).

g E Hom(B, D)

h o (g o f)

h E Hom(D, E)

_ (h o g) o f.

When f E HomC(A, B) we write f : A - B or A - B and use diagrams in an obvious way; for instance, the hypothesis 1. 2

f

Notation.

h

f

of (ii) is 'whenever A - B

D - E'. We sometimes write

Mor C = uA,

BEObCHomC(A,

B) (disjoint union).

1. 3

Exercise.

1. 4

Examples and definitions.

Show that identities are unique.

sets

Letting ObC = class of all

abelian groups topological spaces maps and composition and HomC(A, B) = set of all homomorphisms continuous maps to be composition of maps we obtain the categories Sets, Abgp,

A.

Top.

Similarly for any other type of mathematical structure e. g. category of groups; for a ring R the category of R-modules.

Given a preordered set A (i. e. a relation <_ on A which is reflexive and transitive) we can consider A as a category C with B.

ObC = A HomC(A, µ) = j singleton 0

X :s

if

;t )A A

(composition is then uniquely determined).

32

C.

For X E Ob(Top) we have defined three categories:

Presh (or Presh/X if we wish to emphasise X) has objects the presheaves of abelian groups over X, and morphisms the presheaf morphisms. Shv (or Shv/X) has objects the abelian sheaves over X. Shfsp (or Shfsp/X) has objects the sheaf spaces of abelian groups over X. In each case the composition of morphisms is that with which we are familiar. (Note: from now on we shall mainly be concerned with presheaves and sheaves of abelian groups, so we do not reserve special names for the categories of sheaves and presheaves of sets over X.) 1. 5 Definition.

Given two categories C, D a covariant functor

F : C - D is given by: (a) a map F : ObC - ObD (b)

VA, B E ObC a map F : HomC(A, B) - HomD(FA, FB)

such that (i)

V A E ObC

F(lA)

1FA

Vf, g E MorC F(f o g) = Ff o Fg whenever f o g is defined (so RHS is too). A contravariant functor has instead of (b) (b') VA, B E ObC a map F : HomC(A, B) - HomD(FB, FA) and (ii) is replaced by Vf, g E MorC F(f o g) = Fg o Ff whenever f o g is defined (ii') (so RHS is too). Functors can be composed in an obvious manner. (ii)

1. 6

Examples.

There are functors Abgp - Sets, Top - Sets sending an object to its underlying set (called forgetful functors). A.

B.

There is an inclusion functor Shv/X - Presh/X.

If A is a directed set (1. 3. 1) and we consider A as a category as in 1. 4 B, then a direct system of sets (1. 3. 1) is just a functor : A-Sets; C.

while a direct system of abelian groups (1. 3. 11) is a functor : (Exercise: verify these assertions. ) 33

All the 'constructions' we have made are in fact functors:

D.

F: Shfsp/X -+ Shv/X

(2. 3. 4 and 2. 3. 7)

L: Presh/X - Shfsp/X

(2. 3. 8 and 2. 3. 9).

Fix X EX.

'stalk at x': Presh/X - Abgp : F E Ob(Presh) i- Fx

(1.5.4).

We call the composite I'L: Presh/X - Shv/X the sheafification functor. In fact presheaves themselves can be considered as functors: if X is a topological space, let `U. = set of all open subsets of X. U is pre-ordered by the relation S and so may be considered as a category (1. 4B). A presheaf of abelian groups on X is just a contravariant functor: `U - Abgp (cf. 1. 1. 1). We use this to formalise the generalisation mentioned in 1. 1. 2: E.

if K is any category, a K-valued presheaf on X is a contravariant functor: `U - K. 1. 7

Definition.

If F, G : C -+ D are two functors, a natural

transformation n from F to G is specified by giving for each A E ObC a morphism nA E HomD(FA, GA), in such a way that whenever f E HomC(A, B) the square nA

FA

GA

jG(f)

F(f)III

nB

FB

GB

commutes (naturality) (or the square nA

FA

GA

F(f)l

IG(f)

nB

FB

34

a GB

F, G are contravariant). Natural transformations can be composed ((n o m) A = nA o mA), and the functors F, G are called naturally equivalent (or naturally isomorphic) iff there are natural transformations if

n from F to G and m from G to F such that n o m = idG and m o n = idF (where idF is the natural transformation with (idF)A = idF(A)).

Exercise. Show that a natural transformation n is a natural equivalence iff VA E ObC nA is an isomorphism (cf. 1. 5. 2 and 1. 8

1. 9A below).

1.9 Examples.

If F, G are presheaves, considered as functors: `UL - Abgp as in 1. 6E then a presheaf morphism F - G is just a natural transformation of functors (for the naturality square corresponding to U C V is

A.

F(V)

o G(V)

F(U)

o G(U) ).

If A is a directed set, and F : A - Sets a direct system of sets as in 1. 6C, then a target for F is just a natural transformation from F to a constant functor: A - Sets (Exercise: verify this). B.

2. 3. 10 shows that the functor Lr : Shfsp/X -+ Shfsp/X is naturally equivalent to the identity. C.

D.

2. 4. 1 and 2. 4. 4 show that there is a natural transformation n

from the identity id: Presh - Presh to the sheafification functor rL: Presh - Shv I+ Presh; 2. 4. 3 shows that the sheafification functor rL I Shv : Shv - Shv is naturally equivalent to idShv' 3. 2

The categories of sheaves and presheaves

Our aim in Chapter 3 is to prove categorical properties of the categories Presh/X and Shv/X of presheaves and sheaves of abelian groups; in brief, to show that both are abelian categories and that the inclusion functor is left exact. We shall give definitions when needed, and 2. 1

35

refer to [Macl] and [Mit] for the wider view. Analogous (but different) results could be proved for the categories of presheaves and sheaves of sets (see Exercises). If f e HomShv(F, G), in accordance with 2. 5. 7 we use the notation

r(U, f) : r(U, F) - r(U, G) for the map induced by f on the abelian groups of sections over an open set U. In this way

r(U, -) : Shv/X - Abgp becomes a functor, and we shall also be interested in its exactness properties. 2. 2

If F, G are presheaves, given f, g E Hom(F, G) we can

construct a sum (f + g) a Hom(F, G) by putting (f + g)(U) : F(U) - G(U) : s H f(s) + g(s) (compatibility conditions are easily checked). There is a zero morphism 0 E Hom(F, G) with for each U 0(U) = the zero map : F(U) - G(U) : s H 0.

We can verify easily that in this way Hom(F, G) becomes an abelian group, and that composition is bilinear, i. e. for a diagram of presheaves

F' -' F

G

G' we have

g

q o (f+g)=qof+qog and

(f+g) °p=f op+g° p. 2. 3 We denote by 0 any zero sheaf on X, such that 0(U) is a trivial abelian group as in 2. 4. 6; this is the same as (e. g. the constant sheaf 101 X the constant presheaf 101 X of 1. 2. A). For any presheaf, Hom(F, 0) and Hom(0, F) are each trivial groups. For V open U

36

f c Hom(F, G), clearly

,/ f

f = 0

f factors as

F

G

0

2. 2 and 2. 3 show that both Presh and Shv are (pre-) additive categories (cf. [Macl] 1§8, [Mit] 1§18). 2. 4

Exercise. Read enough about category theory to understand and justify the following concise summary of Chapters 1 and 2: 2. 5

is the category of open sets of X, Presh is just the functor category Abgp`u°p (op = dual). Shv is a full subcategory of Presh If

`U,

and we have a diagram of functors:

Shv -,SShfsp

i \,iPresh /L

in which r is an equivalence of categories (with inverse L o i = L I Shv) and I'L is left adjoint to the inclusion i (so that Shv is a reflective subcategory of Presh). 3. 3

Kernels and monomorphisms 3. 1

Definition.

Given f E Hom(F, G) for F, G E Ob(Presh/X),

let

K(U) = ker f(U) = Is E F(U); f(U)(s) = 0G(U) } (for U open in X)

a subgroup of F(U). If U ? V are open in X and S E K(U) then f(V)pv(s) = pv(f(M )(s)) = 0

F(U)

G(U)

T

F(V)

so pss(s) E K(V). Thus the K(U) with the

PV

G(V)

I K(U) form a presheaf over X, called

37

the kernel of f and denoted Ker(f). We have a natural presheaf morphism Ker(f) -+ F and the composite Ker(f) -+ F - G is zero. Proposition. versal property: 3. 2

(* 32)

If f E Hom(F, G), then Ker(f) has the uni-

if H is a presheaf and g E Hom(H, F) is such that H -+

f

F - G = 0, then g factors uniquely as

(that is exists and is unique making the triangle commute).

Proof.

H.

a

Ker (f) --;-' F

Easy exercise. //

Proposition. If f E Hom(F, G), F is a sheaf and G a monopresheaf, then Ker f is a sheaf. Hence if F, G are sheaves so 3. 3

is Ker f. Proof.

The monopresheaf condition is easy, since it works for

F. For the glueing condition, if U = UXEAUX and sA E Ker(f)(UA)

satisfy the compatibility conditions on all UX n Uµ, then as E F(U) such U (s) = s., since F is a sheaf. Then s' = f(U)(s) E G(U) is that VA P U UX

such that VA pU (s') = 0, and so s' = 0 since G is a monopresheaf. A

Hence S E Ker(f)(U), as required. // Remark. In an arbitrary category (with zero object) the universal property (*32) is used to define the notion of kernel (cf. [Macl] VIII §1, [Mit] I §13). 3. 2 and 3. 3 show that both the categories Presh and Shv have kernels of all morphisms, and the inclusion functor is kernelpreserving. 3. 4

3. 5 Theorem. For a morphism f E Hom(F, G) of presheaves, the following three conditions are equivalent: Ker(f) = 0 (i) (ii) Vopen U in X f(U) is injective (iii) f is a monomorphism i. e. if H is any presheaf and

38

H

g

h

F are such that f- g= f o h then g= h. These imply the condition:

Vx E X fx is injective which is a further equivalent condition if F is a monopresheaf. (iv)

Proof.

(ii) is immediate.

(i)

Given Ker(f) = 0 and H, g, h as in (iii), we see that (g - h) : H - F satisfies H g-+h F f-+ G = 0, so by Proposition 3. 2 (g - h) h factors as H F, and so by 2.3(g- h) = 0 i. e. g = h. (i)

(iii)

0

(iii)

(i)

If f is a monomorphism, we prove that 0 - F has the

universal property of Ker(f) stated in (*32): for if H g F - G = 0 then g

H : F - G give the same result, so g = 0 by (iii) i. e. g factors 0

F. Hence 0 has the universal property of Ker(f)

(uniquely) as H 0

and an easy argument like 1. 3. 6 shows that Ker(f) is a zero sheaf.

(F any presheaf.) Suppose t E Fx is such that fx(t) = 0; then a open U and S E F(U) such that s has germ t at x, so that f(U)(s) has germ 0 at x, so that a open V S U with (ii)

(iv)

0 = pv(f(U)(s)) =

is injective by hypothesis, so pj s) = 0, and thus t = 0. (ii) (assuming F a monopresheaf). Suppose s E F(U) is such that f(U)(s) = 0 E G(U); then Vx E U fx(sx) = (f(U)(s))x = 0, so Vx E U sx = 0 since each fx is injective by hypothesis. Since F is a monopresheaf, s = 0 (cf. 2. 3. 1). // f(V)

(iv)

F, G are presheaves (resp. sheaves) on X and f : F - G is a monomorphism, we say that F is (or more precisely represents) a subpresheaf (resp. subsheaf) of G (via f). Two G are said to define the same sub(pre)sheaf monomorphisms F of G iff 21 an isomorphism F Z" F of (pre)sheaves such that the diagram commutes. If G - H is a morphism and 3. 6

Definition.

,

If

G

39

F

Ker f :

G represent the same subpresheaf of G in this sense, then we see easily that F - G has the universal property of a kernel stated in 3. 2, and we say that F - G is a kernel of G - H. The definitions of monomorphism and of subobject given here make sense in an arbitrary category (cf. [Macl] I §5, V §7; [Mit] I §5).

Corollary. If F is a monopresheaf, then it is a subpresheaf of a sheaf (namely its sheafification); conversely a subpresheaf of a sheaf is always a monopresheaf. 3. 7

(iv) of 3. 5; the converse is easy Proof. 2. 4. 5 and (iii) directly from 3. 5(ii) and 2. 1. 1. // 3. 8

Corollary.

If f : E

E' is a morphism of sheaf spaces

over X, then I'f I'E - I'E' is a monomorphism of sheaves 4 f is injective f is a homeomorphism of E onto an open subspace of E'. :

Proof. 3. 9

2. 3. 6 and 2. 3. 7 and (iii)

Proposition.

(iv) of 3. 5; then 2. 3. 5(c). //

If f : F - G is a morphism of presheaves,

then

(Ker f)x = Ker fx

(equality as subgroups of Fx, which the LHS is by (ii) Ker f - F is a monomorphism).

(iv) of 3. 5, since

Proof.

t E (Ker f)x a open U 3 x and s E Ker(f)(U) such that t = sx H open U 3 x and s E F(U) such that t=sx and f(U)(s) = 0 fx(t) = 0. //

If F, F' are subpresheaves of G, we write F < F' iff a a morphism F - F' (necessarily mono) such that F; `F' 3. 10

Definition.

G

commutes. Hence F < F' <_ F4 F, F' are the same subpresheaf (3. 6).

40

Proposition.

3. 11

If F, F' are subsheaves of a sheaf G, then

F:S F't= VxEX Fx SF'x (as subgroups of G

by 3. 5(iv)).

Proof.

G

F'

x

F

x

x

Suppose Vx E X Fx S F. Given S E I'(U, F) with U open in X, there is a unique section of F' with germs sx at each x E U; hence there is a morphism : F - F' such that the maps F -' F' - G and F - G agree on all stalks. By 2. 1. 10 they are equal and so F < F'. // Corollary.

3. 12

For two subsheaves F, F' of a sheaf G we

have

F=F'<-=4 VxEX Fx=FX Proof. 3. 4

Apply 3. 11 twice. //

Cokernels and epimorphisms 4. 1

Definition.

Given f e Hom(F, G) for F, G E Ob(Presh/X)

let

C(U) = G(U)/f(U)[F(U)] = G(U)/Imf(U)

(for U open in X)

(Imf(U) is a sub-abelian group of G(U) - take the quotient group). If U D V are open in X, the map pv: G(U) - G(V)/Imf(V) kills Imf(U):

for if s E F(U) then pV(f(U)()) = f(V)pV(s) E Im f(V).

Hence we get an induced pv: C(U) - C(V) and C becomes a presheaf, called the presheaf cokernel of f and denoted by PCok(f). We have a natural presheaf morphism: G -' PCok(f) and the composite

F - G - PCok(f) is zero. 41

Proposition. versal property: 4. 2

If f E Hom(F, G), then PCok(f) has the uni-

if H is a presheaf and g E Hom(G, H) is such that (*42)

F

f G+

H = 0 then g factors uniquely as

(that is there is one and only one morphism /making the triangle

H

G->PCok(f)

commute).

Proof. 4. 3

Easy exercise. //

Warning.

If F, G are sheaves and f E Hom(F, G), then

PCok(f) need not be a sheaf (cf. 4. 9).

4.4 Defini#ion. If F, G are sheaves on X and f E Hom(F, G), then the sheaf cokernel of f SCok(f) is the sheafification (2. 4. 1) rLPCok(f) of the presheaf cokernel. Thus SCok(f) is a sheaf, and we have a natural morphism G - SCok(f) (given by G - PCok(f) -+ SCok(f)) such that the composite F - G - SCok(f) is zero. Proposition. If f E Hom(F, G) with F, G sheaves, then SCok(f) has the universal property: 4. 5

if H is a sheaf and g E Hom(G, H) is such that (*45)

f g F - G H = 0, then g factors uniquely as r'H

SCok(f)

Proof. We get unique factorisation of g by presheaf morphisms through PCok(f) by Proposition 4. 2; thus through SCok(f) by Theorem

2. 4. 2, since H is a sheaf. // Remark. The universal properties (*42) and (*45) are the versions in Presh and Shv of the definition of a cokernel in an arbitrary category (with zero object) (cf. [Macl] III §3, VIII §1, [Mit] I, §13). Thus both Presh and Shv have cokernels of all morphisms, but the inclusion 4. 6

42

functor is not cokernel-preserving (cf. 4. 3 and 4. 9). Theorem. Let F, G be presheaves and f E Hom(F, G). Then the following three conditions are equivalent: (i) PCok(f) = 0 4. 7

V open U in X

is surjective (iii) f is an epimorphism in Presh i. e. if H is any presheaf (ii)

f(U)

and G 9 H are -such that g o f= h o f, then h=g. h

Proof. (i)

(ii) is immediate. (iii) is easy (apply 4. 2 to (g - h); cf. 3. 5). // (i)

Theorem. Let f : F - G be a morphism of sheaves. Then the following four conditions are equivalent: 4. 8 (i)

SCok(f) = 0

Vx E X (PCok(f))x = 0 (iii) Vx EX fx is surjective (iv) f is an epimorphism in Shv i. e. if H is any sheaf and (ii)

G

h H are such that g o f= h o f then g= h. Furthermore any of the conditions of 4. 7 implies all of these. Proof.

(i)

(ii)

(i)

(iv) is easy (apply 4. 5 to (g - h)).

SCok(f) = 0 4

(SCok(f))x = 0 since is a sheaf (apply 2. 1. 10 to SCok(f) id the morphisms SCok(f) SCok(f)). Vx E X (PCok(f))x = 0 by 2. 4. 5 V open U 3 x and s E PCok(f)(U) H open V with U? V 3 x and pU(s)=0 V V open U 3 x and s E G(U) a open V with U?V 3 x and pv(s)EImf(V) Vx E X

p-p

4

(iii) PCok(f)X 0

fx surjective. Finally, clearly 4. 7(iii) 4. 9

4. 8(iv) (aliter 4. 7(i)

4. 8(ii)). //

Examples to show the inequivalence of the conditions of

4. 7 and 4. 8 and to justify 4. 3 and 4. 6.

43

Let X = [0, 1] '-+ R and let F be the constant sheaf with stalks Z (with sheaf space X X Z -1. X). Let G be the sheaf whose stalks

A.

are

G

x

Z

=

0

if x=0or1 otherwise

so that, for instance, r(X, G) = Z G Z. Let f : F - G be the unique morphism such that fx = idz if x = 0 or 1. Then f is clearly a sheaf epimorphism, by 4. 8(iii); but r(X, cannot be surjective, so by 4. 7(ii) f is not a presheaf epimorphism.

Let X = C and let Cw be the abelian sheaf of C-valued analytic functions on C. Let d : C4) - C(A be the morphism of differentiation: B.

d(U) : Cw(U) -+ CW(U) : f f"' dz

for U open in C.

For any x e C, an analytic function on a small disc neighbourhood of x

can be 'integrated' and expressed as T with f analytic (look at the power series expansion); hence d is a sheaf epimorphism (4. 8(iii)). But if we take U to be not simply connected, there are analytic functions on U which cannot be expressed as d(U) of an analytic function e. g. 1 on z * 0; hence such d(U) are not surjections and d is not a prez sheaf epimorphism. 4. 10 Theorem. For f : F - G a morphism of presheaves, we have the following equivalent conditions: (i) f is an isomorphism; (ii)

V open U of X f(U) is bijective;

(iii)

is a monomorphism and a presheaf epimorphism. is a morphism of sheaves, we have the further equivalent

If f

f

conditions:

44

(iv)

f is a monomorphism and a sheaf epimorphism;

(v)

Vx E X fx is bijective.

Proof.

(ii) is 1. 5. 2.

(i)

(ii) 4 (iii) is clear from 3. 5 and 4. 7. Now suppose that F, G are sheaves. (v) is 3. 5(iv) and 4. 8(iii).

(iv)

(iv) is 4. 7

(iii)

(v)

4. 8.

(i): If (v) is true, then by the Construction 2. 3. 8 the mor-

is bijective, and it is a local homeomorphism of sheaf spaces Lf : phism by 2. 3. 5(c); hence it is an isomorphism of sheaf spaces, and so

in fLFrLf- rLf (which commutes by 2. 4. 4) IM is an isomorphism. 4F f. G £

Hence f is too by 2. 4. 3. // 4. 11

Proposition.

Let f : F - G be a morphism of presheaves.

Then Vx E X

(PCok f)x = Cok fx (= Gx/Im fx),

(equality as quotient abelian groups of Gx, which LHS is by 4. 8(iii) since G - PCok(f) is a presheaf epimorphism). If f is a morphism of sheaves, then in addition Yx e X (SCok f)x= Cok fX

Proof.

For t E Gx we have a open U 3 x and s E G(U) such that sx = t and

t F- 0 E (PCok(f)) x

s i- 0 E PCok(f)(U)

a open U 3 x and s E G(U) such that sx = t and s E IM f(U) t E Im(fx). PCok(f)

In the sheaf case, we have a commutative diagram G on stalks ,--,,'(P

Gx

+

inducing

SCok(f)

k f) x (

(SCok f) x

45

where the vertical arrow is isomorphic by 2. 4. 5. Hence the result. // 4. 12

Definition.

Dually to 3. 6, if f : F -' G is a presheaf

epimorphism, we say that G is a quotient presheaf of F. If F, G are sheaves and f is a sheaf epimorphism, we say that G is a quotient sheaf of F. In either case G

epi

F epi

G,

define the same quotient object of F iff there is an isomorphism G ' G' such that

F(

G

commutes. As in 3. 6 we can define the notions of a G'

presheaf cokernel and a sheaf cokernel. The definitions of epimorphism and quotient object given here make sense in an arbitrary category (cf. references for 3. 6). 4.13 Theorem. (i) Every subpresheaf is a kernel of some presheaf morphism. (ii) Every subsheaf is a kernel of some sheaf morphism. (iii) Every quotient presheaf is a (presheaf) cokernel of some presheaf morphism. (iv) Every quotient sheaf is a sheaf cokernel of some sheaf morphism. In other words, both Presh and Shv are categories satisfying: (a) every monomorphism is a kernel, and (b) every epimorphism is a cokernel. mono

Proof. (i) Given f : F -. G in Presh, we wish to prove F = Ker(G -+ PCok f). But G - PCok(f) kills (sends to zero) exactly

those sections s E G(U) which arise by f from sections of F(U). mono (ii) Given f : F G in Shv, we have F < Ker(G - SCok(f)) (as subobjects of G); for by the universal property of Ker (3. 2), since F - G - SCok f = 0, F - G factors through the kernel. But for x E X

46

Fx = Ker(Gx -+ (SCok f)x) _ (Ker(G - SCok f)) x

by 4. 11

by 3. 9

by 3. 11. and so F = Ker(G - SCok f) epi (iii) Given f : F -+ G in Presh, we wish to prove G = PCok(Ker f - F), as quotient presheaves of F. By 4. 2 we get

PCok

F Iiy

Ker f

G

f

and i is a presheaf epimorphism since f is, and * is easily seen to be monomorphic, so iy is an isomorphism by 4. 10. epi

Given f : F - G in Shv, we wish to prove G=SCok(KerffF) as quotient sheaves of F; by 4. 5 we get (iv)

Ker f-' and Vx E X /ix is surjective since fx is (4. 8), and Vx E X 4x is injective by 4. 11. Hence by 4. 10 P is an isomorphism of sheaves. //

4.14 Remark.

Given a subobject F - G, we are tempted, by analogy with Abgp, to write G/F for the quotient object of G whose kernel is F, as in 4. 13(i) and (ii). But this notation does not lend itself well to the distinction between the categories Presh and Shv; as 4. 9 shows, G/F may differ in the two. 3. 5

Biproducts, and the abelianness of Presh and Shv

Let F, G be presheaves of abelian groups on a topological space X. We can define a presheaf F ® G (the direct sum, or biproduct of F and G) by 5. 1

Construction.

