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Sheaf Theory
W. D. Gillam
Department of Mathematics, Brown University
E-mail address: [email protected]
Contents
Introduction
7
Chapter I. Sheaves
1. Open sets
2. Presheaves
3. Representable presheaves
4. Generators
5. Sheaves
6. Stalks
7. Sheafification
8. Espaces étalé
9. Sieves
10. Properties of Sh(X)
11. Examples
12. Exercises
9
9
9
10
11
11
13
15
16
18
20
22
22
Chapter II. Direct and Inverse Images
1. Functoriality of Ouv(X)
2. Direct images
3. Skyscraper sheaves
4. Inverse images
5. Sheaf Hom
6. Constant sheaves
7. Representable case
8. Inverse images and espaces étalé
9. Faithfulness of X 7→ Sh(X)
23
23
23
23
24
26
27
28
28
29
Chapter III. Descent
1. Canonical topology
2. Open maps and variants
3. Descent for spaces
4. Descent for sheaves
31
31
31
31
31
Chapter IV. Cohomology
1. Abelian categories
2. Enough injectives
3. Homological algebra
4. Abelian Sheaves
33
33
33
33
33
3
4
CONTENTS
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Ringed spaces
Sheaf Ext
Supports
Sheaf cohomology
Acyclic sheaves
Flasque sheaves
Godement resolution
Soft sheaves
Filtered direct limits
Vanishing theorems
Higher direct images
Leray spectral sequence
Group quotients
Local systems
Examples
Exercises
36
37
38
39
39
39
42
42
42
46
46
47
48
48
48
48
Chapter V. Base Change
1. Proper maps
2. Comparison map
3. Homotopy invariance
51
51
55
57
Chapter VI. Duality
1. Adjoints for immersions
2. Extension by zero
3. Proper maps - Second treatment
59
59
61
63
Chapter VII. Finite Spaces
1. Supports
2. Specialization
3. Coniveau
4. Equivalence
5. Stalk formulas
6. Hom formulas
7. Poincaré duality
8. Examples
9. Exercises
65
65
65
66
67
67
67
68
69
74
Chapter VIII. Čech Theory
1. Čech resolution
2. Refinement maps
3. Alternating variant
4. Cech cohomology for a fixed cover
5. Basic properties
6. Čech cohomology
7. Čech acyclic sheaves
75
75
77
78
78
79
79
80
CONTENTS
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Čech presheaf
Comparison theorems
The unit interval
Simplicial objects
The standard simplex
Truncations and (co)skeleta
Simplicial homotopy
Simplicial sets
Standard simplicial sets
Geometric realization
The abelian setting
Simplicial rings
Simplicial sheaves
Weak equivalences
Properties of coskeleta
Čech theory revisited
Exercises
Bibliography
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107
109
Introduction
The aim of this document is to give a concise treatment of the “classical” theory
of sheaves on topological spaces, with the following philosophical guidelines:
(1) Appeal to standard facts about limits in the category of sets whenever
possible, e.g. exactness of filtered direct limits.
(2) When possible, formulate constructions and results about sheaves on a topological space so that make sense on a site or topos with little modification.
(3) Appeal to general homological algebra whenever possible.
(4) Follow Godement [G] closely as long as this does not conflict with the other
guidelines.
Regarding (1) and (2), my feeling is that the general philosophy of topos theory
is that a topos is a category that behaves like the category of sets, so, for example,
abelian group objects in a topos should behave like abelian groups, etc. Almost
any theorem about sets that can be stated and proved using category theoretic
language should also hold true in a topos. In particular, the category of sheaves
on a topological space should be a lot like the category of sets. This philosophy is
usually manifested by bootstrapping up from the category of presheaves, which is
“very much” like the category of sets from the perspective of category theory. A
related philosophical point is that, since our object of study is the category of sheaves
on a topological space, we should try to phrase our constructions intrinsically using
this category, rather than refering back to the topological space, its category of open
sets, etc. The manifestation of this is the usage of the representable sheaf associated
to an open set, rather than the open set itself.
Regarding (3), my feeling is that basically every spectral sequence arises as a
special case of the Grothendieck spectral sequence. One should never have to write
down a double complex out of the blue and analyse what its horizontal and vertical
cohomology mean, etc. Grothendieck already did that once and for all.
7
CHAPTER I
Sheaves
1. Open sets
For a topological space X, let Ouv(X) denote the category of open subsets of
X, where the only morphisms are inclusions. The category Ouv(X) has arbitrary
direct limits given by the formula
[
lim (f : C → Ouv(X)) =
f (C)
−→
C∈C
and inverse limits over any category with finitely many objects, given by the formula
\
lim (f : C → Ouv(X)) =
f (C).
←−
C∈C
In particular, ∅ ∈ Ouv(X) is initial (the direct limit of the empty functor, or the
empty union) and X ∈ Ouv(X) is final (the inverse limit of the empty functor).
2. Presheaves
A presheaf on a topological space X is a functor
F : Ouv(X)op → Ens
U 7→ F (U ).
Presheaves on X form a category PSh(X) where a morphism of presheaves is a
natural transformation of functors. For U ∈ Ouv(X), elements of F (U ) are called
sections of F over U . For an inclusion of open sets V ⊆ U and s ∈ F (U ) we often
write s|V for the image of s under the morphism FV ⊆U : F (U ) → F (V ). We refer
to these morphisms as restriction morphisms. For U ∈ Ouv(X), the functor
PSh(X) → Ens
F 7→ F (U )
is often denoted Γ(U, ). We sometimes write Γ(U, F ) in lieu of F (U ), especially
when U = X, in which case Γ(X, ) has a special name: the global sections functor.
The category PSh(X), being defined as a category of functors to Ens, inherites
various properties from Ens. For example, PSh has all small direct and inverse
limits. These limits are formed “objectwise.” For example, for a direct limit system
9
10
I. SHEAVES
i 7→ Fi in PSh(X), the direct limit presheaf F = lim Fi is given by
−→
F (U ) = lim Fi (U ).
(2.1)
−→
In particular, notice that the section functors
Γ(U,
) : PSh(X) → Ens
commute with all limits.
3. Representable presheaves
The prototypical presheaf is the representable presheaf hU associated to U ∈
Ouv(X), defined by hU (V ) = HomOuv(X) (V, U ). By abuse of notation, we often
write U instead of hU . Formation of hU is functorial in U and defines a functor
h : Ouv(X) → PSh(X) called the Yoneda functor.
Lemma 3.1. The Yoneda functor h is fully faithful and commutes with arbitrary
inverse limits. There is a bijection
HomPSh(X) (hU , F ) = F (U )
natural in U ∈ Ouv(X) and F ∈ PSh(X).
Proof. This is a straightforward variant of Yoneda’s Lemma.
Remark 3.2. The Yoneda functor does not generally commute with direct limits,
or even with finite direct sums. In particular, the presheaves hU ⊕ hV and hU ∪V do
not generally coincide. Indeed, for W ∈ Ouv(X), by definition of direct sums, we
have
a
HomPSh(X) (hU ⊕ hV , hW ) = HomPSh(X) (hU , hW )
HomPSh(X) (hV , hW )
a
= hW (U )
hW (V )
a
= HomOuv(X) (U, W )
HomOuv(X) (V, W ),
while
HomPSh(X) (hU ∪V , hW ) = HomOuv(X) (U ∪ V, W ).
These two sets do not generally have the same cardinality. Now, we could remedy
this particular failure, as some authors do, by using a variant of Ouv(X) where
we consider not only open sets of X, but all maps f : Y → X which are local
homeomorphisms. From our point of view, there is no reason to do this. We are only
using Ouv(X) as a stepping stone to get to another category; we do not particularly
care about the category Ouv(X) or the behaviour of its Yoneda functor.
5. SHEAVES
11
4. Generators
Recall that a set G of objects of a category C is called a generating set if, for
any parallel C morphisms f, g : C → D, the following are equivalent:
(1) f = g
(2) f h = gh for every G ∈ G and every h ∈ HomC (G, C). That is,
f∗ , g∗ : HomC (G, C) → HomC (G, D)
are equal for every G ∈ G.
If C has direct sums, then G is a generating set iff the “evaluation map”
M
M
G→C
G∈G HomC (G,C)
is an epimorphism for every C ∈ C. An object G of C is called a generator if {G}
is a generating set. Dually, G is called a cogenerating set if G is a generating set for
Cop .
The paradigm example here is: any nonempty set is a generator of Ens.
Lemma 4.1. The representable presheaves {hU : U ∈ Ouv(X)} form a generating set for PSh(X).
Proof. Two morphisms f, g : F → G in PSh(X) are equal iff f (U ), g(U ) :
F (U ) → G (U ) are equal for every U ∈ Ouv(X). On the other hand, f (U ), g(U )
are identified with
f∗ , g∗ : HomPSh(X) (hU , F ) → HomPSh(X) (hU , G )
by (3.1).
5. Sheaves
Let U = {Ui : i ∈ I} be a family of open subsets of X with union U (a cover of
U ). A presheaf F is separated for U if the product of the restriction maps
Y
F (U ) →
F (Ui )
i∈I
is injective (that is, if a section of F is completely determined by its restrictions
to the open sets in a cover). F has the gluing property (or descent property) with
respect to U if, for any set of sections {si ∈ F (Ui ) : i ∈ I} with si |Uij = sj |Uij for
all i, j ∈ I, there is a section s ∈ F (U ) with s|Ui = si for all i ∈ I. F is separated
(resp. has the gluing property) if it is separated (resp. has the gluing property) for
every such family U. A presheaf is a sheaf if it is separated and it has the gluing
property.
12
I. SHEAVES
We can sum this up by saying that F is a sheaf if and only if
Y
Y
(5.1)
F (U ) →
F (Ui ) ⇒
F (Uij )
i∈I
(i,j)∈I×I
is an equalizer diagram of sets for any U ∈ Ouv(X) and any cover U = {Ui : i ∈ I}
of U .
Remark 5.1. If F is a separated presheaf, then the set F (∅) is punctual (has
one element) because the empty set is covered by the empty cover and the empty
product in the category of sets is punctual.
The following criterion is often useful:
Lemma 5.2. For F ∈ PSh(X), the following are equivalent:
(1) F is a sheaf.
(2) F commutes with filtered inverse limits and has the sheaf property for covers
of the form U = {U, V }. That is,
(a) For any filtered union U = ∪i Ui of open sets of X, the natural map
F (U ) → lim F (Ui )
←−
is an isomorphism, and
(b) For any U, V ∈ Ouv(X), the diagram
F (U ∪ V ) → F (U ) × F (V ) ⇒ F (U ∩ V )
is an equalizer diagram.
Proof. (1) =⇒ (2). The natural map F (U ) → lim F (Ui ) is monic because F
←−
is separated for the cover {Ui } of U and epic because F has the gluing property for
this cover.
(2) =⇒ (1). Let U = {Ui : i ∈ I} be a family of open subsets of X with union
U . To prove F is separated for U, suppose s, t ∈ F (U ) have the same restriction to
every Ui . By Zorn’s Lemma and the fact that F has the sheaf property for filtered
unions, there is a maximal J ⊆ I such that s and t have the same restriction to
V := ∪i∈J Ui . If J 6= I, then there is an i ∈ I \ J and, using the separation of F for
the cover {Ui , V } we find that s and t agree on V ∪ Ui , contradicting maximality of
J. The gluing property is proved similarly.
Corollary 5.3. If X is a noetherian topological space, then F ∈ PSh(X) is
a sheaf iff
F (U ∪ V ) → F (U ) × F (V ) ⇒ F (U ∩ V )
is an equalizer diagram for every U, V ∈ Ouv(X).
Proof. By definition, X is noetherian iff any filtered direct limit system in
Ouv(X) is essentially constant, so any presheaf F trivially commutes with filtered
inverse limits.
6. STALKS
13
A morphism of sheaves is defined so that the category Sh(X) of sheaves on X
forms a full subcategory of the category PSh(X).
Lemma 5.4. Every representable presheaf hU is a sheaf and
{hU ∈ Sh(X) : U ∈ Ouv(X)}
is a generating set for Sh(X).
Proof. The first statement is straightforward, and the second is obvious from
(4.1).
6. Stalks
The stalk of a presheaf F at a point x ∈ X is
Fx := lim
F (U ).
−→
U 3x
The direct limit is taken over open neighborhoods of x and restriction maps between
them. Given a section s ∈ F (U ) and a point x ∈ U we let sx ∈ Fx denote the
image of s under the natural map
F (U ) → lim
F (V ) = Fx .
−→
V 3x
Observe that a presheaf F is separated if and only if, for any object U of Top(X)
and any s, t ∈ F (U ), we have s = t if and only if sx = tx for all x ∈ U . A morphism
of presheaves f : F → G induces a morphism fx : Fx → Gx on stalks for each
x ∈ X by taking the direct limit of the morphisms f (U ) : F (U ) → G (U ) over all
open neighborhoods U of x.
For example, the stalk hU,x of the representable sheaf hU at x is a punctual set
if x ∈ U and empty otherwise.
In contrast to the situation for presheaves, the stalks of a map f : F → G of
sheaves are a good reflection of f :
Lemma 6.1. For a morphism f : F → G in Sh(X), the following are equivalent:
(1) f is a monomorphism in Sh(X).
(2) f (U ) : F (U ) → G (U ) is a monomorphism in Ens for every U ∈ Ouv(X).
(3) fx : Fx → Gx is a monomorphism in Ens for every x ∈ X.
The following are equivalent:
(4) f is an epimorphism in Sh(X).
(5) fx : Fx → Gx is an epimorphism in Ens for every x ∈ X.
The following are equivalent:
(6) f is an isomorphism in Sh(X).
(7) f (U ) : F (U ) → G (U ) is an isomorphism in Ens for every U ∈ Ouv(X).
(8) fx : Fx → Gx is an isomorphism in Ens for every x ∈ X.
14
I. SHEAVES
Proof. (3) =⇒ (2) If f (U )(s) = f (U )(t), then fx (s(x)) = fx (t(x)) for every
x ∈ U . Since the maps on stalks are monic, we have sx = tx for all x ∈ U , hence
s = t because a sheaf is separated, so f (U ) is monic.
(2) =⇒ (3) A filtered direct limit of monic maps of sets is monic.
(2)⇐⇒(1) By definition, f is monic in Sh(X) iff
f∗ : HomSh(X) (H , F ) → HomSh(X) (H , G )
is a monomorphism of sets for every H ∈ Sh(X). In any category, it suffices to
check this for H running over a generating set, so by (4.1), we conclude that f is
monic iff
f∗ : HomSh(X) (hU , F ) → HomSh(X) (hU , G )
is monic for every U ∈ Ouv(X). But this map of sets is identified with the map
f (U ) by Yoneda (3.1).
(5) =⇒ (4) Suppose g1 , g2 : G → H satisfy g1 f = g2 f . Then g1,x fx = g2,x fx for
every x ∈ X, hence g1,x = g2,x for every x ∈ X because f is surjective on stalks. We
conclude g1 = g2 using separation for H .
(4) =⇒ (5) Since f is epic, by definition, the map
f ∗ : HomSh(X) (G , H ) → HomSh(X) (F , H )
is a monomorphism of sets for every H ∈ Sh(X). If H = x∗ A (c.f. (3) below) for
A ∈ Ens, then this monomorphism is identified with
fx∗ : HomEns (Gx , A) → HomEns (Fx , A).
Since this is a monomorphism for every x, A, fx is an epimorphism in Ens for every
x.
(8) =⇒ (7) Consider a t ∈ G (U ). Since the maps on stalks are epic, there are
neighborhoods Ux of every x ∈ U and sections s(x) ∈ F (Ux ) such that f (Ux )(s(x)) =
t|Ux . After restricting to U ∩ Ux , we may assume every Ux is contained in U , so that
the Ux form a cover of U . Since f commutes with restriction maps, we have
f (Ux ∩ Uy )(s(x)|Ux ∩Uy ) = t|Ux ∩Uy = f (Ux ∩ Uy )(sy |Ux ∩Uy ).
Since each f (Ux ∩ Uy ) is monic by ((3) =⇒ (2) above), the s(x) agree on overlaps,
so they glue, by the sheaf property for F , to an s ∈ F (U ). It then follows that
f (U )(s) = t because this equality can be checked on each Ux by separation for G .
(6)⇐⇒(7) Obvious.
(7)⇐⇒(8) A map of sets is an isomorphism iff it is both epic and monic, so this
follows from the previous results.
7. SHEAFIFICATION
15
7. Sheafification
The main goal of this section is to prove that Sh(X) is a reflexive subcategory
of PSh(X), i.e. the inclusion functor Sh(X) → PSh(X) has a left adjoint, called
sheafification.
For a presheaf F , we will construct a sheaf F + (the sheafification of F ) and a
morphism of presheaves F → F + , such that any morphism of presheaves F → G
with G a sheaf will factor uniquely through F → F + . We set F + (U ) equal to the
set of all
Y
t = (t(x))x∈U ∈
Fx
x∈U
satisfying the following local coherence condition: There are a cover U = {Ui } of U
and sections s(i) ∈ F (Ui ) such that t(x) = s(i)x ∈ Fx for all x ∈ Ui .
The presheaf F + has natural restriction maps given by restriction of functions
and it is easy to check that F + is a sheaf. Indeed, we may actually glue functions
together in the usual sense if they agree on overlaps. The construction is functorial
because a map of presheaves f : F → G induces maps of stalks fx : Fx → Gx which
we may use to define maps (pullback) between the sets of functions
f + (U ) : F + (U ) → G + (U )
as above. The map f + (U ) takes a function t satisfying the local coherence condition
to another such function because if s(i) ∈ F (Ui ) witness it for t then f (Ui )(s(i)) ∈
G (Ui ) witness it for f + (U )(t).
Notice that the map F → F + induces an isomorphism on every stalk. It then
follows from (6.1) that if F is already a sheaf, then the natural map F → F + is an
isomorphism. In particular, if F → G is a morphism of presheaves with G a sheaf,
then we obtain the claimed factorization F → F + → G + ∼
= G . This factorization
is unique because one can check easily that a morphism of sheaves is determined by
its values on stalks.
The fundamental facts about sheafification are summed up in the following
Theorem 7.1. The inclusion functor Sh(X) → PSh(X) commutes with arbitrary inverse limits and has a left adjoint: the sheafification functor
PSh(X) → Sh(X)
F 7→ F + .
The adjunction morphism F + → F is an isomorphism for any sheaf F .
Proof. We proved the statements about adjointness in the discussion above.
Saying that the forgetful functor commutes with inverse limits means that an inverse
limit of sheaves (taken in the category of presheaves) is already a sheaf, which can
be checked directly from the fact that the sheaf property is defined in terms of a
certain diagram being an equalizer diagram (a certain kind of inverse limit) and the
fact that “inverse limits commute amongst themselves”.
16
I. SHEAVES
Corollary 7.2. The category Sh(X) has all small direct and inverse limits.
The sheafification functor commutes with arbitrary direct limits and finite inverse
limits.
Proof. We have already mentioned that Sh(X) has all inverse limits and that
these coincide with the inverse limit taken in PSh(X). Using the adjointness properties of the sheafification functor, it follows from a formal exercise in category theory
that the direct limit of a direct system of sheaves i 7→ Fi can be obtained by first
taking the direct limit in the category of presheaves (c.f. (2.1)), then applying the
sheafification functor to the result.
Sheafification commutes with arbitrary direct limits simply because it is a right
adjoint (formal category theory). To prove that it commutes with finite inverse
limits, let f : C → PSh(X) be a functor with C a finite category (finitely many
morphisms and finitely many objects). We must check that the natural map
+
lim (f : C → PSh(X))
→ lim (f + : C → Sh(X))
←−
←−
is an isomorphism. By (6.1), it is enough to check this on stalks. On the other hand,
stalks commute with sheafification, so this map is identified with the natural map
lim
lim
f (C)(U ) → lim
lim
f (C)(U ),
−→
←−
←−
−→
U 3x C∈C
C∈C U 3x
which is an isomorphism because finite inverse limits commute with filtered direct
limits in the category of sets.
For example, the representable sheaf hX is the terminal object of Sh(X) (and
PSh(X)). Indeed, hX (U ) is punctual (terminal in Ens) for every U ∈ Ouv(X) so
there is clearly a unique map F → hX as there is a unique map F (U ) → hX (U )
for every U ∈ Ens.
8. Espaces étalé
Let F be a presheaf on a space X. We define a topological space Et F , called
the espace étalé of F and a continuous map π : Et F → X as follows: As a set, we
let
a
Et F :=
Fx
x∈X
be the disjoint union of the stalks of F . The set Et F is given the topology where
a basic open subset is a set of the form
Us := {sx ∈ Fx ⊆ Et X : x ∈ U },
where U ∈ Ouv(X), s ∈ F (U ). To show that these sets form the basic opens for a
topology we should show that every sx ∈ Et F is in one of the Us (this is clear) and
8. ESPACES ÉTALÉ
17
that an intersection Us ∩ Vt of such basic opens is again such a basic open. Indeed,
the set
W := {x ∈ U ∩ V : sx = tx }
is open in U ∩ V (hence also in X) because equality at a stalk by definition implies
equality on a neighborhood, and we have Us ∩ Vt = Wr , where r := s|W , or r := t|W
(note that s, t have the same stalk everywhere in W , but they might not be equal
in F (W ) because F might not be a sheaf). Define π : Et F → X so that π −1 (x) =
Fx ⊆ Et F for each x ∈ F . The map π is continuous because
[
π −1 (U ) =
Vs
V ⊆U,s∈F (V )
for any U ∈ Ouv(X).
Given any U ∈ Ouv(X) and any s ∈ F (U ), we define a map φ(U )(s) : U →
Et F by φ(U )(s)(x) := sx ∈ Fx ⊆ Et F . This map clearly sections π : Et F → X
and is continuous because
φ(U )(s)−1 (Vt ) = {x ∈ U : x ∈ V and sx = tx }
= {x ∈ U ∩ V : sx = tx }
is open in U as we argued above. Our maps
φ(U ) : F (U ) → {s ∈ HomTop (U, Et F ) : πs = IdU }
are clearly compatible with restricting to smaller U , hence define a morphism φ from
F to the sheaf of local sections of π : Et F → X (see (11.2)).
Lemma 8.1. For any space X and presheaf F on X, the map φ from F to the
sheaf of local sections of the espace étalé π : Et F → X identifies the latter sheaf
with the sheafification of F . In particular, if F is a sheaf, then F is isomorphic to
the sheaf of local sections of π : Et F → X.
Proof. By the stalk criterion (6.1) it suffices to prove that φ is an isomorphism
on stalks. To see that φx is surjective, consider a local section f : U → Et F of π on a
neighborhood U of x ∈ U . We must prove that f |V agrees with φ(V )(t) : V → Et F
for some t ∈ F (V ) for some neighborhood V of x in U . By definition of the
stalk, we can find some neighborhood W of x in U and some r ∈ F (W ) such that
rx = f (x) ∈ Fx . Since f is continuous, the set
V := f −1 Wr := {y ∈ W ∩ U : ry = f (y)}
is open in U and clearly contains x, hence V and t := r|V are as desired.
To see that φx is injective, suppose s, t ∈ F (U ) for some neighborhood U of x
and the functions φ(U )(s), φ(U )(t) : U → Et X agree on a neighborhood of X. We
have to show that sx = tx ∈ Fx . But this is obvious from the fact that φ(U )(s) and
φ(U )(t) agree at x because
sx = φ(U )(s)(x) = φ(U )(t)(x) = tx .
18
I. SHEAVES
Evidently then, the theory of the espace étalé gives another means of sheafifying
a presheaf. It also gives a means of translating various sheaf-theoretic questions into
questions of topology.
Remark 8.2. Given a map p : W → X, the espace étalé of the sheaf F of
local sections of p is a topological space π : Et F → X over X that generally has
little to do with the space W . One only knows that the sheaf of local sections of
π is isomorphic to the sheaf of local sections of p. The map π is built solely from
the sheaf of local sections of p, but one cannot hope to recover the map p from
this sheaf. Another very special property of the espace étalé π is that the map
[f ] 7→ f (x) identifies germs of local sections of π near x with the fiber π −1 (x). For a
general space p : W → X over X, the map [f ] 7→ f (x) from germs of local sections
of p near x to the fiber p−1 (x) is neither surjective nor injective.
The formation of the espace étalé is functorial in F , so we may think of Et as a
functor
Et : Sh(X) → Top/X.
(It is actually defined on PSh(X), but we saw above that the éspace étalé of a
presheaf is naturally isomorphic to the espace étale of its sheafification.) Formation
of the espace étalé is also functorial in X, in a sense that we will make precise
in Section 8 once we discuss the functorial dependence of the category of sheaves
Sh(X) on X.
9. Sieves
The following categorical constructions are used to define sheaves in more general settings. The constructions are not needed elsewhere in any serious way, though
some of the terminology will be natural in our study of Čech cohomology in Chapter VIII.
Given a category C, a sieve in C is a full subcategory U ⊆ C such that, for any
U ∈ U and any V ∈ C,
HomC (V, U ) 6= ∅ =⇒ V ∈ U.
Let Crib(C) denote the set of sieves in C. Crib(C) has a natural ordering by
inclusion. A set of objects G ⊆ C determines a sieve—namely the sieve consisting of
all C ∈ C which admit a morphism to some G ∈ G. For a sieve U and subset G ⊆ U,
we say that G is a generating set for U if the sieve generated by G (which is clearly
contained in U) is equal to U. For an object X ∈ C, we set Crib(X) := Crib(C/X).
The trivial sieve is the sieve in Crib(X) generating by {IdX } (i.e. the entire category
C/X).
9. SIEVES
19
Given a sieve U ∈ Crib(Y ), and C morphisms Y → Z, W → Z, let W ×Z U ∈
Crib(W ) be the sieve consisting of all V → W fitting into a commutative diagram
V
/
U
Y
W
/
Z
for some (U → Y ) ∈ U.
For an object Y ∈ C, a sieve U ∈ Crib(Y ), and a family of sieves
V = {V(U ) ∈ Crib(U ) : U ∈ U},
let V/U ∈ Crib(Y ) be the sieve consisting of all morphisms W → Y which can be
factored W → V → U → Y with (U → Y ) ∈ U and (V → U ) ∈ V(U ).
For a topological space X, and U ∈ Ouv(X), notice that U ∈ Crib(U ) is
nothing more than a family of open subsets of U closed under passage to smaller
open subsets. We declare a sieve U ∈ Crib(U ) to be a covering sieve if U is a cover
of U , and we let Cov(U ) denote the set of covering sieves (it inherits an ordering
from Crib(U )). A covering sieve U ∈ Cov(U ) is nothing more than a cover of U
closed under passage to smaller sets, and a generating set for U is a subset of G ⊂ U
such that every V ∈ U is contained in some G ∈ G . This clearly implies that G is a
subcover, but not every subcover is a generating set—G must contain all the largest
open sets in U.
The covering sieves in Ouv(X) enjoy the following properties:
(1) For all U ∈ Ouv(X), the trivial sieve is in Cov(U ).
(2) For all Ouv(X) morphisms Y → Z, W → Z and all U ∈ Cov(Y ), W ×Z
U ∈ Cov(W ).
(3) For all Y ∈ Ouv(X), all U ∈ Cov(Y ), and all families
V = {V(U ) ∈ Cov(U ) : U ∈ U},
V/U ∈ Cov(Y ).
We remark that the sheafification of F ∈ PSh(X) can be constructed as follows.
First define F 0 ∈ PSh(X) by setting


