Download Sheaf Theory (London Mathematical Society Lecture Note Series)

London Mathematical Society Lecture Note Series. 20
Sheaf Theory
B.R.TENNISON
CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE
LONDON NEW YORK MELBOURNE
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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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<y,
pay(ua) = P4
<y and
/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