(F G G)(U) = F(U) ® G(U) pV : F(U)

for U open in X

®G(V) : (s, t)'"'(p(s), p'v(t))

for UDV open in X

47

and we have natural presheaf morphisms:

F®G

G'

2

G

2

where for example J-1(s) = (s, 0) for s e F(U) 7T 2 (s, t) = t for s e F(U), t e G(U). 5. 2

Proposition.

F ® G has the universal property of a bi-

product in Presh: namely (i)

for any presheaf H and morphisms GtiH, there is a

unique morphism F ® G - H such that

FL IL

F ®G

IL

-

H

G

commutes (that is, F (D G is a categorical sum (coproduct) of F and G). (ii)

for any presheaf H and morphisms HyG there is a

unique morphism H - F ® G such that F

__ / G

commutes (that is, F ® G is a categorical product of F and G). Proof.

Easy, using the same result for the category Abgp. //

5. 3 Proposition. If F and G are sheaves, then so is F ® G; hence in this case F ® G has the universal property of a biproduct in Shy (so that 5. 2 holds with H always a sheaf).

Proof.

48

The exactness of

11 0 - H(U) ~ nxEAH(Ux) ~ (x, µ)EAxAH(UA n Uµ)

for H = F, G implies its exactness for H = F ® G. // Remark. Hence both Presh and Shv have biproducts (cf. [Macl] VIII §2, [Mit] I §§17, 18), and the inclusion-functor is biproduct5. 4

preserving. Recall that a category C is said to be abelian iff (a) C is additive: that is, C has a zero object and each set HomC(A, B) has a natural abelian group structure (as in 3. 2. 2); (b) C has a biproduct of each pair of objects; (c) C has kernels and cokernels of all morphisms; (d) in C every monomorphism is a kernel, and every epimorphism a cokernel. (cf. [Macl] VIII §3, [Mit] I §20.)

Both Presh/X and Shv/X are abelian categories, and the inclusion functor Shv I- Presh is kernel-preserving. 5. 5

Theorem.

Proof. 4. 13; 3. 3. // 3. 6

3. 2. 2, 3. 2. 3; 5. 2 and 5. 3; 3. 2 and 3. 3; 4. 2 and 4. 5;

Exact sequences 6. 1

Definition.

(i) If

f : F - G is a morphism of presheaves,

we define the (presheaf) image of f to be PIm(f) = Ker(G - PCok f). (ii) If f : F - G is a morphism of sheaves, we define the (sheaf) image of f to be SIm(f) = Ker(G - SCok f). 6. 2 Exercise. Formulate the universal property that you would like a concept of 'image' to satisfy, and verify that PIm and SIm do in the categories Presh and Shv.

Exercise. Check that PIm(f) is a presheaf whose abelian group of sections over each open U is the image of f(U), while SIm(f) is a sheaf whose stalk at each x E X is the image of fX 6. 3

6. 4

Definition.

f

Let ... -F S. G 9 H -+ ...

be a sequence of

presheaves and morphisms over a space X. We say that the sequence is

49

exact at G as a sequence of presheaves iff PIm(f) = Ker(g) (equal subpresheaves of G); and that it is an exact sequence of presheaves iff it is exact at each point at which this condition makes sense. If the sequence consists of sheaves, we say that it is exact at G as a sequence of sheaves iff SIm(f) = Ker(g) (equal subsheaves of G); and define an exact sequence of sheaves analogously. These definitions make sense in any category with kernels and cokernels, although they are primarily used in abelian categories. In particular we shall use the concept of exactness in Abgp.

(i) F - G -. H is an exact sequence of presheaves iff V open U in X F(U) - G(U) - H(U) is an exact sequence of 6. 5

Theorem.

abelian groups. (ii) F -+ G - H is an exact sequence of sheaves iff

Yx e X Fx - Gx - Hx is an exact sequence of abelian groups. (iii)

g

If F - G - H is a sequence of sheaves which is exact as

a sequence of presheaves, then it is an exact sequence of sheaves. Proof. (i)

F - G 9 H exact 4

Ker g = PIm f

'open U Kerg(U)=Ker(G(U)-+G(U)/Imf(U)) =Im f(U) Vopen U F(U)-G(U)-H(U) exact. f Vx EX (Kerg)X=(SImf)x by 3.12. (ii) F -. G - H exact by definition But (SIm Ox = (Ker(G - SCok(f)))x by 3. 9 = Ker(Gx - (SCok f)X)

= Ker(Gx - Gx/Im fx)

=lmfx. Hence the result.

50

by 4. 11

(iii)

We have Ker g = PIm f; hence Vx E X

(Ker g)x = (PImf)x =

by 4. 11

= (SIm f)x

and so Ker g = SIm f by 3. 12. //

(c)

In each of the categories Presh and Shv: 0 f F - G is exact f is a monomorphism. F - G - 0 is exact 4- f is an epimorphism. For any morphism f : F - G

(e)

0-F G is exact f g F - G H - 0 is exact

6. 6 (a) (b)

Corollary.

f

is a kernel of g. g is a cokernel of f.

Proof. Direct from 6. 5, using 3. 5, 4. 7 and 4. 8, 3. 9 and 4. 11. Aliter: this result holds in any abelian category, and may be proved by using the universal properties of monomorphisms, epimorphisms, kernels

and cokernels. // 6. 7 Definition. A covariant functor T between two categories in each of which the concept of exactness is defined (e. g. two abelian categories) is called exact (resp. left exact, right exact) iff whenever

is an exact sequence, the sequence

is also exact.

(resp.

Exercise. exact sequences. 6. 8

Prove that T is exact 4 T preserves all

51

6. 9

(i) The inclusion functor Shy - Presh is left

Theorem.

exact.

The functor 'sheafify' = rL: : Presh - Shv is exact. (iii) For each open U in X the functor 'sections over U' (ii)

- (U) : Presh - Abgp : F -+ F(U)

is exact. (iv)

For each open U in X the functor 'sections over U'

r(u, -) : Shv -' Abgp : F H- r(U, F) is left exact. Proof.

(i) If

0 - F - G 4 H - 0 is exact in Shv, then f is a

kernel of g, and so 0 - F - G -+ H is exact in Presh (all by 6. 6). (ii) An exact sequence 0 - F - G - H - 0 of presheaves over X gives, for each x E X an exact stalk sequence 0 - Fx-PGx-.Hx-P0 (by, for instance 3. 5 and 4. 11, with 6. 6(a) and (e) in mind). By 2. 4. 4 and 2. 4. 5 there is a commutative diagram

0--F - GIx IX

H -0 IX

S S

o -->(rLF) x o. (rLG)x-

S

(rLH) x -o- 0

with the vertical arrows isomorphisms. Hence the lower sequence is also exact, and so 0 - rLF - rLG -+ rLH - 0 is an exact sequence of sheaves, by 6. 5(ii). (iii) (iv)

Is a restatement of 6.5 (i). f

If 0 -' F -. G

-g H - 0 is exact, then f is a kernel of g

and so for each open U, r(U, f) is a kernel of r(U, g); that is the sequence 0 - r(U, F) - r(U, G) - r(U, H) is exact in Abgp. // 6. 10 Example. Continuing the example of 4. 9B, we have an exact sequence of sheaves over X = C

0 - C - C W d= differentiate70 C W ., 0

(*)

52

where C denotes the constant sheaf on X. Thus for any connected open

set U in X

o - c - r(U, c w )

r(U, d)

r(u, C W )

is an exact sequence of abelian groups; but the last arrow need not be surjective, as in 4. 9B (which shows that (*) is not an exact sequence of presheaves). We shall see later in our study of cohomology that this failure of right exactness can be regarded as a property of the kernel (viz C) as a sheaf on U. Remark. Some authors succumb to the temptation to define exactness of sequences of presheaves and of sheaves by the properties 6. 5(i) and (ii); but this really begs the question of whether this concept of exactness is the usual one in an abelian category, as we have defined it. However, once 6. 5 is known, it is a convenient criterion. 6. 11

3. 7

Change of base space Suppose we are given a continuous map

Construction.

7. 1

$ : X - Y of topological spaces, and a presheaf F on X. We obtain a presheaf $*F on Y, called the direct image of F by $, by putting ($*F)(U) = F($ 1U)

_

U

IpV

$-1U

for U

V open in Y.

- p$-1V

If f : F - G is a morphism of presheaves on X, we get a morphism $*f : $*F - $*G in an obvious way (over an open U, $*f(U) = f($-1U)).

Remark. If we replace X, Y by their categories of open sets U, `U (as in 3. 1. 6E), then the continuous map $ can be regarded as a functor: `U- 91. Regarding F as a functor: 'U, - Abgp (loc. cit.) the 7. 2

functor $*F is just the composite v - 91: - Abgp. 7. 3

Proposition.

Proof.

If F is a sheaf on X, then so is $*F on Y.

Straightforward: reduce the properties (M) and (G) of

53

2. 1. 1 and 2. 1. 3 for a cover (Ui) of U c Y to the same properties for the cover (0 1(Ui)) of 0-1(U) c X. //

Exercise.

7. 4

Verify the functorial properties of 0,k:

$*(idF) = ids F; O*(f ° g) = $*f ° $*g.

Verify the functorial properties of -* : id* = id, (0 ° P)* = ¢* ° q*. Hence given a continuous map 0 : X - Y, 0* is a functor: Presh/X - Presh/Y, and by 7. 3 restricts to a functor 0* : Shv/X-+Shv/Y.

For any X, let Y have one point and 0 be the only map. Then Shv/Y = Abgp are isomorphic categories, and 0* : Shv/X P Shv/Y = Abgp is the same functor as r(X, -). Examples.

7. 5

A.

Let Y be a nice space, such as R2, and X a nice open subset, such as an open disc. Let 0 : X '- Y and F be a constant sheaf on X, say with stalks Z. Let G be the constant sheaf Z on Y. Then we see that for the open set V illustrated B.

Y

r(v, O*F) = Z e Z, while r(V, G) = Z. Hence q*F is not a constant sheaf on Y. In fact the stalks of $*F are easily seen to be given by (O*F)

x_ f

Z 0

if x E X otherwise

so that 0* 'spreads' F out onto the closure of X. C.

Let X=Y=S1 be the circle, and

a double covering (given by z 1- z2 if we represent S1 as 1z c C; z = 1 } ). Let F be the constant

sheaf Z on X. Then if V is a small open interval in Y, we have

¢*F(V) = F( 54

1(V))

=Zez

so that the stalks of 0,kF are (0,kF)y = Z e Z for each y E Y.

But 0,kF is not a constant sheaf, since

r(Y, 0,kF) = r(X, F) = Z. [Note that this gives an example of two non-isomorphic sheaves which nevertheless have the same stalks; compare 3. 12. ] 7. 6

Proposition.

The functor

0* : Shv/X

Shv/Y is left

exact.

0 -+ F - G - H - 0 is exact in Shv/X, then for V open in Y, let U = 1V. By 6. 9(iv) Proof.

If

0 - r(U, F) - r(u, G) - r(U, H) is exact in Abgp, that is 0 - r(V, O*F) -+ r(v, 0,kG) - r(V, 0,kH) is exact in Abgp,

so that 0 - 0*F - O*G - O*H is exact in Shv/Y by 6. 5(i) and (iii). // After example 7. 5A we see that the left exactness of 0* is a generalisation of the left exactness of r(X, -) : Shv/X-.Abgp 7. 7

Remark.

(6. 9 (iv)).

If 0 : X - Y is a continuous map and F, G are presheaves on X, Y respectively, a morphism f : G - F over 7. 8

Definition.

(or p-morphism) is given by a collection of Abgp morphisms f(U, V) : G(V) - F(U)

for all U open in X, V open in Y such that U S $-1 V, subject to the condition that if U 2 U', V ? V' with U', V' open and such that U' S i 1(V') then the square commutes:

55

f(U, V)

pU1pv' G(V)

F(U)

F(U')

G(V')

f(U', V')

The collection of all 0-morphisms G - F is denoted by Hom0(G, F).

If 0 = idX : X - X, then a ¢-morphism: G - F is just a morphism in Presh/X. Hence Homid (G, F)=Hom(G, F). 7.9 Examples.

A.

X

If 0 : X - pt as in 7. 5A, then G is just an abelian group, and to give a ¢-morphism: G - F is to give an Abgp-morphism: G -+ r(X, F); for then all the other morphisms follow from the commutativity of the triangle B.

For instance, if G, F were sheaves of rings (cf. 1. 6E) and f : G -+ F was a ¢-morphism of sheaves of rings (that is, all the f(U, V) are ring morphisms), then to give f is to give F the structure of a sheaf of G-algebras (cf. 1. 6E). C.

For any 0 : X - Y and presheaf F on X, there is a natural

0-morphism $ : 0,kF - F, given by V

$(U, V) = pU

: (0,kF)(V) - F(U)

for U c

1V.

= F(O-1V) 7. 10

Proposition.

Given a continuous 0 : X -+ Y and pre-

sheaves F, G on X, Y respectively, there is a natural bijection HomY(G, 0,kF) = Hom0(G, F)

given by 'composition' with the natural p-morphism Proof.

56

: 0,kF - F.

A 0-morphism f : G - F is in fact determined by the

f(0-1V, V) for V open in Y, since then the f(U, V) for U S 0-1V are

prescribed by the commutativity of the triangle

(putting V' = V in square of 7. 8). But the f(O-1V, V) : G(V) - F(O-1V) = (O*F)(V) just constitute a morphism

G - 0*F in Presh/Y. // Given a continuous map 0 : X - Y, and a presheaf G on Y, we can construct a sheaf 0 G on X, functorially in G (called the inverse image of G by 0), together with natural morphisms: (i) for any presheaf G on Y, G -+ O*O*G 7.11

(ii)

Construction.

for any sheaf F on X, 4*¢*F - F

(that is, natural transformations from idPresh/Y to 0*$* and from 0*0* to idShv/X).

It may help to keep the examples of 7. 5 in mind. First construct the sheaf space LG as in 2. 3. 8 to obtain a diagram of continuous maps and topological spaces: LG

XY

Now let

E = { (e, x) E LG X X; p(e) = O(x) }

with the topology induced from the product topology on LG X X. Thus we have a commutative diagram of continuous maps: E

p'I

1'p

57

(e, x) " x). I claim that (E, p') is a sheaf space over X; that is, p' is a local homeomorphism, for (with p'

:

(e, x) E E

H open neighbourhood W of e and inverse homeo

morphisms W

EnWx0V

W

v

V with V open in Y.

p4-

¢

(V) are inverse homeo-

(qo u, id)

morphisms.

Let $*G = I'E be the corresponding sheaf (2. 3. 6); then in fact we can interpret the sections of O*G: for U open in X, continuous maps v : U -' E; p' o o = idU }

r(U, $*G)

{ continuous maps a' : U - LG; p o v' = 0 I U

and we see from this that ¢*G is an abelian sheaf (2. 5. 2). 7.12

Lemma.

The continuous map E - LG induces isomor-

phisms (O*G) x

G$(x)

for xEX. Proof.

This is clear, since p' -1(x) = p-1(0(x)) x { x } . //

Continuing 7. 11, if f : G - G' is a morphism of Presh/Y, we get a diagram

E'

Ey

I

LG' LG

X

and we obtain easily a sheaf space morphism E - E' and so a sheaf morphism O*G -+ O*G'.

(Checking 0*(id) = id, $*(f o g) = $*f o s*g is

easy, as usual). Now for the natural transformation (i): idPresh/Y

-

*

$*.

We have a natural 0-morphism G - 4*G, given over V open in Y by 58

r(¢-1 V, $*G)

G(V)

where q : a- (o-1 : x-'(a($(x)), x)),

and a' is continuous since it

r(v, I G)

is the composite $-1V

($,

Hence by 7. 10 there is a natural morphism G $*O*G. Now for the natural transformation (ii): 0*0* - idShv/X. If F is a sheaf on X, and V is open in Y, we have maps r($_1v,

r(V, $*F) =

F) -. Fx

if x E 0-1V. If we fix x E X and let V run through all open sets of Y with 4(x) E V, we obtain a target

r(V, $*F) - Fx for the direct system defining (O*F)O(x). Hence there is an abelian group

morphism rx : (O*F)O(x) , Fx (not in general isomorphism; cf. 7.5C). Now apply the 0* construction to O*F: LO*F

x

1p

Y

We can map

r(U, O*$*F)

continuous a : U - LO*F; p o a = o } y r(U, LF)

by

a f+ a' where a'(x) = rx(a(x)) for x E U (so that a(x) E (4*F)O(x)). We must check that a' is continuous: to obtain rx(a(x)) we pick a neighbourhood V of O(x), and some X E r(V, O*F) such that XO (w ) = a(x); since a is continuous we may assume, by shrinking V, that Vx' E $ 1(V) XO(x,) = a(x'); but then a'(x) = rxa(x) = Xx, where

59

we now regard x E r(V, $*F) as a member of r(o-1V, F). However, x will suffice to evaluate or' (x') for any x' E ¢-1(V), so that in $-1(V) a' agrees with X : x''- x x and so a' is continuous at x. It follows easily that we have defined a natural morphism O*O*F - F. The diagram shows the sheaf spaces involved:

/

LF

L* F py

LF p

Y

r is the morphism such that r(e) =

if x = p'(e) where

x : (O*O*F)x - (O*F)O(x) is the natural morphism of 7. 12.

7.13 Theorem. If 0 : X - Y is a continuous map, and G is a presheaf on Y, then the sheaf $*G with the morphism n : G - O*O*G

has the universal property that for any sheaf F on X, the map HomX(O*G, F) - HomY(G, O*F) : f - (O*f) o n

is bijective. Furthermore this property characterises O*G isomorphism in Shv/X. Thus we have natural isomorphisms:

uniquely up to

HomX(O*G, F) :* Hom0(G, F) = HomY(G, O*F)

whenever F is a sheaf on X and G a presheaf on Y. Proof.

n

$*f

Given f : O*G -+ F we obtain G - O*O*G -+ O*F. The

inverse map is: given g : G - 4*F, compose $*G $*$*F - F (7. 11(ii)). That these procedures are inverse may be seen by considering the procedures: given f : O*G -'F we get a diagram for V open in Y, and the dotted map gives G - O*F; 60

r(o-1

V, $*G) -- G(V)

r(O-1V, F)

r(V, $*F)

given g : G- O*F we get VxEX: and the dotted maps give

(*G)

O*G - F (and determine it uniquely since $*G and F are both sheaves).

x

C' $(x)

t

For the uniqueness, if also n : G - O*G has the property, then n, n factor through each other: say $*O*G

*G O*(f)

n

Tl O*(g)

Then g o f: G - G has $*(g o f) o n= O*(id)

n and so g f = idG;

similarly f o g = idO*G, and so O*G = G. // 7.14

In the language of adjoint functors, 7. 13 says

Scholium.

$*

that in the diagram Shv/X ; Presh/Y, 0* is left adjoint to

.

0*

7.15 Remark. If we had worked harder and proved the existence of ¢*G satisfying 7. 13 independently of Chapter 2 and the L construction, w(

could have deduced the sheafification of a presheaf G on Y as (ids,)*G; for if 0 = idyl 7.13 reduces to 2. 4. 2. 7.16 Proposition.

The functor 0* : Shv/Y - Shv/X is exact.

Proof. An exact sequence of sheaves on Y gives by 7. 12 for each x E X a commutative diagram:

0 - OW

G

$(W )- 1$(W )

0 - ($*F)x- ($T*G)x

0

* ($t*H)x--> 0

with the vertical maps all isomorphisms. Since the top line is exact, so is the bottom line, and so 0 -+ 4*F -+ O*G - O*H - 0 is an exact sequence of sheaves by 6. 5(ii). //

61

7.17 Remark. After 7. 14, we should know from the properties of adjoint functors that 0* is left exact and 0* is right exact. The fact that 0* is also left exact is special to this situation. Restriction and extension

3. 8

If X is a subspace of Y, with inclusion map X 4 Y, and G is a sheaf on Y, the sheaf $*G is called the restriction of G to X and is denoted by G I X. If X is open in Y, this co8. 1

Definition.

incides with the notion of 2. Ex. 3. 8. 2 Proposition. If 0 : X c- Y and F is a sheaf on X, then the morphism (O*F) IX - F of 7. 11(ii) is an isomorphism in Shv/X.

Proof. The morphism is given on stalks by the rx : (¢*F)x-D.Fx (for x E X) which are defined by the target r(V, 0* F) Fx

\

of the direct system obtained as V runs // r(o1 V, F) through all open sets of Y such that V 3 x. Hence $-1V = V n X runs through all open neighbourhoods of x

in X, and so this is the direct system defining F. Hence each rx is an isomorphism, and the result follows from 4. 10. // 8. 3

Theorem.

If 0 : X ' Y and G is a sheaf on Y with

sheaf space (LG, p), then G I X is a sheaf with sheaf space 1 pjp-X). 1 GJX has the universal property that for any sheaf F (p X, on X, any morphism g : GIX - F extends to a unique morphism f : G - 0*F whose restriction 4*f to X is the morphism g. The first part follows directly from the construction of O*G (7. 11). The second part is a restatement of 7. 13 in this situation, taking account of 8. 2. // Proof.

If X S Y and F is a sheaf on X, a sheaf F on Y is called an extension of F to Y iff F' IX = F; so 8. 2 shows that O*F is an extension of F. F' is called an extension of F by 8. 4

Definition.

zero iff F' IX = F and Vy E Y\X FY = 0 (so that F' IY\X = 0 by 8. 3). Example 7. 5B shows that O*F need not be an extension of F by zero.

62

Definition. A subspace X of Y is called locally closed iff the following equivalent conditions hold: 8. 5 (a)

Vx E X HU 3 x, U open in Y such that U n X is closed

in U;

HU open in Y and V closed in Y such that X = U n V; (c) X is open in its closure X. For example, R\ {0 } c- R '- R2 is locally closed, but Q '--R is not. If X is locally closed in Y, and Y is locally closed in Z, then X is locally closed in Z. (b)

8. 6

(*86)

Theorem.

X S Y is locally closed iff it has the property:

for any sheaf F on X, there is (up to isomorphism) a unique extension of F by zero, denoted FY. (Some authors denote

itby j!F where j:X4Y.) Suppose X has the property (*86). Then in particular,

Proof.

putting F = ZX the constant sheaf on X, there is a sheaf G on Y with stalks

G={ Z if y E X y

0

if y c Y \X.

Given x e X we can find an open U in Y and a section s E r(U, G) such that s(x) = 1; by shrinking U we can ensure that Vx' E U n X s(x') = 1 (since G I X = ZX). But then U n (Y\X) = t y E U; s(y) = 0 1, and this is open in U (for if s(y) = 0 then s(y') = 0 for y' in some neighbourhood of y). Hence X is locally closed. Now suppose that X is locally closed in Y, and that F' is an extension of F by zero. Then for U open in Y we have an injective map: r(U, F') - r(U n X, F) : s' '- s' I U n x and a section s E r(UnX, F) is in the image of this map iff s remains continuous when it is extended to U by the definition: s(y) = 0 if y E U\(U n X). But this will hold iff {x E U n X; s(x) * 0 } is closed in U, a condition which is independent of

the particular F' chosen. Hence if FY exists, it is unique up to isomorphism.

63

We shall show that F can be extended to Y by first extending F to X, and thence to Y. So assume now that X is open in Y. Let

(LF, p) be the sheaf space of F over X, and set

E=LFJJ-Y/-, where - is the smallest equivalence relation with x - Ox for all x c X, where 0x is the zero element of the stalk p- 1(x) at x. Give E the quotient topology and let p' : E - Y be the natural map. Since X is open in Y, p' is easily seen to be a local homeomorphism. Letting FY = rE be the associated sheaf, we see that FY is an extension of F by zero. The proof is completed by the following Lemma, which enables us

to extend FX from X to Y if X is any locally closed subspace of Y. // 8. 7

Lemma.