Y
Y
F 0 (U, U ) := lim 
F (Ui ) ⇒
F (Ui ∩ Uj )
←−
Ui ∈U
(i,j)
for a covering sieve
U = {Ui : i ∈ I} ∈ Cov(U ).
Next note, for
U 0 = {Ui : i ∈ I 0 } ⊆ U,
20
I. SHEAVES
with U 0 ∈ Cov(U ) (i.e. for a Cov(U )op morphisms U → U 0 ), we have an obvious
map F 0 (U, U ) → F 0 (U 0 , U ) induced by the restriction map
Y
Y
F (Ui ) →
F (Ui ).
i∈I 0
i∈I
Set
F 0 (U ) :=
lim
−→
F 0 (U, U ).
U ∈Cov(U )op
Note that Cov(U )op is filtered: any two covering sieves have a common refinement.
Note that, for an Ouv(X) morphism V → U , we have a natural morphism F 0 (U ) →
F 0 (V ) induced by the functor
Cov(U ) → Cov(V )
U 7→ V ×U U.
We leave it as an exercise to check that
(1) For any F ∈ PSh(X), F 0 is separated.
(2) For any separated F ∈ PSh(X), F 0 is a sheaf.
(3) For any F ∈ PSh(X), there is a natural isomorphism (F 0 )0 = F + .
Notice that this construction avoids the use of stalks.
10. Properties of Sh(X)
In this section we establish some facts about the category Sh(X) that will allow
us to work more directly with Sh(X) as a category per se, without so much explicit
reference to the topological space X.
Lemma 10.1. Epimorphisms in Sh(X) are closed under base change.
Proof. Base change commutes with stalks (finite inverse limits of sets commute
with filtered direct limits), so by (6.1) we reduce to proving that epimorphisms in
Ens are closed under base change, which is easy.
Theorem 10.2. A morphism f : F → G in Sh(X) is an epimorphism iff, for
every H ∈ Sh(X), the diagram
HomSh(X) (G , H ) → HomSh(X) (F , H ) ⇒ HomSh(X) (F ×G F , H )
is an equalizer diagram of sets.
Proof. Injectivity of the left map in the diagram for all H is the definition
of “epimorphism”, so the implication (⇐=) is obvious, and to establish ( =⇒ ), it
remains only to show the following: suppose g : F → H is a Sh(X) morphism
such that gπ1 = gπ2 : F ×G F → H , then g = hf for some h : G → H . The
fastest way to prove this is to use stalks: the theorem we are trying to prove is
clear in Ens and stalks preserve epimorphism (6.1) and commute with base change,
10. PROPERTIES OF Sh(X)
21
so we at least have a (unique) map hx : Gx → Hx (for each x ∈ X) such that
gx = hx fx . Since
Q H is a sheaf, we can view a section t ∈ H (U ) as an element
t = (t(x)) ∈ x∈U Hx satisfying the local coherence condition (c.f. (7)). For U ∈
Ouv(X), and s ∈ G (U ), it remains only to prove that h(U )(s)(x) := hx (sx ) actually
satisfies the local coherence condition (the desired equality g = hf will then follow
immediately by checking on stalks). To check local coherence of h(U )(s) at a point
y ∈ U , we use surjectivity of fy to find an open neighborhood V of y in U and a
section t ∈ F (V ) such that f (V )(t) = s|V (equivalently, fz (tz ) = sz for all z ∈ V ).
Then g(V )(t) ∈ H (V ) witnesses local consistency:
g(V )(t)z = gz (tz ) = hz fz (tz ) = hz (sz ) = h(U )(s)(z)
for all z ∈ V .
In highfalutin language, the next result says that the “usual covers of X are
cofinal in the category of epimorphisms to the terminal object of Sh(X).”
Lemma 10.3. Let G ∈ Sh(X). The morphism G → hX to the terminal object is
an epimorphism iff there is a cover U = {Ui : i ∈ I} of X and a morphism
a
hUi → G .
i∈I
Proof. ( =⇒ ). By (5.4), the hU form a generating set for Sh(X), so we can
certainly find an epimorphism of the form
a
a
hUi → G
Ui ∈Ouv(X) j∈Ji
(take Ji = HomSh(X) (hUi , G ) = G (Ui ) and use the evaluation map, for example). If
x ∈ X is any point, then since
(1) the composition of epimorphisms is an epimorphism,
(2) coproducts (or any direct limits) commute with stalks, and
(3) an epimorphism in Sh(X) induces an epimorphism on stalks (6.1),
we have an epimorphism
a
a
hUi ,x → hX,x .
Ui ∈Ouv(X) j∈Ji
Note hX,x is punctual, so this simply says there is some Ui(x) such that hUi(x) ,x 6= (i.e.
x ∈ Ui(x) , see (6)). The open sets Ui(x) hence clearly cover X and the coproduct of
the hUi(x) clearly maps to G by first mapping to the “big” coproduct above, which
can be done in many ways.
(⇐=). Check on stalks.
22
I. SHEAVES
11. Examples
For example, if X is a one point space, then PSh(X) is isomorphic to the arrow
category of Ens via the map
PSh(X) → Ens·→·
F 7→ (F (X) → F (∅)),
while Sh(X) is equivalent to Ens via the map
Sh(X) → Ens
F 7→ F (X),
(c.f. Remark 5.1).
Example 11.1. For any topological space X, the presheaf C (X, R) which associates to every U ∈ Ouv(X) the set of continuous real-valued functions on U
(with the obvious restriction maps) is clearly a sheaf because continuity is local
in nature (can be checked on each open set in a cover). This sheaf contains the
presheaf C b (X, R) of bounded functions. This latter presheaf is separated, but it
is not generally a sheaf because being bounded is global in nature and one cannot
glue together functions which are locally bounded into a function which is globally
bounded. Indeed, any continuous map f : X → R is locally bounded.
Example 11.2. Generalizing the previous example, for any topological spaces
X, Y , the presheaf U 7→ HomTop (U, Y ) (with the obvious restriction maps) is a sheaf
on X because continuity is local. Similarly, if π : W → X is a map of topological
spaces the presheaf
U 7→ {s ∈ HomTop (U, W ) : πs = IdU }
of local sections of π is a sheaf on X. We will see in Section ?? that every sheaf is
isomorphic to a sheaf of this form.
12. Exercises
Exercise 12.1. Let X be a topological space, F ∈ PSh(X), U a cover of X,
V a refinement of U. Show that, if F is a sheaf for V, then it is a sheaf for U.
Exercise 12.2. A basis is a subset (full subcategory) B ⊆ Ouv(X) such that
every U ∈ Ouv(X) is a union (direct sum) of open sets in B. Set PSh(B) =
HomCat (Bop , Ens). Say F ∈ PSh(B) is a sheaf if, for every U ∈ B and every cover
{Ui ∈ B : i ∈ I} of U , the diagram
Y
Y
F (U ) →
F (Ui ) ⇒
F (Uij )
i∈I
(i,j)∈I×I
is an equalizer diagram of sets. Let Sh(B) be the full subcategory of PSh(B)
consisting of sheaves. Observe that the functor PSh(X) → PSh(B) induced by the
inclusion Bop ⊆ Ouv(X)op takes Sh(X) into Sh(B). Show that Sh(X) → Sh(B)
is an equivalence of categories.
CHAPTER II
Direct and Inverse Images
So far we have only considered the category of (pre) sheaves on a fixed topological
space X. In this section, we study the functoriality of these categories in X.
1. Functoriality of Ouv(X)
First notice that Ouv(X) is contravariantly functorial in X. The functor
Ouv : Espop → Cat
X 7→ Ouv(X)
is given on a morphism f : X → Y in Esp by
Ouv(f ) : Ouv(Y ) → Ouv(X)
U 7→ f −1 (U ).
The functor Ouv(f ) commutes with direct limits (unions) and finite inverse limits
(intersections). Furthermore, Ouv(f ) takes covers to covers. Though we do not
wish to define this, Ouv(f ) is a continuous morphism of sites.
2. Direct images
Let f : X → Y be a continuous map of topological spaces, F a presheaf on X.
The direct image presheaf f∗ F is the presheaf on Y given by
f∗ F : Ouv(Y )op → Ens
U 7→ F (f −1 (U )).
The restriction maps for this presheaf f∗ F (U ) → f∗ F (V ) are given by the restriction maps F (f −1 (U )) → F (f −1 (V )) for F . In other words, f∗ F : Ouv(Y )op →
Ens is the composition of Ouv(f )op : Ouv(Y )op → Ouv(X)op and F : Ouv(X)op →
Ens.
3. Skyscraper sheaves
For example, if x : {x} ,→ X is the inclusion of a point, and A is a set (regarded
as a sheaf on {x} as in (11)), then x∗ A is the sheaf on X given by
A,
x∈U
U 7→
{0},
otherwise
23
24
II. DIRECT AND INVERSE IMAGES
A sheaf on X isomorphic to such a sheaf is called a skyscraper sheaf. The stalks of
x∗ A are given by
A,
y ∈ {x}−
(x∗ A)y =
{0},
otherwise
If F is a sheaf, then so is f∗ F because f∗ F has the sheaf property with respect
to a cover U if and only if F has it with respect to the cover f −1 U, as is clear from
the definitions. The construction f∗ is clearly functorial in F .
Example 3.1. The following example shows why direct limits of sheaves do
not typically commute with sections. Let X be any infinite set with the discrete
topology and let ZX be the constant sheaf. The presheaf direct sum ⊕x∈X x∗ Z of all
stalks of Z is given by
M
U 7→
Z.
x∈U
This presheaf if clearly separated, but it does not have the gluing property for the
cover of X by singleton sets. The sheafification ⊕x∈X x∗ Z of this presheaf is given
by
Q
U 7→ x∈U Z
because these are the functions to the stalks which (trivially) satisfy the local coherence condition. If, on the other hand, X is given the cofinite topology (a noetherian
topology), then this presheaf if already a sheaf.
4. Inverse images
For a continuous map f : X → Y , and a presheaf G on Y , then the inverse
−1
G ∈ PSh(X) is defined by
image presheaf fpre
−1
fpre
G : Ouv(X)op → Ens
lim
G (V ),
−→
U 7→
V ⊇f (U )
where the direct limit is taken over open subsets V of Y containing f (U ) with
respect to the restriction maps of G (this is a filtered subcategory of Ouv(Y )). Note
V ⊇ f (U ) iff U ⊂ f −1 (V ), so the direct limit is over the category U ↓ Ouv(Y ),
whose objects are pairs (V, g) where V ∈ Ouv(Y ) and g : U → Ouv(f )(V ) is
morphism in Ouv(X). A morphism (V, g) → (V 0 , g 0 ) in U ↓ Ouv(Y ) is a morphism
i : V → V 0 in Ouv(X) making the diagram
U
g0
g
y
Ouv(f )(V )
Ouv(f )(i)
/
%
Ouv(f )(V 0 )
4. INVERSE IMAGES
25
in Ouv(Y ) commute.1
For any U ∈ Ouv(X), we have f (f −1 (U )) ⊆ U so a section s ∈ G (U ) yields
−1
−1
a section fpre
(s) ∈ (fpre
G )(f −1 U ) given by the class of s in the direct limit system
−1
−1
defining (fpre
G )(f −1 (U )). It is clear from the definition of fpre
G that its stalks are
given by
−1
G )x = Gf (x) .
(fpre
The stalk of a presheaf F on a space X at a point x ∈ X is nothing but the
inverse limit of F under the inclusion map x : {x} → X.
−1
does not generally take sheaves to sheaves. We are led to define
The functor fpre
f −1 : Sh(Y ) → Sh(X)
−1
G 7→ (fpre
G )+
−1
and the sheafification functor:
by composing fpre
−1
f −1 G := (fpre
G )+ .
As for the presheaf inverse image, a section s ∈ G (U ) determines a section f −1 s ∈
−1
−1
(f −1 G )(f −1 (U )), namely, the image of fpre
s ∈ (fpre
G )(f −1 (U )) under the sheafification morphism
−1
(fpre
G )(f −1 (U )) → (f −1 G )(f −1 (U )).
Examining the construction of sheafification, we see that a section of (f −1 G )(V ) can
be viewed as an element
Y
Gf (x)
s = (s(x)) ∈
x∈V
satisfying the local coherence condition: For every x ∈ V , there is a neighborhood
W of x in V , a neighborhood U of f (W ) in Y , and a section t ∈ G (U ) such that
s(x0 ) = tf (x0 ) for every x0 ∈ W .
−1
The formation of fpre
and f −1 are both functorial in f in the sense that for
f : X → Y and g : Y → Z the functors
−1 −1
fpre
gpre , (gf )−1
pre : PSh(Z) → PSh(X)
are naturally isomorphic, as are the functors
f −1 g −1 , (gf )−1 : PSh(Z) → PSh(X).
(In particular, the case where X = {y} is a point of Y gives the stalk formula
−1
(gpre
G )y = Gg(y) mentioned above.)
−1
Theorem 4.1. The functor fpre
: PSh(Y ) → PSh(X) is left adjoint to f∗ :
−1
PSh(X) → PSh(Y ). The functor fpre
commutes with arbitrary direct limits and
finite inverse limits. The functor f∗ commutes with arbitrary inverse limits.
1Of
course, the usage of the category U ↓ Ouv(Y ) is overly complex, but it generalizes well and
allows us to link up with various general category theoretic techniques (comma categories, Kan
extension).
26
II. DIRECT AND INVERSE IMAGES
Proof. The adjointness is a special case of the Kan Extension Theorem and is
straightforward to check. The other statements are formal consequences of adjoint−1
commutes with finite inverse limits. This is proved by
ness, except the fact that fpre
noting that the category of neighborhoods of any subspace is filtered, and filtered
direct limits commute with finite inverse limits in Ens.
Corollary 4.2. The functor f −1 : Sh(Y ) → Sh(X) is left adjoint to f∗ :
Sh(X) → Sh(Y ). The functor f −1 commutes with arbitrary direct limits and finite
inverse limits. The functor f∗ commutes with arbitrary inverse limits.
Proof. This follows formally from the theorem using the adjointness and exactness properties of the sheafification functor (7.1), (7.2).
Corollary 4.3. Inverse images commute with sheafification: For f : X →
Y and any presheaf F ∈ PSh(Y ) we have a natural isomorphism f −1 (F + ) =
−1
F )+ in Sh(X).
(fpre
Proof. This is a consequence of Yoneda’s Lemma and the adjointness results
just established: For any G ∈ Sh(X) we have
HomSh(Y ) (f −1 (F + ), G ) = HomSh(Y ) (F + , f∗ G )
= HomPSh(Y ) (F , f∗ G )
−1
= HomPSh(X) (fpre
F,G)
−1
= HomPSh(X) ((fpre
F )+ , G ).
Remark 4.4. Suppose i : U ,→ X is the inclusion of an open set and F is a
sheaf on X. Then for any V ∈ Ouv(U ), V is in Ouv(X) and is cofinal among
−1
open neighborhoods of i(V ) in X, hence i−1
pre F (V ) = F (V ). Furthermore, ipre F
is already a sheaf, so we have i−1 F (V ) = F (V ). Evidently then, i−1 F is nothing
but the restriction of the functor F : Ouv(X)op → Ens to the full subcategory
Ouv(V )op ⊆ Ouv(X)op . Accordingly, we often write F |U instead of i−1 F .
5. Sheaf Hom
Consider the functor
(5.1)
H om( ,
) : PSh(X)op × PSh(X) → PSh(X)
(F , G ) 7→ H om(F , G ),
where H om(F , G ) is the presheaf
H om(F , G )(U ) := HomPSh(U ) (F |U , G |U ).
We use the notation F |U of Remark 4.4.
Lemma 5.1. If G ∈ Sh(X), then the presheaf H om(F , G ) is a sheaf.
6. CONSTANT SHEAVES
27
Proof. Say {Ui → U } is a cover. If f, g Hom(F |U , G |U ) agree when restricted
to Ui for every i, then for any V ∈ Ouv(U ) and any s ∈ F (V ), we have f (V )(s)|Ui ∩V =
g(V )(s)|Ui ∩V for all i and {Ui ∩ V → V } is a cover, so f (V )(s) = g(V )(s) by separation for G . This holds for any s, V , so f = g, hence H om(F , G ) is separated.
Given maps fi ∈ Hom(F |Ui , G |Ui ) which agree on overlaps, define f : F |U →
G |U as follows. For any V ∈ Ouv(U ), and any s ∈ F (V ) consider the fi (Ui ∩
V )(s|Ui ∩V ) ∈ G (Ui ∩ V ). Since fi (Ui ∩ Uj ∩ V ) = fj (Ui ∩ Uj ∩ V ), these agree on
overlaps, so by the gluing property for G there is an f (V )(s) ∈ F (V ) restricting to
fi (Ui ∩ V )(s|Ui ∩V ) in G (Ui ∩ V ) for every i. To show that the maps
f (V ) : F (V ) → G (V )
s 7→ f (V )(s)
actually define a map of presheaves F |U → G |U , it remains to show that
/
F (V )
F (W )
/
G (V )
G (V )
commutes for any morphism W → V in Ouv(U ). This follows from separation for
G.
In light of the lemma, we may view (5.1) as a functor
H om( ,
) : Sh(X)op × Sh(X) → Sh(X)
(F , G ) 7→ H om(F , G ).
The sheaf H om(F , G ) is called sheaf hom from F to G .
For a space X and an open set i : U ,→ X, the formulae
H om(hX , F ) = F
H om(hU , F ) = i∗ (F |U )
are clear from the Yoneda Lemma (3.1). For a continuous map f : X → Y , the
adjunction (f −1 , f∗ ) “sheafifies” to yield a natural isomorphism
f∗ H omX (f −1 F , G ) = H omY (F , f∗ G ).
6. Constant sheaves
An extreme case of the formation of inverse images occurs when f is the map
from X to a one point space Y . If we identify Sh(Y ) with Ens as in (11), then the
−1
presheaf inverse image functor fpre
is identified with the functor A 7→ Apre
X , where
pre
AX is the constant presheaf U 7→ A, and the inverse image functor f −1 is the
functor A 7→ AX , where
AX : Ouv(X)op → Ens
U 7→ Aπ0 (U ) ,
28
II. DIRECT AND INVERSE IMAGES
where π0 (U ) is the set of connected components of U . A sheaf isomorphic to some
AX is called a constant sheaf.
7. Representable case
It is instructive to describe f∗ F and f −1 F in the case where F = hU is a
representable sheaf. The equalities
−1
hU
f −1 hU = fpre
= hf −1 (U )
are clear for any morphism f : X → Y and any U ∈ Ouv(Y ). The set (f −1 hU )(V )
is punctual if f (V ) ⊆ U (equivalently V ⊆ f −1 (U )) and empty otherwise. Similarly,
for V ∈ Ouv(X), the sheaf f∗ hV assigns the punctual set to U ∈ Ouv(Y ) if
f −1 (U ) ⊆ V , else (f∗ hV )(U ) is empty.
8. Inverse images and espaces étalé
The formation of inverse images is compatible with the construction of the espace
étalé (§8).
Lemma 8.1. Let f : X → Y be a continuous map of topological spaces, F a
presheaf on Y , π : Et F → Y its espace étalé. Then π1 : X ×Y Et F → X is
−1
naturally isomorphic to the espace étalé of fpre
F . Similarly, if F is a sheaf, π1 is
−1
naturally isomorphic to the espace étalé of f F .
Proof. Recall from Section ?? that, as a set,
a
a
−1
−1
Et fpre
F =
(fpre
F )x =
Ff (x) ,
x∈X
x∈X
which we prefer to think of as the set of pairs (x, s) where x ∈ X and s ∈ Ff (x) .
On the other hand, X ×Y Et F is exactly the same set of pairs (the preimage of
f (x) ∈ Y under π : Et F → Y is Ff (x) by construction of π). This identification
clearly respects the two maps to X, so it remains to prove that it is continuous.
−1
A basic open set in Et fpre
F is a set of the form
−1
Ut = {(x, s) ∈ Et fpre
F : x ∈ U, s = tx ∈ Ff (x) }
−1
where U is an open subset of X and t ∈ (fpre
F )(U ). Such a t is represented by
a V ∈ Ouv(Y ) with f (U ) ⊆ V and a section r ∈ F (V ). For x ∈ U one then
has tx = rf (x) , so this open corresponds to the open subset U ×Y Vr under our
identification.
A basic open set of X ×Y Et F is given by U ×Y Vs where U ∈ Ouv(X),
V ∈ Ouv(Y ), s ∈ F (V ) and Vs := {sy ∈ Fy ⊆ Et F : y ∈ V }. But we have
−1
U ×Y Vs = f −1 (V ) ×Y Vs = f −1 (V )fpre
s
−1
under our identification, so this basic open in X ×Y Et F is also open in Et fpre
F.
9. FAITHFULNESS OF X 7→ Sh(X)
29
The statements about sheaves follow from the statements about presheaves because both inverse images and espaces étalé commute with sheafification.
9. Faithfulness of X 7→ Sh(X)
CHAPTER III
Descent
1. Canonical topology
canonical cover = universal effective epimorphism
2. Open maps and variants
open maps, weakly open maps, universal w.o. maps, basic properties (composition, base change)
3. Descent for spaces
4. Descent for sheaves
31
CHAPTER IV
Cohomology
The purpose of this chapter is to give a concise, self-contained treatment of sheaf
cohomology.
1. Abelian categories
2. Enough injectives
3. Homological algebra
basic theory of delta functors and derived functors, Grothendieck spectral sequence, spectral sequence from complexes
4. Abelian Sheaves
For a topological space X, let Ab(X) denote the category of abelian group
objects in Sh(X). An object F ∈ Ab(X) is often called a sheaf of abelian groups or
just an abelian sheaf. Similarly, we let PAb(X) denote the category of abelian group
objects in PSh(X). Note that PAb(X) is isomorphic to the category of functors
from Ouv(X)op to Ab and that this identifies Ab(X) with the full subcategory
consisting of those F : Ouv(X)op → Ab whose composition with the forgetful
functor Ab → Ens is a sheaf. For F ∈ PAb(X) and any inclusion V ⊆ U of open
subsets of X, the restriction map F (U ) → F (V ) is a homomorphism of (abelian)
groups.
The basic theory of sheaves from Chapter ?? translates to the abelian setting in
the same way that one can translate various statements about sets into statements
about (abelian) groups.
Lemma 4.1. The forgetful functor Ab(X) → PAb(X) has a left adjoint (sheafification) which commutes with arbitrary direct limits and finite inverse limits.
Proof. The analogous statements are true of Sh(X) → PSh(X) and preservation of finite inverse limits implies that sheafification takes (abelian) group objects
to (abelian) group objects.
The categories PAb(X) and Ab(X) clearly carry the structure of additive categories. Kernels and cokernels (in fact all limits) are formed “objectwise” in PAb(X)
just as they were in PSh(X); in particular, PAb(X) has all small direct and inverse
33
34
IV. COHOMOLOGY
limits. Consequently, Ab(X) also has all small direct and inverse limits: inverse limits are formed by taking the inverse limit in PAb(X) and noting that it is already
in Ab(X). Indeed, Ab → Ens commutes with inverse limits, so this is obvious
from the corresponding statement about inverse limits in Sh(X). However, to form
direct limits in Ab(X), one must sheafify (i.e. apply the sheafification functor of the
lemma to) the direct limit taken in PAb(X).
Explicitly, the zero object 0 ∈ PAb(X) is the constant presheaf U 7→ {0}. Note
that this is also a sheaf and that it is also the zero object in Ab(X). For any
morphism f : F → G in PAb(X), the natural map
Cok Ker f → Ker Cok f
is an isomorphism because
Cok Ker f (U ) → Ker Cok f (U )
is an isomorphism for each U ∈ Ouv(X) (Ab is an abelian category!). A similar
statement holds in Ab(X) because the natural map there is obtained by sheafifying
this natural map. We have proved:
Proposition 4.2. Ab(X) and PAb(X) are abelian categories with all small
direct and inverse limits.
Observe that Z is a generator for Ab (c.f. §??), and that this is obvious on
category theoretic grounds from the fact that a one element set is a generator for
Ens and the free abelian group functor is left adjoint to the forgetful functor Ab →
Ens. These observations are easily promoted to Ab(X), by the usual method of
bootstrapping through PAb(X).
Lemma 4.3. The forgetful functor PAb(X) → PSh(X) has a left adjoint, the
“free abelian group” functor
PSh(X) → PAb(X)
F 7→ ⊕pre
F Z,
where ⊕pre
F Z ∈ PAb(X) is given by
op
⊕pre
→ Ab
F Z : Ouv(X)
U 7→ ⊕F (U ) Z.
Proof. Obvious.
Corollary 4.4. The forgetful functor Ab(X) → Sh(X) has a left adjoint, the
“free abelian group” functor
Sh(X) → Ab(X)
+
F 7→ (⊕pre
F Z) ,
Proof. Follows from the lemma and the adjointness property of sheafification
(7.1).
4. ABELIAN SHEAVES
35
Theorem 4.5. {⊕pre
hU Z : U ∈ Ouv(X)} (resp. {⊕hU Z : U ∈ Ouv(X)}) is a
generating set for PAb(X) (resp. Ab(X)). PAb(X) and Ab(X) have all small
direct and inverse limits. Finite inverse limits and filtered direct limits commute in
both PAb(X) and Ab(X).
Proof. The first statement is purely formal; let us prove it for Ab(X) for
concreteness. For a sheaf F , we have
HomAb(X) (⊕hU Z, F ) = HomSh(X) (hU , F )
= F (U ).
This is natural in F , so for parallel arrows f, g : F → G , the maps
f (U ), g(U ) : F (U ) → G (U )
(which are equal for all U iff f = g) are identified with
f∗ , g∗ : HomAb(X) (⊕hU Z, F ) → HomAb(X) (⊕hU Z, G ).
The existence of limits was discussed above. The last statement is certainly true of
PAb(X) since these limits are formed objectwise and this statement is true of Ab
(because it is true of Ens and Ab → Ens is conservative and commutes with filtered
direct limits and inverse limits). Hence this is also true of Ab(X) by adjointness
and exactness properties of sheafification.
Corollary 4.6. Every F ∈ PAb(X) (resp. Ab(X)) admits a monomorphism
into an injective object of PAb(X) (resp. PAb(X)).
Proof. This follows formally from the theorem as in Grothendieck’s Tohoku
paper.
The section functors
PAb(X) → Ab
F 7→ Γ(U, F )
commute with all direct and inverse limits, as these are formed “objectwise” in
PAb(X) just as they were in PSh(X). The section functors
Ab(X) → Ab
F 7→ Γ(U, F )
commute with inverse limits (note Ab → Ens commutes with inverse limits, and the
section functors for sheaves commute with inverse limits), but they do not generally
commute with direct limits. That is, Γ(U, F ) preserves kernels, but not cokernels.
Note that Γ(U, ) does, however, preserve finite direct sums since these coincide
with finite direct products. In other words, Γ(U, ) is a left exact morphism of
abelian categories—its right derived functors will be considered in more generality
in §??.
36
IV. COHOMOLOGY
The forgetful functor Ab(X) → PAb(X) is similarly a left exact morphism of
abelian categories. Its nth right derived functor is denoted F 7→ H n (F ) and is
called the nth cohomology presheaf of F . It will be studied further in Chapter ??.
A morphism of topological spaces f : X → Y induces an exact morphism of
abelian categories f −1 : Ab(Y ) → Ab(X) and a left exact morphism of abelian
categories f∗ : Ab(X) → Ab(Y ). The right derived functors Rn f∗ are called the
higher direct images of f . These will be studied further in Section 15.
5. Ringed spaces
Let An(X) be the category of ring objects of Sh(X). For OX ∈ An(X) we
can consider the category Mod(OX ) of modules (“module objects”) over OX . Concretely, OX ∈ An(X) is a functor OX : Ouv(X)op → An whose composition with
the forgetful functor An → Ens is a sheaf. From this point of view, F ∈ Mod(OX )
is a functor F : Ouv(X)op → Ab (whose composition with Ab → Ens is a sheaf)
such that, for each U ∈ Ouv(X), the abelian group F (U ) is equipped with an
OX (U ) module structure making the restriction maps F (U ) → F (V ) linear over
OX (U ) → OX (V ). That is, for s ∈ F (U ) and t ∈ OX (U ), we require
(t · s)|V = t|V · s|V .
Note that Mod(ZX ) is isomorphic to Ab(X). The results of §4 on Ab(X)
carry over, mutatis mutandis, to Mod(OX ). In particular, Mod(OX ) is an abelian
category with all small direct and inverse limits, a set of generators, etc. A pair
(X, OX ) consisting of a topological space X and a sheaf of rings OX ∈ An(X) is
called a ringed space. If OX is clear from context, we will simply write X for (X, OX )
and Mod(X) for Mod(OX ). We will write “X module” instead of “OX module,”
etc. Note that the stalk Fx of an OX module has a natural module structure over
the stalk OX,x of OX .
The following direct proof that Mod(X) has enough injectives is sometimes
useful, especially if we need to get our hands on a supply of easily understood
injectives.
Lemma 5.1. Let x : {x} ,→ X be the inclusion of a point of a topological space
X with a sheaf of rings OX . Let I be an injective OX,x module. Then x∗ I is an
injective OX module.
Proof. We must show F 7→ HomX (F , x∗ I). Note OX,x = x−1 OX , so by the
adjointness (x−1 , x∗ ),
HomX (F , x∗ I) = HomOX,x (Fx , I),
which is exact because the stalk functor is exact and because HomOX,x ( , I) is exact
by definition of injective.
Theorem 5.2. For any ringed space X, the category Mod(X) has enough injectives.
6. SHEAF EXT
37
Proof. Let F ∈ Mod(X). Since Mod(A) has enough injectives for any ring
A, we can choose, for each x ∈ X, a monomorphism Fx ,→ I(x) from the stalk of
F into an injective OX,x module. The adjunction morphism for the adjoint functors
(x−1 , x∗ ) furnishes a natural morphism
F → x∗ I(x) of X modules. The product of
Q
all
these
maps
is
a
map
F
→
x
I(x)
which is easily seen to be monic. Note
x∈X ∗
Q
x∈X x∗ I(x) is a product of injectives and is hence injective.
6. Sheaf Ext
For G ∈ Ab(X) and F ∈ Sh(X), observe that H om(F , G ) (c.f. (5)) inherits
an abelian group object structure from that of G . We may view sheaf Hom as a
functor
H om( ,
) : Sh(X)op × Ab(X) → Ab(X)
(F , G ) 7→ H om(F , G ).
Similarly, if X is equipped with a sheaf of rings OX , then we have an “internal
Hom” or “sheaf Hom” functor
H om( ,
) : Mod(X)op × Mod(X) → Mod(X)
(F , G ) 7→ H om(F , G ).
For any fixed F ∈ ModX , the functor H om(F ,
functors are denoted
G
) is left exact. Its right derived
7→ E xtp (F , G ).
Lemma 6.1. Let X be a ringed space, F ∈ Mod(X). There is an isomorphism
of δ functors from Mod(X) to Mod(Γ(X, OX )) :
H• (X, F ) = Ext• (OX , F )
Proof. It is straightforward to check that mapping s ∈ Γ(X, F ) to the homomorphism ·s ∈ Hom(OX , F ) given by
(·s)(U ) : OX (U ) → F (U )
f 7→ f · s|U
yields an isomorphism Γ(X, F ) = Hom(X, F ). The result now follows since both
δ functors are effaceable, as Mod(X) has enough injectives.
The proof of the following lemma will require several constructions that we have
not yet discussed, but in any case it is natural to state it now.
Lemma 6.2. Local-to-Global Ext Sequence. For F , G ∈ Mod(X), there is
a natural first quadrant spectral sequence
E2p,q = Hp (X, E xtq (F , G ))
=⇒
Extp+q (F , G ).
38
IV. COHOMOLOGY
Proof. Since Γ(X, H om(F , G )) = Hom(F , G ), this will be obtained as a
special case of the Grothendieck spectral sequence as long as we can prove that
H om(F , G ) is Γ-acyclic for any injective sheaf G ∈ Mod(X). Since flasque sheaves
are Γ-acyclic by (10.4), below, it suffices to prove that H om(F , G ) is flasque, so
what we must show is that, for any open set ι : U ,→ X and any homomorphism
f : ι−1 F → ι−1 G , there is a homomorphism F : F → G in Mod(X) with ι−1 F = f .
To do this, we apply the extension by zero functor ι! (to be discussed in (2), below)
to f to get a diagram of solid arrows
ι! ι−1 _F
/ ι! ι−1 G 
5/
G
F
F
in Mod(X), which completes as indicated by injectivity of G since the vertical
arrow is monic. Since ι−1 ι! = Id, the vertical and second horizontal arrows become
isomorphisms upon applying ι−1 so this F is as desired.
7. Supports
Let F be an abelian presheaf on a topological space X, U an open subset of X,
s ∈ F (U ) a section. Let Supp s := {x ∈ U : s(x) 6= 0 ∈ Fx } be the support of s.
The set Supp s is closed in U , hence locally closed in X. For any s, t ∈ F (U ), the
set {x ∈ U : s(x) = t(x)} is an open subset of U .
The closed set
Supp F := {x ∈ X : Fx 6= 0}−
of X is called the support of F . The support of any direct sum or product of sheaves
is the closure of the union of their supports.
A family Φ of closed subsets of a topological space X is called a family of supports
if it satisfies the conditions:
(1) If A ∈ Φ and B is a closed subset of A, then B ∈ Φ.
(2) If A, B ∈ Φ then A ∪ B ∈ Φ .
The first axiom in particular implies that ∅ ∈ Φ. For example, the family Φ of all
closed subsets of X is a family of supports. Another example is the family Φ of all
closed quasi-compact subsets of X (if X is Hausdorff, then the adjective “closed” is
redundant, but this is not true for a general X). For a subspace Z of X, we could
take Φ to be the family of all closed subsets of X contained in Z. We could also take
Φ to be the family of all closed, paracompact subsets of X. We say that a family of
supports Φ has neighborhoods if, it satisfies the additional condition
(3) Every A ∈ Φ has a neighborhood B with B − ∈ Φ.
If, furthermore, Φ satisfies
10. FLASQUE SHEAVES
39
(4) Every A ∈ Φ has a neighborhood B such that B − ∈ Φ and B − is paracompact,
then we say Φ is paracompactifying.
Recall that a space X is called paracompact iff X is Hausdorff and every open
cover of X has a locally finite (open) refinement.
For a subspace Y ⊆ X, and a family of supports Φ on X, the families
Φ ∩ Y := {A ∩ Y : A ∈ Φ}
ΦY := {A ∈ Φ : A ⊆ Y }
are both both families of supports on Y . We emphasize that, in general, these are
very different, so we caution the reader to take care in keeping track of which family
of supports we are using.
If F is an abelian sheaf on X and i : U ,→ X is an open subset of X, then the
−1
global sections with support in Φ of the abelian sheaf F |X
U = i! i F on X are given
by ΓΦ (X, F ) = ΓΦU (U, F ). This is immediate from the definitions.
8. Sheaf cohomology
For any abelian presheaf F on X, and any family of supports Φ on X,
ΓΦ (X, F ) := {s ∈ F (X) : Supp s ∈ Φ}
is a subgroup of Γ(X, F ) (it contains 0 because ∅ ∈ Φ, and it is closed under addition
because Φ is closed under finite unions and passage to smaller closed sets) called the
group of global sections of F with support in Φ. It is easy to see that F 7→ ΓΦ (X, F )
is a left exact functor Ab(X) → Ab. The corresponding right derived functors are
denoted HiΦ (X, F ) and called the cohomology of F with support in Φ. In case Φ is
the family of supports consisting of all closed subsets contained in a closed subspace
Y ⊆ X, HiΦ (X, F ) is usually denoted HiY (X, F ) and is called the cohomology of F
with support in Y .
Unlike the usual global section functor, ΓΦ does not generally commute with
infinite inverse limits, or even with infinite products.
9. Acyclic sheaves
Let X be a topological space, Φ a family of supports on X. A sheaf F ∈ Ab(X)
is called ΓΦ -acyclic iff HnΦ (X, F ) = 0 for n > 0.
(PUT SOMETHING HERE TO THE EFFECT THAT COHOMOLOGY CAN
BE COMPUTED USING GAMMA ACYCLIC RESOLUTIONS)
10. Flasque sheaves
A sheaf F on a topological space X is called flasque iff the restriction map
F (U ) → F (V ) is surjective for every inclusion V ⊆ U of open subsets of X.
40
IV. COHOMOLOGY
Lemma 10.1. The direct image of a flasque sheaf under any morphism of topological spaces is again flasque.
Proof. This is clear from the formula (f∗ F )(U ) = F (f −1 U ) for sections of the
direct image sheaf.
Lemma 10.2. (Gluing sections) Suppose A = {Ai : i ∈ I} is a locally finite
family of subsets of X with union A and we have sections si of an abelian sheaf F
over Ai for each i ∈ I which agree on overlaps (si |Aij = sj |Aij for all i, j ∈ I). Then
there is a unique section s of F over A such that s|Ai = si for each i ∈ I.
Q
Proof. Certainly there is a unique map s : A → x∈A Fx with
Y
s(x) = si (x) ∈ Fx ⊂
Fa
a∈A
for every i ∈ I with x ∈ Ai . The only question is whether s satisfies the local
coherence condition. To see this, fix a ∈ A and use local finiteness to find an open
neighborhood U of a which meets only Ai1 . . . Ain . The sections sij satisfy the local
coherence condition at a, so there are open neighborhoods Uij of a and sections
tij ∈ F (Uij ) such that tij (x) = sij (x) = s(x) for all x ∈ Uij ∩ Aij . Let V be the
intersection of U and the Uij so V is an open neighborhood of a since there are only
finitely many ij . The set W where the restrictions tij |V all agree is an open subset
of V containing a. Let t ∈ F (W ) be the common value. Then t witnesses the local
coherence of s at a.
Theorem 10.3. Let X be a topological space, Φ a family of supports on X. The
class of flasque sheaves satisfies the following properties:
(1) For any short exact sequence of sheaves
0 → F 0 → F → F 00 → 0
on X with F 0 flasque, the map ΓΦ (X, F ) → ΓΦ (X, F 00 ) is surjective.
(2) For any short exact sequence of sheaves
0 → F 0 → F → F 00 → 0
on X with F 0 and F flasque, F 00 is flasque.
(3) A direct summand of a flasque sheaf is flasque.
(4) Any sheaf admits a monomorphism into a flasque sheaf.
If Φ is paracompactifying, then the class of Φ soft sheaves satisfies the same properties.
Proof. The fact that summands of flasque or Φ soft sheaves are again flasque or
Φ-soft follows from the fact that (F ⊕ G )(U ) = F (U ) ⊕ G (U ) for any open U ⊆ X.
Any sheaf embeds in a flasque and Φ-soft sheaf by taking the sheaf of discontinuous
sections (see ).
10. FLASQUE SHEAVES
41
Now assume
0 → F 0 → F → F 00 → 0
is a short exact sequence of sheaves on X with F 0 flasque. We will first show that
Γ(X, F ) → Γ(X, F 00 ) is surjective. Let s ∈ Γ(X, F 00 ). Consider the set A of all
pairs (U, t) where U is an open subset of X, and t ∈ F (U ) maps to s|U under
F (U ) → F 00 (U ). Order A by declaring (U, t) ≤ (U 0 , t0 ) iff U ⊆ U 0 and t0 |U = t. By
the sheaf properties for F and F 0 , any chain in A has an upper bound, so by Zorn’s
Lemma, A has a maximal element (U, t). If U = X we’re done. If not, there is
some x ∈ X \ U . Since F → F 00 is surjective on stalks, we can find a neighborhood
V of x in X and a section t0 ∈ F (V ) such that t0 7→ s|V under F (V ) → F 00 (V ).
Note t|U ∩V − t0 |U ∩V ∈ F (U ∩ V ) actually lies in F (U ∩ V ) because both t|U ∩V and
t0 |U ∩V map to s|U ∩V under F (U ∩ V ) → F 00 (U ∩ V ). Since F 0 is flasque, there
is a w ∈ F 0 (V ) restricting to t|U ∩V − t0 |U ∩V on U ∩ V . It follows that t ∈ F (U )
and t0 + w ∈ F (V ) agree on U ∩ V , hence glue to a section t̃ ∈ F (U ∪ V ). The
section t̃ maps to s|U ∪V because this can be checked on U and V by separation for
F 00 and adding w ∈ F 0 (V ) to t0 doesn’t change its image in F 00 (V ). We conclude
that (U ∪ V, t̃) contradicts maximality of (U, t).
Now we show that ΓΦ (X, F ) → ΓΦ (X, F 00 ) is also surjective. Let s ∈ ΓΦ (X, F 00 ).
By what we just proved, there is some t ∈ F (X) mapping to s, but t may not have
support in Φ, so we wish to find a t0 ∈ F 0 (X) such that t − t0 has support in Φ—
subtracting off t0 from t doesn’t change its image in F 00 (X), so this will complete
the proof. Note t|X\Supp s ∈ F (X \ Supp s) is actually in F 0 (X \ Supp s). Take t0
to be any lift of this to F 0 (X) (such a t0 exists since F 0 is flasque).
Corollary 10.4. A flasque sheaf is ΓΦ -acyclic for any family of supports Φ and
a Φ-soft sheaf is acyclic for any paracompactifying Φ. Any injective sheaf is flasque.
Proof. Follows from the theorem by standard homological algebra.
Lemma 10.5. Let F → G • be a resolution of F ∈ Ab(X) by ΓΦ -acyclic sheaves.
Then for all p ≥ 0 we have
Hp (ΓΦ G • ) = HpΦ (X, F ).
Proof. By standard homological algebra, there is a spectral sequence with
E2p,q = HpΦ (X, G q )
q
is acyclic, this degenerates to yield the
converging to Hp+q
Φ (X, F ). If every G
desired isomorphism.
According to the corollary and the lemma, sheaf cohomology (with supports)
can be computed using resolutions by flasque sheaves (or Φ soft sheaves when Φ is
paracompactifying).
Lemma 10.6. (Flasque is local) Let F be a sheaf on X, U an open cover of
X. Then F is flasque if and only if F |U is a flasque sheaf on U for every U ∈ U.
42
IV. COHOMOLOGY
Proof. The implication =⇒ is obvious since (F |U )(V ) = F (V ). For the
other implication, suppose s ∈ F (V ) and we want to show there is a lift of s
to F (X). Consider the set of pairs (U, t) where U is a neighborhood of V and
t ∈ F (V ) restricts to s on V . Argue as in the proof of the above theorem, using
Zorn’s Lemma.
Lemma 10.7. A flasque sheaf is Φ soft for any paracompactifying Φ.
Proof. Follows immediately from (1.7).
11. Godement resolution
Given a sheaf F on a space X, let C 0 (F ) be the sheaf of discontinuous sections
Y
U 7→
Fx ,
x∈U
with the obvious restriction maps. There is a natural morphism of sheaves ε : F →
C 0 (F ) taking t ∈ F (U ) to the map (tx )x∈U ∈ C 0 (F )(U ). This map is injective
because if tx = t0x for all x ∈ U then t = t0 because F is separated. Taking
C 1 (F ) := C 0 (Cok ε) and continuing yields a resolution ε : F → C • (F ) called
the Godement resolution or (by Godement) canonical resolution of F . It is clear
that from the formula for sections that, for any sheaf F and any n ≥ 0, the sheaf
C n (F ) is flasque. A morphism of sheaves f : F → G induces a natural morphism
of canonical resolutions.
Theorem 11.1. For a sheaf F on a topological space X, and any family of
p
supports Φ, there is a canonical isomorphism Hp (ΓΦ G • (F )) ∼
= HΦ (X, F ).
Proof. The Godement resolution is, in particular, a flasque resolution, and a
flasque resolution is ΓΦ acyclic, so this follows from standard homological algebra.
12. Soft sheaves
A sheaf F on a space X is called soft iff the natural map
Γ(X, F ) → Γ(Y, i−1 F )
is surjective for every closed subset i : Y ,→ X. More generally, if Φ is a family of
supports on X, then F is called Φ-soft iff, for every closed embedding i : Y ,→ X
with Y ∈ Φ, the natural map F (X) → Γ(Y, i−1 F ) is surjective.
13. Filtered direct limits
The formation of direct limits of sheaves on a space X, which involves sheafification, can often be difficult to understand. However, for filtered direct limits, the
situation simplifies greatly when X is compact or noetherian. In this section, we
13. FILTERED DIRECT LIMITS
43
begin by showing that global sections commute with filtered direct limits on a compact or noetherian space. We use these results to prove that appropriate classes of
Γ-acyclic sheaves are also preserved under filtered direct limits on such spaces. Combining this with some general nonsense, we find that sheaf cohomology commutes
with filtered direct limits on a compact or noetherian space.
Theorem 13.1. Let X be a topological space, c 7→ Fc a direct limit system of
sheaves on X indexed by a filtered category C. If X is quasi-compact, then the
natural map
(13.1)
lim Γ(X, Fc ) → Γ(X, lim Fc )
−→
−→
is injective. If, furthermore, X is Hausdorff (i.e. compact), or noetherian, then it
is an isomorphism.
Proof. Recall that the direct limit sheaf F := lim Fc is given by the sheafifi−→
cation of the presheaf direct limit F pre , whose sections are given by
F pre (U ) = lim Fc (U ).
−→
The map (13.1) is the map on global sections induced by the sheafification map
F pre → F .
The injectivity when X is quasi-compact is not hard: Recall from the usual
construction of filtered direct limits that an element [s] ∈ lim Fc (X) is represented
−→
by an index c ∈ C and an element s ∈ Fc (X). Two such pairs [s] = (c, s), [t] = (d, t)
are equivalent iff there are C-morphisms f : c → e and g : d → e so that
F (f )(s) = F (g)(t) ∈ Fe (X).
Unwinding the sheafification construction, we see that [s] and [t] map to the same
element of Γ(X, F ) iff there is a cover {Ui : i ∈ I} of X, a function e : I → C, and
C-morphisms fi : c → e(i), gi : d → e(i) for each each i ∈ I so that
(13.2)
F (fi )(s|Ui ) = F (g)(t|Ui ) ∈ Fe(i) (Ui ).
If X is quasi-compact, we can assume that I is finite, hence, since C is filtered,
we can find a single index e ∈ C and maps hi : e(i) → e so that hi fi = hj fj and
hi gi = hj gj for every i, j ∈ I. Call the respective common maps f : c → e and
g : d → e. From the equality (13.2) we see that
F (f )(s)|Ui =
=
=
=
=
=
=
F (f )(s|Ui )
F (hi fi )(s|Ui )
F (hi )(F (fi )(s|Ui ))
F (hi )(F (gi )(t|Ui ))
F (hi gi )(t|Ui )
F (g)(t|Ui )
F (g)(t)|Ui .
Since F is a sheaf, we conclude F (f )(s) = F (g)(t), hence [s] = [t].
44
IV. COHOMOLOGY
We now prove surjectivity of (13.1) when X is compact. In general, a section
[s] ∈ Γ(X, F ) is represented by an open cover {Ui : i ∈ I}, a function c : I → C,
and sections si ∈ Fc(i) (Ui ) which satisfy the following compatibility condition on
each pairwise intersection Uij = Ui ∩ Uj : For every x ∈ Uij , there are: an index
c(i, j, x) ∈ C, C morphisms f = f (i, j, x) : c(i) → c(i, j, x), g = g(i, j, x) : c(j) →
c(i, j, x), and a neighborhood W = W (i, j, x) of x in Uij such that
(13.3)
F (f )(si |W ) = F (g)(sj |W )
∈ Fc(i,j,x) (W ).
The reader can spell out the details of when two such collections of data represent
the same section—we will implicitly replace one such data set with an equivalent
one in what follows.
First, since X is compact, we can assume I is finite, hence, since C is filtered, we
can assume that c := c(i) is independent of i ∈ I. (Choose an index c so that there
are C morphisms fi : c(i) → C for each i ∈ I and replace si by F (fi )(si ).) Now, by
a standard topological argument1 we can find a cover {Vi : i ∈ I} (same indexing set
I!) of X such that V i ⊆ Ui for all i ∈ I. Now, using compactness of V ij and the fact
that C is filtered, we can find an index c(i, j) ∈ C, a neighborhood W (i, j) of V ij in
Uij , and a map f (i, j) : c → c(i, j) so that c(i, j) (resp. f (i, j), f (i, j), W (i, j)) can
serve as c(i, j, x) (resp. f (i, j, x), g(i, j, x), W (i, j, x)) in the compatibility condition
for each x ∈ V ij . In fact, since there are only finitely many pairs (i, j), we can
(using C filtered) even arrange that d := c(i, j) and the map f := f (i, j) : c → d
are independent of the pair (i, j). By the same kind of calculation we made in the
injectivity argument, the compatibility condition (13.3) now implies that
F (f )(si )|Vij = F (f )(sj )|Vij .
Hence the local sections F (f )(si )|Vi ∈ Fd (Vi ) glue to a global section s ∈ Fd (X).
The image of s in lim Fc (X) clearly maps to our [s] under (13.1).
−→
The argument in the case where X is noetherian is similar, but easier. In this
case, since an open subset of a noetherian space is quasi-compact, the intersections
Uij are quasi-compact. Using this quasi-compactness and the fact that C is filtered,
we find an index c(i, j) and a map f (i, j) : c → c(i, j) so that c(i, j) (resp. f (i, j),
f (i, j), Uij ) can serve as c(i, j, x) (resp. f (i, j, x), g(i, j, x), W (i, j, x)) in the compatibility condition for each x ∈ Uij . The rest of the argument is identical (replace
U with V everywhere).
Corollary 13.2. Let X be a noetherian topological space, c 7→ Fc a filtered
direct limit system of sheaves on X. The direct limit sheaf F := lim Fc coincides
−→
with the direct limit presheaf:
F (U ) = lim Fc (U )
−→
for every open subset U ⊆ X.
1This
uses in an essential way the fact that X is Hausdorff. The statement we want is an easy
exercise in our case, but it is also a very special case of a general “Shrinking Lemma” (c.f. [Eng,
1.5.18]).
13. FILTERED DIRECT LIMITS
45
Proof. An open subset of a noetherian topological space is noetherian, so this
follows from the previous Theorem.
Alternatively, one can just directly prove that the direct limit presheaf is a sheaf
by using Corollary 5.3 and the fact that a filtered direct limit of equalizer diagrams
of sets is an equalizer diagram. (Filtered direct limits commute with finite inverse
limits of sets.)
Corollary 13.3. A filtered direct limit of flasque sheaves on a noetherian topological space is flasque. A filtered direct limit of soft sheaves on a compact topological
space is soft.
Proof. Since a filtered direct limit of surjective maps (of sets) is again surjective, the first statement is immediate from Corollary 13.2, and the second is
immediate from Theorem 13.1 (note Z is compact when X is compact).
Theorem 13.4. Sheaf cohomology commutes with filtered direct limits on a compact or noetherian space X. That is: Let c 7→ Fc be a filtered direct limit system
of sheaves of abelian groups on a compact or noetherian space X. Then the natural
map
(13.4)
lim Hp (X, Fc ) → Hp (X, lim Fc )
−→
−→
is an isomorphism for all p ≥ 0.
Proof. Let C be the filtered category indexing the direct limit system c 7→ Fc .
Let A := Hom(C, Ab(X)) be the abelian category of C-indexed (hence filtered)
direct limit systems of sheaves of abelian groups on X. For a fixed p, both sides
of (13.4) are natural in (c 7→ Fc ) and may viewed as functors A → Ab. These
functors agree when p = 0 by Theorem 13.1, so, by general nonsense, they will agree
for all p once we prove that both sides are effaceable δ-functors A → Ab. To see
that both sides are δ-functors, first note that the formation of filtered direct limits
of abelian groups
lim : Hom(C, Ab) → Ab
−→
is an exact functor (this is an easy exercise and is well-known). It follows easily from
this fact and the fact that sheafification is exact that
lim : Hom(C, Ab(X)) → Ab(X)
−→
is also exact. Now the fact that both sides are δ-functors follows from the fact that
cohomology H• is a δ-functor.
It remains only to prove that both sides are effaceable. For any F ∈ Ab(X), we
can find an injection F ,→ F 0 from F into a flasque sheaf, functorial in F (use the
first step of the Godement resolution, for example), so we can find an injection from
any limit system (c 7→ Fc ) into a limit system (c 7→ Fc0 ) where each Fc0 is flasque. It
now remains only to prove that, for p > 0, both sides of (13.4) vanish when the Fc
are flasque. The left side vanishes (with no hypotheses on X) because a flasque sheaf
46
IV. COHOMOLOGY
has no higher cohomology. In the noetherian case, the right side vanishes because a
filtered direct limit of flasques is flasque (Corollary 13.3). In the compact case, the
right side vanishes because a flasque sheaf is soft (Lemma ??), a direct limit of soft
sheaves is soft (Corollary 13.3), and a soft sheaf on a (para)compact space has no
higher cohomology.
14. Vanishing theorems
Recall that a closed subset Z of a topological space X is called irreducible iff
it cannot be written as a union of two proper closed subsets. The dimension of a
topological space X (denoted dim X) is the maximal length n of a strictly increasing
chain
∅=
6 Z0 ( Z1 ( · · · ( Zn
of non-empty irreducible closed subsets of X. We set dim X = ∞ if X contains
arbitrarily long such chains.
Theorem 14.1. (Grothendieck’s Vanishing) Let X be a noetherian topological space of dimension n, Φ a family of supports on X. Then HpΦ (X, F ) = 0 for
every p > n and every abelian sheaf F on X.
Proof.
The following cohomology vanishing theorem sometimes comes in handy.
Theorem 14.2. Let X be a zero dimensional, second countable topological space,
F an abelian sheaf on X. Then Hp (X, F ) = 0 for every p > 0. Indeed, the global
section functor is exact.
Proof. It is enough to prove that, for any surjection f : F → G of sheaves on
X, the map Γ(X, F ) → Γ(X, G ) is surjective. Let s ∈ Γ(X, G ). Since F → G is
surjective, there are a cover U = {Ui : i ∈ I} and sections ti with f (Ui )(ti ) = s|Ui
for every i ∈ I. Since X is zero dimensional and second countable, we may assume,
after possibly passing to a refinement and restricting sections that U = {U1 , U2 , . . . }
is countable and that every Ui is open-and-closed in X. By replacing Ui with the
open set
i−1
U i := Ui \ (∪j=1
Uj )
and restricting sections, we may even assume that the Ui are pairwise disjoint. But
then the si automatically agree on overlaps, so by the sheaf property on F they lift
to a section t ∈ Γ(X, F ) and f (X)(t) = s by the sheaf property on G , since this is
true on each Ui .
15. Higher direct images
Let f : X → Y be a morphism of topological spaces. The direct image functor
f∗ : Ab(X) → Ab(Y ) is easily seen to be left exact (this left exactness is inherited
16. LERAY SPECTRAL SEQUENCE
47
from left exactness of the section functors). Since Ab(X) has enough injectives, we
can form the right derived functors Rp f∗ : Ab(X) → Ab(Y ).
Lemma 15.1. For a sheaf F ∈ Ab(X), the sheaf Rp f∗ F ∈ Ab(Y ) on Y is naturally isomorphic to the sheaf associated to the presheaf U 7→ Hp (f −1 (U ), F |f −1 (U ) ).
In particular, the stalks of Rp f∗ F are given by
(Rp f∗ F )y = lim Hp (f −1 (U ), F |f −1 (U ) ),
−→
where the filtered direct limit system is over neighborhoods U of y in Y .
Proof. Certainly this is true for p = 0, so by general nonsense it suffices to
prove that the indicated sheafifications form an effaceable δ-functor from Ab(X)
to Ab(Y ). To see that they form a δ-functor, first observe that before sheafifying,
the presheaves defined by the above formula certain form a δ-functor Ab(X) →
PAb(Y ). But sheafification is exact, so the sheafified versions also form a δ-functor.
To see that it is effaceable, it will suffice to show that the presheaves defined by
the above formula are zero when p > 0 and F is injective. This is clear from the
fact that F |f −1 (U ) is also injective in Ab(f −1 (U )) by (??), hence has no higher
cohomology.
16. Leray spectral sequence
Theorem 16.1. (Leray) Let f : X → Y be a continuous map of topological
spaces, F a sheaf on X. There is a first quadrant spectral sequence
E2p,q = Hp (Y, Rq f∗ F )
=⇒
Hp+q (X, F ).
Proof. We have Γ(Y, f∗ F ) = Γ(X, F ) for any sheaf F on X, and by (10.1)
f∗ takes injective sheaves on X (which are flasque) to flasque sheaves on Y (which
are Γ(Y, ) acyclic by (10.4)), so this spectral sequence is a special case of the
Grothendieck spectral sequence.
One consequence of the Leray spectral sequence is the following celebrated theorem.
Corollary 16.2. (Vietoris-Begle) Let f : X → Y be a proper map with
acyclic fibers:
Z,
p=0
p
−1
H (f (y), Z) =
0,
p > 0.
Then Hp (Y, F ) → Hp (X, f −1 F ) is an isomorphism for every p.
Proof. First of all, by a “universal coefficients” arguement, the “acyclic fibers”
assumption implies an analogous formula for cohomology with coefficients in any
constant sheaf. We will show that the Leray spectral sequence (??) degenerates by
48
IV. COHOMOLOGY
showing that f∗ f −1 F = F and Rp f∗ f −1 F = 0 for p > 0. This follows from the
formula
(Rp f∗ ZX )y = Hp (f −1 (y), (f −1 F )|f −1 (y) )
of (15.1) and the fact that (f −1 F )|f −1 (y) is just the constant sheaf on f −1 (y) associated to the abelian group Fy , as is clear from functoriality of inverse images and
commutativity of the cartesian diagram
/
f −1 (y)
{y}
/
X
f
Y.
17. Group quotients
18. Local systems
For an abelian group A, an A-local system or an A-locally constant sheaf on a
space X is a sheaf F ∈ Ab(X), which, locally on X, is isomorphic to the constant
sheaf AX . The Leray spectral sequence is most easily understood when f : X → Y
is a fiber bundle with fiber F (i.e. Y can be covered with open sets Ui so that
f −1 (Ui ) is isomorphic, over Ui , to Ui × F ). In this case, the sheaf Rp f∗ ZX will be
an Hp (F, Z)-local system on Y .
19. Examples
20. Exercises
The exercises below are intended to show that one cannot prove (1.7) without
some assumptions along the lines of (2) or (1).
Exercise 20.1. Construct an example where f : X → Y is a closed embedding
of finite topological spaces, but the natural monomorphism of (1.7) is not surjective.
Exercise 20.2. Let S be the Sorgenfrey line (R with basic open sets [a, b)
for a < b), let Y = S × S, and let f : X → Y be the inclusion of the skew
diagonal {(−x, x) : x ∈ R} in the subspace topology it inherits from Y . Let F be
the constant sheaf associated to the two element set {0, 1}. Show that X has the
discrete topology, hence
Y
Γ(X, f −1 F ) =
Fx
x∈X
= {0, 1}R .
Show that the characteristic function of the irrational numbers f ∈ {0, 1}R is not in
the image of the natural map of (1.7). Hint: Baire category theorem.
20. EXERCISES
49
Exercise 20.3. Let X be a locally compact Hausdorff space. Show that the
family of supports Φ consisting of compact subsets of X is paracompactifying. Cohomology with support in Φ in this situation is usually called compactly supported
cohomology.
Exercise 20.4. Let A be an abelian group, X a topological space, F a subsheaf
of AX . Show that for any a ∈ A, the set
Ua := {x ∈ X : a ∈ Fx ⊆ AX,x = A}
is open in X. Now suppose that A is a ring and I is an ideal sheaf of AX (a subobject
of AX in the category of AX -modules). Show that, for a finitely generated ideal I
of A, the set
UI := {x ∈ X : I ⊆ Ix }
is open in X.
Exercise 20.5. Let F be a subsheaf of ZX for a quasi-compact space X. Show
that there are only finitely many n ∈ Z>0 such that Un (notation from the previous
exercise) is nonempty. For x ∈ X, let n(x) be the smallest non-negative integer
such that n(x) ∈ Fx . Show that the set V := {x ∈ X : n(x) > 0} is open in X.
Show that the sets Vs := {x ∈ V : n(x) ≤ s} are open in V and that Vs = V for
s sufficiently large. Then Zs := Vs \ Vs−1 is locally closed subspace of X. := Show
that the sheaves Fi := (F |Vi )X define a finite filtration of F with Fs /Fs−1 ∼
= ZX
Zs .
Exercise 20.6. Let X be an infinite set with the cofinite topology, and let x ∈ X
be a point. Show directly that HomX (x∗ Z, ZX ) = 0. Show that x! ZX = 0 (c.f. Theorem 1.4). Conclude that H om(x∗ Z, ZX )x is not equal to HomAb ((x∗ Z)x , (ZX )x )
so “sheaf Hom doesn’t commute with stalks.”
Exercise 20.7. Recall that the family of constructible subsets of a topological
space X is the smallest family of subsets of X containing the open subsets, and
closed under finite intersections and complements (hence also containing the closed
subsets and closed under finite unions). If X is noetherian, show that a union of
two locally closed subsets is locally closed, and hence the complement of a locally
closed subset is locally closed. Conclude that the locally closed subsets coincide
with the constructible subsets. On the other hand, if X = R, show by example that
the union of a closed subset and a locally closed subset need not be locally closed.
Hint: Take a sequence of positive real numbers
x1 > x2 > x3 > · · ·
with xi → 0 as i → ∞. Let
A := [x1 , x2 ) ∪ [x3 , x4 ) ∪ · · ·
and let B = {0}. Note that A is locally closed, but A ∪ B is not.
Exercise 20.8. Show that a direct limit of flasque sheaves need not be flasque
in general.
Exercise 20.9. Let i : U ,→ X be an open set. Show that the sheaf ⊕hU Z on
X considered in Section 4 coincides with the sheaf i! ZU .
50
IV. COHOMOLOGY
Exercise 20.10. Give an example of a sheaf F on R such that the set {x ∈ R :
Fx 6= 0} is not locally closed in R. Hint: Recall that a subspace of a locally compact
space is locally closed iff it is locally compact. Let α be the smallest ordinal with
infinitely many limit ordinals less than it. Show that the space α + 1 (with the order
topology) embeds in R (as does any countable ordinal with its order topology), but
is not locally compact. One can arrange that Fx 6= 0 if and only if x is in the image
of this embedding.
Exercise 20.11. If f : F → G is a surjective morphism of flasque sheaves then
the induced maps f p : C p (F ) → C p (G ) are surjective for every p ≥ 0. Hint: Induct
on p using 10.3.
Exercise 20.12. Supp F ⊆ Supp C p (F ) for every p.
Exercise 20.13. For any nonnegative integer n, the functor Ab(X) → Ab(X)
given by F 7→ C n (F ) is exact. Hint. Check on stalks for p = 0 and induct using
the 3x3 Lemma.
Exercise 20.14. Construct an example of a topological space X, a basis B ⊆
Ouv(X) and a sheaf F ∈ Ab(X) such that F (U ) → F (V ) is surjective for every
B morphism V ,→ U , but F is not flasque.
Exercise 20.15. Show that the Sorgenfrey line S is zero dimensional, but not
second countable. Is there a sheaf of abelian groups on S (or S × S) with nonvanishing higher cohomology? c.f. (14.2).
CHAPTER V
Base Change
1. Proper maps
Some of the most topologically intricate arguments in sheaf theory occur in the
study of the “compatibility” of direct and inverse images with base change—the
subject of this section and the next.
Definition 1.1. Let X be a topological space, Y ⊆ X a subspace, {Ai : i ∈ I}
a family of subsets of X. We say that Y is relatively Hausdorff in X iff any two
distinct points of Y have disjoint neighborhoods in X. We say that {Ai } is locally
finite iff any point x ∈ X has a neighborhood U in X such that {i ∈ I : U ∩ Ai 6= ∅}
is finite. We say that X is paracompact iff every open cover of X has a locally finite
open refinement.
Definition 1.2. We say that a map f : X → Y of topological spaces is proper
iff it is closed and each fiber f −1 (y) is compact and relatively Hausdorff in X.
The condition that the fibers of f be relatively Hausdorff is equivalent to saying
that the diagonal X ⊆ X ×Y X is closed (i.e. f is separated ). It is easy to see that
this condition is closed under base change. The property of being a closed map is
certainly not preserved preserved under base change. For example, π1 : R2 → R
is not closed even though it is a base change of the closed map from R to a point.
Never-the-less, we have:
Lemma 1.3. Proper maps of topological spaces are closed under base change and
composition. Being proper is local on the base.
Proof. Everything is straightforward except perhaps the statement that the
base change π1 : Z ×Y X → Z of a proper map f : X → Y along an arbitrary
map g : Z → Y is a closed map. To see this, suppose W ⊆ Z ×Y X is closed, and
z∈
/ π1 (W ). We want to see that z has a neighborhood disjoint from π1 (W ), so that
π1 (W ) is closed. Since z ∈
/ π1 (W ), we know that, for any x ∈ f −1 (g(z)), (z, x) ∈
/ W.
Since W is closed we can find a basic open neighborhood Ux ×Y Vx of (z, x) disjoint
from W . Since f −1 (g(z)) is compact, we only need finitely many such Ux ×Y Vx to
cover π1−1 (z), so we can intersect the finitely many Ux that we need to find a single
open neighborhood U of z ∈ Z and a neighborhood V of f −1 (g(z)) in X so that
U ×Y V is a neighborhood of π1−1 (z) in Z ×Y X. The set T := Y \ f (X \ V ) is
open in Y since f is closed, and T contains g(z) because V contains f −1 (g(z)), so
g −1 (T ) is an open neighborhood of z in Z. I claim that U ∩ g −1 (T ) is disjoint from
51
52
V. BASE CHANGE
π1 (W ). Indeed, if z 0 ∈ U ∩ g −1 (T ), then g(z 0 ) ∈ T , so f −1 (g(z 0 )) ⊆ V , hence no
point (z 0 , x) ∈ Z ×Y X can be in W because U ×Y V is disjoint from W .
Lemma 1.4. A map f : X → Y of locally compact Hausdorff spaces is proper iff
f (K) is compact for every compact subspace K ⊆ Y . If f : X → Y and g : Y → Z
are maps of locally compact Hausdorff spaces such that gf and g are proper, then f
is proper.
−1
Proof. (=⇒) Suppose {Ui : i ∈ I} is a family of open subsets of X covering
Z := f −1 (K). Since f is proper, each fiber of f is compact, so for each x ∈ K, there
is a finite subset Ix ⊆ I such that Ux := ∪i∈Ix Ui covers f −1 (x). Since the proper
map f is closed, Vx := Y \ f (Z \ Ux ) is an open neighborhood of x in Y , and it is
clear that f −1 (y) ⊆ Ux for every y ∈ Vx ∩ K. Since K is compact, we can find a
finite subset S ⊆ K so that {Vx : x ∈ S} covers K. Then J := ∪x∈S Ix is a finite
subset of I and such that the Uj cover Z.
For (⇐=), the only issue is to prove that f is closed. Suppose Z ⊆ X is closed
and y ∈
/ f (Z), but y is in the closure of f (Z). Choose a neighborhood W of y in Y
with compact closure W . Then y is still in the closure of W ∩ f (Z). Since f −1 (W )
is compact and Z is closed, Z ∩ f −1 (W ) is compact, so K := f (Z ∩ f −1 (W )) is also
compact, hence closed. But this K is a closed subset of Y containing W ∩ f (Z) with
y∈
/ K, which contradicts y being in the closure of W ∩ f (Z).
For the final statement, note that by the first part of the lemma, it suffices to
prove that f −1 (K) is compact for any compact K ⊆ Y . Certainly g(K) is compact
in Z, so (gf )−1 (g(K)) is compact because gf is proper. But f −1 (K) is a closed
subspace of (gf )−1 (g(K)), so it is also compact.
The following technical lemmas are the main topological input for the proper
base change theorem:
Lemma 1.5. Let X be a topological space, Y ⊆ X a compact, relatively Hausdorff
subspace, {Ui : i ∈ I} a family of opens in X covering Y . Then there is a finite
family {Vj : j ∈ J} of opens in X whose union V contains Y and a map φ : J → I
such that (V ∩ V j ) ⊆ Uφ(j) for all j ∈ J.
Proof. Fix y ∈ Y . Choose φ(y) ∈ I so that y ∈ Uφ(y) . By the relatively Hausdorff assumption, we can choose, for each y 0 ∈ Y \Uφ(y) , disjoint open neighborhoods
Sy,y0 ⊆ Uφ(y) and Wy,y0 of y and y 0 respectively. By compactness of Y , there is a
finite subset Iy ⊆ Y \ Uφ(y) so that
Ty := Uφ(y) ∪ ∪y0 ∈Iy Wy,y0
is an open neighborhood of Y in X. Set Sy := ∩y∈Iy Sy,y0 . Then it is easy to see
that (S y ∩ Ty ) ⊆ Uφ(y) . Again using compactness of Y , find a finite J ⊆ Y so that
∪j∈J Sj contains Y . Set T := ∩j Tj . Then it is easy to see that {Vj := Sj ∩ T } is as
desired.
1. PROPER MAPS
53
Lemma 1.6. Let X be a paracompact topological space, {Ui : i ∈ I} an open
cover of X. Then there exists a locally finite open cover {Vj : j ∈ J} of X and a
map φ : J → I so that V j ⊆ Uφ(j) .
Proof. Fix x ∈ X and choose µ(x) ∈ I so that x ∈ Uµ(x) . For each y ∈ X \Uµ(x) ,
choose disjoint open neighborhoods Sx,y ⊆ Uµ(x) and Wx,y of x and y respectively.
By paracompactness of X, we can find a locally finite family of opens {Tk : k ∈ K}
and a map ψ : K → X \ Uµ(x) so that Tk ⊆ Wx,ψ(k) and so that the Tk and Uµ(x)
cover X. Find an open neighborhood Rx ⊆ Uµ(x) so that
K(x) := {k ∈ K : Tk ∩ Rx 6= ∅}
is finite and let Sx := ∩k∈K(x) Sx,ψ(k) . Then it is easy to see that S x ⊆ Uµ(x) . Choose
a locally finite refinement {Vj : j ∈ J} of the cover {Sx : x ∈ X}, so we have a map
ν : J → X with Vj ⊆ Sν(j) . Then this {Vj } is as desired if we set φ := µν.
Lemma 1.7. Suppose X is a topological space and Y ⊆ X is a subspace. Then
the natural map
lim
F (U ) → Γ(Y, F |Y )
−→
U ⊇Y
is monic (the direct limit is over open neighborhoods U of Y in X). Suppose furthermore that at least one of the following assumptions holds:
(1) Y is compact and relatively Hausdorff in X.
(2) There is a paracompact neighborhood U of Y in X so that Y is closed in U .
Then the natural map above is an isomorphism.
f
−1
−1
Remark 1.8. The “natural map” is Γ of the sheafification morphism fpre
F →
F.
Proof. To prove the first statement, suppose U, V ⊇ Y are open neighborhoods
and s ∈ F (U ) and t ∈ F (V ) agree in Γ(Y, F |Y ). Then sy = ty ∈ Fy for any y ∈ Y ,
so we can find a neighborhood Uy of y in X (contained in U ∩ V ) with s|Uy = t|Uy .
Then W := ∪y∈Y Uy is a neighborhood of y in X on which s and t agree, hence they
are equal in the (filtered) direct limit.
For the second statement it remains only to prove surjectivity. By definition of
the inverse image F |Y , a section s ∈ Γ(Y, F |Y ) can be viewed as an element
Y
s = (s(y)) ∈
Fy
y∈Y
so that there is a family of opens {Ui : i ∈ I} in X with Y ⊆ U := ∪Ui and sections
si ∈ F (Ui ) so that s(y) = siy for all y ∈ Ui ∩ Y . We can always pass to a smaller
neighborhood of Y in U and restrict our cover Ui and our sections si , so under
either assumption, we can assume that {Ui } is a locally finite cover of U . When
the second assumption holds, we can (and will) assume without loss of generality
that X itself is paracompact and Y is closed in X. Our goal is to produce (after
54
V. BASE CHANGE
possibly shrinking U to a smaller neighborhood of Y in X) a section t ∈ F (U ) so
that ty = s(y) for all y ∈ Y .
I next claim that we can further assume that there is a locally finite open cover
{Vj : j ∈ J} of U and a map φ : J → I with (V j ∩ U ) ⊆ Uφ(j) . To see this under the
first assumption, apply Lemma 1.5 and then make the replacements U 7→ U ∩ V ,
Ui 7→ Ui ∩ V , Vj 7→ Vj ∩ U , etc. to assume V = U . (We can even assume that J is
finite in this case.) To see this under the second assumption, apply Lemma 1.6 to
the cover of X given by the Ui and X \ Y .
Now let {Tx : x ∈ U } be neighborhoods in U witnessing the local finiteness of
{Ui } and {Vj }, so the sets
I(x) := {i ∈ I : Tx ∩ Ui 6= ∅}
I 0 (x) := {i ∈ I : x ∈ Ui }
J(x) := {j ∈ J : x ∈ Vj }
are all finite and φ(J(x)) ⊆ I 0 (x) ⊆ I(x). For any x ∈ U , the sections {si : i ∈ I 0 (x)}
all have the same stalk (namely s(x)) at x, so, since I 0 (x) is finite, we can find a
neighborhood
Wx00 ⊆ ∩i∈I 0 (x) Ui ∩ Tx
of x in U so that the sections {si : i ∈ I 0 (x)} all agree on Wx00 . Call this common
section t0 (x) ∈ F (Wx00 ). Set
Wx0 := Wx00 ∩ ∩j∈J(x) Vj
Wx := Wx0 ∩ ∩j∈J(x):φ(j)∈I(x)\I 0 (x) (X \ V j )
= Wx0 \ ∪j∈J(x):φ(j)∈I(x)\I 0 (x) V j
so that Wx is a neighborhood of x in U because
φ(j) ∈ I(x) \ I 0 (x)
=⇒
x∈
/ Uφ(j)
=⇒
x∈
/ V j.
Set t(x) := t0 (x)|Wx .
I claim that t(x1 )|(Wx1 ∩ Wx2 ) = t(x2 )|(Wx1 ∩ Wx2 ) for all x1 , x2 ∈ U so that the
t(x) glue to a global section over U (which will then clearly map to s as desired).
We can check this on stalks at a typical point u ∈ Wx1 ∩ Wx2 . Since u ∈ Wx1 , in
particular u ∈ Tx1 . Since u ∈ Wx2 there is j2 ∈ J(x2 ) so that u ∈ Vj2 . Then φ(j2 ) is
in I(x1 ) because
u ∈ Tx1 ∩ Vj2 ⊆ Tx1 ∩ Uφ(j2 ) ,
but φ(j2 ) must actually be in I 0 (x1 ) ⊆ I(x1 ), because otherwise u wouldn’t be in
φ(j )
Wx1 since it is in V j2 . Since φ(j2 ) ∈ I 0 (x1 ) ∩ I 0 (x2 ), we have t(x1 )y = sy 2 = t(x2 )y
by definition of t(x1 ), t(x2 ).
2. COMPARISON MAP
55
2. Comparison map
Now we begin our discussion of base change. Consider a commutative diagram
X0
f0
Y0
g
h
/
/
X
f
Y
of topological spaces. For any sheaf F on X, there is a natural map
h−1 f∗ F → (f 0 )∗ g −1 F
(2.1)
of sheaves on Y 0 , defined as follows. First of all, by the adjointness (h−1 , h∗ ), it is
equivalent to give a map
f∗ F → h∗ (f 0 )∗ g −1 F
(2.2)
of sheaves on Y . For any open set U ⊆ Y , we have (f∗ F )(U ) = F (f −1 U ) and
−1
−1
(h∗ (f 0 )∗ gpre
F )(U ) = (gpre
F )((f 0 )−1 h−1 U )
=
lim
V
−→
⊇g((f 0 )−1 h−1 (U ))
F (V ).
But commutativity implies
g((f 0 )−1 h−1 U ) = g(g −1 f −1 U )
⊆ f −1 (U ),
so f −1 (U ) is one of the V ’s in the direct limit system, hence we have a natural map
−1
(f∗ F )(U ) → (h∗ (f 0 )∗ gpre
F )(U )
given by the structure map to the direct limit. This is clearly natural in U since the
restriction maps are induced by the restriction maps for F on both sides, hence we
−1
obtain a map f∗ F → h∗ (f 0 )∗ gpre
F . Composing with h∗ (f 0 )∗ of the sheafification
−1
map gpre
F → g −1 F yields the map (2.2).
There are variants for pushforward with proper support, as follows: If the diagram is cartesian, then there is a natural map
(2.3)
h−1 f! F → (f 0 )! g −1 F
defined in the same way. The key point is that, if s ∈ F (U ) has proper support Z
over Y , then g −1 s ∈ F (g −1 (U )) has proper support over Y 0 because
Supp(g −1 s) = g −1 Supp s = g −1 Z,
56
V. BASE CHANGE
and we have a cartesian diagram
/
g −1 Z
X0
Y0
Z
/
/
X
Y
where Z → Y is proper, hence g −1 Z → Y 0 is also proper (Lemma 1.3).
One also has the map (2.3) when the spaces X, X 0 , Y, Y 0 in the diagram are
locally compact (and Hausdorff) and g and h are proper. In this case the point is
that f 0 |g −1 Z : g −1 Z → Y 0 is proper when f |Z : Z → Y is proper. To see this,
first note that g|g −1 Z → Z is proper (since it is a base change of g), hence the
composition (f |Z)(g|g −1 Z) is proper (Lemma 1.3) and we have
h(f 0 |g −1 Z) = (f |Z)(g|g −1 Z)
by commutativity, hence f 0 |g −1 Z is proper by the 2-out-of-3 property of proper maps
from Lemma 1.4.
An important special case of the map (2.5) occurs when h : Y 0 → Y is the
inclusion of a point y ∈ Y and the diagram is cartesian so that X 0 = Xy is the fiber
of f over y ∈ Y . In this case, h−1 is the stalk functor F 7→ Fy and f 0 is the functor
of global sections over Xy , so (2.5) can be viewed as a map
(f∗ F )y → Γ(Xy , F |Xy ).
(2.4)
The next result is a special case of the general base change theorem (Theorem ??), but it actually turns out that the more general base change theorem reduces
to this special case rather easily.
Theorem 2.1. Suppose f : X → Y is a proper map. Then the natural map (2.4)
is an isomorphism for every y ∈ Y and every sheaf F on X. For any F ∈ Ab(X),
the natural map (Ri f∗ F )x → Hi (Xy , F |Xy ) is an isomorphism for all i.
Proof. The natural map in question is the natural map
lim
F (f −1 (U )) → Γ(Xy , F |Xy ),
−→
U
where the direct limit runs over all neighborhoods U of y in Y . But the fact that f
is closed implies that the f −1 (U ) are cofinal in the set of all neighborhoods of the
fiber Xy , so the map in question is the same as the map
lim
F (V ) → Γ(Xy , F |Xy ),
−→
V
where V runs over all neighborhoods of Xy in X. Since the fiber Xy is compact and
relatively Hausdorff in X, this map is an isomorphism by Lemma 1.7(1).
3. HOMOTOPY INVARIANCE
57
The second statement follows from the first by general “universal δ-functors”
nonsense.
Theorem 2.2. Suppose
X0
f0
g
/
X
f
/Y
Y0
is a cartesian diagram of topological spaces with f proper. Then the natural map
(2.5)
h
h−1 f∗ F → (f 0 )∗ g −1 F
is an isomorphism for every sheaf F on X.
Proof. We can check this on stalks at any y 0 ∈ Y 0 . But f 0 is also proper
(Lemma 1.3) and f and f 0 have the same fibers, so we reduce formally to the
situation of the previous theorem.
3. Homotopy invariance
Let I := [0, 1] denote the unit interval. For a continuous map H : X ×I → Y and
t ∈ I, let ιt : X ,→ X × I denote the map x 7→ (x, t) and set Ht := Hιt : X → Y .
Maps f, g : X → Y are called homotopic if there is a homotopy f to g: a map
H : X × I → Y such that H0 = f and H1 = g. Being homotopic is an equivalence
relation on maps preserved by compositions.
Sheaf cohomology enjoys the following homotopy invariance property:
Theorem 3.1. Let f, g : X → Y be homotopic maps of topological spaces. For
any abelian group A, the maps f ∗ , g ∗ : Hp (Y, A) → Hp (X, A) coincide.
Proof. Let H : X × I → Y be a homotopy from f to g. The maps ι0 , ι1 induce
isomorphisms on cohomology by the “2-out-of-3” property of isomorphisms, the fact
that π1 ιi = Id : X → X, and the fact that π1 : X × I → X induces isomorphisms
on all constant sheaf cohomology by Vietoris-Begle (use the fact that the interval is
acyclic—see the computations in (??)). The result is now immediate since Hι0 = f
and Hι1 = g.
Warning: Do not read too much into the “homotopy invariance” property above.
Many people falsely believe that a sheaf of abelian groups on a contractible topological space can have no higher cohomology groups (this is true for constant sheaves).
In fact, the very existence of the Leray spectral sequence ensures that this cannot
be the case:
Example 3.2. Let S 1 ⊆ R2 be the unit circle and let f : S 1 → I be the projection from S 1 onto the interval I = [−1, 1] so that f is a double cover branched
over {±1} ⊆ I. Certainly f is a proper map by (3.4), so we easily conclude that
58
V. BASE CHANGE
R>0 f∗ = 0 by the stalk formula (3.5) and the fact that no sheaf has higher cohomology on a space with at most two points. The Leray spectral sequence for f therefore
yields an isomorphism
H1 (S 1 , Z) = H1 (I, f∗ ZS 1 ).
Note that this group is Z by the computations in (??). To get a feel for the sheaf
F := f∗ ZS 1 , note that
F ([−1, 1)) = Z
F ((−1, 1]) = Z
F ((−1, 1)) = Z2
and both restriction maps
F ([−1, 1)), F ((−1, 1]) → F ((−1, 1))
are the diagonal ∆ : Z → Z2 .
CHAPTER VI
Duality
1. Adjoints for immersions
Let f : X → Y be a morphism of topological spaces. We always have a left
adjoint f −1 : Ab(Y ) → Ab(X) to f∗ : Ab(X) → Ab(Y ). It would also be
desirable, for many reasons, to have a right adjoint
f ! : Ab(Y ) → Ab(X)
to f∗ . In general, we know this is impossible because the existence of a right adjoint
f ! to f∗ would imply that f∗ commutes with all direct limits, which it generally does
not (even when Y is a point). In the general case then, one is forced to look for the
right adjoint f ! only in an appropriate derived category sense. However, in the case
where f is a closed embedding (so f∗ does preserve direct limits) we will construct
the desired right adjoint f ! in this section by elementary means.
In fact, it is natural to work not only with closed embeddings, but with immersions. Recall that f : X → Y is an immersion iff f can be factored f = ji as an
open embedding i followed by a closed embedding j. In other words, f is an immersion iff it is an isomorphism onto a subset A ⊆ X which can be written A = Z ∩ U
where Z is closed in X and U is open in X. Such a subset A is called locally closed,
and an immersion is hence sometimes called a locally closed embedding. The term
“locally closed” is used because of the following:
Lemma 1.1. A ⊆ X is locally closed iff, for each x ∈ A there is a neighborhood
Ux of x in X such that A ∩ Ux is closed in Ux .
Proof. The implication =⇒ is clear. If A satisfies the second condition, then
A will be closed in U := ∪x∈A Ux because being closed is local, hence A = U ∩ Z for
some Z closed in X, i.e. A is locally closed in our original sense.
Here are the basic facts about immersions, all of which follow easily from the
lemma and discussion above.
Proposition 1.2. An open embedding is an immersion, and a closed embedding
is an immersion. Immersions are the smallest class of morphisms containing open
embeddings and closed embeddings and closed under composition. Immersions are
closed under base change, and the property of being an immersion is local on the
codomain (base).
59
60
VI. DUALITY
In a noetherian topological space, locally closed subsets are the same as constructible subsets, but this is not true in general (Exercise 20.7).
Lemma 1.3. Let j : Z ,→ X be the inclusion of any subspace. Then j∗ :
PAb(Z) → PAb(X) is fully faithful, hence j∗ : Ab(Z) → Ab(X) is also fully
faithful.
Proof. We want to show that for any F , G ∈ PAb(Z), the map
j∗ : HomPAb(Z) (F , G ) → HomPAb(X) (j∗ F , j∗ G )
is bijective. Given any morphism PAb(X) morphism f : j∗ F → j∗ G and any
U ∈ Ouv(X), the fact that the f (U ) respect the restriction maps implies that
the morphism f (U ) : F (U ∩ Z) → G (U ∩ Z) depends only on U ∩ Z: Indeed,
if V ∩ Z = U ∩ Z, then both are also equal to (V ∩ U ) ∩ Z and the restriction
maps for the inclusions (V ∩ U ) ,→ V and (V ∩ U ) ,→ U are the identity for
both presheaves j∗ F , j∗ G . In other words, for U ∈ Ouv(Z), we can define a
map f (U ) : F (U ) → G (U ) unambiguously by choosing any U 0 ∈ Ouv(X) with
U = U 0 ∩Z and setting f (U ) := f (U 0 ). The maps f (U ) clearly respect the restriction
maps since this is true for the maps f (U 0 ) and any inclusion of opens in Z lifts to
an inclusion of opens in X. The function f 7→ f hence defines an inverse to our j∗
above.
Theorem 1.4. Let j : Z ,→ X be a closed subspace of a topological space X.
(1) The direct image functor j∗ : Ab(Z) → Ab(X) commutes with all direct
and inverse limits and is fully faithful. Its essential image consists of those
F ∈ Ab(X) where Supp F ⊆ Z.
(2) For F ∈ Ab(X), define j ! F ∈ Ab(Z) by
U 7→
ΓY ∩U (U, F )
= {s ∈ F (U ) : Supp s ⊆ Z ∩ U }
on X (with the obvious restriction maps inherited from those of F ). The
functor j ! : Ab(X) → Ab(Z) is right adjoint to j∗ and the adjunction
morphism F → j ! j∗ F is an isomorphism for any F ∈ Ab(Z).
Proof. First of all, any section s of j∗ F over U ∈ Ouv(X) trivially satisfies
Supp s ⊆ Z ∩ U since the stalks of j∗ F at any x ∈ X \ Z are zero, so the final
statement of the theorem is clear. Also, if f : j∗ F → G is any Ab(X) morphism,
U ∈ Ouv(X), s ∈ (j∗ F )(U ), then obviously
Supp f (U )(s) ⊆ Supp s ⊆ Z ∩ U,
so f factors uniquely through the subsheaf j∗ j −1 G ⊆ G , hence we have
HomAb(X) (j∗ F , G ) = HomAb(X) (j∗ F , j∗ j −1 G ).
But by Lemma 1.3, j∗ is fully faithful, so
HomAb(X) (j∗ F , j∗ j −1 G ) = HomAb(Z) (F , f −1 G )
2. EXTENSION BY ZERO
61
and putting the two together yields the desired adjunction. The functor j∗ commutes
with all limits because it is both a left adjoint (to j ! ) and a right adjoint (to j −1 ). Theorem 1.5. Let i : U ,→ X be an open subspace of a topological space X. For
a F ∈ Ab(U ), let i! F be the sheaf associated to the presheaf ipre
! F defined by
F (V ),
V ⊆U
V 7→
{0},
otherwise.
The functor i! : Ab(U ) → Ab(X) is left adjoint to i−1 (i.e. the restriction functor
F 7→ F |U ).
Proof. The functor ipre
: PAb(U ) → PAb(X) is clearly left adjoint to i−1 :
!
PAb(X) → PAb(U ), so the desired adjointness follows formally from the adjointness property of sheafification.
Remark 1.6. Unravelling the local coherence condition in the sheafification
construction (7), we see that the sheaf i! F can be explicitly
described as follows:
Q
A section s ∈ (i! F )(V ) is an element s = (s(x)) ∈ x∈U ∩V Fx satisfying the two
conditions:
(1) For every x ∈ V \U there is a neighborhood W of x in V such that s(x0 ) = 0
for all x0 ∈ W ∩ U .
(2) For every x ∈ V ∩ U , there is a neighborhood W of x in V ∩ U and a section
t ∈ F (W ) such that tx0 = s(x0 ) for x0 ∈ W .
2. Extension by zero
Extension by zero will allow us to construct a kind of “wrong way” map which
is the basis for Poincaré and Verdier duality.
Lemma 2.1. Let i : U → X be the inclusion of an open subset, F a sheaf on U .
The sheaf i! F associated to the presheaf
F (V ),
V ⊆U
pre
i! F : V 7→
0,
otherwise
is given by
(i! F )(V ) = {s ∈ F (V ∩ U ) : Supp s is closed in V }.
The stalk of i! F at x ∈ X is Fx if x ∈ U and zero otherwise.
Proof. First note that i! F is a sheaf because being closed is a local property.
If V ⊆ U , then U ∩ V = V and the support of any s ∈ F (V ) is certainly closed
in V , so there is a natural morphism ipre
! F → i! F . To see that this induces an
+
isomorphism (ipre
F
)
→
i
F
,
it
suffices
to check that it is an isomorphism on
!
!
pre
stalks. Certainly if x ∈ U ⊆ X, then (i! F )x → (i! F )x is an isomorphism because
both sheaves coincide on all sufficiently small neighborhoods of x (namely those
contained in U ). It remains only to show that (i! F )x = 0 for x ∈ X \ U . Indeed, if
V is any neighborhood of x and s ∈ (i! F )(V ) any section, then s restricts to zero
on the neighborhood V \ Supp s of x and is hence zero in the stalk.
62
VI. DUALITY
Corollary 2.2. The functor ι! : Ab(U ) → Ab(X) is exact for any open set
ι : U ,→ X.
Proof. If 0 → F 0 → F → F 00 → 0 is exact on U , then one easily checks that
0 → ι! F 0 → ι! F → ι! F 00 → 0
is exact on X by checking exactness at stalks using the lemma.
The sheaf ι! F is called the extension by zero of F (it is sometimes denoted
F X ). It is manifestly a subsheaf of i∗ F , though the two do not generally coincide.
For example, if i : U → X is the inclusion of the open interval (0, 1) in the closed
interval [0, 1] and F = AU is a constant sheaf on U , then i∗ F is just the constant
sheaf AX , which has stalk A at every point of X, while the stalk of i! F is zero at
0, 1 ∈ X.
Whenever i : U ,→ X and j : V ,→ X are open sets with U ⊆ V and F ∈
Ab(X), we have a natural morphism i! i−1 F → j! j −1 F . On W ∈ Ouv(X), this
map sends a section s ∈ F (W ∩ U ) with Supp s closed in W (hence also in W ∩ V )
to the section of F (W ∩ V ) obtained by gluing s and 0 ∈ F ((W ∩ V ) \ Supp s).
Note that the support of this glued section is the same as the support of s, hence it
is still closed in W and this is a well-defined map
(i! i−1 F )(W ) → (j! j −1 F )(W )
which clearly respects restriction.
Theorem 2.3. (Mayer-Vietoris Sequences) Let i : U ,→ X be the inclusion
of an open set, j : Z ,→ X its closed complement. There is a short exact sequence
0 → i! i−1 F → F → j∗ j −1 F → 0
in Ab(X) natural in F ∈ Ab(X). Let k : V ,→ X be another open set and let
l : U ∪ V ,→ X and m : U ∩ V ,→ X be the inclusions. There is a short exact
sequence
0 → m! m−1 F → i! i−1 F ⊕ k! k −1 F → l! l−1 F → 0.
Proof. Check exactness on stalks by using the previous lemma.
Theorem 2.4. (Duality for an open embedding) Let i : U ,→ X be the
inclusion of an open subset of a topological space X. The functor i−1 : Ab(X) →
Ab(U ) is right adjoint to the exact functor i! : Ab(U ) → Ab(X) and i−1 i! = Id.
The functor i−1 : Ab(X) → Ab(U ) takes injectives to injectives.
Proof. Checking the adjointness is a straightforward exercise and the equality
i i! = Id is clear from the definitions. The final statement then follows formally:
Suppose F 0 ∈ Ab(X) is injective and
−1
0 → ι−1 F 0 → F → F 00 → 0
3. PROPER MAPS - SECOND TREATMENT
63
is an exact sequence in Ab(U ). Applying ι! and (2.2), we obtain a diagram of solid
arrows with exact row
0
/ ι! ι−1 F 0
_
F
0
z
/ ι! F
/ ι! F 00
/
0
s
in Ab(X), which completes as indicated by injectivity of F 0 ∈ Ab(X). Applying the
functor ι−1 and using i−1 i! = Id, the vertical arrow above becomes an isomorphism
and ι−1 s provides a splitting of the original sequence.
3. Proper maps - Second treatment
To better understand the stalk formula in (15.1), we will need to prove some
rather technical lemmas. It turns out that the stalk formula behaves very well for
proper maps, which we now introduce.
Definition 3.1. A subspace Z ⊆ X is called relative Hausdorff is any two points
of Z have disjoint neighborhoods in X. A map of topological spaces f : X → Y is
called proper if f is closed (the image of any closed set is closed) and the fibers of
f are compact and relative Hausdorff in X.
For example, the inclusion of a closed subspace Z ,→ X is always a proper map.
A space X is proper over a point iff X is compact Hausdorff. The basic properties
of proper maps are summed up in the following theorem.
Theorem 3.2. The composition of proper maps is proper. Proper maps are
preserved under arbitrary base change. A map f : X → Y is proper iff for some
(equivalently any) cover {Ui } of Y , the maps f : f −1 (Ui ) → Ui are proper. If
f : X → Y is proper and y ∈ Y , then the neighborhoods f −1 (U ) of f −1 (y) (as U
ranges over all neighborhoods of y ∈ Y ) are cofinal in the set of neighborhoods of
f −1 (y) in X.
Proof. We first prove that proper maps are closed under base change. Let
g : Z → Y be arbitrary, and let Z ×Y X be the fibered product. To check that
π1 : Z ×Y X → Z is closed, it is enough to check that π1 (A ×Y B) is closed whenever
A ⊆ Z and B ⊆ X are closed, since closed sets of this form form a basis for all
closed sets. On the other hand, we have π1 (A ×Y B) = A ∩ g −1 (f [B]), which is
(
closed when f is closed. Similarly, π1 z) = {z} × f −1 (g(z)) is compact if f −1 (g(z)) is
compact and if p, q ∈ f −1 (g(z)) have disjoint neighborhoods U, V in X, then Z ×Y U
and Z ×Y V are disjoint neighborhoods of (z, p) and (z, q) in Z ×Y X.
The fact that being proper is local on the codomain follows from the fact that
being closed is local; the “compact relative Hausdorff fibers” condition is clearly
local on the codomain.
64
VI. DUALITY
The last statement is immediate from the fact that f is closed: if V is an open
neighborhood of f −1 (y) in X, then f [X \ V ] is closed in X since f is closed, so
Y \f [X\V ] is an open neighborhood of y in Y and we have f −1 (Y \f [X\V ]) ⊆ V . Lemma 3.3. A map f : X → Y of locally compact Hausdorff spaces is compact
iff f −1 (K) is compact for every compact subset K ⊆ Y .
Proof. Exercise ??.
Corollary 3.4. If f : X → Y and g : Y → Z are maps of locally compact
Hausdorff spaces such that gf is proper, then f is proper. It follows that any map
between compact Hausdorff spaces is proper.
Proof. By the lemma, it suffices to prove that f −1 (K) is compact for any
compact K ⊆ Y . Certainly g[K] is compact in Z, so (gf )−1 (g[K]) is compact
because gf is proper. But f −1 (K) is a closed subspace of (gf )−1 (g[K]), so it is also
compact.
Lemma 3.5. Let f : X → Y be a proper map, F a sheaf on X. For every y ∈ Y ,
the natural map (Rp f∗ F )y → Hp (f −1 (y), F |f −1 (y) ) is an isomorphism.
Proof. Apply (1.7) to the inclusion of the subspace f −1 (y) ⊆ X using the
assumption (1) and use (3.2) to note that the f −1 (U ) are cofinal in the neighborhoods
of f −1 (y).
CHAPTER VII
Finite Spaces
This section is devoted to sheaf theory on finite, typically non-Hausdorff, topological spaces. We begin with some preliminaries on the topology of a finite topological
space X.
1. Supports
For any family of supports Φ on X, there is a closed subspace Y such that Φ
consists of all closed subsets of Y . Indeed, just take Y to be the union of every closed
set in Φ: then Y is in Φ because Φ is closed under finite unions and every closed
subset of Y is in Φ because Φ is closed under passage to smaller closed subsets. As
usual, we will write ΓY (X, F ) instead of ΓΦ (X, F ).
2. Specialization
The points of any topological space X are quasi-ordered by specialization: x ≤ y
iff x ∈ {y}− . We are only interesting in making statements about Sh(X), so there
would be no harm in replacing X with its sobrification, thereby ensuring that ≤ is,
in fact, a partial order (equivalently: X is T0 ). The specialization ordering is particularly important when X is finite, as it completely determines the topology of X.
A set is closed iff it is closed under passage to smaller elements (“specializations”):
this is because any point in the closure of a finite set S must be in the closure of
{s} for some s ∈ S. A subset of X is open iff it is closed under passage to larger
elements.
In particular, the set U1 consisting of ≤ maximal points of (finite) X is open.
Let Z1 be its closed complement. Similarly, the set U2 of ≤ maximal points of Z1 is
open in Z1 and Z2 := Z1 \ U2 is hence closed in X. Continuing in this manner, we
obtain a strictly decreasing sequence
X = Z0 ) Z1 ) · · · ) Zn ) ∅
of closed subspaces of X.
Lemma 2.1. Let X be a sober noetherian space. The noetherian dimension n of
X is equal to the length of the longest chain x0 < x1 < · · · < xn in (X, ≤) (n = ∞
iff there are arbitrarily long such chains).
Proof. Suppose ∅ ( Z0 ( · · · ( Zn is a chain of irreducible closed subspaces
in X. Since X is sober, each Zi has a generic point xi . We have x0 < x1 < · · · < xn
65
66
VII. FINITE SPACES
in (X, ≤). Conversely, given such a chain in (X, ≤), we get a chain of irreducible
closed subspaces ∅ ( {x0 }− ( · · · ( {xn }− .
3. Coniveau
Let X be any ringed space, not necessarily finite, and let F ∈ Mod(X). For any
specialization x ≤ y, we have a natural map fx,y : Fx → Fy which is OX,x → OY,y
linear. It is nothing but the direct limit of the maps F (U ) → Fy over neighborhoods
U of x, all of which contain y since x ∈ {y}− . We view fx,y as an OX,x linear map
by regarding Fy as an OX,x module by restriction of sclars along OX,x → OY,y .
Applying x∗ , we obtain a Mod(X) morphism
x∗ fx,y : x∗ Fx → x∗ Fy .
If we have a sequence of specializations x ≤ y ≤ z, then it is clear that
fx,z = fy,z fx,y ,
so we may in fact view the stalk functor as a functor
Ab(X) → HomCat ((X, ≤), Ab).
The coniveau complex is the complex
Y
Y
0→F →
x0∗ Fx0 →
x0∗ Fx1 →
x0
x0 <x1
Y
x0∗ Fx2 → · · ·
x0 <x1 <x2
in Mod(X), where the products are over strictly increasing chains in (X, ≤) and
where the boundary maps are given by
(dn s)(x0 , . . . , xn+1 ) := x1∗ fx1 ,xn+1 s(x1 , . . . , xn+1 )
n
X
(−1)i x0∗ fx0 ,xn+1 s(x0 , . . . , x̂i , . . . , xn+1 )
+
i=1
+(−1)n+1 x0∗ fx0 ,xn s(x0 , . . . , xn ).
By abuse of notation, we will often simply write
(dn s)(x0 , . . . , xn+1 ) :=
n
X
(−1)i s(x0 , . . . , x̂i , . . . , xn+1 )|xn+1 .
i=1
Observe that this complex “depends on X” even when Sh(X) does not. For
example, the coniveau complex computed on X is not the same as the coniveau
complex computed on its sobrification X sob even though Sh(X) = Sh(X sob ). It is
generally more natural to work with the coniveau complex computed on X sob , where
the specialization ordering better reflects the topology of X.
6. HOM FORMULAS
67
4. Equivalence
When X is finite, every x ∈ X has a smallest neighborhood Ux and Fx = F (Ux ).
We will write ix : Ux → X for the inclusion of this open set throughout. Note x ≤ y
iff Uy ⊆ Ux . When x ≤ y, the restriction map F (Ux ) → F (Uy ) may be viewed as a
map fx,y : Fx → Fy (we will usually denote this s 7→ s|y ). The restriction maps fx,y
completely determine the sheaf F , as we will now show. Note that U = ∪x∈U Ux for
any U ∈ Ouv(X) and
[
Ux ∩ Uy =
Uz ,
z≥x,y
so, by using the sheaf property for F , we can recover F (U ) as the equalizer