If

0 : X - Y and X is closed in Y, then ¢*F

is the extension of F by zero. Proof. By 8. 2 O*F I X = F. If y c Y\X, then since for any open U with y E U C= Y\X we have

r(u, $*F) = r(o-1U, F) = r(U n X, F) = r(i, F) = 0 we see that (O*F)y = 0 as required. //

Let X be a subspace of Y and F a sheaf on

8. 8

Corollary.

(a)

If X is closed in Y, then for U open in Y

X.

r(U, FY)

r(UnX, F):v''ofU nX

is an isomorphism.

If X is open in Y, then LF is an open subspace of LFY (namely p -'X, if p : LFY - Y is the structure map). (b)

Proof. (a) By 8. 7 FY = O* F and r(U,*F) = r(u n x, F) by definition of $*. Aliter: by the proof of 8. 6 the image of the map contains s e r(U n X, F) iff { x E U n X; s(x) 0 } is closed in U;

but this set is always closed in U n X, and so if X is closed it is also

64

closed in U.

This follows from the topology placed on LFY = E in the proof of 8. 6. // (b)

8. 9

Proposition.

If X is a locally closed subspace of Y, the

functor

Shv/X - Shv/Y : F -b FY

is exact. Proof. The functoriality of F - FY follows easily from the constructions of 8. 6 and 8. 7. An exact sequence of sheaves 0-'F-'G-'H-'0

over X gives a sequence 0 -* FY - GY - HY - O

(*)

and the stalk sequence of this over y e Y is

if y EX and

0-0-0-00 if y% X each of which is exact, so (*) is exact in Shv/Y. // 8. 10

Corollary.

If X is a locally closed subspace of Y, the

functor

Shv/Y - Shv/Y : G - (G I X) Y

is exact (some writers denote (GIX)Y by GY).

0:X

Proof. It is the composite of the functors -Y and 0* where Y, each of which is exact (by 8. 9 and 7. 16). // 8. 11

Theorem.

If U is an open subspace of X and F is a

sheaf on X, there is a short exact sequence in Shv/X:

65

Proof. That there is a morphism (F I U)X - F follows from the fact that the inclusion L(F IU)X `- LF is open and so a morphism of sheaf spaces (2. 3. 5(c)); for L(F I U) X ={ 0x E Fx; x E X) Up- 1 (U)

where p : LF - X is the projection and by 8. 8(b) p-1U has the subspace topology inside LF. Putting C = X\U, by 8. 8(a) there are maps (for V open in X) r(V, F) - r(V n C, F I C) = r(V, (F I C)X) a F- aI V n C; hence there is a sheaf morphism F -+ (F I C)X. The exactness of the sequence follows from the fact that the stalk

given by

sequence for x E X is id x

-'Fx -0-0

if XEU

and

id 0- 0 -'Fx-'Fx-'0

if x EX\U

and each of these is exact. // Suppose that X is an open subspace of Y and F is a sheaf on X. Then for each extension F' of F to Y there are unique morphisms g, h such that the diagram 8.12 Proposition.

$*F

commutes, and gives on restricting to X a commutative diagram FY IX

SIX l F'IX

F M

66

O.FIX

gIX

where n is the isomorphism of 8. 6, m is the isomorphism of 8. 2 and

is the isomorphism which shows that F' I X = F.

1

Proof. By the universal property of 0* (8. 3 or 7. 13), the n morphisms F' Ix f F and F Y Ix n F give rise to unique morphisms F' + $*F and FY - 0*F such that the triangles Z

FY IX

commute.

and

Z

For U open in Y we have maps

r(U, FY) - r(U, F') a" a' where a' (y)

a(y) 0

if y E X if y It X

for y E Y (using the identification 1); a' is continuous since if y E X, a' agrees on the open neighbourhood X of y with a, and hence a' is continuous at y, while if y E Y\X and al(y) = T(y) with T E r(U, F') then T is zero on some neighbourhood of y, and so is a' since a is continuous, so that a' is continuous at y. Hence there is a sheaf morphism FYhh F' such that the diagram FYIX

n/ \ hIX F/*

iF'IX

commutes. The result follows. //

8.13 Remark. In categorical terms, 8. 12 shows that if X is open in Y and 0 : X ti Y, the category of extensions of a sheaf F on X to Y has a final object O*F (this is in fact true for any subspace X of Y, as the proof shows), and an initial object FY. This in fact shows that for open subspaces as well as having a right adjoint 0*, the functor

67

X : Shv/Y - Shv/X has a left ad joint

$*

Y

(which is exact, by

8.9).

4

8.14 Example. Let X = R \ { 0 } R = Y and let G be a constant sheaf on Y with typical fibre an abelian group A. Then

F = GIX is the constant sheaf with fibres A over X, and FY has stalks

A

(FY) y

if y # 0

{A

if y=0

0

Lr Y

OA

+

while $*F has stalks

Y

U

A

(O*F)y

{A

if y

A

0

®A if y = 0

(since if U is a small open interval around 0, U n X has two components). Y 0

Exercises on Chapter 3

Let P be the category of pointed sets, whose objects are the pairs (A, a) with a e A E Ob(Sets), and whose morphisms (A, a) - (B, b) are the maps of sets f : A - B such that f(a) = b. Show that P is a category with a zero object, kernels and cokernels, but in which not every epimorphism is a cokernel. 1.

For X any topological space, show that the following presheaves

2.

of sets over X are in fact sheaves: (a) Fixing an open V C X, let hV(U) _

singleton

if

tc V

n

}

for U open in X,

with the unique restrictions. (b) Let f2(U) = { W; W open in X and W C U } for U open in X with restriction: Q(U) - 1(V) : W'- W n V. Interpreting presheaves on X as contravariant set-valued functors on `U, the category of open sets of X (1. 6E), show that the hV defined by (a) are the representable functors

68

hV = HomC(-, V) : U F- HomC(U, V).

Interpret the Yoneda Lemma ([Macl] III. 2, [Mit] IV. 2) as saying that

HomP/X (hV, F) = F(V)

for any presheaf F on X, where P/X denotes the category of setvalued presheaves on X (that is, the functor category Sets` top, where `U.op is the dual of `U). Putting F = hU, this shows that V I-' hV is a full and faithful embedding of `U, into P/X. 3. For a topological space X, let P/X and SIX be the categories of presheaves and sheaves of sets over X. Imitate the results of Chapter 3 as follows:

Show that F -+ G is a monomorphism in P/X (or in S/X) iff for all open U F(U) - G(U) is injective is injective. iff for all x E X Fx - Gx (b) Show that F - G is an epimorphism in P /X iff for all open U F(U) - G(U) is surjective; while F - G is an epimorphism in S/X (a)

iff Vx e X Fx - Gx is surjective. (c)

Discover an appropriate definition of 'equivalence relation'

on a presheaf F so that you can define a presheaf F/- (respectively a sheaf F/- if F is a sheaf) together with an epimorphism F - F/(in the appropriate category) with a universal property. [This is an analogue of cokernel; if necessary, see [G] Ch. II,

1. 9.]

(d) Show that each of the categories P/X and S/X has all limits and all colimits (in particular all equalisers and coequalisers); that in each category every monomorphism is an equaliser and every epimorphism is a coequaliser; and that every morphism has a unique epimorphism-

monomorphism factorisation. [For definitions, see [Macl] III, 3 and 4; V. 2. These results are the analogue of the abelianness of Presh/X and Shv/X. See also Q5 below. ] (e)

Show that for a continuous map 0 : X - Y there are induced

functors SIX « S/Y (as in §3. 7) with the adjointness property 7. 13, $*

69

and such that 0* preserves all finite limits (analogue of '0* is left exact'). (f) Is there an analogue of extension by zero?

Let F, G be presheaves of sets on a topological space X. Define a new presheaf Hom(F, G) with 4.

Hom(F, G)(U) = HomSets

(F(U),

G(U))

for U open in X. Show that if F and G are both sheaves, then so is Hom(F, G). Prove that this construction has the following universal property:

for F, G, H presheaves (respectively sheaves) of sets on X, there is a bijection Hom(H, Hom(F, G)) = Hom(H X F, G)

(*4)

natural in F, G and H. Here H X F is the product object provided by Q3(d); it is constructed 'pointwise'. [This shows that for each presheaf (respectively sheaf) F, the functor Hom(F, -) is right adjoint to - x F; compare the situation in Sets, where the same is true for the functor

Hom(F, -) : GNGF=set of maps : Show that, if we want the property (*4) to hold, then the definition of Hom is forced on us, for presheaves at least. [Hint: put H = hU and use the Yoneda lemma (Q2). ]

Reinterpret (*4) as requiring the existence of an evaluation map

Hom(F, G) X F - G with a suitable universal property.

Show that each of the categories P/X, S/X of Q3 has a subobject

5.

classifier; that is, an object 52 with the equivalent properties (prove their equivalence): (a) there is a natural bijection

Hom(F, 62) = set of subobjects of F

70

for any object F;

t S2 has a special subobject 1 - f2 such that any monomorphism G - F is the pullback of t over a unique morphism F -+ SZ (called the classifying map of G - F). [Hint: for P/X, put F = hU and use the Yoneda lemma (Q2) to (b)

discover 62; for S/X use the

62

of Q2(b). ]

Compare the situation in Sets = S/X for X a one-point space, where 62 is a two-point set and the classifying map is the characteristic map of a subset. [Q3(d), Q4 and Q5 show that P/X and S/X each satisfy the axioms of an elementary topos; see Kock and Wraith: Elementary Toposes (Aarhus Lecture Notes No. 30, 1971); Freyd: Aspects of Topoi (Bull. Aust. Math. Soc. , 7 (1972) 1-76). ]

Since any category of presheaves is a functor category, we can define it for categories other than those of open sets of a topological space. 6.

For C any category, let P/C be the category SetsC°p of contravariant functors C -Sets (and natural transformations) (here Cop denotes the dual of C, which has the same objects and morphisms, but has composition defined in the reverse order to that in C). As a special case, any group G can be regarded as a category with one object whose endomorphisms are the elements of G, with composition defined as in G (hence every morphism is an isomorphism). Show that the category P/G can be regarded as the category of setswith-a-GOP- action, that is the category of permutation representations of G.

If A and B are abelian categories, a functor F : A - B is called half exact iff whenever 0 - P - Q - R - 0 is exact in A, FP -+ FQ - FR is exact in B. F is called additive iff all the maps 7.

F : HomA(P, Q) - HomB(FP, FQ) are abelian group homomorphisms. Consider the conditions:

F is half exact F preserves biproducts (that is F(P G Q) = FP G FQ) (b) (c) F is additive. Show that (a) (b) <- (c). (a)

71

Let K be an abelian category and X a topological space. Let K-Presh/X denote the category of K-valued presheaves on X (1. 6E). Show that it is an abelian category. 8.

Show that we can define the full subcategory K-Shv/X of K-valued

sheaves on X by the following condition: a presheaf F is a sheaf iff for each object T of K the presheaf of sets U)- Hom(T, F(U))

(for U open in X)

is a sheaf of sets. (Verify that if K has the appropriate products, this condition coincides with that of 2. 1. 4 or 2. 1. 8. )

What extra conditions do you need on K to ensure that K-Shv/X is an abelian category? To ensure that the statement of 2. 5 holds for K in place of Abgp? (Consider, for example, K= the category of all finite abelian groups.)

Let 0 - P - Q -+ R - 0 be an exact sequence of presheaves of

9.

abelian groups over a topological space. Show that (a)

if Q is a sheaf and R is a monopresheaf, then P is a

(b)

if P is a sheaf and Q is a monopresheaf, then R is a

sheaf; monopresheaf.

A sheaf F of abelian groups on X is called locally free if and only if each point x E X has an open neighbourhood U in X such that the sheaf F I U is isomorphic to a constant sheaf with typical stalk a free abelian group (of finite rank). Show that if X is connected, a locally free sheaf has a well-defined rank. Show by example that a locally free sheaf on X need not be isomorphic to a constant sheaf even if X is connected, and that 0* does not preserve the property of being locally free. Prove however that the inverse image of a locally free sheaf of rank n is locally free and of the same rank. [Recall that a group G is free abelian if and only if there is a natural number n (called the rank of G) such that G is isomorphic to Zn = ®in1Z. You may assume that 'rank' is well-defined for such groups. ] 10.

72

4 Ringed spaces

This chapter brings us to the essential core of geometry, as expressed in the language of sheaf theory. We study spaces equipped with a sheaf of rings, and particularly the geometric spaces, where the stalks are all local rings: we show that there is some justification for this name, since morphisms between geometric spaces specialise to the appropriate kinds of maps between several types of manifolds (differentiable, analytic, and so on). We construct a universal geometric space associated with each commutative ring, and this leads us to the definition of schemes, which are central in modern algebraic geometry. We then consider sheaves of modules over a ringed space, which generalise the idea of vector bundles, and globalise the idea of a module over a ring. The module constructions of direct sum and product, tensor product and module of homomorphisms also globalise to these sheaves, with appropriate universal properties. Similarly, change of base space by a morphism of ringed spaces gives rise to direct and inverse image functors. Finally, we define the picard group of a ringed space; we shall see later that this can be interpreted as a cohomology group. Throughout this Chapter the word 'ring' will mean commutative ring with a one, and ring morphisms are required to preserve ones. 4. 1

The category of ringed spaces over a ring R

Recall that if R is a ring, an R- algebra is a ring S together with a ring morphism a : R - S called the structure map. A morphism of R-algebras 0 : S - S' is a ring morphism making the triangle 1. 1

`` a

R

at

S

la

it.. S'

73

commute. Hence we have a category R-alg of all R-algebras. For instance, for any ring S there is a unique ring morphism Z - S (taking n - n1S) and any ring morphism S - S' makes the triangle

commute. Hence any ring is a Z-algebra in a unique way, and indeed the category Z-alg is isomorphic to the category of rings. If R is a field,

such as R, C or the p-adic numbers Qp for a prime number p E N, then an R-algebra is an R-vectorspace together with an R-linear multiplication.

If R is a ring, a ringed space over R is a pair (X, OX) where X is a topological space and 0X is a sheaf of 1. 2

Definition.

R-algebras on X. Of course a sheaf of R-algebras is a presheaf of R-algebras (defined as in 1. 1. 2, or 3. 1. 6E with K =R-alg) which satisfies the sheaf conditions (M) and (G) of 2. 1. 1 and 2. 1. 3; a morphism of sheaves of R-algebras is defined in the obvious way, following 2. 1. 13. We can regard OX as a sheaf of abelian groups (forgetting the multiplicative structure) and apply the ideas and constructions of Chapters 2 and 3. By abuse of language, we often say 'X is a ringed space (over R)' and call OX the structure sheaf. If no mention is made of R, it is often

assumed that R = Z, so that after 1. 1 OX is just a sheaf of rings. It is easy to see that the direct limit of a direct system of Ralgebras and morphisms has a natural R-algebra structure; in particular the stalks 0X x for x e X are R- algebras. Suppose that (E, p) is a sheaf space over X such that each fibre p-1(x) (x a X) has a given R-algebra structure. Then E is the sheaf space of a sheaf of R-algebras iff equivalently 1. 3

Proposition.

(a)

for each open U in x, I'(U, E) is an R- algebra under

pointwise addition and multiplication of functions;

let E v E = { (e, e') e E X E; p(e) = p(e') 1; then the maps (e, e') I- e + e' and (e, e') h" eel (ring operations in p-1(p(e))) are (b)

74

continuous: E 7T E - E, and for every r e R the map E - E : e'- re is continuous.

Proof.

Similar to 2. 5. 2. //

Corollary. If X and Y are ringed spaces over a ring R, and 0 : X - Y is a continuous map, then the sheaves 4)*OY and $*OX 1. 4

are sheaves of R-algebras (on X and Y respectively), and the natural morphisms OY - 0*0*0Y' 0*0*OX - OX are morphisms of sheaves of R-algebras. Proof.

Easy verification from 3. 7. 1 and 3. 7. 11, using 1. 3 where

necessary. // 1. 5

Examples.

For any topological space X, letting the structure sheaf OX = ZX, the constant sheaf (2. 4. 6), makes X into a ringed space

A.

over Z. Let R be a topological ring, that is a ring endowed with a topology such that the ring operations (of subtraction and multiplication) are continuous; for example R, C, the p-adic numbers Zp, or any ring with the discrete topology. For any topological space X, let OX = CR be the sheaf of continuous functions on X with values in R (1. 2. B and B.

Then (X, OX) is a ringed space over R; the R-algebra structures R - r(U, OX) for U open in X are given by the constant func2. 2. B).

tions.

For R = Z with the discrete topology, this gives example A. C.

Let X be a banach space over R (for instance Rn for n e N)

and let OX = Cr be the sheaf of r-times continuously differentiable R- valued functions on X (cf. 1. 2. C, 2. 2. B). Then (X, OX) is a ringed space over R. Note that for U open in X, the sheaf Cr defined on U as in loc. cit. is just OX I U (3. 8. 1 and 1. 4). References for differentiability: Bourbaki, Varietes differentielles et analytiques, §2; [L] Chapter 1.

Let K be a complete non-discrete valued field, such as R, C or the p-adic numbers Q ; let X be a banach space over K. Then the

D.

p

75

structure sheaf OX = C ' of all K-valued analytic functions on X (cf. 1. 2. D and 2. 2. B) makes X into a ringed space over K. References for analytic functions: Bourbaki, Varietes differentielles et analytiques, §§3, 4; Serre, Lie algebras and Lie groups, Chapter LG2. Definition. Let R be a ring. A morphism : (X, OX) - (Y, OZ,) of ringed spaces over R is given by a continuous 1. 6

4?

map 0 : X - Y of the underlying topological spaces together with a 0-morphism of sheaves of R-algebras 0Y -+ OX; as in 3. 7.10 this may be regarded as a morphism OY - 0,0 X of sheaves of R-algebras on Y. Given morphisms (X, OX) - (Y, OY) - (Z, 0Z) of ringed spaces over R, we get a composite morphism (X, OX) - (Z, OZ) by composing lp the underlying continuous maps X $ Y Z and combining 0Y-4-0, 0X and OZ - zp,k0Y to get OZ - tp, oy ~ *$*CX = (Vi ° $)*OX. Hence we have defined a category of all ringed spaces over R. 1. 7

Proposition.

4;

: (X, OX) - (Y, OY) is an isomorphism

of ringed spaces over R iff 4i is invertible (that is a* such that 4; ° 4, = id, '1 ° 4f = id) iff the underlying map 0 : X -+ Y is a homeomorphism and the map O Y - $*0X is an isomorphism of sheaves of R-algebras. Proof. The first equivalence is by definition, and the second follows easily. //

Let (X, OX) and (Y, 0Y) be ringed spaces of the kind described in 1. 5B (resp. 1. 5C, 1. 5D), and let 0 : X - Y be a continuous (resp. differentiable, resp. analytic) map. Then there is a morphism of ringed spaces (X, OX) - (Y, 0Y) with underlying continuous map 0, given by 'composition with 0': the 0-morphism OY OX is 1. 8

Example.

defined by

1' (V, OY) - r($-1 V, OX) : s - s ° o

for V open in Y. We shall see a qualified converse to this observation in U. 76

Proposition. Any morphism '1 : (X, OX) -+ (Y, 0Y) of ringed spaces over R can be factored uniquely as 1. 9

(X, 0X)

(X, O*OY) -* (Y, 0Y)

with 1F having underlying continuous map idX and 6 having as morphism of sheaves of R-algebras the natural morphism OY -- O*O*EY (of 3. 7. 11 and 3. 7. 13).

Moreover this factorisation has the universal property that if (X, OX) is another ringed space over R through which 4 factors as (X, O X

(X, O) X ''

(Y, O Y)

with I' having idX as underlying continuous map, then there is a unique morphism 0*0 Y-4 O.

making the triangle

(X, OX),*OY)

commute.

Proof. The existence and uniqueness of I follow from 3. 7. 13: the given morphism OY -+ O*OX associated with 4) corresponds uniquely

to a morphism $*OY- OX. The universal property of ' also follows from 3. 7. 13, using the bijection Hom(0Y, 0*0X)

Hom(0*OY, 0X).

In each use of 3. 7.13 we must make the easy observation that the bijection restricts to a bijection between sets of morphisms of sheaves of R-alge-

bras. // Proposition. Any morphism fi : (X, OX) - (Y, 0 Y) of ringed spaces over R can be factored uniquely as 1. 10

(X, OX

(Y, O,.)

'

(Y, OY) 77

with 0 having as underlying continuous map idY and morphism of sheaves OY -+ id* 0Y = OY a sheaf epimorphism, while the sheaf mor-

phism 0' - $*0X associated with * is a monomorphism. e is then characterised by the subsheaf (of abelian groups) I = Ker(0Y - OY) of 0Y, which is a sheaf of ideals in 0 that is, :

equivalently

V y E Y Iy is an ideal in 0 Y, y

and V open V in Y r(v, I) is an ideal in r(V, OY). Proof. Set 0Y = SIm(0Y - 0*0X) (3. 6. 1); this is easily seen to be a sheaf of R-algebras with the desired properties, since 0Y-+¢*0X factors as OY -, 0Y - 0*0X. The exact sequence of sheaves of abelian groups

0-1

OY91- OY-+ 0

shows that I = Ker(g) and 0Y = SCok(f) determine each other (3. 6. 6), and the stalk sequence over y E Y shows that Iy is an ideal of 0Y, Y The equivalence of the second condition follows from the sheaf space interpretation (1. 3). // Proposition. If 0 : (X,OX) -+ (Y, OY) is a morphism of ringed spaces over R such that the underlying map 0 : X - Y is injective 1. 11

and the morphism 0Y - 0*0X is a sheaf epimorphism, then 0 is a monomorphism in the category of ringed spaces over R. Proof.

(Z, 0Z)

We must show that given a diagram Ip

0 (X, OX) -' (Y, OY)

such that 0 0 = 0 0 F, we can deduce The diagram of underB lying continuous maps is Z X Y and since 0 is injective, 0 = V/. The two composites

0Y - (e*0X)

78

I

(0 ° 0)*OZ

are equal by hypothesis. Applying 3. 7. 13 we get a diagram of sheaves

on Z (6 0 0)*0Y = $*6* OY - $*EX

OZ

in which the composites are equal too. The stalk sequence over z E Z is ((0 U 0)*0 Y)z ~ (O*OX)z ~ OZ, z S4

a

OY, NOW)

by 3. 7. 12, and by hypothesis a is surjective. Hence the two maps X + OZ are equal, and by 3. 7. 13 so are the maps OX $*OZ defining 4 and %F. Hence 4 = T. // 1. 12

Scholium.

Any morphism 4) : (X, OX) - (Y, 0Y) of

ringed spaces over R can be factored as (Y OY)'

(X,

.

O* O Y) --i

,1,

OX(Y, OY

(X,

such that 6 is a monomorphism with underlying continuous map idY and ' has underlying continuous map idX and is final among those ringed space structures on X through which 4 factors. 1. 13 Hard exercise. Prove or refute: the ringed space (Y, 0Y) of 1. 10 and 1. 12 is the image of 4' in the category of ringed

spaces over R. For definiteness we record here the definition of image in an arbitrary category (cf. 3. 6. 2). The image of f : A -' B is the smallest subobject of B through which f factors: that is, a factorisation A

f

B

79

with g monomorphic, f

A

having the property that if

B

is another factorisation with h monomorphic, then sarily uniquely) through h as

1. 14

g factors (neces-

Let X = {x } be a one point space and Y = R2

Example.

be the real plane. Give X and Y the structure sheaves OX, OY of continuous R-valued functions (1. 5B), and let 0 : X - Y map x to the point y 0 e Y. The associated morphism of ringed spaces over R (1. 8) has stalk map OY, yo ~ (O*OX)y

= 0X, x = R

assigning to the germ of a function f its value f(y0) at yo. The sheaf O Y of 1. 10 has stalks OY, y

_ R {

{0}

if if yy

YO

y0

while the sheaf of ideals I of 1. 10 has stalks

Iy =0 Y, y if y#y

o

and

I

yo

= {g e 0 Y, yo ; g is the germ of a function f such that f(y 0 )=0}

The sheaf 0*0Y on X of 1. 9 has stalk ($*OY)x = 0Y, yo

by 3. 7. 12 and the map 0 Y,yo

value at y 80

0.