Y
Y
Y
(4.1)
F (U ) = lim 
Fx ⇒
Fz  ,
←−
(x,y)∈U ×U z≥x,y
x∈U
where the two maps are (sx ) 7→ (fx,z (sx )) and (sx ) 7→ (fy,z (sx )). There is an
equivalence of categories
Sh(X) = HomCat ((X, ≤), Ens)
F 7→ (x 7→ Fx )
natural in X, whose inverse is given by taking a functor f to the presheaf (in fact
sheaf) whose sections are defined by the formula (4.1).
5. Stalk formulas
The formulae
A, y ≤ x
0, otherwise
A, x ≤ y
0, otherwise
(x∗ A)(y) =
(ix! A)y =
are clear from the formulae for stalks (3), (2.1).
6. Hom formulas
Similarly, we have
HomX (F , x∗ A) = Hom(Fx , A)
HomX (ix! A, F ) = Hom(A, Fx ).
For example, the first isomorphism is given by mapping g : F → x∗ A to gx : Fx →
A. This is an isomorphism because gy : Fy → (x∗ A)y must be given by gx fx,y if
x ≤ y (or zero otherwise).
68
VII. FINITE SPACES
There is a kind of duality for finite spaces obtained by inverting the specialization
ordering (replacing (X, ≤) with the opposite category). This exchanges x∗ and
ix! ( ), etc.
Lemma 6.1. Let X be a finite ringed space, x a point of X. If I is an injective
OX,x module, then x∗ I ∈ Mod(X) is injective. If P is a projective OX,x module,
then ix! P ∈ Mod(X) is projective.
Proof. Both statements follow from the formulas for Hom above and the exactness of the stalk functor.
Theorem 6.2. Let X be a finite ringed space of dimension n. For every F ∈
Mod(X), the coniveau complex of (3) is a resolution
M
M
M
0→F →
x0∗ Fx0 →
x0∗ Fx1 →
x0∗ Fxn → 0
x0
x0 <x1
x0 <···<xn
by finite direct sums/products of skyscraper sheaves. There is a “dual” resolution by
direct sums of sheaves of the form ix! F x .
Proof. We will show the the stalk of this complex at a point z ∈ X is a contractible complex. The key point is that we can commute the finite direct products
in the definition of the coniveau complex with stalks (finite inverse limits commute
with filtered direct limits). Therefore, by (3), this stalk complex is given by
M
M
M
0 → Fz →
F x0 →
F x1 → · · · →
Fxn → 0.
z≤x0
z≤x0 <x1
z≤x0 <···<xn
Define
cn :
M
M
Fxn+1 →
z≤x0 <···<xn <xn+1
F xn
z≤x0 <···<xn
by
(cn s)(x0 , . . . , xn ) :=
s(z, x0 , . . . , xn ), z < x0
0,
otherwise
with the convention that c−1 : ⊕x Fx → Fz sends s to s(z). One can check that
dnz cn + cn+1 dn+1
= Id .
z
Indeed, the computation is formally similar to that in the proof of Theorem 1.1. The resolution of the theorem will be called the coniveau resolution.
7. Poincaré duality
We now prove an instance of the Poincaré Duality theorem. Let X be a finite
topological space with the constant sheaf of rings k associated to a field k. Let Db (k)
be the bounded derived category of Mod(k) and let Db (k) be the bounded derived
category of k vector spaces.
8. EXAMPLES
69
◦
Theorem 7.1. There is a complex ωX
∈ Db (k) and a Db (k) isomorphism
◦
R Hom•k (R ΓF • , V • ) = R Hom•k (F • , ωX
⊗ V •)
natural in F • ∈ Db (k) and V • ∈ Db (k).
Proof. By (6.1), the coniveau resolution F → C (F )• is an injective resolution
of X (every vector space Fx is injective). Hence, we have an isomorphism R ΓF =
ΓC (F )• , natural in F . For a k vector space V , we have
R Hom•k (R ΓF , V ) = Hom•k (ΓC (F )• F , V ).
This latter complex looks like
M
M
Homk (
Fxn , V ) → · · · → Homk (
Fx0 , V ),
x0 <···<xn
x0
sitting in the interval [−n, 0]. On the other hand, by the Hom formulae (6), this is
naturally isomorphic to the complex
M
M
Homk (F , xn∗ V ) → · · · →
Homk (F , x0∗ V ),
x0 <···<xn
x0
which is nothing but the complex
◦
Hom•k (F , ωX
⊗ V ),
◦
where ωX
is the complex
M
xn∗ k → · · · →
M
x0∗ k
x0
x0 <···<xn
◦
(sitting in the interval [−n, 0]). Furthermore, ωX
⊗ V is a complex of injective k
modules, so this is also
◦
R Hom•k (F , ωX
⊗ V ).
This discussion is easily extended to complexes F • , V • by applying the coniveau resolution termwise, and making the same arguments with the resulting double/triple
complexes.
8. Examples
In this section we work out some sheaf cohomology groups on finite topological
spaces. Finite (non-Hausdorff) topological spaces are a rich source of examples in
the theory of sheaves.
Our main examples will involve the following spaces. The Sierpinski space S
is the unique topology on {1, 2} where {1} is closed, but {2} is not. The space
X = {1A, 2, 1B} has three points with basic open sets
U1A := {1A, 2}
U2 := {2}
U1B := {2, 1B}.
70
VII. FINITE SPACES
The space Y = {0A, 1A, 2, 1B, 0B} has basic open sets those of X, plus the sets
U0A := {0A, 1A, 2, 1B}
U0B := {1A, 2, 1B, 0B}.
Let us start with the Sierpinski space S. Here every presheaf F ∈ PAb(S)
is a sheaf as long as F (∅) = 0 (the sheafification functor does nothing but set
F (∅) = 0) because S has only two interesting open sets S, {2} and neither of these
can be covered nontrivially. A sheaf F ∈ Ab(S) is nothing but the data of the
restriction map F (S) → F ({2}), which is also the data of the specialization map
F1 → F2 (note 0 < 1 is the specialization ordering of S), so Ab(S) is nothing but
the arrow category Ab•→• of Ab. The global section functor is identified with the
stalk functor F 7→ F1 , hence it is exact. However, sections Γ1 with support in the
closed set {1} is identified with the kernel functor
Γ1 (S, F ) = Ker(F1 → F2 ).
This is not exact. Indeed, an exact sequence in Ab(S) is just a morphism of exact
sequences in Ab, and the sequence of kernels of such a morphism need not form an
exact sequence; they must be linked up with the cokernels as in the Snake Lemma.
In fact it is straightforward to check, using the obvious flasque resolution
/
0
0
/
F1
F2
/
F1 ⊕ F 2
/
/
π2
F2
/
F2
/
0
/
0
0,
that