- 0 X,x = R is again given by taking the

Exercise. is not injective. 1. 15

4. 2

Analyse an example like 1. 14, but where 0

The prime spectrum of a ring 2. 1

Let R be a commutative ring with a one. We can associate

with R a ringed space (Spec R, 0) over R (or over Z) as follows. An ideal p of R is called prime iff the ring R/ p is integral (= has no zero divisors and is not the zero ring). Let Spec R be the set of prime ideals of R, and for any ideal a of R let V(a) = I p E Spec R; p D a}.

Then the V(a), as a runs through all ideals of R, form the closed sets of a topology: this follows from the easy observations: Lemma. (i) V( {0 }) = Spec R; V(R) _ 0. (ii) V(F1iai) = niV(ai). (iii) V(ab) = V(a) U V(b).

2. 2

Proof.

The only difficulty is V(ab) c V(a) U V(b); but if p E (Spec R)\(V(a) u V(b)) then Hf E a \ p and ag E b \ p so that fg

but fg E a b; therefore p

p

a b. //

2.3 For f E R, Iet D(f) = I p E Spec R; f

p } = SpecR\V(Rf).

Then the D(f) for f E R form a basis for the topology on X = Spec R; for p E X\V(a) then picking f E a \p, we have p E D(f) S X\V(a). if Also D(f) n D(g) = D(fg), so that the basis {D(f); f E R I is closed under finite intersections. 2.4 Examples. A. If k is any field, Spec k has just one point, 10 }. Let k be an algebraically closed field (e. g. C), and let B. R = k[x] be the polynomial ring in one variable. The only prime ideals

of R are

{0 }

and the R(x - a) for a E k.

Hence Spec(k[x]) = {{0}} u M

81

bij

k; the topology induced on M has the finite sets as its where M closed sets, and in Spec R the closure of the point {0 } is the whole space.

If k is an algebraically closed field, then

C.

Spec k[xl, ... , xn] D M where M is the set of closed points (= maximal ideals), and by the bij n Weak Nullstellensatz M . k , since any maximal ideal is of the form

(xl - all x2 - a2, ... , xn- an) for some ai E k (1 < i

n).

The topology induced on kn is the Zariski topology, having as its closed sets all the sets of common zeros of a collection of polynomials: that is, all sets of the form { (al,

... ,

an) E kn;

V1 s i < m fi(all a2, ... ,

where fl, ... , fm E k[xl, ... ,

an) = 0 }

xn].

Spec Z has one closed point for each prime number p E N, and one other point. D.

2. 5

To each basic open set of Spec R we can assign a ring:

D(f) I- R f

(where Rf is the ring of fractions of R with respect to the multiplicative system 11, f, f2, ... }), and to any containment D(f) D D(g) a restriction ring morphism Rf - Rg; namely

D(f) D D(g) = an E N, a E R such that gn = af; then define Rf

f

-

r

g fm :

.amr

gnm

(mnemonic for this:

r fm

82

am r _ amr

(af)m - gmn

In particular, if D(f) = D(g), then Rf 4 Rg, so that the assignment D(f) I- Rf is 'well-defined'. (See also [EGAI] 1. 3. 1. ) Since D(f) n D(g) = D(fg), if D(f) = u AEAD(f ) is a cover of a X basic open set by basic open sets, the sequence of 2. 1. 6 becomes Rf - RAEARfA

~ U(x, µ)EAxARfXfµ

and we can check by commutative algebra that this sequence is an equaliser. (For details, see Macdonald, Algebraic geometry, Prop. 5. 1 or Mumford, Introduction to algebraic geometry, Chapter 2, §l. ) Hence we have the data for a 'sheaf defined only on the open sets of a basis', and we need the following sheaf-theoretic Lemma. Lemma. Suppose that X is a topological space and `a is a basis for the topology which is closed under finite intersections. Let 2. 6

F be the data of a presheaf (sets of sections and restriction maps) given only for open sets of the basis, which satisfies the condition that whenever U = UXEAUX with U E (U and VA E A Ua E `U., the sequence F(U) - UAEAF(UX) "' II(A, µ)EAXAF(UA n Uµ

is an equaliser diagram (maps as in 2. 1. 6). Then there is a sheaf G on X, unique up to isomorphism, such that VU E U

r(U, G) = F(U)

and VU, V E `1, if U D V the restriction maps F(U) - F(V) and

r(U, G) - r(V, G) agree. Sketch of one possible proof: Step 1: If such a G exists, it must have stalks

G = lim F(U) x

cL13U3X

so that the underlying set of the sheaf space LG is determined. Step 2: The topology on LG is also determined, since the s[U] with

83

U e `U. and s e F(U) must form a basis for it (cf. 2. 3. 8 and 2. 3. 5). Hence the sheaf G is unique if it exists. Step 3: Construct G by forming the stalks Gx = lim F(U) and thus `113U3x

a sheaf space E _ xEXGx, and check as in 2. 4. 3 that F(U) -' r(U, E) for U E `U;, // An alternative construction is to set, for V open in X,

r(V, G) = lim

F(U).

VDU E`d

2. 7

Applying Lemma 2. 6 to the situation of 2. 5 we see that

X = Spec R has on it a sheaf of rings, denoted by 0 or OX, with the property that for f e R r(D(f), 0) = Rf.

In particular, putting f = 1 we have

r(X, 0) = R so that we can recover the ring R from the ringed space (Spec R, (D) (at least up to isomorphism). At the point p c Spec R, 0 has stalk 0P

=D(f)3p li R =1W Rf f UP

=Rp

(by commutative algebra),

where RP is the localisation (R\p)-1R of R at the prime ideal p Rings of the form RP are all local rings; that is, they each have a unique maximal ideal. In fact, as the stalk of the structure sheaf 0 at the point p , RP describes the nature of the ringed space Spec R 'near' the point p , and this is the origin of the term 'local ring'. 2. 8 Definition. A ringed space (over Z) isomorphic to (Spec R, 0) for some ring R is called an affine scheme. As the construction of 2. 5 and 2. 7 shows, (Spec R, 0) has a natural structure of

84

ringed space over R; indeed for any S such that R is an S-algebra (that is, we are given a ring morphism S - R), Spec R is a ringed space over S. 2.9 Examples. A. Although for all fields k the topological spaces Spec k are homeomorphic, they are distinguished by their structure sheaves, since

k = r(Spec k, 0) = OP where P is the point of Spec k. B. For any ring R, the affine scheme Spec R[xi, x2, ... , xn ] of the polynomial ring in n variables is called affine n-space over R, and is denoted by AR. For R = k, an algebraically closed field, as in 2. 4B and C, and S = k[x1, ... , xn], all the Sf (for f E S) can be considered as subrings of

k(x1, ... , xn) = the field of fractions of

S.

At a point (x1 - al, ... , xn - an) of M S Spec S, f E S has the value f(a1, ... , an) E k; this corresponds to taking the image of f in the residue field Rp /pRp of the local ring at p = (x1 - al, . ' x n- an Then

M n D(f) = { (al, ... , an) E kn;

f(a1, ... , an) * 0 }

and for s E k(x1, ... , xn) and U open in Spec S = Ak we have s E r(U, 0)

V (a1, ... , an) E M n u s can be written in the form s = g with f, g E S and g(a1, ... , an 0.

Thus if we look just at 0 IM, we see the sheaf of algebraic functions on kn. Compare 2. Ex. 6.

Spec is a contravariant functor: that is, a ring morphism r : R - S gives rise in a natural way to a continuous map 2. 10

Proposition.

0 : Spec S - Spec R

and a 0-morphism of sheaves of rings 0 Spec R

-0

Spec S'

Furthermore

85

for each x E Spec S the stalk morphism 0 Spec R, O(x)

-.

0Spec S, x

is a local morphism of local rings; that is, it takes the maximal ideal into the maximal ideal. Proof. If p is a prime ideal of S, let gy(p) = r-1 [p]; then since the ring morphism

R/r-1[p] - S/p is injective, O(p) is a prime ideal of R. Since for f E R -1[D(f)]

p e Spec S; 4(p) e D(f) }

p E Spec S; r-1[p] I f I = D(rf)

the map 0 is continuous. Under the map r

R- S-S r(f)

f becomes a unit, so we get induced maps

r(D(f), 0) = Rf - Srf = r(D(rf), 0) giving the 0-morphism required. Similarly, since for

p c Spec S

R-r S -Sp sends every f ¢(p) into a unit, the induced map RO(p)

Sp

of stalks sends each member of the maximal ideal O(p).

maximal ideal pSp. //

RO(p)

into the

A ring morphism R - S makes S into an R-algebra, and induces a morphism Spec S - Spec R of ringed spaces. over R. We may choose to regard it as a morphism of ringed spaces over Z, since each of R, S is a Z-algebra in a unique way (cf. 1. 1, 1. 2 2. 10

and 2. 8). 86

Scholium.

4. 3

Geometric spaces and manifolds

Let R be a ring. A ringed space (X, OX) over R is called a geometric space (over R) iff all the stalks 0X x , (for x E X) are local rings (that is, each has a unique maximal ideal, called mx). A morphism of geometric spaces over R (X, OX) - (Y, OY) 3. 1

Definition.

(or R-morphism) is a morphism of ringed spaces, with underlying continuous map 0 : X - Y and 0-morphism of sheaves of R-algebras Ji : OZ, - OX, with the additional property that for each x E X the stalk map

'Ix : OY, $(x)

-

OX, x

is a local morphism of local rings; that is Px(m¢(x)) S mx. Thus we have defined a (non-full) subcategory of the category of

ringed spaces over R. Examples. By 2. 7 and 2. 8 any affine scheme Spec R is a geometric space 3. 2

A.

(over R or Z), and any ring morphism R - S gives rise by 2. 10 to a morphism of geometric spaces Spec S - Spec R (again over R or Z). B. Let R be any ring, and F an R-algebra which is also a topological field (so that the operations of subtraction and division are continuous); for instance R = F = R, or R = Z, F = C. For any topological space X, let CF be the sheaf of continuous F-valued functions on X (cf. 1. 5B). Then (X, CF) is a geometric space over R (see 3. 3). The example (X, Cr) of 1. 5C (with X a banach space over R) C. is a geometric space over R (see 3. 3). The example (X, C('O) of 1. 5D is a geometric space over K (see 3. 3).

D.

In each of the cases 3. 2B, 3. 2C, 3. 2D the stalks 0X x are local rings with maximal ideal 3. 3

Mx= { g E O X x; g is the germ of a function f such that f (x)=0 1. ,

87

For this is clearly an ideal, and OX x/mx = F or R or K (according to the case) by g r- value of g at x, so that mx is maximal (since F, R, K are each fields). Also g nix implies that g is the germ of a function f such that f(x) * 0, and so in a small enough neighbourhood of x, exists; hence g is a unit in OX x, so that m x is the unique maximal ideal of 0X X, Note that in each case the R-algebra structure on 0X is given by

the maps R - r(u, OX) (for U open in X) which send r E R to the constant function with value r. 3.4 When we look at geometric space morphisms between geometric spaces of the types of 3. 2B, C, D, a remarkable thing happens. Let us examine for example the typical case of an R- morphism (X, 0X) - (Y, OY) where

X = Rn, OX = Cr is the sheaf of differentiable functions on X;

Y = Rm, OY=Cr is that on Y. The underlying continuous map 0 : Rn -R m is given by say O(x) = (Pi (x), ... , Pm(x)) E Rm

where if ni : Y -R is projection onto the ith coordinate, we have Pi-7T i0 0.

Now consider the 0-morphism y : OY - X, This has the property that each stalk map

0 Y,O(x)~0X,x for xEX is local; hence by the above interpretation of the maximal ideals, if f is a differentiable function on some open set V in Y, we have Vx E $-1(V) (i. e.

f(0(x)) = 0

fO(x) E mO(x))

T/i(f)(x) = 0

(i. e.

'x(fO(x)) E mx).

Applying this to the differentiable function (f - c) with c E R we get 88

f(0(x)) = c

Vx E $-1(V)

ip(f)(x) = c

(*34);

ifi(f)(x) = f(O(x)), so that

in other words

41(f) = f O q.

Thus the sheaf morphism ip is just 'compose with 0':

X

U

$IU

$

Y-i

(f)

-'V

R f

as in 1. 8. In particular, ip(77i) = 'ff i o ¢ = pi; but each Hi is differen-

tiable and so each ip(w) = pi is differentiable, so that 0 is actually an (r times continuously) differentiable map: Rn -+ Rm. Thus the effect of insisting that the a priori merely continuous has an associated 0-morphism CY -,. CX is to ensure that 0 is actually differentiable. Conversely, a differentiable map 0 : X - Y clearly defines a ¢-morphism CZ, - CX by the above procedure of composing with 0, and hence a geometric space morphism: X - Y. A similar argument shows that any K-morphism of geometric spaces (X, C (0) - (Y, C W) (spaces as in 3. 2D) has an underlying map which is analytic. In each case the step (*34) is valid, since the morphism is over R = R or K and so takes the constant function with value r E R to the constant function with value r.

Exercise. Show that any geometric space morphism between affine schemes Spec S - Spec R arises from a ring morphism R -+ S as in 2. 10. Deduce that the category of affine schemes and geometric space morphisms is contravariant- equivalent to the category of rings. 3. 5

Fix a ring R. Suppose we have a geometric space (M, 0M) over R (to be regarded as a 'model'). We say that a geometric space (X, 0X) over R is locally isomorphic to (M, 0-M) iff V x E X there is an open neighbourhood U of x in X and an open 3. 6

Definition.

89

set V in M such that there is an isomorphism (U, 0XIU)

(V, OMIV)

as geometric spaces over R. Given a class `h of model spaces (M, OM), we say that (X, 0X is a manifold of type M (over R) iff X can be covered by open sets U` such that each OX I U is locally isomorphic to some (M, OM) E M. A morphism between manifolds of type 3 is just a morphism of geometric spaces over R. Some authors use the word variety in place of manifold. 3. 7

Examples.

Let R = R.

A.

Putting `.)7C = { (Rn, CR) } we obtain topological manifolds of

dimension n. Putting On = { (Rn, Cr) } we obtain differentiable manifolds of

class r and dimension n. Putting M= J (Ft , C ); n e N) we obtain real analytic manifolds. B. Let R = C. Putting ` t = 3 (C, CW) } we obtain Riemann surfaces. Putting `_11t = { (Cn, Ca'); n E N I we obtain complex analytic manifolds.

Let R = Q .

C.

Putting Mt = {(Qp, CW); n E N } we obtain p-adic analytic manifolds. Putting ¶ _ {(E, C°'); E is a hilbert space over Qp } we obtain

p-adic analytic manifolds of hilbert type.

Let R = Z. Putting i = class of all affine schemes, we obtain schemes.

D.

Remark. After 3. 4 and 3. 5 each of the above definitions is in accordance with the more usual definitions in terms of atlases of charts with transition maps of the appropriate kind (see the references for 3. 9 below), with two possible exceptions. Some authors may require that X have a countable base of open sets. Other authors may insist that (X, OX) satisfy a separation (hausdorff) condition of some kind to avoid 3. 8

90

examples like

(=RUR/- where 'x - y iff x = y and x * 0'). This condition is just that the underlying space X be a hausdorff space in 3. 7A, B, C but is a little more complicated in the case 3. 7D of schemes (it gives separated schemes). Exercise. Compare 3. 6 and 3. 7 with any other definition of manifold you may have met; for example as in Bourbaki, Varietes differentielles et analytiques; [H] Chapter I, 2. 5; [L] Chapter 2. In particular, verify (using 3. 4) that the geometric space morphisms between topological (respectively differentiable, analytic) manifolds are just the continuous (respectively differentiable, analytic) maps that are usually considered between manifolds. What about piece-wiselinear (PL) manifolds? 3. 9

3. 10 Aside. In proving in 3. 4 that the geometric space morphisms between geometric spaces of types 3. 2C and 3. 2D were precisely the maps we desired between the underlying spaces, it was vital that the sheaf morphism tP preserved constant functions (*34). We used the supposition that was a morphism of sheaves of R-algebras to justify this. However, it is an elementary fact of analysis that for R = R any morphism of sheaves of rings preserves all the (R-valued) constant functions. This follows from the Lemma below, which we prove for the convenience of the reader. For R-valued functions f, g on a set U we write

f>g

Vx EU

f(x)> g(x)

and we denote the constant function with value c E R again by c. Lemma.

Let U, V be sets and let R (resp. S) be a subring

of the ring of R-valued functions on U (resp. V) such that: (i) c E R (resp. S) Vc ER (ii) Vf E R (resp. S) f > 0 ag E R (resp. S) such that

f=g2. 91

Then if

ip

: R -+ S is any ring morphism, we have

Vc c R

4'(c) = c.

Proof.

Since /i(1) = 1 we have for n E N

ti(n)=,(F 11)=Ii 1 and since

(1)=n,

(-n) + a/(n) = 0, we have '(-n) = -n. For m, n E Z

n*(n) = Cm) = m so Vq E Q 4,(q) = q. We have for f E R

f> 0

ag E R such that f = g2 Hg E R such that

(f) = (4i(g))2

Of) - 0. Now suppose we are given c E R. For any e > 0 a q, q' E Q with

q>c>q' and q-q'<e 0, 4/(c-q')?0 so

and then

q'

and q-q'<e.

Letting a - 0 we see that y(c) = c. // Corollary. Proof.

R has no non-trivial ring automorphism.

Let U = V have one point. //

We note that condition (ii) is satisfied for the subrings of differentiable (or analytic) functions on Rn since we can compose with the square root function which is analytic away from zero. Hence if we were interested only in geometric spaces and manifolds over R, we need only have insisted that the structure sheaves be sheaves of rings (that is Z-algebras) in order to get the 'correct' notion of m orphis m.

However this is no longer true if we wish to consider other types of manifold. For instance if a is any automorphism of C (for instance

92

complex conjugation), then for an analytic f : C - C, v o f o Cr- 1 is again analytic, and f - or o f o 0 1 is an automorphism of the ring of analytic C-valued functions on C which does not preserve the constant functions. 3. 11 The Spec construction has a universal property among all geometric spaces. Theorem. Let (X, OX) be a geometric space over Z, and let R be a ring. For each ring morphism t : R - r(X, OX) there is a

unique morphism of geometric spaces

G : (X, OX) - (Spec R, 0) such that the induced map

R =' r (Spec R, 0) - r(X, OX) is t. In other words the map HomGsp((X, 0X), (Spec R, 0)) - HomRing(R, r(X,

4 )'

is bijective (where Gsp is the category of geometric spaces over Z). Given t, we first construct the underlying continuous map g of G. For x E X, let tx be defined to make the diagram Proof.

-r(x, ox t commute. Let g(x) = t-1(m x) where m x is the maximal ideal of 0 X x' then g(x) a Spec(R). For f E R we have g-1(D(f)) = { x E X; g(x)

f}

= {x E X; tx(f) it mx }

and this is open in X, for f' = tx(f) m x holds iff f' is a unit in 0X x' , so Hg' E 0x x such that f'. g' = 1, and this equality still holds in some ,

neighbourhood of x. This calculation also shows that the map

R - r(x, OX) ~ r(g 1(D(f)),

(Y x)

93

inverts f, so it factorises as r(D(f), 0) = Rf -' r(g-1(D(f)), OX)

and this defines the morphism G, which clearly has t as its associated global section map. The uniqueness follows from the easily verified fact that this construction is inverse to the process of taking global sections (so that the map given in the Theorem is bijective). // 3.12 Remark. 3. 11 shows that Spec and r(-, 0 ) are adjoint functors. It follows that to give (X, OX) the structure of a ringed space

over R (that is to make r(X, OX), and hence all the r(U, OX) for U open in X, into R-algebras) is the same as to give a morphism of geometric spaces (over Z) X -' Spec R. Modules over ringed spaces

4. 4

Recall that if R is a ring with a one, a (left) R- module is an additive abelian group M equipped with a map 4. 1

RXM-M:(r, (that is, an operation of R on M) satisfying the conditions: (i)

(ii)

yr E R ym, n c M yr, s E R ym E M

r(m + n) = rm + rn (r + s)m = rm + sm

(rs)m = r(sm) (iii) ym E M lm = m. A morphism f : M - N of R-modules is an abelian group morphism which satisfies the additional condition: and

Yin E M V r E R

f (rm) = rf (m)

(so that f respects (or preserves) the operation of R). The category R-mod of all R-modules is abelian (cf. 3. 5. 4) (even

if R is not commutative); indeed, the kernels, cokernels and biproducts are constructed as if in the category Abgp, and are then seen to carry a natural R-module structure.

94

For example, for R a field we get the category of all R-vector spaces; while for R = Z we get the category of all abelian groups (for

to prescribe a - b is to prescribe na for all n E Z). More generally, if p : R - S is a (one-preserving) ring morphism, and M is an R-module and N is an S-module, then a map f : M - N is called a morphism over p (or dihomomorphism) iff f is 4. 2

an abelian group morphism and

Vr ER Vm EM

f(rm) = p(r)f(m).

If R' is an R-algebra, with structure map a : R - R', then R' is an R-module with the operation of R defined by 4. 3

Example.

rr' = a(r). r' (product in R') for r E R, r' E R'. A commutative square of ring morphisms: R'

at

R

S'

P'

P

?a S

may be regarded as giving a module morphism p' : R' - S' over p. 4. 4 Definition. Suppose that (X, 0) is a ringed space over a ring R. An 0-Module (note the use of the capital letter), or sheaf of 0-modules, M is a sheaf of abelian groups on X with the additional structure that (i)

(ii)

for each open U in x, F(U, m) is a I'(U, 0)-module; and whenever V c U are open in X the restriction map

r(u, m) -+ r(v, m) is a module morphism over the ring morphism (restriction) F(U, (as in 4. 2).

There is an obvious definition of morphism for 0-Modules: f : M - N must be a morphism of abelian sheaves, and for each open U

in X 95

r(u, f) : r(u, m) - r(u, N) must be a r(U, 0)-module morphism. Hence we have defined a category O-Mod of all 0-Modules on X. 4. 5

Examples.

For any topological space X, let 0 be the constant sheaf Z on X (as in 1. 5A). Then any abelian sheaf on X is an 0-Module, and the category '9-Mod is just the category Shv/X we dealt with in Chapter 3. Let R be a ring, X = Spec R and 0 be the structure sheaf on B. X as in 2. 7. Starting with an R-module M, we can construct an 0Module M on Spec R as follows. To each basic open set D(f) of Spec R we associate the Rfmodule Mf = S 1M where S = {fn; n E N }. The restriction maps

A.

follow from Commutative Algebra, and the sequence Mf

HAEAMfX

H(X,N)EAXAMfXf

corresponding to an open cover D(f) = uXEAD(fx) is an equaliser. Hence by Lemma 2. 6, M defines a sheaf 91 on Spec R which is easily seen to be an 0-Module. We remark in passing that R = 0, and that it can be shown that

the category R-mod is equivalent to a full subcategory of 0-Mod by the assignment M I- M. An OX Module on a scheme (X, OX) which is given locally (on an open cover by affine schemes) by Modules of the form ick is called quasi-coherent. Suppose that (X, OX) is a differentiable or analytic manifold in C. the interpretation of 3. 7. Then for p E N the assignment

U )- set of p-forms on U (for U open in X) defines an OX-Module.