 Ker(F1 → F2 ), p = 0
Cok(F1 → F2 ), p = 1
Hp1 (S, F ) =
 0,
p>1
Notice that, furthermore, the forgetful functor Ab(S) → PAb(S) is exact, so
Γ1 is not even exact as a functor Γ1 : PAb(X) → Ab. Even worse, there is no hope
of “making it exact” (computing its derived functors) by using Čech cohomology.
Indeed, the trivial cover {S} refines every other cover, and, since we may compute
with alternating Čech cohomology, the Čech cohomology groups in this case are
simply
p
Γ1 (S, F ), p = 0
Ȟ1 (S, F ) =
0,
p>0
This shows the necessity of the “neighborhoods” assumption in (7.3).
The specialization ordering (Y, ≤) and the category Ouv(Y )op look like
8. EXAMPLES
71
Y
0A
0B
'
w
1A
2
and
}
!
U0A
!
1B
~
U1A
X
U0B
}
}
!
!
U2
U1B
}
∅
respectively. The sheafification functor
PAb(Y ) → Ab(Y )
is given by:
F (Y )
{
F (U0A ) ×F + (X) F (U0B )
#
F (U0A )
w
F (U0B )
{
'
F (X)
{
F (U1B )
{
F (U2 )
F (∅)
w
F (U0A ) ×F (U2 ) F (U0B )
#
F (U1A )
F (U1B )
7→
#
#
'
F (U1A )
w
'
F (U1A )
'
F (U1B )
F (U2 )
w
0
Note the “double fiber product” formula for global sections F + (Y ). The fact that
sheafification preserves stalks is manifest in the fact that F (U ) → F + (U ) is always
an isomorphism for
U ∈ {U0A , U0B , U1A , U1B , U2 }
as these are the smallest neighborhoods of the five points of Y .
72
VII. FINITE SPACES
The open cover U = {U0A , U0B } of Y refines any other open cover, hence we can
use ordered Čech cohomology with U to compute the absolute Čech cohomology of
any F ∈ PAb(Y ). The ordered Čech complex Č(U, <, F ) is simply
F (U0A ) ⊕ F (U0B ) → F (X),
where the map is the difference of the restriction maps. For example, if F is the
presheaf
Z
1
1