Similarly the sheaves of sections (cf. 2. 2. C) of vector bundles over a topological, differentiable or analytic manifold (X, 0X) can be regarded as 0X Modules. (References: [H] Ch. I, §§3. 2, 3. 6; [L] Ch. III, V; Atiyah, K-Theory,

Ch. I.) 96

It is easy to see from the construction of direct limits that if M is an 0-Module on X and x E X, then the stalk Mx has a natural 0x-module structure. 4. 6

Applying the definitions of kernel (3. 3. 1), cokernel (3. 4. 4) and biproduct (3. 5. 1) to 0-Modules and 0-Module morphisms, we get Modules and morphisms again. All the results of § §3. 2-3. 6 hold for the

category 0-Mod in place of Shv/X. In particular, 0-Mod is an abelian category, and a sequence K -+ L - M in 0-Mod is exact iff it is exact as a sequence of sheaves of abelian groups (which holds iff each stalk sequence is exact, by 3. 6. 5). 4. 7

Moral (after 4. 5A). We should have been dealing with

®-Modules all along.

Constructions. Recall that if R is a ring and (MA)XEA a family of R-modules, then the direct product IIAEAMX is the R-module defined by giving the product set pointwise operations. The direct sum eXEAMA is the sub-R-module of IIAEAMX generated by the images of all the injection maps 4. 8

iµ : Mµ y IIXEAMX . m N (mX)XEA

'=µ where ma = { m0 if if X* P

(cf.

EHM 1. 3. 19) (so that (m) X X EA X EA

is X

an element of the direct sum iff for all but finitely many of the A E A, mA = 0). Each of these constructions enjoys a universal property. For any R-module N there are bijections: (*48)

f

Hom(N, HXEAMX) Z* IIAEAHom(N, MX) : f - (pX U f) XEA _

Hom(eAEAMx, N)

HAEAHom(MX, N) : f H (f o iX)XEA

where pp : IIXEAMA - Mµ is the pth projection. If now (X, 0) is a ringed space over R and (MX)XEA is a family of 0-Modules, it is easily verified that the following presheaves are sheaves:

97

MA)

U

open in X

and so define 0-Modules called the direct product BAEAMA and the direct sum ®XEAMX of the family. There are natural projection morphisms pµ : HAEAMA -0. Mµ and injection morphisms iµ : and the universal properties (*48) hold as stated, in the category 0-Mod. Hence the abelian category 0-Mod satisfies Grothendieck's axioms AB3 and AB3* (existence of infinite sums and products) ([T] R. 5). Of course, the case 0 = ZX, A having two elements was dealt with in U. 5.

Since direct sums and stalks are both defined as colimits, we have for x E X the stalk: lim

U ,4x

(®athr(u MA)) '

= ®xEA li_m 1J 3X

r(u, M

®AEA(MX)x

as can easily be verified directly. However there is no corresponding expression for the stalks of the direct product.

Let (X, 0) be a ringed space over a ring R. If L, M are 0-Modules we can define a presheaf on X by 4. 9

Construction.

U - r(U, L) ®r(u, o)r(u, m) for U open in X (tensor product of r(U, 0)-modules). The sheafification of this presheaf is an 0-Module called the tensor product of L and M over 0 and denoted by L ®0M. See 5. 8 for an example where the presheaf is not a sheaf. Since sheafification does not change the stalks (2. 4. 5), and tensor product commutes with direct limits (see for example Bourbaki, Algebra, Ch. II §6. 7), L ®0M has stalk (L ®0M)x

98

1. U3x r(u, -rn

L) ®r(U' 0)r(u,

M)=Lx

®0xMx

at x e X; also if s, t are sections of L, M respectively over an open set U then the map

xI-sx ®tx

for x E U

is a section of L ®0M over U. It is easy to see from the definition that for any 0-Module M we have M ®p C. = M.

Tensor product of Modules inherits many of the properties of tensor product of modules; for instance, tensor product with a fixed 0-Module M gives a right exact covariant functor 4. 10

M ®0 - : 0-Mod - 0-Mod.

There is also a universal property as follows. If L, M, N are 0Modules, a bilinear map from L, M to N is a map of sheaves of sets f : L G M - N such that for each open set U the map

r(U, f) : r(U, L) e r(u, m) - r(U, N)

is a bilinear map of r(U, 0)-modules. Such maps form a set Bilin(L, M; N) which is covariant-functorial in N. Proposition. L ®0 M represents the functor Bilin(L, M; -) 0-Mod -Sets; that is, there is an isomorphism

Bilin(L, M; N) = Hom,. L 00M, N) which is natural in the 0-Module N. This property characterises the 0-Module L ®0 M up to isomorphism. Proof. Direct from the corresponding proposition for modules over a ring, and the universal property of sheafification (2. 4. 2). // 4.11 Suppose that (X, 0) is a ringed space over a ring R, and U is open in X. Then U is a ringed space with structure sheaf 0 I U (3. 8. 1 and 1. 4), and if M is an 0-Module, it is easy to see (using

3. 8. 3) that M I U is in a natural way an (0 I U)-Module. 99

Proposition. If L, M are 0-Modules and U is open in X, then there is a natural isomorphism of (0 I U)-Modules (L I U) ®0 U(M I U) - (L ®0M) I U.

Proof.

There is a natural bilinear map

(L I U) ® (M I U) '' (L ®e.M) I U

obtained by restricting L ® M - L ®0M to U. By 4. 10 this induces the required map, which is an isomorphism since on the stalk at x E U it reduces, after the identification of 4. 9,to the identity map

Lx ®Mx -'L x®M x (using 3. 7. 12 or 3. 8. 3). // 4. 12 Construction. Let (X, 0) be a ringed space over a ring If M, N are 0-Modules, define the presheaf Hom(M, N) by R.

Hom(M, N)(U) = HomO l U(M I U, NIU) for U open in X,

where the right-hand side is the set of (r) I U)-Module morphisms of M I U

into NIU, and has a natural structure of r(U, 0)- module. If U ? V are open, we have a map Homo lU(MIU, NIU) -* Hom0IV(MIV, NIV)

where a : V 4 U is the inclusion (3. 7. 11), and a* is easily seen to be a module morphism over the restriction r(U, 0) - r(V, 0). If we reinterpret the elements of Homo U(M I U, NIU) as continuous maps between certain subspaces of the sheaf spaces LM and LN (cf. 3. 8. 3) we can check readily that Hom(M, N) is a sheaf of 0-modules, called the sheaf of germs of homomorphisms from M to N, or sheaf-hom for short; if we wish to emphasise the structure sheaf we can write I

Homo (M, N).

100

4.13 Suppose that 4i : (X, OX) - (Y, 0Y ) is a morphism of ringed spaces over a ring R, with underlying continuous map 0 : X - Y and morphism of sheaves of R-algebras iy : OY - 0*0X. If M is an OX Module, the definition

r(v, $*M) = r(q 1V, m) for V open in Y shows that O*M is naturally a (0*0X)-Module (cf. 1. 4). The morphism

/ gives for each open V in Y a change-of-rings morphism r(v, OY) - T(V, ¢*OX) which enables us to regard O*M as an OY Module. Hence 4i induces a functor 4i

: OX Mod - OY Mod.

If N is an OY Module, then O*N is a sheaf of abelian groups on X (3. 7. 11), and an easy examination of its groups of sections shows that c6*N has a natural structure of (0*0Y)-Module. We also have a change-ofby 3. 7. 13, and we define rings morphism O*OY - OX deduced from 4;*N = $*N ®0*0 OX

to get a functor (D* : OY Mod - OX-Mod.

4.14 Theorem. If 4) : (X, OX) - (Y, OY) is a morphism of rimmed spaces over a ring R, then 4?* is left adjoint to 4;*; that is there is a natural bijection Homo (4i*N, M) 4 Hom0 (N, 4) *M) X

Y

whenever M c Ob OX Mod and N E Ob OY Mod.

Proof. Let 0 : X - Y be the underlying continuous map of Then we can construct the following diagram:

4i.

101

HomZ (O*N, M) -

f

XI

Hom'.*0 (O*N, M)

HomZ (N, $*M)

Y

Home (N, 4i*M) g

Y4

Y

h

Hom0X(O*N®O*0YOX, M)

where the bijection f is given by 3. 7.13, and is easily seen to induce the bijection g, while the map h is given by composition with the morphism

$*N"$*N®0*0

OX

Y

(given on sections by s I- s ® 1) and has inverse sending a morphism u : O*N - M to the composite v u®id $*N ®0*0 OX --im- - M ®0*O OX -+ M, Y

Y

where the multiplication morphism v is given on sections by s ®t- is (since M is an OX-Module). // 4. 15

If

(X, OX) is a ringed space over a ring R and U is an

open subspace of X, then OX I U (3. 8. 1) is a sheaf of R-algebras on U which makes U into a ringed space over R and gives a morphism of ringed spaces 4b

: (U, OX I U) + (X, OX)

over the inclusion map U `i X (the map of sheaves is given by the natural map of 3. 7. 11(i)). If M is an OX Module, then 4i*M (as in 4.14) is the (OX J U)-Module M I U.

Locally free Modules

4. 5

(X, 0) is a ringed space over a ring R, an 0-Module M is called locally free of rank n E N iff X can be covered by open sets U such that 5. 1

102

Definition.

If

M I U= (0 I U)n as (0 I U)-Modules

where for a Module N, Na = ®i 1N is the biproduct of n copies of N. 5. 2

If (X, 0) is a ringed space over a ring R, the endomor-

phism ring of 0 End(O) = HomO(O, 0)

is the set of 0-Module morphisms: 0 - 0, with composition as multiplication. Its group of units is the group Aut(O) of 0-Module automorphisms of 0. We have a map

r(X, 0) - End(O) sending s E r(X, 0) to the endomorphism which is given over an open U by multiplication by the restriction pt(s): U

r(u, o) - r(u, o) : t H pU(s). t. 5. 3

Proposition.

In the situation of 5. 2, the map

r(X, 0) 4 End(O)

is a ring isomorphism. Hence Aut(O) =* r(x, 0)* (the group of units

of the ring r(X, 0)) and for n E N End(On) Z* Mn(r(X, 0)) = Endr(X, O)(r(X, O)n)

can be identified with the ring of n x n matrices over r(x, 0), while Aut(On) = GLn(r(X, 0)) = Autr(X, O)(]QX, O)n). Proof.

The map of 5. 2 is clearly a ring morphism, and it is injective, since if s E r(X, 0) gives the zero morphism 0 - 0, then in particular s = pX(s). 1 = 0.

103

Given f E End(O), each r(U, f) : r(U, 0) - r(U, 0) is a r(U, 0)-module morphism, and so is multiplication by some tUEr(U, 0) (namely tU= r(u, f)(1)); then the commutativity of the square r(X, 0)-

- r(X, 0)

I r(u, 0)0) 1

shows that tU= pU(tX). Hence f i-+ tX = r(u, f)(1) gives an inverse to the morphism of 5. 2. If multiplication by s E r(x, 0) is an automorphism of 0, then t H s. t is surjective on r(X, 0), so s is a unit; the converse is clear. The last result follows from the usual procedure of considering the various composites

0-0 to obtain the matrix components. // 5.4 Examples If X is a manifold of the kind described in 3. 7A, B, C with

A.

structure sheaf 0, then the locally free 0-Modules of rank n are just the sheaves of sections of rank n vector bundles. For it is easy to see that such a sheaf is locally free, by considering its restriction to the open sets of a trivialisation. Conversely, given a locally free Module M, we can patch together copies of Rn over the intersection U A V of two open sets over which M is free by means of the isomorphism (0 IUn V)n(OIU)nIV=MIUnV='(0 IV)nIU = (0 !U n V)n which by 5. 3 is given by a member of the general linear group GLn(r(U n V, 0)) (that is, an invertible n X n matrix of R-valued functions on U n V). B. It can be shown that the locally free Modules over an affine scheme

Spec R are the Modules of the form P (4. 5B) where P is a projective R-module (of constant finite rank). (References: Bourbaki, Algebre Commutative, Ch. II, §§5. 2 and 5. 3, especially Th. 2; [K], 3. 3. 7. ) 104

5. 5

Definition.

The locally free Modules of rank 1 over a

ringed space (X, 0) are called the invertible 0-Modules. We shall see later (5. 4. 16) (or we could check directly) that there is only a set of isomorphism classes of invertible 0-Modules; taking this on trust, this

set is called the picard group Pie X of the ringed space (X, 0). The justification for these terms is given by the following result. 5. 6

Theorem.

If (X, 0) is a ringed space over a ring R, then

PicX is a group under tensor product, with identity the isomorphism class of the free module 0, and with the inverse of the class of an invertible sheaf M being given by the class of Hom(M, 0) (the 'dual' of M).

Proof.

We saw in 4. 9 that for any 0-Module M, M ®O 0 = M.

If M, N are locally free of rank 1, then for all members U of a sufficiently fine open cover we have (after 4. 11)

(M(& 0N)IU=MIU®OIUNIU=OIU®OIU0IU=0IU

so that M ®0N is also invertible. We must show that there is an isomorphism f : Hom M, 0) 0 0 M - 0. We have a map from the presheaf defining the LHS to 0 by HomOIU(MIU, 0IU) ®r(U, O)r(U, m) -+r(U,

g)(s)

and so a morphism f exists by 2. 4. 2. But if U is such that M is trivial (free) over U, using M I U = 0 I U we have the isomorphism HomO I U(0 I U, 0 I U) Z. r(U, 0)

of 5. 3; hence Vx E X the stalk morphism fx is an isomorphism, and so by 3. 4. 10 f is an isomorphism. // Exercise (for Commutative Algebraists and Number Theorists). Show that if A is an integral ring, Pic(Spec A) is the 5. 7

105

group of classes of fractional ideals of A; in particular, if A is the ring of integers of a number field, Pic(Spec A) = the ideal class group of A and so is finite.

Let U, V be two copies of C with coordinate functions u : U - C and v : V - C. Then U1 = { x E U ; u(x) * 0 } and V1 = {x E V; v(x) 0 1 are isomorphic by the recipe uv = 1; that is, 5. 8

Example.

by the map 0 : U1

Vl

x H y where v(y) = fi1x)

Glueing U, V together by 0 we obtain a space X with open subsets U', V' and maps U

U, U V' = X

C

U

V'

u

(z -Z

such that U' n V' C

C*

v ". C 1

commutes. This determines the topology on X. In fact X is P (C) (the Riemann sphere) with homogeneous coordinates

(u, 1) = (u, uv) = (1, v) (at any point one of these expressions is well-defined). The map 0 is an isomorphism for any reasonable structure on U, V. Let us drop the distinction between U and U', and V and V', and for the sake of definiteness let OU, 0V be the sheaves of C-valued analytic functions on U, V (so that (U, OU) = (C, Cam) = (V, OV)). Then

these define a unique sheaf of rings OX on X giving X the structure of C-analytic manifold. Indeed, for W open in X r(w, oX>

(f, g) E r(w n U, ®U) x r(w n V, 0V); Vx E U n V f(u(x)) = g(v(x)) }.

By Liouville's theorem, r(X, OX) = C consists of just the constant 106

functions.

For n c Z we can define an invertible OX-Module 0X(n) on X as follows. For W open in X, let

r(w, oX(n)) = {(f, g) E r(w n u, o) x r(w n v, ov Vx E U n V f(u(x)) = u(x)ng(v(x)) }.

In other words, 0X(n) is obtained by glueing together OX I U and OX IV = OV by means of the isomorphism over U n V

0XIVnU

0XIUnV IIS

0

C

U

multiply

IC*

by zn

IIS

0 C

IC

where C*= {z EC; Z#01. Then we have

r(X, OX(n)) =

for n< 0 10 (abelian group of homogeneous poly- for n>0 nomials of degree n in x, y)

(where notionally u=x/y and v=y/x), and it is easy to check that OX(m) = 0X(n + m)

0X(n) ®0 X

Hom(OX(n), 0X)

= OX(-n)

so that OX(n) = (0X(1))n in Pic X. Taking for example n = m = 1 we r(X, 0X(1)) IM OX(2)) (they are see that r(X, 0X(1)) ®r(X, 0 X)

C-vectorspaces of dimensions 2. 2 = 4 and 3); this provides the example advertised in 4. 9. We have shown that for X = P1 (C), PicX has an infinite cyclic subgroup generated by 0X(1); it can be shown that this is all of Pic X.

Exercises on Chapter 4

Let X be a topological space and R be a ring. The category of sheaves of R-modules (compare 3. Ex. 8) on X is (isomorphic to) the 1.

107

category of 0-Modules, where 0 is the constant sheaf R. (a) If L, M are two sheaves of R-modules on X, then L ®R M denotes the sheafification of the presheaf u )- r(U, L) ®RI'(U, m) (U open in X), in conformity with 4. 9. If Y is another topological space, and N a sheaf of R-modules (b) on Y, show that there is a unique sheaf L ®R N (called the total tensor product) on the product space X X Y with the properties (i)

V (x, Y) E X X Y

(ii)

if U is open in X and V is open in Y, and S E r(u, L)

(L ®R N)

(X, y) - LX ®R NY

and t E I'(V, N), then (x, y) I- sX ® ty E LX ®R NY

=- (L ®R N) (X, Y)

is a section of L ®R N over the open set U X V. (c) IZ

(d)

Show that L ®R N = (7i*L) ®R (ii N), where v1 : X X Y - X and : X X Y - Y are the projections. Show that L ®R M = d*(L ®R M) where d : X - X X X is the

diagonal map.

Discover the universal property satisfied by the total tensor product construction. (Compare [G] 2. 10.) (e)

Show that we can remove the condition in 2. 6 that the basis `a be closed under finite intersections, provided we replace the condition on the 'presheaf F defined only on the basis I1' by the following condition:

2.

whenever U = Ux EAUx with U E U and V A E A Ux E 'It,

for any set (or abelian group or ring as appropriate) T, the map Hom(T, F(U)) - UREA Hom(T, F(Ux))

f

U

(P US

° f) XEA

is an injection, with image { (fx)xEA; V A, µ E A, VV E `U such that V S UX n Uµ U

PV °fPV fµ 108

Show that, even in this more general setting, the alternative construction indicated in 2. 6 (setting r(V, G) = lim F(U, G)) works. (Compare [EGAI] 0, 3. 2 and 3. Ex. 8. )

Draw a picture of Spec Z[t]. (Compare Mumford, Introduction to Algebraic Geometry, p. 141.) 3.

Show that for any ring R, Spec R is compact.

4.

Let R be a ring and f E R. Show that the natural ring morphism R +Rf = R[f ] = S 1R with S = {fn n E N } induces an isomorphism of 5.

ringed spaces between (Spec Rf' 0Spec R f ) and (D(f), $*0Spec R

Spec RID(f))

where 0 : D(f) '- Spec R is the inclusion of the open subspace D(f).

Let R be a ring and a an ideal of R. Show that the natural

6.

ring morphism R - R/ a induces a morphism of ringed spaces Spec(R/a) -+ Spec R which is a homeomorphism of Spec(R/a) with the closed subspace V(a) of Spec R; but show by example that it need not induce an isomorphism of ringed spaces between Spec(R/a) and (V(a), ®Spec R I V(a))

For k E N, let Sk = Ix E Rk+1;

7.

II

xII=1}.

Letting 0 be the

sheaf of continuous (or differentiable, or analytic) R-valued functions on define a sheaf of ideals I in 0 I Sk = 0*0 (where 0 : Sk . Rk+1) with stalks Rk+1,

I = { f E (0 y

15k) y ; f

is the germ of a function f' such that

f'ISk = 0}.

and show that putting 0' to be the quotient of 0 1 Sk by I, the ringed space (Sk, 0') is a k-dimensional topological (or differentiable, or analytic) manifold, the k-sphere. (Projective n-space) (Topological version) Let K = R or C, and let V be an (n+l)-dimensional vectorspace over K. Let P = Pk be the set of hyper8.

(a)

109

planes (= n-dimensional subspaces) in V. Show that P is a quotient of a sphere Sk of dimension k = n (if K = R) or 2n + 1 (if K = C) by an equivalence relation whose equivalence classes have two elements if K = R, or biject with S1 if X = C. Giving P the quotient topology show that P has a natural structure of analytic (or differentiable, or topological) manifold.

[Hint: the map Sk - P is a covering map; cover Sk by hemispheres and use these to give P a structure sheaf. ] (b) (Algebraic version) Let K be any field. Use the ring isomorphisms

x

x x K[x.' X.' 1

1

K[Xx

nX 1

x

a, Xl, J

J

... ,

J

x

n][% ] for Osi, jSn J

1

to glue together the n + 1 copies Ui = Spec R. of AK, where

Ri = K[-o, ... , Xn] = K[Y1, ... Yn] , 1

1

into a scheme PK. Show that if K is algebraically closed, the closed points of PK biject with the (n+l)-tuples (o' ... , n) a Kn+1 considered modulo the relation (a0,

".9

an) - (Aaa,

... ,

Aan) for A e K\ { 0 }.

(Homogeneous coordinates.) Show that r(PK, 0) = K. Working by analogy with 5. 8, construct on PK an invertible sheaf 0(1) with the property that r(PnK, 0(1)) = the set of homogeneous linear forms in X0, ... , Xri

If you are ambitious, show that the points of PK biject naturally with the homogeneous prime ideals of the graded ring K[X0, ... , Xn] which do not contain the ideal (X0, ... , Xn). Deduce an analogous construction of a scheme Proj(S) for any graded ring S. (Reference [EGA II], 2. 3. )

110

9.

(Algebraic curves)

(Topological version) you are ambitious) given by (a)

Show that the subset of R2 (or of C2 if

C = { (x, y); y2 = (x + 1)x(x - 1) ) is a manifold (topological, differentiable or analytic as you wish), whereas that given by the equation y2 = x2 (x + 1) is not. (Draw a picture. ) (b) (Algebraic version) Let K be any field and f (x, y) a K[x, y] = R.

By Q6 the morphism R - R/Rf identifies the space Spec(R/Rf) with a closed subspace C of Spec R = AK = X say. Find the sheaf of ideals I in OX I C such that, letting 0' be the quotient of OX I C by I, the morphism

Spec(R/Rf) -' (C, 0') is an isomorphism of ringed spaces (and so of affine schemes). Q7.)

(Compare

Putting f(x, y) = y2 - x3 + x or y2 - x3 - x2 we get the algebraic analogues of the curves of part (a). What is different about the two cases? Other curves for your amusement: y2

=

x2(x - 1)

y=x. 2

3

Show that the construction of the prime spectrum of a ring can be generalised as follows. Let X be a scheme, with structure sheaf 0. 10.

Let A be a sheaf of 0-algebras (an 0-Algebra), that is an 0-Module such that for each U open in X, the r(U, 0)-module T(U, A) has a structure of r(U, 0)-algebra, in a way compatible with restriction maps (aliter, there is a given multiplication A 00 A - A and a section 1 E I'(X, A) satisfying the usual laws for an algebra). Suppose that A is quasi-coherent as an 0-Module. For each open affine U = Spec R of X, A I U ib an 0 1 U- Algebra and so we get a scheme morphism

Spec r(U, A) - Spec R = U; 111

for different U, these fit together into a scheme morphism

Spec A -X

(*)

where Spec A is obtained by glueing the different Spec I'(U, A) by means of the isomorphisms provided by the structure of A. (a) If X = Spec R is affine, then so is Spec A; indeed it is the spectrum of an R-algebra. For this reason, morphisms of the type (*) are called affine morphisms. (b) Show that a scheme morphism f : Y -' X is affine iff X can be covered by affine open sets U such that each f-1(U) is affine. (Hint:

put A= f*(0 ); see [EGA I] §9. 1. ) (c) For example, for any X let A = 0[t] be defined by having sections

r(U, A) = r(u, 0)[t]

for U open in X

(the polynomial ring). Then Sec A = Spec 0[t] is called the affine line AX over X. For X = Spec R, AX = Spec R[t]. Show that for any scheme X, AX is the pullback in the category of schemes of the diagram Al = Spec Z[t] Z

XSpec Z (that is, the product X X AZ1 , since Spec Z is final in the category of schemes). (d) Generalising (c), show that any 0-Module E gives rise to an 0-Algebra SE constructed locally out of symmetric algebras, and hence to a scheme morphism V(E) = Spec SE - X.

(The example of (c) comes by putting E = 0.) If E is locally free (of rank n) show that V(E) is a 'vector bundle' over X, in the sense that it is locally of the form U x An (where An = Spec Z[tl, ... , tn]). (Compare [EGA I] §9. 4. )

112

11.

Let (X, 0) be a ringed space and M an 0-Module. M is said to

be of finite type iff each x E X has an open neighbourhood U such that for some n E N (*)

f

there is an exact sequence of (0 I U)-Modules

(0IU)n''(MIU)''0 )i

(where (0 I U)n = 1 (0 I U)). Show that (*) is equivalent to

there are sections s1, s2, ..., sn E I'(U, M) such that for each Y E U the Oy-module My is generated by the germs (s1)y,

..., (sn)y.

Suppose that M is of finite type, that U is open in X with x E U, and that t1, t2, .... tm E r(U, M) are such that the germs (t1)x' ... , (tm)x generate the 0X module MX Prove that there is an open set V with x E V S U and such that for all y c V, the germs (t1)y, ..., (tm)y generate the Oy-module My. [Hint: Express the (si)x of (**) as an 0X linear combination of the (t.)X ] Now suppose that k is a field and that (X, 0) is in fact a ringed space over k with the property that for all x E X

0x= k. Let M be an 0-Module of finite type. Deduce that for each x E X, Mx is a k-vectorspace of finite dimension. Deduce also that the function

e:X-N: is upper semi-continuous (that is, for each n E N, {x E X; e(x) j n } is closed in X). This function provides X with a 'stratification' into locally closed subspaces {x E X; e(x) = n I over which the stalks of M have constant dimension. What further properties of this stratification can you see? If in addition X is compact? Use this to give a simple example of an 0-Module M which is not of finite type.

113

Let (X, 0) be a ringed space and M an 0-Module. M is said

12.

to be of finite presentation iff each x E X has an open neighbourhood U such that for some m, n E N there is an exact sequence of (0 I U)Modules

(0IU)m - (0 IU)n . (MIU) .. 0

(so that M is of finite type, and so are suitably chosen 'sheaves of relations' Ker((0 I U)n - (M I U))).

Show that for any 0-Modules F, G and for each point x E X there is a natural morphism (Home (F, G))x .r. Home (Fx, Gx) x

(*)

which is in general neither injective nor surjective. Prove however that if F is of finite type, then the morphism (*) is always injective, while if F is of finite presentation it is bijective.

Let (X, 0) be a ringed space, and M, N, P be 0-Modules.

13.

Show that there is a natural bijection Hom0(M, Home(N, P)) = Hom0(M (&0N, P).

[Compare 3. Ex. 4 and 4. 10; this adjunction can be interpreted as saying that Homo is an internal hom-functor in 0-Mod; see [Macl] VII. 7, Schubert, Categories, 17. 8. 1. ] Use this adjunction to simplify the proof of 5. 6.

If M is a Module over a ringed space (X, 0) over a ring R,

14.

we can use the notation End0(M) for the 0-Module Home(M, M). Show that End0(M) has a natural structure of sheaf of R-algebras (multiplication being given by 'composition of endomorphisms'). (Compare 5. 2. ) Show that if N is a sheaf of R-modules, then to give N a structure of 0-Module is the same as to give a morphism 0 - EndR(N)

of sheaves of R-algebras (here R also denotes the constant sheaf). Note especially the case R = Z, when N starts just as a sheaf of abelian groups. 114

5 Cohomology

We now wish to measure the lack of exactness of the global section

functor I'm, -); we have seen that it is left exact, but need not take a sheaf epimorphism into a surjective map of sections. We first consider the problem in the general setting of homological algebra: we wish to mend the lack of right exactness of a left exact functor between abelian categories. This leads us to define injective objects, and to show that they can be used to define the right derived functors of our functor, which fit into a long exact sequence extending the left exact sequence it produces. The right derived functors have a suitable universal property, which is used to obtain identities concerning composite functors. We next apply this procedure to the case of sheaves. Having verified that there are enough injective sheaves, we deduce the existence of cohomology functors fitting into a long exact sequence. The general method also yields the higher direct images of a morphism, which generalise the cohomology groups, but may be expressed in terms of them. We investigate the processes of changing structure sheaves and base rings, and summarise an alternative approach to this universal cohomology theory, using flasque sheaves. Finally we give an alternative and more computable version of cohomology, the Cech theory, which agrees with the universal theory in some useful cases. In particular we obtain a reinterpretation of the picard group of a ringed space as a cohomology group. 5.1

Injective objects

1. 1 Let K be an abelian category (such as Abgp, R-mod, or 0-Mod). For E E ObK there is a contravariant functor

Hom(-, E) : K - Abgp : F I- Hom(F, E).

115

The functor Hom(-, E) is always right exact; that is, if 0 - A - B - C - 0 is exact in K, then the derived sequence Proposition.

0 - Hom(C, E) - Hom(B, E) - Hom(A, E)

is exact in Abgp. Hom(-, E) is exact iff E has the equivalent properties: (i) given a monomorphism A - B and a morphism A - E, there is an extension A - B of f to a morphism B - E making the E

triangle commute (but not necessarily unique);

every short exact sequence 0 - E - A - B - 0 splits (that is, is isomorphic to (ii)

H1

77

2

O-'E -, E(DB - B40). Proof. Easy verification, using the universal properties of epimorphisms and cokernels, in particular. Note that a sequence

B--C-+0 splits iff ah : B - A such that h o g = idA (see [Mit] I. 19, or Freyd, Abelian Categories 2. 68). //

1.2 Definition.

If K is an abelian category and E E ObK is such that Hom(-, E) is exact, we say that E is an injective object of K. We say that K has enough injectives iff VA E ObK there is an injective E and a monomorphism A - E; in other words, iff every object of K can be embedded in an injective. We shall see in §5. 2 that this is a desirable property of K, and in §5. 3 that it holds for the category of Modules over any ringed space. We show first that it always holds for the category R-mod of modules over a ring R. (The result, and indeed the proof, does not require R to be commutative; we should then talk of the category of left (or right) R- modules instead). 1. 3

injective. 116

Lemma.

(i) A product IIiEIEi is injective

Vi E. is

An abelian group G is injective (in Abgp) divisible (that is Vg E G `o'n E Z* 21h E G nh = g). (ii)

Proof.

(i) Hom(F, HiEIEi) ~

1iEIHom(F,

G is

Ei), and the RHS is

exact in F iff each factor Hom(-, E i) is exact. (ii)

:

Given g E G, n E Z*, define: n Z - G : n I-' g. The exten-

sion Z - G given by 1. 1(i) sends 1 Z. to a suitable h. ,t=: Given F 0 G with f monomorphic, we aim to apply Zorn's f Lemma to the set of extensions of 0 to subgroups H of H containing F. Certainly any chain of such extensions (under S) is dominated by another, so there is a maximal extension, to F c- F' C- H say. If h E H \F', consider Zh n F'. If this = {0 } , $\ I we may define i/i(h) arbitrarily, and obtain an extension G of to F' + Zh = F' ®Zh. If it is not 101, then it is of the form Znh for some 0*nEZ; i. e. nhEF'; so t(nh)=ni/i(h)EG, and since G is divisible we may pick g E G such that ng = iy(nh) and define qi(h) = g to extend Vi to F' + Zh. In either case this contradicts the maximality of (F', iy) unless F' = H. Hence this is the case, and by 1. 1(i) G is injective. // i1..'

For reasons which will become clear, we shall be interested in the abelian group G = Q/Z; by 1. 3(ii) this is injective in Abgp. Now let R be a ring, and for an abelian group G (such as Q/Z) let HomZ(R, G) be the R-module of abelian group morphisms R - G, with the action of r given by 1. 4

(rf)(s) = f(rs) for r, s ER and f E HomZ(R, G). Lemma. (i) For F E R-mod, G E Abgp there is a natural isomorphism of abelian groups 1. 5

HomR(F, HomZ(R, G)) = HomZ(F, G). (ii)

If G is an injective abelian group, then HomZ(R, G) is an

injective R-module.

117

Suppose E is an injective R-module, such that for any R-module F we have: (iii)

Yf E F f * 0 = ao E HomR(F, G) such that 4>(f) * 0.

(*15)

Then any R-module can be embedded in an injective (namely a product of

copies of E). Proof.

It is 0'- (f - 4>(f)(1)) for f e E, with inverse * - (V- (r - zp(rf))) for f E F, r ER. By (i), for E = HomZ(R, G), the functor HomR(-, E) is (ii) naturally isomorphic to the functor HomZ(-, G), which is exact since G (i)

is injective in Abgp. (iii) The embedding we wish to use is (D

: F '' rI OEHomR(F, E)E :f "*

($(f))OEHom(F, E)

The RHS is injective by 1. 3(i), and the R-module morphism 4) is monomorphic iff its kernel is zero i. e. iff Vf E F f * 0 f >E Ker 4), which is the given condition. // Remark. Those familiar with Commutative Algebra will recognise (i) as a special case of HomR(-, HomS(-, -))ZHomS(-OR , -)' 1. 6

1. 7

Theorem.

For a ring R, the category R-mod has enough

injectives.

Proof. We shall apply 1. 5(iii) to the R-module E = HomZ(R, Q/Z). By 1. 3(ii) and 1. 5(ii) E is injective. To check (*15), let 0* f E F E Ob(R-mod). We first produce a Z-module morphism 4i : F - Q/Z such that i(i(f) * 0. We can find a non-zero Z- module morphism Zf -+ Q/'Z, for if f has infinite order then f )- i + Z will work, while if f has order n c N* then f F- + Z will work; by the injectivity

of Q/Z this extends to a Z- module morphism 4' : F - Q/Z such that n 4'(f) * 0.

But now the formula of 1. 5(i) produces a corresponding 0 E HomR(F, E) such that 4>(f) * 0. Hence 1. 5(iii) applies and we are done. //

118

5. 2

Derived functors

The advantage of abelian categories with enough injectives is that we can deal well with left exact functors between them. We shall now consider some results which belong to a course on Homological Algebra. The results are obtained in an abelian category, since we wish to apply them to the category of sheaves and the section

functor, but they may be easier to prove in a category like R-mod, where objects have elements; in general such a proof will suggest the proof appropriate to a more general abelian category.

Let K be an abelian category. A right-co complex L' (or just complex) in K is an N-indexed family of objects Ln+l (n E N) such ILn ; n E N J together with morphisms do : Ln that Vn E N do+l o do = 0. L' is often written L° -0. L1 - LZ - ... . Definitions.

2. 1

The cohomology of the complex L' consists of the objects

Hn(L') = Ker dn/Im do-1 (for n E N; convene L-1 = 0) of K; we sometimes write H*(L') = {Hn(L' ); n E N}. The complex L' is exact (or ac clic iff Yn E N* (that is, n ? 1) Hn(L') = 0; that is, iff

-...

-,L1

is an exact sequence. If A E ObK, a complex over A is a complex L'

with H°(L') = A; if this L' is exact, it is called a resolution of A, and if also Vn E N Ln is injective, then L' is an injective resolution of A. A morphism of complexes g : L' - M' is given by K-morphisms gn : Ln -+ Mn for each n E N such that do E N the square Ln gn

do

-

Ln+l

I

I

gn+1

0 Mn+1

Mn

d'n

commutes (that is, d'g = gd). Two morphisms g, h : L' - M' are (chain) homotopic, written g h, iff there are K-morphisms 119

kn :L n+1 -, Mn for n e N such that do e N

d'

kn-l+kndn gn hn

(mnemonic:

Mn-1Mn Proposition. A morphism of complexes g : L' - M' induces morphisms of cohomology H(g) = g* : H*(L') - H*(M') in a functorial way. Homotopic maps induce the same cohomology morphism. 2. 2

Proof. This is easy to prove for K a module category, and such a proof can then be used to construct a proof in any abelian category. In the diagram Im do-1

- Ker do

h'

h

Mn-1- Im d'

io Ker do

Ln

I

gn

Mn

the morphisms h, h' can be constructed in turn, using the universal properties of Ker and Im; hence h induces g* : Hn(L') - Hn(M. ). If g ~ g' : L' - M' , then in the diagram Ker d n a

1

Ln

Mn-1

n

-

Ln+l

- Mn Mn/Im do-1

we have j(g' -g)i = j(kd + dk)i = jkdi + jdki = 0 (omitting subscripts for ease of writing), whence the result. // 2. 3

Proposition.

If L' (resp. M') is an injective resolution of

A (resp. B), then any morphism f : A - B can be lifted to a morphism g : L' - M' (such that f = g* : H°(L') H°(M') up to isomorphism),

120

and any two such liftings are homotopic.

Proof. We construct gn by induction on n. If this has been done up to gn-1 (n > 0) we have a commutative diagram Ln-1i Mn-1

in which i and j are injective by the exactness of L' and M' and g' exists since dn-lgn-ldn-2 do-ld' 2gn-2 = 0. Hence since Mn is an

injective object, jg' can be extended to a morphism gn : Ln - Mn by 1. 1. The outside square commutes, so g is a morphism of complexes. To start the induction we use the same argument on the diagram

0H°(L')

La

A

fI

0

H°(M' )

0 MO

If also h = {hn; n e N) makes all the diagrams commute, we can construct a homotopy h g by induction, using the diagram Ln/Im do-i Ln

Ln-1 k kn- 2 Mn- 2

_ Mn-1

i,_

Ln+1

ti

n-

gn'i hn

_ Mn -A'

_

_

k

_ Mn+1

If we assume that dkn-2 + kn-1d = gn-1 - hn-1' then the map t = gn - hn - do-1kn-1 is such that

td = gd - hd - dkd = gd - hd - d(g - h - dk) = 0;

hence t factors through Ln/Im do-1 as t'. Since Mn is injective,

121

t' extends to kn such that k d = gn - hn - dkn-1 n

as required. // 2.4 Remark. A closer examination shows that we have used only the exactness of L' and the injectivity of M' , so that a more general proposition holds.

If K has enough injectives (1. 2), then every object A of K has an injective resolution, and any two injective resolutions L' and M' of A are homotopy equivalent: that is, there are morphisms L' 4- M' such that g o h idM. and h o g = idL. 2. 5

Corollary.

Proof.

By 1. 2 A embeds in an injective object L 0, with co-

kernel A° say:

0- A-'L0

\A o

Now embed A° in an injective L1 with cokernel diagram

A2,

say, to get a

in which the top row is exact; proceeding by induction, we construct an injective resolution L' of A. If M' is another, then idA lifts by 2. 3 M' and h : M' - L', and both hog and id1 . to morphisms g : are lifts of idA to morphisms L' -+ L' ; hence by 2. 3 h o g ~ idL. Similarly g o h idM.. //

Let F : K - K' be a left exact functor, where K, K' are abelian categories, and K has enough injectives. Given A E ObK, let L' be an injective resolution of A (2. 5), and let 2.6 Construction.

122

(RnF)(A) = Hn(FK' )

be the cohomology of the complex { F(Kn), F(dn); n e N). Then the

RnF : K - K' for n E N are the right derived functors of F. To define RnF on a map f : A - B in K, lift f to a morphism of complexes g : L' - M' , where V, M' are injective resolutions of A and B respectively (2. 3), and obtain a map Rnf : RnF(A) = Hn(FL

F(g)*

-.

Hn(FM') = R"F(B)

by 2. 2.

Proposition. (a) The RnF (n ? 0) are well defined; that is, they are independent of the choices of injective resolutions and of maps between them. 2. 7

(b)

(c)

R°F is naturally isomorphic to F. If A is injective, then for n > 0 RnFA) = 0.

Proof. (a) Any two liftings of f : A - B to morphisms of injective resolutions g, g' : L' - M' are homotopic by 2. 3; hence so are F(g) and F(g') (use the F(kn)), which thus give the same map

RnFA - RnFB by 2. 2. The functoriality of RnF (using fixed resolutions) follows.

Apply this to A = B, f = idA, L' and M' two injective resolutions of A; we deduce that idA lifts to unique isomorphisms Hn(FL' ) ' Hn(FM') (they are isomorphisms by functoriality, since f is). Hence RnFA is well-defined up to canonical isomorphism, as required. (b) Since F is left exact, its effect on an injective resolution 0 - H°(L') - L° - L1 -' ... of A = H°(L') is to produce an exact se-

quence 0 - FA - FL° - FL1. Hence H°(FL-)=Ker(FL°-.FL 1)=FA. (c)

If A is injective, it has the injective resolution id

0 - A -- A - 0 - 0 II

L°

II

L1

II

L2 123

On applying F to L' we get the complex FA - 0 - 0 -... which has FA

RnFA = Hn(FL')

n= 0 n > 0. //

Remark. We must be functorial, and ensure in 2. 7(a) that the RnFA are well-defined up to canonical isomorphism; hence for example automorphisms of A will induce automorphisms of RnFA so as to give a group homomorphism 2. 8

Aut(A) -' Aut(RnFA).

Let K, K' be abelian categories and let a E N U { -) (where by convention Vn e N n < 00). A a-functor T' : K -' K' is a sequence of functors { Tn : K -' K'; 0:5n< a) together with an assignment to each short exact sequence 2. 9

Definition.

(*S)

in K of a collection of morphisms a = a,l,

:

Tn-1C -+TnA (0 < n < a)

such that (i)

if

0-A-B-C-0

if

0 - A' - B' -

commutes in K and has exact

Ig - 0

rows, then the corresponding diagrams Tn-1C - TnA commute Tn-1g l Tnf T n1 C'

;

TnA'

(in other words, a is 'natural'); and (ii)

whenever (*S) is exact, the associated long sequence 0-'TEA-'TSB-FT0

(for n < a) is a complex (the composite of two successive morphisms is zero). The a-functor is called exact iff for any sequence (*S) the corresponding sequence (*L) is always exact. A morphism of a-functors (with the same a) S' - T' is given by

124

a sequence of natural transformations { Sn - Tn; 0 s n < a I such that for any short exact sequence (*S) the diagrams

Sn-C T

n-1

a

S

SnA

In C air-T A

commute; hence this gives a morphism of complexes between the long sequences (*L). If F : K -+ K' is a functor, a a-functor over F is a a-functor

{ Tn, aT } together with a natural isomorphism F - T 0; hence if T is

exact, F is left exact. 2.10 Theorem.

If F : K - K' is a left exact functor between

abelian categories, where K has enough injectives, then the sequence of functors R' F = {RnF; n e N) forms an exact a-functor over F. Proof. We have to show that there is a natural assignment to each short exact sequence

in K of a long exact sequence 'FA-'... -'Rn- I We first need two Lemmas. Lemma. If F K -+ K' is a left exact functor between abelian categories and 0 -1- A B - C - 0 is a split exact sequence in K, then 2. 11

0

is also (split) exact. (Hence F(A q3 C) = FA e FC. ) Proof.

f

0 - A -+ B 9 C - 0 is split exact iff A - B - C is exact

and a a diagram A

B

- C such that f' o f = idA and g o g' = idC.

125

Then 0 -+ FA -' FB - FC - 0 also has these properties, and so is split exact too. //

2.12 Lemma.

Suppose L' - M' -+ N' are morphisms of complexes in an abelian category K such that V n e N 0

is exact (we call this a short exact sequence of complexes). Then there is a collection of morphisms a : Hn(N') - Hn+1(L') (n E N) such that the sequence (*LH)

0-4H

a

0(L'

)-'H0(N' )-'Hl(L' )-+...-+Hn(N'

a

(L')-. ..

is exact. Moreover a is natural in the sense that if 0 - L' - M' -'N' -'0

0 - L'1 - M' 1-+N'1 -0 is a commutative diagram of morphisms of complexes, with exact rows, then the induced morphism between the long exact sequences (*LH) is a morphism of complexes; that is Vn E N

Hn(N' aHn 1(L' ) i Hn(N-1)

commutes (where

a

1

a1

Hn+1(Li)

is the morphism constructed from the lower exact

row).

Proof. This result should be familiar to those who have learnt some Algebraic Topology, perhaps only in the category K = Abgp or R-mod for some ring R. The proof is rather tedious; it may be recon-

structed for an arbitrary K from the proof for K = Abgp in Spanier, Algebraic Topology, 4. 5. 4, using the techniques of 2. 1-2. 5 and if necessary, [Macl] VIII. 4. // Proof of 2. 10.

Let L' (respectively N') be an injective reso-

lution of A (respectively C). We first construct a complex of injective objects M' over B and a short exact sequence of complexes

126

0 - L' - M' - N' -+ 0. If this has been done up to the situation in the diagram Mn-1 -

Ln-1

0

Nn-1

0

0 - Ln - Mn - Nn - 0 for n e N (where we make the convention L-1 = A, M-1 = B, N 1 =C) then we can construct Mn-1

0 --o Ln-1

IBM Nn

g

I

0

m

Nn

0

1 1 o. cok g - aio cokh

a

0 - Ln+1

am

h

Mn

cok f

1

Ln+1 ®Nn+1 1L

N' 1-*-0 7f

1

2

(f', h' arising from f, h by the universal property of cokernels), and it is (fairly) easy to check that the sequence of cokernels is short exact (for instance, using 'members' as in [Macl] VIII §4). We let Mn+1 = Ln+1 ® which is injective by 1. 3(i), and define g' : cok g -+ Mn+1 by letting the diagram Nn+l,

cok g gi

L n+I L

n+1

1

,n+l \ h'o/ `

N

n+l

commute, where f" is an extension of f' to cok g, which exists by 1. 1(i) since Ln+l is injective and a is monomorphic. It is easy to see that M' is a complex and that 0 - I' - M' -, N' - 0 is a short exact sequence of complexes. Now since L' and N' are exact, the long exact sequence of cohomology of 2. 10 reduces to a collection of exact pieces

127

0 - Hn(M') - 0 for n E N* which shows that V n ? 1 Hn(M') = 0, so that M' is exact and thus is an injective resolution of B; hence we may use M' to compute RnF(B). Since each of the short exact sequences

4n_0

(n EN)

is split exact (as it must be by 1. 1(ii)), by Lemma 2. 11 the sequence

0-'FL' -'FM' -FN' -'0 of complexes is exact and so yields by Lemma 2. 12 a long exact sequence

... -+ Hn(FL-) -+ Hn(FM') -+ Hn(FN') - Hn+1(FL') - .. . which is the required sequence by 2. 6 and 2. 7. The naturality of follows from the corresponding property in 2. 12. //

a

2. 13 Theorem. Let F : K - K' be a left exact functor between abelian categories, where K has enough injectives. Then the a-functor R' F has the following universal property. Suppose that { Gn, 0-5n< al is a a-functor over F (where a E N U { co } ); then there is a unique morphism of a-functors

n < a } - { Gn; 0!5n< al

{RnF; 0

such that the triangle

R ° F -- D G 0

\I-

/I,

F

commutes. If furthermore G' is exact and effaceable, that is for any

injective object E of K we have

GnE = 0 for 0< n < a, then the morphism R' F - G' is an isomorphism of a-functors over F.

128

2.13; Remark. Hence the derived functor R' F is characterised up to natural isomorphism as the exact effaceable a-functor over F. Conversely any exact effaceable 2-functor T' (sometimes called a cohomological a-functor) is (up to natural isomorphism) the derived functor of its T0, and is the universal 2-functor over T 0 (in the sense of the theorem). Proof of 2. 13. For an object A of K we construct the morphisms RnFA - GnA by induction on n; for n = 0 we compose the isomorphisms which show that R' F and G' are each a-functors over F. Embed A in an injective object E of K to obtain a short exact

sequence

say. 2. 7(c), the induction hypothesis and the long sequences of this exact sequence give us a diagram

0 -Rn1

f

. RnFA -- - 0

FB h

g

- GnA

with an exact top row; hence f is anisomorphism, and so there is a unique map g making the diagram commute. Straightforward arguments show that this is independent of the choice of E and that it defines a morphism of 8-functors. If G' is exact and effaceable, the same diagram shows, by induction on n, that g is an isomorphism (since then h is). // F

G

2.14 Corollary. Let K - K' K" be functors between abelian categories where K and K' each has enough injectives. Suppose that

G is left exact and F is exact and transforms injectives in K into G-acyclic (ii) objects; that is, whenever E is injective in K we have (i)

RnG(FE) = 0 for n > 0.