Z
Z
(1,0)

(1,1)
Z⊕Z
(1,0)
(1,0)

Z
Z
1

1
Z
0
this Čech complex is
1 −1
: Z2 → Z2 ,
0 −1
which has no homology at all. On the other hand, F + (or even F sep ) is just the
constant sheaf ZY and the sheafification map F → F + is an isomorphism on every
open set except that F (X) → F + (X) is given by (1, 0) : Z⊕Z. The monomorphism
0
Ȟ (Y, F ) → Γ(Y, F + )
of Lemma 8.4 is simply 0 ,→ Z. The “reason” this is not surjective is because the
two sections 1 ∈ F (U0A ) and 1 ∈ F (U0B ) “should” glue to form a global section
since their difference in F (X) is zero when restricted to F (U1A ) and F (U1B ) and
U1A and U1B cover X.
Higher Čech cohomology of sheaves on Y is also poorly behaved. We will now
construct a sheaf F 0 ∈ Ab(Y ) where the natural monomorphism
2
Ȟ (Y, F 0 ) → H2 (Y, F 0 )
2
of Theorem 5.1 is given by 0 ,→ Z. Of course, we already know Ȟ (Y, F 0 ) vanishes
for any F 0 ∈ PAb(Y 0 ) as the Čech complex is supported in degrees 0, 1, so we
simply seek an F 0 ∈ Ab(Y ) with nonvanishing H2 . Given that Y has noetherian
dimension two, it is not terribly surprising that we can find such a sheaf.
8. EXAMPLES
73
Our sheaf F 0 is part of the short exact sequence
0 → F 0 → F → F 00 → 0
(8.1)
in Ab(Y ) given below.
0
Z
Z
=
0