129

Then there is a natural isomorphism of 2-functors R' (G o F) = (R' G) o F.

Proof. Apply 2. 13: by the conditions, (R' G) o F is an exact effaceable a-functor over G o F and so is its derived functor. // F G Let K -+ K' -+ K" be functors between abelian categories where K and K' each has enough injectives. Suppose that

Corollary.

2. 15 (i)

and

(ii)

F is left exact G is exact.

Then there is a natural isomorphism of a-functors 1:r°(G o F) = G o R' F.

Proof.

Apply 2. 13: G o R' F is an exact effaceable a-functor

over G o F. // Example. We already have enough machinery to give the definition of one cohomology theory. Let G be a group and ZG its (integral) group ring. The category 2. 16

(ZG)-mod is the category of G-modules (abelian groups with an action of G), and has enough injectives by 1. 7. The functor

{aeA; VgEG ga=a} is left exact, and its derived functors are the cohomology of G with coefficients in A: Hn(G, A) = Rn(-)G. (A)

(n c N)

(see Shatz, Profinite Groups, Arithmetic and Geometry; or Lang, Rapport sur la cohomologie des groupes). More generally, let k be any (commutative) ring and kG the group algebra of G over k (k is often a field). We have a diagram of functors

130

S'

kG-mod S

k-mod IF Z- mod

where SA = AG = { a E A; Vg E G ga = a), S'A is the same Z-module considered with its k-module structure, and F is the forgetful functor.

Then F is exact, so by 2.15

R'S=FoR'S'. Furthermore, for A E Ob(kG-mod) we can recover the k-module structure on RnS(A) by considering the endomorphisms induced by the

A - A : a'- Xa for A E k. Hence we may compute the cohomology of G with coefficients in k: Hn(G, A) = RnS'(A)

by taking a resolution of A by injective ZG-modules (that is by computing RnS(A)). 5. 3

Sheaf cohomology

3. 1 We have now constructed a general method of measuring the lack of exactness of a left exact functor; in order to apply this to the functor

r(X, -) : 0-Mod - R-mod for a ringed space (X, 0) over a ring R, we must check that 0-Mod has enough injectives.

Let (X, 0) be a ringed space over the ring R. Let (Mx)xEX be a family such that for each x E X, Mx is an 0X module. Then there is an 0-Module M such that whenever N is 3.2

Lemma.

another 0-Module,

Hom(N, M) - H Homo (Nx, Mx) : f F' (fx)xEX xEX x is a bijection. 131

Proof.

We can define M by its sections:

r(u, m) = HxEUMx for U open in X; this is easily seen to define a sheaf. Alternatively, for X E X let Mx be the 0-Module with stalks (Mx) = { 0

y#x

Mx y=x.

y

Then M = Hx-XMx is the product in 0-Mod of the Mx (4.4.8). // 3. 3

Lemma.

Under the hypotheses of 3. 2, if for each x E X,

Mx is an injective O-module, then M is an injective object in 0-Mod. Proof.

Easy from 3. 2 and 1. 1(i). //

(X, 0) is a ringed space over a ring R, then the category 0-Mod has enough injectives. 3. 4

Theorem.

If

Proof. Given an 0-Module A, for each x E X we can find an embedding (= monomorphism) Ax - Ex with Ex an injective 0X module (1. 7). Then 3. 2 provides an embedding A - E with E an injective 0-Module by 3. 3. // 3. 5 Theorem. If (X, 0) is a ringed space over a ring R, there is a universal 8-functor (see 2. 132 )

{ Hn(X, -) : 0-Mod - R-mod; n E N}

over the functor r(X, -) : 0-Mod - R-mod. Hence an exact sequence of 0-Modules 0 - A - B - C - 0 gives rise in a natural way to a long exact sequence:

0-.r(X, A)-'r(X, B)-r(X, C)

A) - Hn(X,

Hn(X, C) - .. .

(Hn(X, A) is called the cohomology of X with coefficients in A).

132

Proof.

Set Hn(X, -) = Rn(r(X, -)) and apply 2. 10 and 2. 13. //

Corollary. Suppose that A is an 0-Module with H1(X, A)=0. Then for any exact sequence of 0-Modules 0 -+ A - B -+ C - 0 the induced map r(X, B) - r(x, c) is surjective. // 3. 6

3. 7

Theorem.

Let 4> : (X, OX) - (Y, 01,) be a morphism of

ringed spaces over a ring R. Then there is a universal 0-functor {Rn4;* : OX Mod -+ 0Y Mod I

over the left exact functor 4>*. Hence an exact sequence

0-A-B-C-0 of OX-Modules gives rise in a natural way to an exact sequence of OY Modules:

0-4i*A-+4i*B-+(D*C- Rl4) Proof.

is left exact by 4. 4. 13 and 3. 7. 6. By 3. 4 we may apply 2. 10 and 2. 13. // 4i*

Let P be a topological space with just one point. Then a sheaf over P is given by just one set of sections (since for a sheaf 3. 8

Remark.

F of abelian groups r(F, 0) _ {0 }) which is also the stalk at p e P. Hence we can make P into a ringed space over R, the structure sheaf OP having stalk R. Then there is an equivalence of categories OP-Mod y R-mod

and as in 4. 7. 5A, if (X, 0) is a ringed space over R and 4; : X - P is the (only) morphism, the diagram 4.*

0-M

0-Mod

r(X,

/"rp, -)

R-mod

133

commutes. Thus the functors Rnqt * and Hn(X, -) are the derived functors of essentially the same functor, and by the uniqueness (2.13 and 2. 14) we have a natural isomorphism for A e 0-Mod

r(P,

Hn(X, A).

In fact, as the next result shows, we can describe Rn4)* for general

' in terms of cohomology too.

If + : (X, OX) - (Y, OY) is a morphism of ringed spaces over a ring R, with underlying continuous map 0 : X - Y, and A is an 0X Module, then Rn+*A is (up to isomorphism) the sheaf associated (by sheafification) to the presheaf 3. 9

Theorem.

V -Hn(¢-1(V), A) for V open in Y. (V is considered as a ringed space with structure sheaf 0 Hn(0-1(V), A) is a r(V, OY)-module).

I V,

so

Proof. For n e N let FnA be the sheafification of the presheaf described. Then Fn : OX Mod - OY Mod is easily seen to be a functor. We have F ° = *, and for an exact sequence 0 -+ A - B - C - 0 in OX Mod, the exact sequences

... -Hn-1($-1(V), C)_,Hn(0-1(V),

A)-'Hn($-1(V),

B)_Hn(¢-1(V), C)-. .

of 3. 5 give an exact sequence of presheaves and so by 3. 6. 9(ii) an exact sequence of sheaves

... Hn(0- 1(V), A) instead of Note that we are abusing language by writing Hn( 1(V), p*A) (where : 1(V) '- X); since p* is an exact functor (3. 7. 16) the application of 3. 5 is valid.

Hence { Fn; n e N } forms an exact a-functor over +*; also by

Lemma 3. 10 below, if E is injective in OX Mod and U is open in X, then E I U is injective in (OX I U)-Mod, so for n E N* Hn(U, E I U) = 0, so that FnE is the zero sheaf. Hence F' is also effaceable, and 2. 13

134

shows that it is isomorphic to the derived functor R' 4,* of

4,k.

//

is a ringed space and U is open in X, then for E injective in 0-Mod, E IU is also injective in (0 IU)-Mod. Lemma.

3. 10

If (X, 0)

We give two proofs, since each illustrates a useful tech-

Proof. nique. (a)

Let A be an (0 I U)-Module. Then by 3. 8. 12 if AX denotes

the extension by zero of A to X, there is a bijection (*10)

Homo (AX, E)

HomO U(A, E I U) I

which is natural in A. (For the map is given by restriction to U, and has inverse given by (_) X

Hom o I UTA , VI U) -,

Hom O (AX , (E I U)X) --b. Hom o (AX , E)

where the last map is derived from the morphism (E I U)X - E of 3. 8. 12; all the maps are easily seen to respect the Module structures). But now the right-hand side of (*10) is exact in A by 3. 8. 9 and the hypothesis on E; hence the left-hand side is exact in A and so by 1. 2 (E I U) is injective. (b) If E is an injective of the kind constructed in 3. 3, clearly E I U is also of this type and so is injective by 3. 3. For general E, 3. 4 shows that we can embed E in an injective E' of the type of 3. 3; but

then the inclusion E I E' splits by 1. 1(ii), so E' = E e E" for some E". Hence (E' I U) = (E I U) a (E" I U)

and (E' IU) is injective as already remarked; hence E IU is injective by 1. 3(i). //

Let (X, 0) be a ringed space over R, and let 0 : S - R be a ring morphism. Let r' denote the composite functor 3. 11

Proposition.

t

G

0-Mod - R-mod - S-mod

135

where r = r(x, -) and G is the change-of-structure functor associated with

6.

Then there is a natural isomorphism of a-functors: Rnr' = G o R n r = G(Hn(X, -))

In other words, we may compute the cohomology of an 0-Module as if its section sets were only S-modules. Proof.

G is exact; apply 2. 15. //

3.12 Remark. The R-module structure on Hn(X, -) may be recovered as follows. For an 0-Module A and for r E R, the operation

'multiply by r' induces an endomorphism of A and hence an endomorphism of Hn(X, A) (as S-module) which shows how r operates on Hn(X, A).

Applied to the special case S = Z, 3. 11 shows that we may always regard modules of sections as just abelian groups for the purposes of computing cohomology.

Proposition. Let (X, 0) be a ringed space over a ring R, which satisfies the hypothesis: 3. 13

(FL)

for any open U in x, r(U, 0) is a torsion free abelian group.

Let r' be the composite functor 0-Mod

Zx-Mod

- Z-mod

where r = r(x, -), ZX is the constant sheaf Z on X and F is the forgetful functor (cf. 4. 4. 5A). Then there is a natural isomorphism of a-functors

Rnr' =' (Rnr) ° F = Hn(X, F(-)). In other words, we may compute the cohomology of an 0-Module as if it were just a sheaf of abelian groups; in particular we may use resolutions of injective sheaves of abelian groups.

136

We wish to apply 2.14; since F is exact we have the required result provided that F transforms injectives in 0-Mod into r-acyclic objects. We shall show that under the hypothesis (FL), F transforms injectives into injectives. In fact, F is the 'direct image part' of a morphism of ringed spaces (over Z): Proof.

4 : (X, 0) - (X, Z X) with underlying continuous map idX and the unique possible morphism ZX - (idX)*0. This induces an adjoint pair (4. 13 and 4. 14):

4*=F 0-Mod . - ZX Mod. 4; *

Now let E be injective in 0-Mod, and A aZX Module. Then by 4.14 there is a natural bijection Hom(A, FE) = Hom()*A, E) and by 4. 13 the functor 4)* is given by

*A=A®Z 0. X

Under the assumption (FL), each r(U, 0) is a flat Z-module (see for example [K], 2. 2. 3. 1), and so (using 3. 6. 9(ii)) '* is an exact functor. Hence Hom(A, FE) is exact in A and so FE is injective by 1. 2. // As in 3.12, the R-module structure on the Hn(X, -) can be recovered. 3. 14

Remark.

3.15 Remark. In fact the conclusion of 3. 13 holds even without the hypothesis (FI) on (X, 0). This is most easily seen by using flasque sheaves, and since the details of the proof have other applications, we give a sketch for the convenience of the reader. A sheaf F of abelian groups on a topological space X (or more generally an 0-Module F on a ringed space (X, 0)) is flasque (= soggy)

iff for every open set U of X the restriction map r(X, F) - r(u, F)

137

is surjective (and hence so is every restriction map of F). Then: (a) If 0 - F - G - H - 0 is an exact sequence of sheaves and F, G are flasque, then so is H, and the sequence

H)- 0

O-'r(X,

is exact. (Proof by direct manipulation. ) (b) A direct summand of a flasque sheaf is flasque.

(Direct

proof. )

Every sheaf F can be embedded in a flasque sheaf F°; indeed F can be made a functor of F as follows. Let E P X be the sheaf space of F, and let E ° be the set E with the coarsest topology such that p remains continuous (that is, with open sets all the p 1(U) for U open in X). Let F 0 be the sheaf of sections of E ° p X (2. 2. C); thus for U open in X, r(U, F°) is the set IIxEUFx of not-necessarily continuous sections of p : E - X. (Compare 3. 2.) The natural map F - F 0 is the required embedding. (d) Every injective sheaf is flasque. (First proof: embed E in a flasque E ° by (c); then E is a direct factor of E ° by 1. 1(ii) and so is flasque by (b). Second proof: for U open in X apply the exact functor Homo(-, E) to the exact sequence (c)

0

0 - (0IU)X-0 of 4. 8. 11. )

After (c), by the technique of 2. 5 every sheaf F has a flasque resolution (e)

-4,

Apply r(X, -) to this complex and take cohomology. Then it can be shown that the resulting functors Hf (X1 F) = H*(r(X, F' ))

form an exact a-functor with Hf (X, F) = ker(r(X, F°) + r(X, F1))

= r(X, F). 138

(X, -) is effaceable, by (d), since any flasque F has the Also H*(X, id resolution 0 - F - F - 0 - 0 - ... Hence by 2.13 Hf (X, F) = H*(X, F). (This is the definition of sheaf cohomology used by Godement in [G]. ) (f) We deduce that for any flasque sheaf F, Hn(X, F)

r(X,0 F)

n n

= 0;

that is, F is acyclic. In fact this follows directly from (a), (b) and (c); see [T] 3. 3. 1.

But now, in the situation of 3. 13 (without (FL)), if M is an injective 0-Module, then it is flasque by (d), and hence F(M) is a flasque ZX Module, and so acyclic by (f). But this is enough to prove 3. 13 by applying 2. 14.

Exercise. Formulate and prove the generalisation of 3. 13 to the 'change of sheaf of rings from 0 to 0' ' situation (giving 3. 13 when 0' = ZX). Under what situations does it generalise fully to morphisms of ringed spaces, and with Rn4i* in place of Rnl' = Hn(X, -)? (Hint: the spectral sequence of a composite functor: [T] 2. 4. 1. ) 3. 16

3. 17

For some applications we may be interested in left exact

subfunctors of r and their derived functors; in particular, we can obtain

cohomology with supports as follows.

We formulate the results for

sheaves of abelian groups; the extension to ringed spaces is clear.

A set 4? of closed subsets of a topological space X is called a system of supports iff it satisfies the conditions: (a)

A, BE(b =AUBE4,

(b)

A S B E 4) and A closed = A E 4).

For example 4) = {{x }, 0 } where x is a closed point; or if X is

hausdorff 4 = {A; A is compact }. For F a sheaf on X, the group of sections of F with 4i-support is

139

r11) (F) = is E r(X, F);

Is I

E

}

where Is I = {x E X; sx * 0 1. Then r4) is a left exact functor: Shv/X - Abgp and its derived functors are the cohomology of X with 4;-support Hn (X, F) = Rnr4)(F).

For instance we obtain in this way the cohomology of X with compact For more details, see Swan, Theory of sheaves, [G] or support. Bredon, Sheaf Theory. Cech cohomology

5. 4

'r'

: Presh/X - Abgp : F - F(X) is exact (3. 6. 9(iii)) and so has zero right derived functor. However there is a a-functor Presh/X - Abgp which is of interest as it may sometimes aid in the The functor

computation of sheaf cohomology, whose existence is guaranteed by 3. 5, but in a way that makes evaluation difficult.

4.1 Let I be a set. For n E N, let [0, n] = {n E N; 0<_ m<_ n }. An n-simplex on I is a function v : [0, n] - I, and the set of n-simplices is denoted by In.

For 0 - m <_ n+1 there are maps ('omit the mth vertex') am

:I n+1

In

o(k) if k < m : v - (v : k - { a(k+1) if k ? m

4. 2 Let X be a topological space and let F be a presheaf of abelian groups on X (that is an object of the category Presh/X). Let 91- (U.)iEI be an open cover of X. For an n-simplex a E In let

Uo = n { Uo(m); M E [0, n] } ,

which is an open set of X. For n E N let

Cn(u, F) _ rUEIn F(Ua. This is an abelian group, and we use notations like (sa) for the element whose oth coordinate is S. 140

The maps am of 4. 1 induce am : Cn(U., F) -. Cn+l(`U, F)

(so) ' (tat)

where tat = sa m

or,

and we set for each n 1O(-1)mam : Cn(`U, F) _ Cn+1(U, F).

do =

m

It is easy to check that in this way C' (U, F) becomes a complex, called the Cech complex belonging to 91 and F, and its cohomology (2. 1) Hn(U, F) = Hn(C. (U, F))

(n c N)

is the Cech cohomology of F with respect to the covering `U. The constructions of C' (`U., F) and H*('U, F) are functorial in the presheaf F.

For example, H°(U, F) is computed as the kernel of

4. 3

C°(`U, F) - C' (U, F) II

H j.EI F(U

i

) - II (1,

.) I

EIXI

F(U i n U

where the lower map sends (si) to Ui

Uj

(PUnu,(sj) - PUnU(si (compare 2. 1. 8). Hence if F is in fact a sheaf, we have H°(`U, F) = F(X)

for any open covering `U.

4.4

= (V1)jJ, we say that `U is a refinement of `U iff there is a refinement map r : J -I with the property that Given two open coverings `U.= (U.)iEI' .D

Vi e j

Vi C_

Ur(j)'

Such a refinement map r induces a morphism of complexes 141

r'

: C' (91, F) - C. (`U, F)

derived from the maps Heel F(Uo) -

n

n

F(VT) .

(se)'-

(tT)

where tT _ ur(T) (sr(T)), and where r also denotes the map induced by T

composition from Jn to In Hence a refinement map induces (2. 2) a morphism of Cech cohomology

r : Hn(`U, F) - Hn(U, F)

(n a N).

Lemma. If `U is a refinement of `U, and r1, r2 : J - I are two refinement maps, then they induce the same morphism of Cech 4. 5

cohomology

rl = r2 : Hn(`U, F) - Hn(`U, F). Proof. In fact, in the notation of 4. 4, r' and r' are homotopic morphisms of complexes (2. 1), by the homotopy: F)

Cn(D, F)

(se) ~

(tT)

Cn+l(cU.,

kn :

UT

where tT = k_0(-1)kPV

(s Tk T

rl (T(M))

if m

r2(T(m-1))

if m > k

where Tk(m) _

k

for m E [0, n+l]. The result

follows by 2. 2. //

Hence the abelian groups Hn('U, F) forma directed system as `U. varies over the open covers of X, and we define the Cech cohomology of the presheaf F on X to be 4. 6

Hn(X, F) = lim Hn(`u, F) 91

(over finer and finer covers). There is a set-theoretic difficulty in that 142

the class of open covers of X is not a set, but we may avoid this either by allowing only covers indexed by subsets of a suitably large set such as the power set of X, or by proceeding as in 4. 7 below. 4. 7

If `0 is a refinement of `U, the morphism

r' : C' (l, F) - C' (`U, F) depends on the choice of the refinement r, although r does not. Hence we have difficulty in forming a direct system of the C' (`U., F) so as to obtain an exact cohomology sequence. This can be solved by using a neat trick due to Godement ([G] §5. 8).

Let (R(X) be the set of open covers (U) indexed by X in such a way that VX E X

XE U

of X which are

X

On I3(X) we define a preorder by

V?`U iff VXEX Vx CUx id

and so obtain a canonical refinement map X - X when 'D > `U. As in 4. 3, `D ? 91 implies that we have a morphism C' (`U, F) - C' (`U, F)

of complexes, and we define

C' (X, F) = lim

C. (91, F).

`U E(R(X)

Since lim is exact (see Bourbaki, Algebre, Chapter II, 6. 6. 8 and 1. Ex. 8) we find

Hn(X, F) = Hn(C' (X, F)).

4.8 Theorem. Presh/X - Abgp.

{Hn(X, -); n E N} is an exact Z-functor:

143

Proof. An exact sequence 0 - P - Q -+ R - 0 in Presh/X gives for each `U. E 63(X) an exact sequence

P)-C'(`u, Q)''C'(`U, as is easily checked. Applying the exact functor lim we get an exact sequence of complexes 0.

0 - C' (X, P) - C' (X, Q) '' C' (X, R) - 0 which gives by 2. 12 the required long exact sequence. //

Exercise. Deduce from 4. 8 that H*(X, -) is the right derived functor of H°(X, -). [Hint: Use 2. 13; to check that Presh/X has enough injectives, mimic 3. 2 and 3. 3 (compare Artin, Grothendieck 4. 9

Topologies, I, 2.7 and 3. 2). ] 4. 10

Exercise.

Show that F '- F+ where for U open in X

F+(U) = H°(U, F I U)

is a functor Presh/X - Presh/X with the properties: (i) for any presheaf F, F is a monopresheaf; (ii) if F is a monopresheaf, F+ is a sheaf; (iii) for any presheaf F there is a natural morphism F -+ F+ such that if G is a sheaf then any morphism F - G factors through

F"'F+. Deduce that F

++

is (isomorphic to) the sheafification of F.

If we consider the restriction of Hn(X, -) to the category Shv/X = ZX-Mod of sheaves of abelian groups on X, we find that it is not necessarily a 2-functor (exact sequences in Shv/X are not necessarily exact in Presh/X). By 4. 3 we know that if P is a sheaf on X, then 4. 11

H°(X, P) = r(x, P) T H°(X, P). An exact sequence f

144

in Shv/X has R = SCok(f) and so gives a diagram R

f where the horizontal line is exact in Presh/X and the vertical map is the natural map of sheafification. Thus by 4. 8 we obtain diagrams

r(X, P) -' r(X, Q) -' r(X, R) = H°(X, R)

0-'H'(X, P) -'AV , Q)''H'(X,

R)- 1(X, P)-'...

Hn(X, R)

.. i Hn(X, Q)

' Hn(X,

R') y Hn+l (X, P) -' .. .

with exact bottom rows. 4. 12

Lemma.

If E is an injective sheaf of abelian groups on

X then for n > 0 Hn(X, E) = 0

(that is, H*(X, -) is effaceable on Shv/X). Proof. We show in fact that if E is any sheaf of the kind constructed in 3. 2, then for each open covering `U of X, the Cech complex C' (`U., E) is contractible; that is, it admits a homotopy (defined below) between the identity and zero endomorphisms. Hence id = 0 : Hn(`U, E) - Hn(`1, E) for n > 0, and so fn (X, E) = 0 for n > 0. By 3. 3 any injective sheaf E' embeds in a sheaf E of this form,

and so E = E' ® E" for some E" since E' is injective (1. 1). Hence

for n > 0 0=

fn(X, E) = Hn(X, E') ® Hn(X, E")

(this follows easily from the exact cohomology sequence of the split exact

sequence 0 - E' - E - E" - 0) and the result follows. Given E of the type described in 3. 2 and an open covering of X, pick j E I. A suitable homotopy is then given by the U= morphisms 145

kn :

Cn+1(U,

E) '' Cn(u, E) : (so)

(s,r')TEI n

m=0 where T'(m) = { T(m-1) for for m > 0 for m E [0, n+1], j

and for T E In, s T , is the image of s T, under the natural map

XEUT,Ex

-

ex if xEU

XEU

(e x ~ ( { 0

if X ¢

, })

(we have UT, S UT, and we may have to enlarge the 'domain' of the section sT,).