0
Z
Z
=
0
=
Z
=
Z
∆
Z
π2
|
"
=

|
Z
Z⊕Z
π1
Z
"
∆
=

0
#
/
Z
=
{
Z
=

/
0
=
=
#
Z
0
Z
{
Note that exactness in Ab(Y ) can be checked on stalks, which is to say it can
be checked on U0A , U0B , U1A , U1B , and U2 . To compute the cohomology groups of
F 00 , we consider the SES of sheaves
0 → F 00 → G → G 00 → 0
(8.2)
given by the diagram below.
Z⊕Z
Z
=
=
=
Z⊕Z
{
~
Z
Z
~
∆
/
{
π1
~
Z
0
#
Z
~
Z
Z
/
~
0
π2
{
~
π2
=
Z⊕Z
π2
Z
Z⊕Z
=
Z⊕Z
π1
#
Z⊕Z
∆
π1
=
#
#
0
Z
0
~
{
Here the maps on global sections
F 00 (Y ) = Z → G (Y ) = Z ⊕ Z → G 00 (Y ) = Z ⊕ Z
0
0
~
74
VII. FINITE SPACES
are given by ∆ and ∆π2 , respectively. The sheaves G and G 00 are flasque, hence
have no higher cohomology, so by examining the LES in cohomology, we compute:
Z p = 0, 1
p
00
H (Y, F ) =
0 p>1
The sheaf F in the appearing in the sequence (8.1) is also flasque, so we can now
take the LES in cohomology there to show that
Z
p=2
p
0
0
H (X , F ) =
0 otherwise
as claimed.
Note that the exact sequence
0 → F 0 → F → G → G 00 → 0
obtained by concatenating the exact sequences (8.1) and (8.2) (which we have used
to work out the cohomology of F 0 ) is nothing but the coniveau resolution of F 0
constructed in (6.2).
9. Exercises
Exercise 9.1. Let Y ⊆ X be a closed subspace of a finite space X. Write down
a formula for ΓY (X, F ) in the langauge of (4) (i.e. making reference only to the
stalks of F and the maps between them arising from specializations). Specialize
your formula to the case where F is a skyscraper.
Exercise 9.2. Extend the above Poincaré Duality to cohomology with support
in a closed subspace Y ⊆ X (c.f. (1)).
Exercise 9.3. A ringed space (X, OX ) is specialization flat if the natural ring
map OX,x → OX,y is flat for any specialization x ≤ y. Show that every scheme
is specialization flat, but show that every ring map arises in this manner for some
appropriate (X, OX ).
CHAPTER VIII
Čech Theory
Čech cohomology is named for the Čech mathematician Eduard Čech (18931960), after whom the Stone-Čech compactification is also named.
1. Čech resolution
Let X be a topological space, F ∈ PAb(X). It will be important to develop
a theory of Čech cohomology for presheaves as well as sheaves. For an open set
i : U ,→ X, we set FU = i∗ i−1
pre F . Note that FU is given by
V 7→ F (U ∩ V ),
−1
and that this is a sheaf if F is a sheaf (i−1
in this situation). Furthermore,
pre = i
FU is clearly a flasque sheaf if F is a flasque sheaf.
If V ⊂ U then there is a natural morphism FU → FV of abelian presheaves
on X induced by the restriction maps of F . Note that the stalk of this map is an
isomorphism at any x ∈ V (both stalks are just Fx ). Let
U = {Ui : i ∈ I}
be a family of open sets of X. For a finite subset {i0 , . . . , ip } ⊆ I, or any vector
(i0 , . . . , ip ) ∈ I p+1 , we use the following notation:
U (i0 , . . . , ip ) := Ui0 ∩ · · · ∩ Uip
F (i0 , . . . , ip ) := FU (i0 ,...,ip ) .
The convention
(1.1)
F () := F
is convenient. For j ∈ {0, . . . , p} the natural inclusion U (i0 , . . . , ip ) → U (i0 , . . . , îj , . . . , ip )
of open sets induces a PAb(X) morphism
F (i0 , . . . , îj , . . . , ip ) → F (i0 , . . . , ip )
s 7→ s|U (i0 ,...,ip ) .
For each nonnegative integer p, define
Y
Cˇp (U, F ) :=
(i0 ,...,ip
)∈I p+1
75
F (i0 , i1 , . . . , ip ).
76
VIII. ČECH THEORY
The presheaves Cˇp (U, F ) ∈ PAb(X) can be assembled into a complex Cˇ• (U, F )
whose boundary maps
dp : Cˇp (U, F ) → Cˇp+1 (U, F )
are given by
(1.2)
p+1
X
(−1)j s(i0 , . . . , îj , . . . , ip+1 )|U (i0 ,...,ip+1 ) .
(d s)(i0 , . . . , ip+1 ) :=
p
j=0
There is a natural augmentation morphism
d−1 : F → Cˇ0 (U, F )
given by setting s(i0 ) := s|U (i0 ) (the usual formula using convention (1.1)). The
abelian presheaves Cˇp (U, F ) are called the Čech presheaves associated to the presheaf
F and the covering sieve U.
Theorem 1.1. If U is a cover of X, then d−1 : F → Cˇ• (U, F ) is a resolution
for any F ∈ Ab(X).
Proof. We have already noted that, since F is a sheaf, each F (i0 , . . . , ip ) is
a sheaf, and hence Cˇp (U, F ) (being a product of such sheaves) is also a sheaf. It
therefore suffices to check exactness of the augmented complex F → Cˇ• (U, F )
when restricted to each of the open sets Ui , since the Ui form a cover.1 In fact, we
will show that
(F → Cˇ• (U, F ))|U
i
is null-homotopic. Define a contraction map
Y
F (i0 , . . . , ip )|Ui →
cp :
i0 ,...,ip
Y
F (i0 , . . . , ip−1 )|Ui
i0 ,...,ip−1
by
(cp s)(i0 , . . . , ip−1 ) := s(i, i0 , . . . , ip−1 ).
In making this definition we are using the obvious identification
F (i, i0 , . . . , ip )|Ui = F (i0 , . . . , ip )|Ui .
By our convention (1.1), c0 : Cˇ0 (U, F )|Ui → F |Ui simply sends s to s(i).
Now we just check that the cp form a retraction for Cˇ• (U, F )x . That is,
dp−1 cp + cp+1 dp = Id .
1It
would even be good enough to check exactness at all stalks; we are proving a stronger statement.
2. REFINEMENT MAPS
77
Indeed, we have
(d
p−1
cp s)(i0 , . . . , ip ) =
p−1
X
(cp s)(i0 , . . . , îj , . . . , ip )
j=0
=
p−1
X
(−1)j s(i, i0 , . . . , îj , . . . , ip )
j=0
and
(cp+1 dp s)(i0 , . . . , ip ) = (dp s)(i, i0 , . . . , ip )
p−1
X
(−1)j+1 s(i, i0 , . . . , îj , . . . , ip ).
= s(i0 , . . . , ip ) +
j=0
The resolution of the theorem is called the Čech resolution of F ∈ Ab(X).
2. Refinement maps
Suppose that U = {Ui : i ∈ I} and V = {Vj : j ∈ J} are two families of open
sets in X. Suppose r : J → I is a refinement in the sense that Vj ⊆ Ur(j) for all
j ∈ J. Then we have a natural restriction map FU (r(j)) → FVj for every j, and in
fact a map of complexes
r∗ : Cˇ• (U, F ) → Cˇ• (V, F )
s 7→ s(r(j0 ), . . . , r(jp ))|V (j0 ,...,jp ) .
Lemma 2.1. Suppose r1 , r2 : J → I are two refinements. Then the maps
r1∗ , r2∗ : Cˇ• (U, F ) → Cˇ• (V, F )
are homotopic.
Proof. The maps
hp : Cˇp (U, F ) → Cˇp−1 (V, F )
given by
(hp s)(j0 , . . . , jp−1 ) :=
p−1
X
(−1)k s(r1 (j0 ), . . . , r1 (jk ), r2 (jk ), . . . , rp−1 (jk ))
k=0
provide a homotopy from r1∗ to r2∗ .
78
VIII. ČECH THEORY
3. Alternating variant
The alernating (or ordered) Čech complex is a “smaller” version of the Čech
complex which is convenient for computations, and has theoretical implications (relations between covering dimension and cohomological dimension). Let {Ui : i ∈ I}
be a family of open sets in X. Fix a total ordering ≤ of I. The sheaves
Q
Cˇp (U, <, F ) :=
F (i0 , . . . , ip )
i0 <···<ip
form a complex Cˇ• (U, <, F ) whose boundary map is defined by the same formula
(1.2) as the boundary map for Cˇ• (U, F ). There is an obvious projection map
(3.1)
Cˇ• (U, F ) → Cˇ• (U, <, F )
which is a map of complexes.
Lemma 3.1. The map (3.1) is a homotopy equivalence.
Proof. We will obtain this later from the simplicial point of view.
For now, we just note that the alternating Čech complex certainly provides a
resolution of any F ∈ Ab(X). This can be proved directly (without the above
lemma) just as in the proof of (1.1) using the contraction operator
s(i, i0 , . . . , ip−1 ), i < i0
(cp s)(i0 , . . . , ip−1 ) :=
0,
otherwise.
4. Cech cohomology for a fixed cover
Finally we define Čech cohomology. Let U = {Ui : i ∈ I} be a cover of X. Let
F ∈ PAb(X), let ČΦ• (U, F ) be the complex in Ab(X) obtained by applying ΓΦ to
the Čech complex Cˇ• (U, F ).
Warning. The obvious inclusion
Y
ČΦp (U, F ) ⊆
ΓΦ∩U (i0 ,...,ip ) (U (i0 , . . . , ip ), F )
(i0 ,...,ip )∈I p+1
is not generally an equality because ΓΦ does not generally commute with products.
This formula is, however, valid if ΓΦ = Γ, or if I is finite. In general, ČΦp (U, F ) is
the subcomplex of Č p (U, F ) consisting of those
Y
s = (s(i0 , . . . , ip )) ∈
F (U (i0 , . . . , ip ))
(i0 ,...,ip )∈I p+1
such that there is a Y ∈ Φ such that
Supp s(i0 , . . . , ip ) ⊆ Y
for all (i0 , . . . , ip ) ∈ I p+1 .
The abelian groups
n
ȞΦ (U, F ) := Hn (ČΦ• (U, F ))
6. ČECH COHOMOLOGY
79
are called the Čech cohomology 2 groups of F with respect to the cover U.
5. Basic properties
We now prove some of the basic properties of the Čech cohomology groups
n
ȞΦ (U, F ).
Theorem 5.1. For any F ∈ Ab(X), there is a first quadrant spectral sequence
p+q
E2p,q = Hp (HqΦ (Cˇ(U, F )• )) =⇒ HΦ
(X, F ).
p
In particular, there are natural maps E2p,0 = ȞΦ (U, F ) → HpΦ (X, F ) for every p.
Proof. Since the Čech complex is a resolution by (1.1), this is just a special
case of the spectral sequence associated to a resolution of an object in an abelian
category.
When each of the sheaves Cˇp (U, F ) is ΓΦ -acyclic, we say that U is a good cover
for F . We say U is a good cover if it is a good cover for every abelian sheaf F .
Corollary 5.2. If U is a good cover for F , then the natural map
p
ȞΦ (U, F ) → HpΦ (X, F )
is an isomorphism for every p.
Proof. The spectral sequence of (5.1) degenerates.
6. Čech cohomology
In this section we introduce the (absolute) Čech cohomology groups, which have
no dependence on a choice of cover.
Set
p
p
ȞΦ (X, F ) := lim
ȞΦ (U, F ),
−→
U
where the direct limit is over the the category of covers U = {Ui : i ∈ I} where there
is a unique morphism from U → V = {Vj : j ∈ J} iff there is a refinement map
r : J → I. The restriction maps are defined by taking the maps Hp (r∗ ) induced on
cohomology and noting that there are independent of the choice of r.
An alternative means of calculating the Čech cohomology groups is by means
of covering sieves (c.f. (9)). Let U = {Ui : i ∈ I} be the set of all open subsets of
X (the trivial sieve on X ∈ Ouv(X)). By slight abuse of notation, say J ⊆ I is a
covering sieve if {Uj : j ∈ J} is a covering seive. Order the set of covering sieves
2Our
definition agrees with Godement’s [G, 5.2].
80
VIII. ČECH THEORY
by reverse inclusion. The resulting set is filtered and we may compute the Čech
cohomology as the cohomology of the direct limit complex:
!
p
ȞΦ (X, F ) := Hn lim
ČΦ• ({Uj : j ∈ J}, F ) .
−→
J
This is because filtered direct limits commute with cohomology and the set of covering sieves is cofinal in the set of all covers, ordered by the existence of a refinement.
p
The abelian groups ȞΦ (X, F ) are called the (absolute) Čech cohomology groups
of F . We will revisit these constructions are first proving some “classical” facts
about Čech cohomology.
7. Čech acyclic sheaves
As with usual sheaf cohomology, we will need a supply of presheaves with vanishing higher Čech cohomology. Unfortunately, our supply of such presheaves will
be rather lame.
Theorem 7.1. Let U be an open cover of X, F ∈ PAb(X). The Čech cohomology groups HpΦ (U, F ) vanish for p > 0 under either of the following hypotheses:
(1) F is flasque and is in Ab(X)
(2) F is Φ soft, Φ is paracompactifying, and F ∈ Ab(X)
Proof. We first prove the statement when F is flasque. Then it is clear from
the definition that ι∗ ι∗ F is flasque for anyQinclusion ι :Q
U ,→ X. Furthermore, a
product of flasque sheaves is flasque (since ( i Fi )(U ) = i Fi (U ) and a product of
surjections is surjective). Hence the Čech resolution ε : F → Cˇ• (U, F ) is a flasque
resolution of F , so
p
ȞΦ (U, F ) = Hp (ΓΦ Cˇ• (U, F ))
= HpΦ (X, F )
by 10.5, but this vanishes for p > 0 by 10.4 because F is flasque.
The above theorem says that Čech cohomology is, in some sense, effaceable.
However, it turns out that Čech cohomology does not generally form a δ-functor
from Ab(X) to Ab: a short exact sequence of sheaves does not generally induce
a long exact sequence of Čech cohomology groups. This is the only obstruction to
identifying Čech cohomology with sheaf cohomology.
If we replace sheaves with presheaves, Čech cohomology is better behaved:
•
Theorem 7.2. For any cover U of X, the Čech cohomology groups Ȟ (U, )
n
0
form an effaceable δ-functor from PAb(X) to Ab, hence Ȟ (U, F ) = Rn Ȟ (U, F )
for every F ∈ PAb(X).
7. ČECH ACYCLIC SHEAVES
81
Proof. Suppose
F = 0 → F 0 → F → F 00 → 0
is a short exact sequence in PAb(X). Then
0 → F 0 (U ) → F (U ) → F 00 (U ) → 0
is exact for every U ∈ Ouv(X) (this is why it is important to use presheaves).
Consequently,
0 → Č • (U, F 0 ) → Č • (U, F ) → Č • (U, F 00 ) → 0
is a short exact sequence of complexes of abelian groups, since, in each degree it is
just a product of short exact sequences of the above form. The associated long exact
cohomology sequence is natural in F hence the first claim is proved. To see that
•
Ȟ (U, ) is effaceable it is enough to check that it is left exact and preserves finite
direct sums (clear) because then it can be erased by injectives, which PAb(X) has
enough of by (4.6).
With supports, we will need to be able to refine covers. The best we can do is:
Theorem 7.3. Let X, be a topological space, Φ a family of supports with neigh•
borhoods. The absolue Čech cohomology groups ȞΦ (X, ) form an effaceable δn
0
functor from PAb(X) to Ab, hence ȞΦ (U, F ) = Rn ȞΦ (U, F ) for every F ∈
PAb(X).
Proof. Let U = {Ui : i ∈ I} be the set of all open subsets of X. By the same
argument as in the previous proof, we reduce to proving that
0 → lim
ČΦp (I 0 , F 0 ) → lim
ČΦp (I 0 , F ) → lim
ČΦp (U, F 00 ) → 0
−→
−→
−→
I0
I0
I0
(the filtered direct limit is over all I 0 ⊆ I such that {Ui : i ∈ I 0 } is a covering sieve,
ordered by reverse inclusion) is exact for any exact sequence of presheaves
0 → F 0 → F → F 00 → 0.
Exactness on the left and in the middle follows from exactness of filtered direct
limits and left exactness of ΓΦ , so the issue is to prove surjectivity on the right. As
mentioned in (??), an element [(s, I 0 )] on the right is represented by a pair (s, I 0 )
where I 0 indexes a covering seive and s is a set of sections
s(i0 , . . . , ip ) ∈ F 00 (U (i0 , . . . , ip ))
such that there is a Y ∈ Φ with
Supp s(i0 , . . . , ip ) ⊆ Y
for all (i0 , . . . , ip ) ∈ (I 0 )p+1 .
Since Φ has neighborhoods, we can find an open set V ∈ Ouv(X) with Y ⊆ V
and V − ∈ Φ. By choosing a common refinement of the cover indexed by I 0 and
the cover {X \ Y, V } we can an I 00 ⊂ I 0 indexing a covering seive, such that, for
82
VIII. ČECH THEORY
every i ∈ I 00 , either Ui ⊆ V or Ui ∩ Y = ∅. In particular, for (i0 , . . . , ip ) ∈ (I 00 )p+1 ,
s(i0 , . . . , ip ) = 0 unless U (i0 , . . . , ip ) ⊆ V . The presheaf exactness implies that
F (U (i0 , . . . , ip )) → F 00 (U (i0 , . . . , ip ))
is always surjective, so we can find, for each (i0 , . . . , ip ) ∈ (I 00 )p with U (i0 , . . . , ip ) ⊆
V , a t(i0 , . . . , ip ) ∈ F (U (i0 , . . . , ip )) mapping to s(i0 , . . . , ip ). By setting t(i0 , . . . , ip ) =
0 for those (i0 , . . . , ip ) ∈ (I 00 )p where U (i0 , . . . , ip ) is not contained in V , we obtain
a preimage [(t, I 00 )] for [(s, I 0 )] = [(s|I 00 , I 00 )]. Note that
Supp t(i0 , . . . , ip ) ⊆ V − ∈ Φ
for all (i0 , . . . , ip ) ∈ (I 00 )p+1 .
The assumption that Φ has neighborhoods cannot be removed. See (??).
8. Čech presheaf
Let X be a topological space, Φ a family of supports on X, F ∈ Ab(X). For
n ≥ 0, let HΦn (F ) ∈ PAb(X) be the presheaf
HΦn (F ) : Ouv(X)op → Ab
U 7→ HnΦ∩U (U, F |U ).
As usual, when Φ is the family of all closed sets, we drop it from the notation.
Note that H 0 : Ab(X) → PAb(X) is just the usual forgetful functor.
Lemma 8.1. The functor
HΦn : Ab(X) → PAb(X)
F 7→ HΦn (F )
coincides with the nth right derived functor of HΦ0 : Ab(X) → PAb(X).
Proof. The functors F 7→ HΦn (F ) form an effaceable (by flasque sheaves, say)
δ-functor which manifestly has the right degree zero term.
Remark 8.2. The functor HΦ0 does not necessarily take a flasque sheaf to a
sheaf. If X is an infinite discrete space, Φ is the family of compact (i.e. finite)
subsets of X, and F is the flasque sheaf of continuous (i.e. arbitrary) Z valued
functions, then HΦ0 is not even a sheaf.
The restriction maps for the presheaf HΦq (F ) can be described concretely as
follows. Fix an injective resolution ε : F → I • of F . Then an inclusion of open
sets V ,→ U induces a morphism of complexes
ΓΦ∩U (U, I • ) → ΓΦ∩V (V, I • )
and thus a morphism on the cohomology of these complexes. On the other hand,
we have
HΦq (F )(U ) = Hq (ΓΦ∩U (U, I • ))
8. ČECH PRESHEAF
83
and similarly with U replaced by V . The abelian presheaves HΦq (F ) are called the
local cohomology presheaves of F with support in Φ.
Lemma 8.3. Assume Φ has neighborhoods. Then the sheaf (or even the separated
presheaf ) associated to the abelian presheaf HΦq (F ) is zero for all q > 0.
Proof. Fix a q > 0, an injective resolution F → I • , an open set U , and an
element s ∈ HΦq (F )(U ). Then s is represented by some q-cocycle s̃ in the complex
ΓΦ∩U (U, I • ), in other words by an element s̃ ∈ I q (U ) where the closure S of Supp s̃
in X is in Φ. Since Φ has neighborhoods, there is a B ∈ Φ such that S is contained
in the interior V of B. Since the complex I • is exact in positive degrees, the qcocycle s̃|U ∩V is a “coboundary on a cover.” That is, there is an open cover {Ui } of
U ∩ V and sections si ∈ I q−1 (Ui ) such that dq−1 (Ui )(si ) = s|U ∩V for all i. Since Ui
is contained in V , the closure in X of Supp si is contained in B, hence is in Φ, so
si ∈ ΓΦ∩Ui (Ui , I q−1 ). The Ui together with the set U \ Supp s̃ form an open cover
of U , and taking the zero section of I q−1 on the latter open set we find that s̃ is a
coboundary on each set in this open cover, hence the restriction of s to every set in
this open cover is zero in cohomology.
Lemma 8.4. Let X be a topological space, Φ a family of supports. The natural
map
0
ȞΦ (X, F ) → ΓΦ (X, F + )
is injective for any F ∈ PAb(X) and an isomorphism if F is separated.
Proof. Let {Ui : i ∈ I} be the open sets of X. The natural map is induced
by the natural map of presheaves F → F + , naturality of the the Čech complex
associated to a presheaf, and the fact that, for any G ∈ Ab(X), we have a natural
isomorphism
0
ȞΦ (X, G ) = ΓΦ
given by sending s ∈ ΓΦ (X, G ) to the class of (s|Ui , I) in s ∈ ΓΦ (X, G ) (this is
injective by separation for G and surjective by gluing for G , taking care to keep
track of supports by noting that the support of the section obtained by gluing
is the unions of the supports of the sections being glued). Now let us prove the
lemma. Making the natural identification just established, this map send a class
0
0
[(s, I 0 )] ∈ ȞΦ (X, F ) to the corresponding class [(s+ , I 0 )] ∈ ȞΦ (X, F + ). Viewing
F + as the sheaf of maps to the stalks with the local coherence condition, it is clear
that [(s+ , I 0 )] is zero iff we can find a finer cover (hence a finer covering seive) I 00 ⊆ I 0
on which (s, I 0 ) is zero.
Now let us try to prove surjectivity. We
Qwill ignore the supports for simplicity.
+
A section sΓ( X, F ) is an element s ∈ x∈X Fx satisfying the local coherence
condition (c.f. (??)), so we can find an I 0 ⊂ I indexing a cover and sections s(i) ∈
F (Ui ) such that s(i)x = s(x) for all x ∈ Ui . Of course, by adding in every set which
is a subset of some Ui and restricting sections, we may assume I 0 indexes a covering
seive, so (I 0 , s) ∈ Č 0 (I 0 , F ) is a class mapping to s ∈ Γ(X, F + ) under the natural
84
VIII. ČECH THEORY
map... but the difficulty is in showing that (I 0 , s) is in the kernel of d0 of the Čech
complex, and this is where we need separation of F . When F is separated, we
simply note that the stalk of s(i)|U (i,j) − s(j)|U (i,j) is zero at every point x ∈ U (i, j)
(because it is just s(x) − s(x) by local coherence), hence s(i)|U (i,j) − s(j)|U (i,j) = 0
by separation.
0
We encountered an example where Ȟ (X, F ) → Γ(X, F + ) is not surjective in
(8).
Theorem 8.5. Let X be a topological space, F ∈ Ab(X), U a cover of X.
There is a first quadrant spectral sequence
p
E2p,q = Ȟ (U, H q (F )) =⇒ Hp+q (X, F ).
If Φ is a family of supports with neighborhoods, then there is a similar spectral
sequence
p
E2p,q = ȞΦ (U, H q (F )) =⇒ Hp+q
Φ (X, F )
using the absolute Čech cohomology.
Proof. The functor Γ : Ab(X) → Ab is the composition of the forgetful
0
functor H 0 : Ab(X) → PAb(X) and the functor Ȟ (U, ) : PAb(X) → Ab,
hence we have the Grothendieck spectral sequence
0
E2p,q = Rp Ȟ (U, Rq H 0 F ) =⇒ Hp+q (X, F ).
This spectral sequence is identified with the desired one by (8.1) and (7.2). The
second statement is proved similarly using (7.3).
Note that, for the existence of the Grothendieck spectral sequence we need to
0
0
know that H 0 takes injectives in Ab(X) to Ȟ (U, ) acyclics (ȞΦ (X, ) acyclics
for the second statement); this follows from (7.1).
Theorem 8.6. For any topological space X and any abelian sheaf F on X, the
natural maps
p
Ȟ (X, F ) → Hp (X, F )
are isomorphisms for p = 0, 1 and injective for p = 2.
Proof. This follows from the exact sequence of low order terms
1
2
0 → Ȟ (X, F ) → H1 (X, F ) → H0 (X, H 1 F ) → Ȟ (X, F ) → H2 (X, F )
in the spectral sequence of (8.5), together with (8.4) and (8.3), which together imply
H0 (X, H 1 F ) = 0.
9. COMPARISON THEOREMS
85
9. Comparison theorems
We conclude our treatment of classical Čech theory with two theorems about
the relationship between Čech cohomology and the usual sheaf cohomology.
Theorem 9.1. (Cartan’s Criterion) Let X, be a topological space, Φ a family
of supports, F ∈ Ab(X). Suppose U is an open cover of X satisfying:
(1) If U, V ∈ U, then U ∩ V ∈ U.
(2) U contains arbitrarily small neighborhoods of any x ∈ X.
p
(3) ȞΦ (U, F |U ) = 0 for any p > 0 and any U ∈ U.
Then the natural map
p
ȞΦ (X, F ) → HpΦ (X, F )
is an isomorphism for every p.
p
Proof. We first prove, by induction on p, that the natural map ȞΦ (U, F |U ) →
is an isomorphism for all p, and all U ∈ U. The case p = 0 is trivial.
Suppose we know this for all q with 0 ≤ q < p. Because of our assumptions (1),(2)
about U, we note that, for any U ∈ U, the covers of U by open sets from U are cofinal
in the family of all open covers of U , so we may compute the Čech cohomology
p
groups Ȟ (U, F |U ) using only such covers. It then follows from assumption (3)
and the induction hypothesis that HΦq (F )(U ) = 0 for every U ∈ U and every
i
q with 1 ≤ q < p, so that Ȟ (U, HΦq (F )) = 0 for every such q because we can
compute Čech cohomology only using covers by open sets from U. Furthermore,
0
Ȟ (U, H p (F )) = 0 by (8.3) and (8.4). Consider the spectral sequence of 8.5 on U .
The natural map
p
E2p0 = ȞΦ (U, F |U ) → HpΦ (U, F |U )
in this spectral sequence must be an isomorphism because we have shown that
E2i,j = 0 in this spectral sequence for every i, j with i + j = p and j > 0.
HpΦ (U, F |U )
Finally, we conclude the desired result by again considering the spectral sequence of 8.5, using the first part of the proof and the assumption (3) to know that
Hp (X, HΦq (F )) = 0 for every q > 0, so that the SES degenerates.
Theorem 9.2. Suppose Φ is a paracompactifying family of supports (§7) on a
topological space X. Then for every p ≥ 0 and every F ∈ Ab(X), the natural map
p
ȞΦ (X, F ) → HpΦ (X, F )
is an isomorphism.
Proof.
Corollary 9.3. For a paracompact topological space X, the natural map
p
Ȟ (X, F ) → Hp (X, F )
is an isomorphism for every p.
86
VIII. ČECH THEORY
10. The unit interval
p
It is possible to compute the Čech cohomology groups Ȟ (X, F ) directly from the
definitions for reasonably simple topological spaces X and sheaves F . For example,
the interval I = [0, 1] can be covered by connected open subinvervals U1 , . . . , Un of
roughly equal length where Ui intersects only Ui−1 and Ui+1 (and this intersection is
connected). (Note U1 only meets U2 and Un only meets Un−1 and we assume n ≥ 2.)
These sort of covers are cofinal in all covers of the interval. Global sections of the
Čech complex for this cover and the constant sheaf Z give the complex d : Zn → Zn−1
(in degrees 0, 1) where the boundary map is given by the matrix:


1 −1 0
0 ··· 0
0
0 1 −1 0 · · · 0
0


0 0
1 −1 · · · 0
0
d = 
..
..
..
.. 
 ... ...
.
.
.
. 


0 0
0
0 · · · −1 0 
0
0
0
0
···
1
−1
This map has cohomology only in degree zero, given by the diagonal ∆ : Z ,→ Zn .
Since the interval is certainly paracompact, Čech cohomology and the usual sheaf
cohomology coincide by Corollary 9.3, so we find
Z,
p=0
p
(10.1)
H (I, Z) =
0,
p > 0.
11. Simplicial objects
For a nonnegative integer n, let [n] := {0, . . . , n}. Let ∆ denote the category
whose objects are the finite sets [0], [1], . . . and where a morphism [m] → [n] is a
nondecreasing function. For each i ∈ [n], let sni : [n − 1] → [n] be the unique ∆
morphism whose image is [n] \ {i} (the pneumonic naming sni is for “skip i”). For
i ∈ [n], let eni : [n + 1] → [n] be the unique nondecreasing map which is surjective
and hits i ∈ [n] twice.
Let C be a category. A simplicial object in C is a functor ∆op → C. For example,
any object C ∈ C determines a constant simplicial object (also denoted C ∈ sC) via
the constant functor [n] 7→ C (which takes every ∆op morphism to IdC ). Simplicial
objects in C form a category
sC := HomCat (∆op , C),
which evidently contains C as a full subcategory by identifying C with the constant
simplicial objects in sC. For C ∈ sC, it is customary to write Cn for C([n]) and
dni : Cn → Cn−1 for C(sni ). The maps dni are called the boundary maps of C.
Similarly, we will write eni : Cn → Cn+1 as abuse of notation for C(eni ). The maps
eni are called degeneracy maps.
More generally, given any surjective ∆ morphism σ : [m] → [n], the map C(σ) :
Cn → Cm is often called a degeneracy map. Notice that every surjective ∆ morphism
13. TRUNCATIONS AND (CO)SKELETA
87
σ has a section (any set-theoretic section of a ∆ morphism is again a ∆ morphism),
so every degeneracy map C(σ) of a simplicial object has a retract.
12. The standard simplex
⊆ Rn+1 denote the convex hull of the
For a non-negative integer n, let ∆top
n
standard basis vectors e0 , . . . , en :
X
∆std
:= {t0 e0 + · · · + tn en ∈ Rn+1 : ti ∈ [0, 1],
ti = 1}.
n
A morphism σ : [m] → [n] in the simplicial category ∆ induces a morphism of
topological spaces
→ ∆top
∆top (σ) : ∆top
n
m
X
X
(t0 e0 + · · · + tm em ) 7→ (
ti e0 + · · · +
ti en ),
i∈σ −1 (0)
i∈σ −1 (n)
so we can regard ∆top as a cosimplicial topological space called the standard (topological) simplex.
The algebraic analog of the standard simplex is the following: For a non-negative
integer n, let An denote the ring
An := Z[t0 , . . . , tn ]/h1 − t0 − t1 − · · · − tn i.
A ∆-morphism σ : [m] → [n] induces a ring homomorphism
A(σ) : An → Am
X
ti 7→
tj ,
j∈σ −1 (i)
so that the An form a simplicial ring A, and hence the ∆alg
:= Spec An form a
n
cosimplicial scheme ∆alg called the standard (algebraic) simplex.
13. Truncations and (co)skeleta
For a nonnegative integer n, let ∆n denote the full subcategory of ∆ whose
objects are [0], . . . , [n]. An n-truncated simplicial object in C is a functor ∆op
n → C.
Let
sn C := HomCat (∆op
n , C)
op
be the category of n-truncated simplicial objects in C. The inclusion ∆op
n ,→ ∆
induces a restriction functor
trn : sC → sn C
C 7→ C|∆op
,
n
which we call the n truncation.
88
VIII. ČECH THEORY
Let ∆mon (resp. ∆mon
n ) denote the subcategory of ∆ (resp. ∆n ) with the same
objects, but where a morphism [m] → [k] is also required to be monic. Define ∆epi
and ∆epi
n similarly.
Assume C has finite inverse limits. By the Kan Extension Theorem, the truncation trn : sC → sn C admits a right adjoint
Coskn : sn C → sC
called the n coskeleton functor. Specifically,
(Coskn C)m := lim {Ck : ([m] → [k]) ∈ [m] ↓ ∆op
n }
←−
= lim {Ck : ([k] → [m]) ∈ ∆n ↓ [m]}
←−
= lim {Ck : ([k] → [m]) ∈ ∆mon
↓ [m]}.
n
←−
The first equality is the usual “formula” for Kan extension; the second is a translation of it. To be clear, we emphasize that the inverse limit is the inverse limit of
the functor
(∆n ↓ [m])op → C
([k] → [m]) 7→ Ck .
The third equality is by cofinality of ∆mon
↓ [m] in ∆n ↓ [m]: every object ([k] → [m])
n
in ∆n ↓ [m] admits a map to an object ([l] → [m]) where [l] → [m] is monic.
In particular, notice that Cm = (Coskn C)m for m ≤ n. Indeed, when m ≤ n,
Id : [m] → [m] is the terminal object of ∆n ↓ m.
The adjunction isomorphism
Homsn C (trn C, D) → HomsC (C, Coskn D)
takes a map f : trn C → D to the map C → Coskn D given in degree m by the map
Cm → (Coskn D)m obtained as the inverse limit of the maps
fk X(σ) : Xm → Yk
over all σ : [k] → [m] in ∆n ↓ [m].
For C ∈ sC, the abuse of notation
Coskn C := Coskn (trn C)
is standard.
Dually, if C has finite direct limits, then the Kan Extension Theorem implies
that truncation trn admits a left adjoint
Skn : sn C → sC
called the n skeleton functor, given on C ∈ sn C by the formula
(Skn C)m = lim {Ck : ([m] → [k]) ∈ [m] ↓ ∆n }
−→
= lim {Ck : ([m] → [k]) ∈ [m] ↓ ∆epi
n }.
−→
14. SIMPLICIAL HOMOTOPY
89
The structure map
(Skn C)(σ) : (Skn C)m → (Skn C)l
induced by a ∆ morphism σ : [l] → [m] is the map on direct limits induced by the
following commutative diagram of functors:
τ 7→σ ∗ τ
([m] ↓ ∆n )op
(τ :[m]→[k])7→Ck
%
/
([l]→[k])7→Ck
y
C
([l] ↓ ∆n )op
Note that s0 C is naturally isomorphic to C, so we can regard Sk0 and Cosk0 as
functors C → sC. With this understanding, Sk0 C is the constant simplicial object
and Cosk0 C is the simplicial object with (Cosk0 C)n = C [n] (the n + 1 fold product
of C) whose boundary maps are the natural projections C [n] → C [n−1] “forgetting
one coordinate” and whose degeneracy maps C [n] → C [n+1] are the diagonals given
by “repeating one coordinate”.
14. Simplicial homotopy
Let f, g : X → Y be sC morphisms. A homotopy h from f to g consists of a C
morphism h(φ) : Xn → Yn , defined for each morphism φ : [n] → [1] of ∆, satisfying
the conditions:
(1) For any commutative triangle
ψ
[m]
/

φ
[n]
θ
[1]
in ∆, the square
Xn
h(θ)
/
Yn
X(ψ)
Y (ψ)
Xm
h(φ)
/
Ym
commutes in C.
(2) For any n, if φ : [n] → [1] is the constant map to 0 ∈ [1], then h(φ) = fn
and if φ is the constant map to 1 ∈ [1], then h(φ) = gn .
The relation consisting of pairs
(f, g) ∈ HomsC (C, D) × HomsC (C, D)
defined by “there is a homotopy from f to g” is not generally an equivalence relation.
It is clearly reflexive, but not generally symmetric or transitive. However, if there is
a homotopy h from f to g, and k, l : D → E are sC morphisms with a homotopy h0
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VIII. ČECH THEORY
from k to l, then there is a homotopy from kf to lg, obtained by composing h and
h0 :
(φ : [n] → [1]) 7→ (h0 (φ)h(φ) : Cn → En ).
We let ∼ be the equivalence relation generated by this relation and we say f is
homotopic to g if f ∼ g. In other words, f ∼ g means there is a finite string of
morphisms f = f0 , f1 , . . . , fn = g such that, for every i, either there is a homotopy
from fi to fi+1 or there is a homotopy from fi+1 to fi .
An sC morphism f : C → D is a homotopy equivalence if there is an sC
morphism g : D → C such that f g ∼ IdD and gf ∼ IdC .
Let HotsC denote the category with the same objects as sC where
HomHotsC (C, D) := HomsC (C, D)/ ∼ .
Composition is defined by choosing representatives of equivalence classes and composing them. This is well defined because of the remarks about composition of homotopies made above. There is an obvious forgetful functor sC → HotsC. Evidently
f : C → D is a homotopy equivalence iff its image in HotsC is an isomorphism.
Lemma 14.1. Suppose C has products and C ∈ C is such that the unique map
f : C → 1 to the terminal object (empty product) admits section s (i.e. there is some
map 1 → C). Then
Cosk0 f : Cosk0 C → 1
is a homotopy equivalence in sC with homotopy inverse
Cosk0 s : 1 → Cosk0 C.
Proof. Certainly (Cosk0 f )(Cosk0 s) = Cosk0 (f s) is the identity of Cosk0 1 = 1
because Cosk0 is a functor and f s = Id1 by definition of “section”. We need to show
that
(Cosk0 s)(Cosk0 f ) = Cosk0 (sf )
is homotopic to the identity of Cosk0 C. The map
(Cosk0 (sf ))n : C n+1 → C n+1
is given by
(Cosk0 (sf ))n (c)(i) = sf c(i)
(i = 0, . . . , n). We define a homotopy to the identity by associating to φ : [n] → [1]
the “straight line homotopy” map
h(φ) : C n+1 → C n+1
given by
h(φ)(c)(i) :=
sf c(i), φ(i) = 0
c(i),
φ(i) = 1.
15. SIMPLICIAL SETS
91
When φ is the constant map to 0 ∈ [1], obviously h(φ) = (Cosk0 (sf ))n and when
φ is the constant map to 1 ∈ [1] obviously h(φ) is the identity. To finish the proof
that h is the desired homotopy, we must show that, for any commutative square
ψ
[m]
/