4. 13 We see from 2. 13, 4. 11 and 4.12 that for a fixed space X and a E N U { -1, we can assert that for any sheaf F of abelian groups

on X we have

(*413) Hn(X, F) = Hn(X, F) for 0 < n < a provided that {Hn(X, -); 0 - n < a } forms an exact o-functor, which will be true if the following condition holds: (a)

if R' is any presheaf on X with sheafification R, then

the induced Cech cohomology map Hn(X, R') .4 Hn(X, R)

is an isomorphism for 0 s n < (a - 1) (where - - 1 = 00). If we let S be a presheaf such that the sequence is exact in Presh/X, we see that S has as sheafification the zero sheaf, and there is an exact sequence

... -

Hn-1(X,

S) _ Hn(X, R') _ Hn(X, R) - H°(X, S) - .. .

It follows that the condition (a) is equivalent to (b)

if S is a presheaf on X with zero sheafification, then for

0< n< (a- 1) Hn(X, S) = 0.

This condition holds, and hence so does (*413), in a number of interesting cases:

146

for any X for a = 2 (see 4. 14 below); (ii) for a = providing X is paracompact (see [G], 5. 10. 1); [or more generally, for a = and any X providing we replace Hn, fin by their versions 'with supports in a paracompactifying family' (cf. 3.16; see also Swan, Theory of Sheaves, VIII)]; (iii) for a = - if X is a scheme and provided we restrict our (i)

attention to quasi-coherent 0X Modules (see [EGA III] 1. 4. 1). More generally, we have in any case a spectral sequence Hp(`U, _Cq(F)) = Hp+q(X, F)

for any open cover of X, where jCq(F) denotes the presheaf U '- Hq(U, F); and (i)-(iii) above can be regarded as cases where it degenerates. See [T], 3. 8. 1; [G] 5. 9. 1; Artin, Grothendieck Topologies, CU

II §3.

For the connections between singular cohomology, AlexanderSpanier cohomology and sheaf cohomology (Cech or Grothendieck (derived functor) version), the reader is referred to Spanier, Algebraic Topology; particularly 6. 8. 8, 6. 9. 1, 6. 9. 5, 6. 9. 7. In general terms these results state that for suitably nice topological spaces X (such as topological manifolds), the cohomology of a constant sheaf G on X is isomorphic to the singular cohomology of X with coefficients in G. More generally, if G now denotes a locally constant sheaf, then the sheaf cohomology of G gives the cohomology of X with the local

system of coefficients G (see Spanier, Chapter 6, Exercise F and Chapter 5, Exercise J).

If X is any topological space and S is a presheaf of abelian groups on X with zero sheafification, then 4. 14

Theorem.

9°(X, S) = 0.

Hence as in 4. 13, (H0, H) form an exact effaceable 8-functor on Shv/X, and so for any sheaf F of abelian groups on X Hn(X, F) = Hn(X, F) for n = 0, 1.

147

Any f1 E H°(X, S) is represented by some

Proof.

f E ker(d0 : C°(`U, S) -C1(`ll, S))

for some open cover `U = (Ui)iEI of X. That is, f = (fi)iEI E BiEIS(Ui).

But S has stalk zero everywhere; hence for each i, U. has an open cover (Vj)jEJ such that 1

U.

Vj E Ji PV1(fi) = 0. J

Then `U = (Vj)jEU1J1 is an open cover of X, refining `U., such that f'- 0 under the refinement map H° (`u' S) - H°(`U, S). //

4.15 Theorem. If X is a topological space and U, 'U are two open covers of X with 'U a refinement of `U, then for any sheaf F of abelian groups on X, the refinement map

H' (`U, F) - H'(U, F)

is injective. Hence H1(X, F) = H'(X, F) is a union of subgroups isomorphic to the H1(U, F). Proof. Let `U = (Ui)iEI' U = (Vj)jEJ and let refinement map.

r : J-'I

be a

Given S E ker(d1 : C'(`U, F) - C2 (`U., F)) such that

a f c Im(d0 : C °(`U, F) -' C'(`U, F)) with Vj, k E J V

U

PVrj'rk s(rj, rk) = pV

(f (j)) jk

jk

Vk

(f(k))

jk

we wish to show that of E Im(do

C°(`U, F)

C'(`U, F)) with Vi, I E I

s(i, 1)=f(i)-f(1); here we are writing Vjk, s(j, k) for VQ,sa where a is the 1-simplex

148

0I- j,

and similarly for 0-simplices. Suppose we are given i E I: to construct f (i) E r(Ui, F) we consider for each x E Ui, and for each j e J such that x e Vi, the element _

U .

V.

f = PUiAV,(1(j)) - PUlnVJ (s(rj, i)) j

E r(Ui n V., F).

(For the rest of the proof we omit the restriction maps and instead name the domain on which equations hold.)

fj

If k E J is also such that x c Vk, then - f k = f(j) - f(k) + s(rk, i) - s(rj, i) V.

= s(rj, rk) + s(rk, i) - s(rj, i)

=0 on UinVjnVk since s E ker d1. Thus the f (j E J) glue j to give an element f(i) E I'(U., F). Furthermore, on U. n U1 (i, 1 E I) we have s(i, 1) = T(i) - Y'(1)

since Vx E Ui n UZ, whenever x E UiZ n Vjk for some j, k E J

f(i) - ?-(I) = f(j) - s(rj, i) - f(k) + s(rk, 1) = s(rj, rk) - s(rj, i) + s(rk, 1)

_ -s(rk, i) + s(rk, 1)

= s(i, 1) on UiZ n Vjk . The result follows, and the last part follows from the construction of the direct limit (cf. 1. Ex. 5). // Cech cohomology has a connexion with the picard group (4. 5. 5 and 4. 5. 6), given as follows. 4. 16

Let (X, 0) be a ringed space. An invertible e-Module M is given by the following data: (a) an open covering MIUi =- 0jUi);

`U. =

of X (such that Vi E I

149

(b)

for each i, j E I an isomorphism of 0 I (UinU1)-Modules

o I (Ui n U.) =. 0 I (Uj n Ui)

(each is isomorphic to M I (Ui n U.)). By 4. 5. 3 the data of (b) is equivalent to giving, for each (i, j) E I X I, a unit

fij E r(Ui n U., 0) (so that the isomorphism of (b) is 'multiply by fij') Now the assignment U'-+ (r(U, 0))* = group of units of r(U, 0) defines a sheaf 0* of abelian groups (under multiplication), and the fij give an element f = (fij)(i, j)EI1 E C1(91, 0*).

Since the isomorphisms in (b) are compatible on the triple intersections Ui n Uj n Uk, f is in fact a cocycle, that is f E ker(d1 : C'(`U, 0*) - CZ(U, 0*)) = Z1(`U, 0*) say.

Conversely, given f E Z1(`lt, 0*) we can construct an 0-Module M by glueing the copies of 0 I (Ui n U.) by the recipe given by f in (b). Hence we have defined a map 1 : Z1(91, 0*) _ Pic X

which has as image the set of isomorphism classes of invertible sheaves which are trivialized by `U (that is, are such that V i E I M I Ui = 0 I Ui). It is also clear from the construction that C is a morphism of abelian groups, that is it takes the composition of cocycles (written naturally as multiplication in this case) to the operation on Pic X induced by . Suppose now f = (fij) is a member of the kernel of C ; then the

invertible sheaf M constructed from f as above is trivial: M = 0.

150

Now we have a global section 1 E r(X, 0*); let gi E I'(U., M)

be the corresponding section. Then by (b)

Vi, j E I gj = fijgi on U. n U.,

so f is a coboundary, that is f E Im(d0 : C°(`ll, 0*) -'' Cl(`U, 0*)).

Hence C induces an injection H1(`U, 0*) - Pic X

with image that of

(cf. 4. 14).

Since every invertible sheaf is trivial over some covering, and the refinement maps are easily seen to be compatible with the maps we obtain

Theorem.

There is a natural isomorphism of abelian groups

H'(X, 0*) = k (x' 0*) = Pic X. // 4. 17 As an application, let (X, 0) be a complex manifold (continuous, differentiable or analytic). Then there is a sheaf morphism 0 - 0* which sends a C-valued function f to the function exp(2sif); this provides a short exact sequence of sheaves of abelian groups:

where Z denotes the constant sheaf (of integer-valued functions). The associated cohomology sequence contains the map

2: Pic X=HI(X,

Z)

and by 4. 13 the target group can be interpreted as the topological (singular) cohomology of X with integral coefficients. When reinterpreted as a map on invertible sheaves, 8 is called the Chern class map. 151

Exercises on Chapter 5 Use 4. 15 to show that HI(C*, C) * {0 }, where C* = C\ {0 }, and C denotes the constant sheaf; this 'explains' 3. 6. 10. 1.

Let G : K - K' be a functor between abelian categories; we say that G is effaceable iff for each object A of K we can find a monomorphism u : A - E in K such that G(u) = 0. A a-functor { Gn : 0 < n < a) is called effaceable iff Gn is for 0 < n < a. Show that if K has enough injectives, this is equivalent to the 2.

definition given in 2. 13.

Show that, for any abelian category K, an exact, effaceable (in this new sense) o-functor F* over a left exact functor F : K - K' has the universal property of 2.13 (for R' F).

Let X be a topological space, considered with the structure sheaf 0 = ZX, the constant sheaf Z, so that 0-Mod = Shv/X is the category of sheaves of abelian groups on X. Let A be a closed subspace of X, with inclusion map j : A '- X. (a) Let G be a sheaf of abelian groups on A and GX = j*G its extension by zero to X. Show that there is a natural isomorphism 3.

H*(X, G5 = H*(A, G). (b)

Deduce that if F is a sheaf of abelian groups on X, there

is an exact cohomology sequence FU)-+Hn(X,

FIA)-'Hn+1(X,

where U = X\A and FU= (F I U)X.

Establish that for any topological space X and sheaf F of abelian groups on X there is a natural map 4.

H2(X, F) - H2(X, F) which is always injective, in two ways: (a) directly, rather as in 4. 13 and 4. 14 (b) by first constructing the spectral sequence

152

Hp(X,

JCq(F))

Hp+q(X, F)

(where 3Cq(F) is the presheaf U'- Hq(U, F) mentioned in 4.13). [One way to do this is to obtain it as the spectral sequence of a composite functor ([T], 2. 4. 1), using 4. 10; compare Artin, Grothendieck Topologies, II §3. ] 5.

Use the ideas of 4. 5 and 4. 12 (chain homotopies) to show that H*(`U, F) = H*(C1(`U, F))

where Ci(91, F) is the subcomplex of C' (9L, F) consisting of the alternating cochains, namely those (s0.) for which (a) s. = c sat if the values of a' form a permutation of sig-

nature c of the values of a; and sa = 0 if a takes two equal values. Suppose that X admits a covering 91 = (Ui)iel such that for some n E N, Ua = 0 for all those a E In which take distinct values (we then say dim X < n). Deduce that for any sheaf F on X we have (b)

m > n = Hm(X, F) = 0. [Compare Serre, Faisceaux algebriques coherents (Annals 61 (1955) 197-278) I. 18 and I. 20; and [G] H. 5. 12. ] 6. Suppose that 91 is an open cover of X which is acyclic for a given sheaf F; that is, for each simplex a,

r > 0 =4, Hr(Uc, F I Ua) = 0. Show that H*(`U., F) c--- H*(X, F).

[This is hard; see [G] Cor du Th. 5. 4.1, or Serre, FAC (see Q5) I. 29. ]

153

The way ahead: further reading

The grounding in sheaf theory given by this course should enable the reader to proceed with the study of a number of subjects. Some suggestions are given below: there is no particular significance in the ordering. (a) In topology, one can study cohomology operations (cup and cap products), the Leroy and Serre spectral sequences, and Borel-Moore homology. See for example G. E. Bredon, Sheaf Theory (McGraw-Hill 1967); R. G. Swan, The Theory of Sheaves (Chicago U. P. 1964); and [G].

In algebraic geometry, only the basic machinery has been indicated. One must now do more work on affine schemes to understand the local nature of an algebraic variety, and use sheaf theory to connect this with the global properties. A very good introduction is D. B. Mumford, Introduction to Algebraic Geometry (mimeographed notes from Harvard). See also I. G. Macdonald, Algebraic Geometry (Benjamin 1968); and [EGA I-IV], although this cannot be recommended wholeheartedly as reading material. For specific examples of the applications of sheaf theory, there is the excellent paper J-P. Serre, Faisceaux algebriques coherents (Annals, 61 (1955) 197-278), which uses an older definition of algebraic variety; D. B. Mumford, Lectures on Curves on an Algebraic Surface (Princeton U. P. , 1966), which relies heavily on cohomology, and incidentally gives a rapid outline introduction to scheme theory; and Y. I. Manin, Lectures on the K-functor in Algebraic Geometry (Russian Math. Surveys, 24 (1969), No. 5, pp. 1-90). The applications of sheaf-theoretic topology to (classical) (c) algebraic geometry (such as the various generalisations of the RiemannRoch Theorem) are well represented in [H]; see also Atiyah, K-Theory (Benjamin, 1967). (b)

154

In working out the abstract machinery necessary to attack the Weil conjectures, Grothendieck and his school were led to a generalisation of topological spaces, over which one can still do sheaf theory: these are the Grothendieck topologies. A good introduction, showing how neat and ultimately categorical sheaf theory can be made is in M. Artin, Grothendieck Topologies (Harvard Lecture Notes, 1962); see also Mumford's chapter in Arithmetical Algebraic Geometry (edited by Schilling; Harper and Row, 1965). The bible (or elbib?) of this sect is SGA4 (Springer Lecture Notes 269, 270 and 305); it is by no means easy reading. See also H. Schubert, Categories, Chapter 20 (Springer, 1972); M. Hakim, Topos anneles et schemas relatifs (Springer, 1972). (e) The latest abstraction from (d) is the theory of elementary topoi; it is an ambitious attempt to unite geometry and set theory, and is at an early stage of development. See A. Kock and G. C. Wraith, Elementary Toposes (Aarhus Lecture Notes No. 30, 1971); P. Freyd, Aspects of Topoi (Bull. Aust. Math. Soc. , 7 (1972) 1-76); F. W. Lawvere, Quantifiers and Sheaves (Proceedings of the International Congress of Mathematicians, Nice 1970; Vol. I, pp. 329-34). (d)

155

References

A. Grothendieck and J. A. Dieudonne, Elements de gdometrie algebrique I (Second Edition); Springer, 1971. [EGA U, III] A. Grothendieck and J. A. Dieudonne, Elements de geometrie algebrique U, III; IHES Publ. Math. 8, 11, 17. [G] R. Godement, Topologie algebrique et theorie des faisceaux; Hermann, 1964. [H] F. Hirzebruch, Topological methods in algebraic geometry (Third Edition); Springer, 1966. [K] J. T. Knight, Commutative algebra; Cambridge University Press, 1971. [L] S. Lang, Differential manifolds; Addison-Wesley, 1972. [Macl] S. Maclane, Categories for the working mathematician; Springer, 1971. [Mit] B. Mitchell, Theory of categories; Academic Press, 1965. [T] A. Grothendieck, Sur quelques points d'algebre homologique; Tohoku Math. Journal, IX (1957), pp. 119-221. [EGA I]

156

Hints and answers to some exercises 1. Ex. 3(iii) 1. Ex. 5 2. 1. 9

No: only torsion groups are so obtainable. All the maps to the direct limit are injective (respectively surjective). Use the empty cover of the empty set. Aliter, apply 2. 3. 1 (or its proof).

2. Ex. 1

r(I, F) = Z e Z; just one; as many as there are pairs of abelian group morphisms, each Z - Z.

3. 6.2

See 4.1.13.

3. 6. 8

F - G - H is exact iff for some objects K, L and morphisms F - K - G - L - H the three sequences

0 - L -H 3. Ex. 7

are exact. This condition is preserved under an exact T. s First show that a short exact sequence 0 -. P -+ Q - R - 0

is split (that is Ht : R - Q such that s o t = idR) iff Q = P e R with s corresponding to the natural projection. Hence (a) (b). Now for s, t : P - P, factorise (s - t) as (s, t)

P - P e P - PeP - P. Hence (b) - (c). For (c) there are morphisms P`

(b) show that X = P e Q iff

P X

Q 157

such that ca = idp, db = idQ da = 0, cb = 0 and 4. 3. 5

ac + bd = idX. Spec S - Spec R induces a morphism of global sections

R = r(Spec R, OR) - r(Spec S, OS) = S. Compare 4. 3. 11. 5. Ex. 3

Use 3. 8. 8(a) to see that H*(X, -X) is an exact c-functor over r(A, -), and 3. 7. 13 to show that it is effaceable; apply 2. 13. (a)

(b) Use 3. 8. 11.

158

Index of terminology

abelian category 35, 49 abelian sheaf 15 acyclic 119, 129, 153 additive category 37, 49 additive functor 71 adjoint functors 37, 60, 61, 94, 101, 114, 117

affine morphism 112 affine scheme 84, 87, 89, 104, 109, 112 affine space 85, 111, 112 algebra

73, 111

algebraic curve

106, 111

algebraic functions

29, 85

2, 27, 30, 44, 52, 76, 88, 89, 93 associated sheaf 22, 134 associativity 32 analytic functions

automorphisms 103, 124

bilinear map biproduct 47 category C ech

99

31, 37, 49

141, 142

classifying map coboundary

151

cochain

153

cocycle

150

coequaliser

69

cohomology

119, 129, 130, 132, 139, 140, 141, 142, 147

cokernel 41, 42 colimit 69, 98 complex 119 composition 10, 31, 76 constant functor 35 constant presheaf

constant sheaf 24-5, 28 continuous functions 2, 17, 18, 25, 27, 58, 75 conventions vii, 27, 33, 36, 73 contravariant functor 33 coproduct 48, 97 covariant functor

curves

33

111

8-functor 124, 129, 132, 143, 147, 152

derived functor 123, 144, 152 differentiable functions 2, 27, 75, 87, 88 dihomomorphism 95 directed set 3, 35 direct image

71

2, 9, 24, 36

53, 54, 60-1, 68, 69,

75, 80, 101, 133, 137 direct limit 4-8, 98 direct product 48, 97 direct sum 8, 47, 97-8 direct system 3, 7, 33, 35 disjoint union vii, 6 159

dual category 37, 69, 71 dual sheaf 105

identity 32, 35 image 49, 79 injective object 116, 118, 119, 123, effaceable 128, 145, 152 132, 135, 138, 144, 145 endomorphisms 103, 114, 131, 136 inverse image 57, 60-1, 68, 69, 75, 80, 101 epimorphism 10, 43, 46, 69, 78 inverse limit 12 equaliser 15, 69 invertible 105, 149 equivalence relation 69 isomorphism 10, 44, 76 evaluation map 70, 105 exact complex 119 kernel

exact a-functor

146, 147, 152, 158 exact functor

38, 40

124, 128, 143, left exact

35, 51, 55, 62, 122, 125

51, 61, 62, 65,

limit 12, 69 local homeomorphism exact sequence 12, 16, 50, 53 extension by zero 62-3, 68, 135, local morphism 86 152 local ring 29, 84 localisation 84 fibre 17, 19 locally closed 63 finite presentation 114 locally connected 28 finite type 113 116, 129-30, 143

locally free

flasque

137

forgetful functor

functor

33

33

functoriality 11, 20, 21, 54

geometric space 87 germ 9, 27, 100 glueing condition

14

half exact functor 71 homogeneous coordinates

110

homotopic 119 homotopy equivalent 122 Hom-sheaf 70, 100, 105, 114

160

72, 102, 104, 112

locally isomorphic

flat 137

18

89

manifold 90-1, 151 module 94, 95 monomorphism 10, 38-9, 69, 78 monopresheaf 14, 72, 144 morphism 31 morphism of complexes 119 morphism of a-functors 124, 143

morphism of geometric spaces morphism of manifolds 90 morphism of modules 94, 95 morphism of presheaves 9, 33 morphism of R-algebras 73 morphism of ringed spaces 76

87

morphism of sheaf spaces 18, 26 , 33

55

morphism over a ring morphism 95

natural 23, 34 natural equivalence 35 natural isomorphism 5, 23, 35, 100

ring

73

ringed space

4

scheme 90 , 147 sections 1, 27, 28 ,

separated presheaf separated scheme

31

105, 107, 149-51

preadditive category preordered set

37

3, 32

presheaf 1, 34, 69, 71, 72 presheaf cokernel 41 presheaf image 49 presheaf morphism 9 prime ideal 81 prime spectrum 81 product 48, 97-8 projective 104 projective space

106, 109-10

quasi-coherent 96, 111, 147 quotient 46, 69

14 91

42

sheaf image 49 sheaf of ideals 78, 80 sheaf of sections 17, 18 sheaf space 18, 26, 74 sheafification 22, 34, 35, 52, 61, 144 sheaves of functions 2, 9, 14, 17, 27,

29, 75, 76, 85, 87

simplex

140

singular cohomology

54, 109, 110

sphere

split stalk

147

139, 147, 152

spectral sequence

116, 125, 157 8, 10, 19, 21, 23, 40, 45, 58, 80

structure map 18, 73 structure sheaf

subobject

74

40

subobject classifier

subpresheaf

rational functions 29 refinement 141 representable functor 68 resolution 119, 138 restriction 28, 62, 99, 102 restriction map 1, 95

52

15, 69, 72, 74

sheaf

sheaf cokernel

natural transformation 34, 57

picard group

51, 62, 116

right exact

morphism of sheaves 17, 33 morphism over a continuous map

objects

riemann surface 90, 106

subsheaf

sum

39

39

8, 36, 47, 97-8

supports target

70

139, 147 4, 7, 35

tensor product

98-9, 105, 108

161

topos torsion

71 12

total tensor product 108 variety 90 vector bundle

96, 104, 112

Z ariski topology

zero morphism zero sheaf 36

162

82

36

Index of notation

N; Z; Q; R; C vii

lim

= vii

F+;

7T

3, 32, 40, 143

Sets; Abgp; Top

11.

32

5

25, 48, 74 vii, 48

Ob

31

R-mod 94

Mor

R-alg 74 Presh; Shv; Shfsp 33, 37

Homc(-, -) 31 Uop 37, 69, 71

S/X; P/X 69, 70

nA

K-Presh 34, 72

idF 35 by 68

0-Mod 96 Gsp pV p«a

F I U 28, 62, 99 3

Fx 8, 34 8

x fx 10 s 21 nF 22, 34

r

18, 34

r(U, E) 18, 27, 28, 36 1'(U, f) 36 L 21, 34 EvE 25, 74 AX

c P1 P2

;

34

93 1

s

32

0*

53, 69

0*

57, 69

4) *; 4i*

101

FY; j, F

63

GI, 65, 152

Hom(-, -) 31, 36 HomX(-, -) 56 Homo(-, -) 56 Hom(-, -) 70, 100, 105 114

End

2, 25

0

Cr; CW 2 3, 9, 14, 18 3, 13, 17

0*

29, 74, 84

150

0X(n)

107, 110

M 96 163

Ker(f) 38 PCok(f) 41

G

47, 97

H

97

PIm(f)

49

® 98 ® 108

SIm(f)

49

Bilin(-, -; -)

SCok(f)

42

Spec R; V(a); D(f) Rf

119, 141 H(g); g* 120 R n F 123 R' F 125

84

87

mx

112

Spec

V; Hn(L' ); H*(L') do

82

RP

81

End; Aut 103 Pic 105 Sn 54, 109, 110

T'; a 8

124

124, 126

ZG; kG; AG 130

P

107, 109-10

I'46

A

85, 111, 112

Rn4,*

140 133

V 112

[0, n]; In; 8m

U; (s

140

Hn(G, A)

130

)

140

Hk(G, A)

131

Cn(LL, F)

140

Hn(X, -)

132

CiF) 153

Hf(X, -)

138

Z1(`tt, -)

150

Hnn(X, -)

140

141

147, 153

Hn(`U, -) fin (X, -)

F+

99

144

cCq(F)

164

142

119