φ
[n]
θ
[1]
in ∆, the square
C n+1
ψ∗
C m+1
h(θ)
h(φ)
/
/
C n+1
ψ∗
C m+1
commutes in C.
Here ψ ∗ is short for (Cosk0 C)(ψ). It is given by (ψ ∗ c)(i) = c(ψ(i)). This is a
simple computation:
sf c(ψ(i)), θ(ψ(i)) = 0
∗
(ψ ◦ h(θ))(c)(i) =
c(ψ(i)),
θ(ψ(i)) = 1
sf c(ψ(i)), φ(i) = 0
=
c(ψ(i)),
φ(i) = 1
sf (ψ ∗ c)(i), φ(i) = 0
=
(ψ ∗ c)(i),
φ(i) = 1
= (h(φ) ◦ ψ ∗ )(c)(i).
15. Simplicial sets
In this section we study the category sEns of simplicial sets. For a simplicial set
X, we refer to the elements of Xn as simplicies of X of dimension n. Fix a simplex
x ∈ Xn . For any injective ∆ morphism σ : [k] → [n], the simplex X(σ)(x) ∈ Xk
is called a face of x. For any surjective ∆ morphism σ : [k] → [n], the simplex
X(σ)(x) ∈ Xk is called a degeneracy of x. Recall that the degeneracy map X(σ)
corresponding to a ∆ surjection σ always has a retract, so in particular it is injective.
The simplex x is called non-degenerate if it is a degeneracy only of itself. It is easy
to see that any simplex x ∈ Xm is a degeneracy of a unique non-degenerate simplex
y: just choose l minimal such that there is a ∆ surjection σ : [m] → [l] and a y ∈ Xl
with X(σ)(y) = x.
The n skeleton functor of §13 behaves like the one the reader may be familiar
with from topology:
Lemma 15.1. For a simplicial set X and a non-negative integer n, the image of
the adjunction morphism Skn X := Skn (trn X) → X consists of the simplices of X
92
VIII. ČECH THEORY
which are degeneracies of some k simplex with k ≤ n. This adjunction morphism is
an isomorphism iff all non-degenerate simplices of X have dimension ≤ n.
Proof. According to the general formulas of §13, an element [σ, x] of (Skn X)m
is represented by a pair (σ, x) consisting of a surjective ∆ morphism σ : [m] → [k],
with k ≤ n, and an element x ∈ Xk . Given any map τ : [k] → [k 0 ] from σ
to σ 0 : [m] → [k 0 ] in [m] ↓ ∆n and any x0 ∈ Xk0 we introduce the relation
(σ 0 , x0 ) ∼ (σ, X(τ )(x0 )). Two such pairs (x, σ) and (x0 , σ 0 ) represent the same element of (Skn X)m iff they are equivalent under the smallest equivalence relation
containing the relations ∼. The adjunction morphism (Skn X)m → Xm takes [σ, x]
to X(σ)(x) ∈ Xm . The description of the image of the adjunction morphism is now
clear.
It remains only to prove that the adjunction morphism is injective when all nondegenerate simplices of X have dimension ≤ n. Suppose [x, σ], [x0 , σ 0 ] ∈ (Skn X)m
have X(σ)(x) = X(σ 0 )(x0 ) =: y ∈ Xm and we wish to show that [x, σ] = [x0 , σ]. By
hypothesis, we can write y = X(ρ)(z) for some surjective ∆ morphism ρ : [m] → [l]
with l ≤ n and some z ∈ Xl . We can complete the diagram
[m]
σ0
σ
~
[k]
[k 0 ]
ρ
τ
~
τ0
[l]
in ∆ as indicated (for example, by choosing sections of the surjections σ, σ 0 ). We
have X(τ )(z) = x because this is true after applying the monic map of sets X(σ)
(both sides are then y), so (σ, x) ∼ (ρ, z). Similarly we have X(τ 0 )(z) = x0 , so
(σ 0 , x0 ) ∼ (ρ, z). From transitivity of the equivalence relation induced by ∼, we
conclude [σ, x] = [σ 0 , x0 ] ∈ (Skn X)m .
16. Standard simplicial sets
There are several special simplicial sets that will be useful later. We collect them
here as a series of examples.
Example 16.1. For each n ∈ N we have the n-simplex ∆[n] ∈ sEns defined by
setting
∆[n]m := {(i0 , . . . , im ) : 0 ≤ i0 ≤ · · · ≤ im ≤ n}.
The boundary maps for ∆[n] are given by
djm : ∆[n]m → ∆[n]m−1
(i0 , . . . , im ) 7→ (i0 , . . . , ij−1 , ij+1 , . . . , im )
17. GEOMETRIC REALIZATION
93
and the basic degeneracy maps for ∆[n] are given by
sjm : ∆[n]m → ∆[n]m+1
(i0 , . . . , im ) 7→ (i0 , . . . , ij−1 , ij , ij , ij+1 , . . . , im ).
It is easy to check that for any X ∈ sEns, the map
HomsEns (∆[n], X) →
7
Xn
f →
7
fn ((0, 1, . . . , n))
is bijective.
Example 16.2. For each n ∈ N we also have the boundary ∂∆[n] of ∆[n], which
is the sub-simplicial set of ∆[n] generated by
(1, 2, . . . , n), (0, 2, . . . , n), . . . , (0, 1, . . . , n − 1) ∈ ∆[n]n−1 .
The map
HomsEns (∂∆[n], X) → HomEns ([n], Xn−1 )
f 7→ (i 7→ fn−1 (0, . . . , i − 1, i + 1, . . . , n))
is a bijection onto the set of f ∈ HomEns ([n], Xn−1 ) satisfying
din−1 f (j) = dj−1
n−1 f (i) for all 0 ≤ i < j ≤ n.
Example 16.3. For example, the simplicial n sphere S n ∈ sEns is the simplicial
set with exactly two non-degenerate simplices: one in dimension zero and one in
dimension n. Using Lemma 15.1 it is not hard to show that such a simplicial set
exists and is characterized up to isomorphism (unique isomorphism unless n = 0)
by this property.
17. Geometric realization
Given a simplicial set X, we define a topological space |X| called the geometric
realization of X as follows. We start with the space
aa
X :=
∆top
n ,
n
Xn
∆top
n
where
is the standard topological n simplex of §12. For clarity, we write ∆top
n (x)
for the component of the coproduct indexed by an n simplex x ∈ Xn . Given any ∆
morphism σ : [m] → [n], we have the map of sets X(σ) : Xn → Xm and the map
top
of topological spaces ∆top (σ) : ∆top
m → ∆n reflecting the cosimplicial structure of
top
∆ . For any x ∈ Xn we glue the component ∆top
m (X(σ)(x)) of X to the component
∆top
(x)
of
X
via
the
map
n
top
∆top (σ) : ∆top
m (X(σ)(x)) → ∆n (x).
That is, we introduce the relation ∼ on X where P ∼ ∆top (σ)(P ) for all P ∈
∆top
m (X(σ)(x)). We let |X| be the space obtained from X by making all such gluings. That is, we let |X| be the quotient of X (with the quotient topology) by the
equivalence relation generated by the relations ∼ introduced above.
94
VIII. ČECH THEORY
The construction is clearly functorial in X, hence defines a functor || : sEns →
Top. In fact, the standard n simplex ∆top
n is compact, so X is a compactly generated
space (a set is open iff its intersection with each compact set is open), hence so is
|X| since it is a quotient of X. We can therefore view || as a functor || : sEns → K,
where K is the category of compactly generated topological spaces. As such, the
functor || preserves finite inverse limits [Hov, 3.2.4].
For more on geometric realization, see [Mil]
18. The abelian setting
Here we briefly review the theory of simplicial objects in an abelian category A.
The basic theorems of Dold-Kan-Puppe assert that simplicial objects in A are the
same thing is chain complexes in A supported in non-negative degrees.
Given an A ∈ sA and an n ∈ N, we set
n−1
Nn (A) := ∩i=0
Ker(din : An → An−1 ).
(Note that the intersection is over the kernels of all but the last boundary map
dnn : An → An−1 . By convention, N0 (A) := A0 . It is easy to check that the last
n
boundary map dnn takes Nn (A) into Nn−1 (A) and that dn−1
n−1 dn = 0, so that we have
a chain complex
N(A) := (N• (A), d•• )
in A supported in non-negative degrees called the normalized chain complex of A.
This is functorial in A and defines a functor
sA → Ch≥0 A,
which is in fact an equivalence of categories. Furthermore, this functor takes homotopies in the simplicial sense of §14 to homotopies in the usual sense of chain
complexes. When A is the category of modules over a ring R, then the simplicial
set underlying any simplicial R module M is fibrant and the homotopy groups of
this simplicial set (with any choice of basepoint; the natural one being 0 ∈ M ) are
identified with the homology of the normalized chain complex Nn (A).
19. Simplicial rings
Let An denote the category of commutative rings with unit. A simplicial object
in An is called a simplicial ring. Given an A ∈ sAn, a module over A is a simplicial
abelian group M such that each Mn is equipped with the structure of an An module
compatibly with the simplicial structures: that is, for each ∆-morphism σ : [m] →
[n], the map of abelian groups M (σ) : Mn → Mm should be A(σ) : An → Am linear:
M (σ)(a · m) = A(σ) · M (σ)(m)
for all a ∈ An , m ∈ Mn . For example, if A → B is a morphism of simplicial rings,
then the modules of Kähler differentials ΩBn /An ∈ Mod(Bn ) fit together to form a
B module ΩB/A .
19. SIMPLICIAL RINGS
95
For a simplicial ring A, we set
i
Nn (A) := ∩n−1
i=0 Ker(dn : An → An−1 )
Zn (A) := ∩ni=0 Ker(din : An → An−1 )
Hn (A) := Nn (A)/dn+1
n+1 Zn+1 (A).
Note that Nn (A), Zn (A) ⊆ An are ideals of An . The subset dn+1
n+1 Zn+1 (A) ⊆ An is
n+1
n
also an ideal because dn+1 : An+1 → An has a section sn : An → An+1 so for a ∈ An
and b ∈ Zn+1 (A) we have
n+1 n
n+1
adn+1
n+1 (b) = dn+1 (sn (a))dn+1 (b)
n
= dn+1
n+1 (sn (a)b)
and snn (a)b ∈ Zn+1 (A) because Zn+1 (A) ⊆ An+1 is an ideal. Consequently, Hn (A),
being a quotient of two ideals of An , becomes an An module.
If An is a noetherian ring, then the An modules Zn (A), dn+1
n+1 Zn+1 (A), Hn (A) are
all finitely generated.
The homology groups Hn (A) fit together into a graded-commutative ring
H∗ (A) := ⊕∞
n=0 Hn (A).
The fastest way to define the multiplication maps
Hm (A) ⊗ Hn (A) → Hm+n (A)
is via geometric realization: we have
Hm (A) = πm (A)
= πm (|A|),
where |A| is the compactly generated topological space obtained as the geometric
realization §17 of the simplicial set underlying the simplicial ring A. The space |A| is
of course pointed by 0 ∈ |A|. Since A is a ring object in simplicial sets and geometric
realization (viewed as a functor to K) preserves finite inverse limits, |A| is a ring
object in K. An element [f ] of πm (|A|) is represented by a map f : I m → A taking
the boundary of the cube I m to 0 ∈ |A| and an element [g] of πn (A) is represented
by a map g : I n → A taking the boundary of I n to 0. Using the ring structure on
A, we can define a map
f g : I m+n → |A|
(x, y) 7→ f (x)g(y)
which also clearly takes the boundary of I m+n to 0 ∈ |A|, hence represents a class
[f g] ∈ πm+n (|A|). The multiplication [f ][g] := [f g] is clearly well-defined; it is
graded commutative because the automorphism (x, y) 7→ (y, x) of I m+n acts by
(−1)mn on the orientation of I m+n .
Similarly, the normalized chain complex N(A) of a simplicial ring A carries a
natural differential graded ring structure. The multiplication maps
Nm (A) ⊗ Nn (A) → Nm+n (A)
96
VIII. ČECH THEORY
are defined using the “shuffle product” map
Am ⊗ An → Am+n
X
a ⊗ b 7→
Sign(σ, τ )A(σ)(a)A(τ )(b).
(σ,τ )
The sum here....
20. Simplicial sheaves
Let X be a topological space, Sh(X) the category of sheaves on X. The category
sSh(X) is called the category of simplicial sheaves on X.
For F ∈ Sh(X), we will write s ∈ F to mean s ∈ F (U ) for some U ∈ Ouv(X),
and we say that some property of s holds “locally” if there is a cover {Ui } so that
this property holds for each s|Ui . For example, we can say that f : F → G is an
epimorphism in Sh(X) iff, for every s ∈ G , there exists (locally) a t ∈ F such
that f (t) = s. Let 1 be the terminal object of Sh(X) (the sheaf represented by the
terminal object X ∈ Ouv(X)).
Remark 20.1. The discussion to follow makes sense in an arbitrary topos. In
this more general setting, s ∈ F means s ∈ Hom(G , F ) for some G and “locally”
means “after precomposing with an epimorphism H → G .”
A simplicial sheaf F ∈ sSh(X) is called fibrant in degree n if, for every k ∈ [n+1]
and every
x0 , . . . , x̂k , . . . , xn+1 ∈ Fn
such that dni xj = dnj−1 xi for all i < j in [n] both distinct from k, there is (locally)
a y ∈ Xn+1 with dn+1
y = xi for all i 6= k in [n]. A simplicial sheaf is fibrant if it is
i
fibrant in every degree.
The definition is motivated as follows: Label the ith codimension one face {ti = 0}
of the standard topological n + 1 simplex
n+1
∆top
: t0 + · · · + tn+1 = 1}
n+1 := {(t0 , . . . , tn+1 ) ∈ R
with xi , so the k th face remains unlabelled; the labelled faces form the so called
k-horn ∆top
n+1 [k] of the standard n + 1 simplex (I would call it the closed star of the
vertex ek opposite the k th codimension one face). The condition dni xj = dnj−1 xi says
that the labellings “agree” on the codimension two faces and the fibrancy condition
asserts the existence of a “map on the whole n+1 simplex” whose restriction to each
of the faces in the k-horn is the map indicated by the labelling. Note that we make
no demands on the restriction of this “filling map” to the k th face. Such a filling
top
map always exists in topology because the inclusion of the k-horn ∆top
n+1 [k] ,→ ∆n+1
admits a retract r, hence any map f from the k-horn to a topological space X can
be lifted to a map f r : ∆top
n+1 → X.
20. SIMPLICIAL SHEAVES
97
We can abstractly define the k-horn of a simplicial sheaf F to be
Fn+1 [k] := lim {Fm : φ : [m] ,→ [n + 1], m ≤ n, k ∈ Im φ}.
←−
Then a section of Fn+1 [k] is nothing but a tuple x0 , . . . , x̂k , . . . , xn ∈ Fn as above.
There is an obvious restriction map Fn+1 → Fn+1 [k]; the fibrancy condition says
that this should be surjective.
A map from the terminal object 1 (the terminal object of sSh(X) is the terminal
object 1 of Sh(X) regarded as a constant simplicial sheaf) to a simplicial sheaf F
will be called a basepoint of F . To give such a map is the same thing as giving a
map 1 → F0 , for then the map 1 → Fn must be the composition of 1 → F0 and
any degeneracy map F0 → Fn , since this degeneracy map is an isomorphism in the
constant simplicial object 1. We will systematically use “1” as abuse of notation
for the map 1 → F . The existence of a basepoint implies, in particular, that each
map Fn → 1 is an epimorphism (since it has a section). The category of based or
pointed simplicial sheaves is the category of simplicial sheaves under 1.
Let 1 → F be a based fibrant simplicial sheaf, n a nonnegative integer. Let
Zn F be the subsheaf of Fn consisting of those x ∈ Fn with dni = 1 for all i ∈ [n].
The sheaf Zn F is called the sheaf of n cycles in F . Define a relation ∼ on Zn F
by declaring x ∼ y iff, locally, there is some z ∈ Fn+1 such that
(1) dn+1
z = 1 for i ∈ [n + 1] \ {n, n + 1} and
i
n+1
(2) dn = x, dn+1
n+1 z = y.
I claim ∼ is an equivalence relation.
Reflexive: (This is the most difficult property to establish.) Given x ∈ Zn Fn , we
seek, locally, a z ∈ Fn+1 such that
(1) dn+1
z = 1 for i ∈ [n − 1] and
i
(2) dn+1
z = x for i ∈ {n, n + 1}.
i
First check that
(1, . . . , 1, x) ∈ Fn+1 [n],
so by fibrancy in degree n we can find, locally, a y ∈ Fn+1 such that dn+1
y = 1 for
i
n+1
y
=
x
(we
know
nothing
about
d
y).
Next
check
that
i ∈ [n − 1] and dn+1
n+1
n
(1, . . . , 1, y, y) ∈ Fn+2 [n + 2],
so by fibrancy in degree n + 1 there is (further locally) a w ∈ Fn+2 such that
(1) dn+2
w = 1 for i ∈ [n − 1] and
i
n+2
(2) di w = dn+2
w = y for i ∈ {n, n + 1}.
i
98
VIII. ČECH THEORY
I claim z := dn+2
n+2 w ∈ Fn+1 is as desired. Indeed, for i ∈ [n − 1], we compute
dn+1
z = dn+1
dn+2
n+2 w
i
i
n+2
= dn+1
w
n+1 di
= dn+1
n+1 1
= 1,
and for i ∈ {n, n + 1}, we compute
dn+1
z = dn+1
dn+2
n+2 w
i
i
n+2
= dn+1
w
n+1 di
= dn+1
n+1 y
= x.
Symmetric: Suppose x, y ∈ Zn F are such that (locally), there is some z ∈ Fn+1
with
(1) dn+1
z = 1 for i ∈ [n − 1] and
i
n+1
(2) dn+1
n z = x, dn+1 z = y.
By reflexivity, further locally, there is r ∈ Fn+1 such that
(1) dn+1
r = 1 for i ∈ [n − 1] and
i
n+1
(2) dn+1
n r = dn+1 r = y.
Since dn+1 r = dn+1
n z it follows that
(1, . . . , 1, z, r) ∈ Fn+2 [n + 2],
so there is, further locally, some u ∈ Fn+2 with
(1) dn+2
u = 1 for i ∈ [n − 1] and
i
n+2
(2) dn+2
n z = z, dn+1 u = r.
Now we check, just as above, that w := dn+2
n+2 u satisfies
(1) dn+1
w = 1 for i ∈ [n − 1] and
i
n+1
(2) dn w = y, dn+1
n+1 w = x.
Transitive: Suppose x, y, z ∈ Zn F have x ∼ y and y ∼ z. We know ∼ is symmetric
so, since y ∼ x, locally there is some u ∈ Fn+1 with
(1) dn+1
u = 1 for i ∈ [n − 1] and
i
n+1
(2) dn+1
n u = y, dn+1 u = x
and since y ∼ z there is locally some v ∈ Fn+1 with
(1) dn+1
v = 1 for i ∈ [n − 1] and
i
n+1
(2) dn v = y, dn+1
n+1 v = z.
21. WEAK EQUIVALENCES
99
n+1
From the equality dn+1
n u = dn v it follows that
(1, . . . , 1, u, v) ∈ Fn+2 [n + 2].
By fibrancy, there is locally some w ∈ Fn+2 with
(1) dn+2
w = 1 for i ∈ [n − 1] and
i
n+2
(2) dn w = u, dn+2
n+1 w = v
n+2
and one checks easily that dn+2
w ∈ Fn+1 witnesses x ∼ z.
Because of the judicious insertions of the word “locally,” the quotient
πn (F ) := (Zn F )/ ∼
is easily seen to be a sheaf, which we will call the nth homotopy sheaf of F .
The sheaf π0 (F ) does not depend on the choice of basepoint. In fact, one can
see easily from the definition that
!
π0 (F ) = lim
−→
F1
d10
d11
/
/
F0
.
Theorem 20.2. Let f, g : F → G be two maps of based fibrant simplicial sheaves
which are simplicially homotopic. Then
πn (f ) = πn (g) : πn (F ) → πn (G )
for every n.
Proof. Exercise.
21. Weak equivalences
A morphism f : F → G of based fibrant simplicial sheaves induces a morphism
Zn F → Zn G for all n and a morphism
πn (F ) : πn (f ) → πn (G )
for all n. Indeed, πn is a functor from based fibrant simplicial sheaves to sheaves.
Such a morphism f is called a weak equivalence if πn (f ) is an isomorphism for every
n.
A morphism of (unbased) fibrant simplicial sheaves f : F → G is called a weak
equivalence if, for every U ∈ Ouv(X), every basepoint 1 → F |U , and every n,
the map πn (f |U ) is an isomorphism.3 It is necessary to look at all locally defined
basepoints in order to have a notion of weak equivalence that is local (even the
existence of a global basepoint is a serious restriction, so otherwise too many maps
would vacuously be weak equivalences).
here that f |U : F |U → G |U can be viewed as a map of based fibrant simplicial sheaves on U
for a unique choice of basepoint on G |U , namely the one given by the composition 1 → F |U → GU .
3Note
100
VIII. ČECH THEORY
Lemma 21.1. A morphism of fibrant simplicial sheaves f : F → G is a weak
equivalence iff fx : Fx → Gx is a weak equivalence of fibrant simplicial sets (bijection
on homotopy sets for every basepoint) for every point x ∈ X.
Proof. f is a weak equivalence iff πn (f |U ) : F |U → G |U is an isomorphism for
every basepoint 1 → F |U . This can be checked on stalks by (6.1). Formation of
homotopy sheaves commutes with stalks since the construction of πn (F |U ) can be
phrased in terms of finite inverse limits (to construct Zn F |U ) and finite direct limits
(to take the quotient (Zn F |U )/ ∼) and these limits commute with stalks.
A fibrant simplicial sheaf F ∈ sSh(X) is called acyclic (or a hypercover ) if the
map to the terminal object F → 1 is a weak equivalence. It is clear that πn (1) = 1
for all n, so F is a hypercover iff πn (F |U ) = 1 for all n for all U ∈ Ouv(X) and
for all basepoints 1 → F |U .
The category HCov(X) of hypercovers of X is the full subcategory of HotsSh(X)
consisting of hypercovers. Note, by (20.2), that the property of being a hypercover
is invariant under homotopy equivalences.
22. Properties of coskeleta
The coskeleta functors play well with fibrancy and homotopy sheaves. In order
to streamline the proofs in this section, we begin with some notation for coskeleta.
Like any inverse limit, (Coskn C)m sits inside the corresponding product, so we may
view a section of (“map to”) (Coskn C)m as a section
Y
s = (s(φ)) ∈
Ck
φ:[k],→[m],k≤n
satisfying the condition: whenever φ : [k] ,→ [m] can be factored in ∆ as
[k]
ψ
/
[l]
θ
/
=
[m]
φ
with l ≤ n, we have s(φ) = C(ψ)(s(θ)).
Of course, if m ≤ n, then such a section s is uniquely determined by s(Id : [m] ,→
[m]) ∈ Cm and conversely any such s(Id : [m] ,→ [m]) determines a unique section
of (Coskn C)m by setting
s(φ : [k] ,→ [m]) := C(φ)(s(Id : [m] ,→ [m]))
and we see that Cm = (Coskn C)m as we already noted in (11).
Evidently then, we should concentrate on the case m ≥ n. Here, s is determined
by the s(φ : [n] ,→ [m]) since any map [k] → [m] with k ≤ n factors through such a
22. PROPERTIES OF COSKELETA
101
φ. The condition that
Y
s = (s(φ : [n] ,→ [m])) ∈
Cn
φ:[n],→[m]
actually determines a section of (Coskn C)m is the following: For any two monic ∆
morphisms φ, ψ : [n] ,→ [m], define an integer k by
k + 1 := | Im φ ∩ Im ψ|.
If k = −1, then no condition is imposed. Otherwise, there is a (unique) commutative
diagram
θ
[n]
?
φ
[k]
>
η
[m]
ψ
[n]
of ∆ monomorphisms and the requirement is:
C(θ)(s(φ)) = C(η)(s(ψ)).
Lemma 22.1. For any C ∈ sC, and any 0 ≤ m ≤ n, the adjunction morphism
Coskm C → Coskn (Coskm C)
is an isomorphism.
Proof. Follows directly from the definitions and the fact that inverse limits
commute amongst themselves.
Lemma 22.2. For any F ∈ sSh(X), Coskn F is fibrant in degrees > n. If F
is fibrant, then Coskn F is fibrant.
Proof. Suppose m > n and consider
x = (x1 , . . . , x̂k , . . . , xm+1 ) ∈ (Coskn F )m+1 [k].
Note each xi is in (Coskn F )m and we view the latter as being contained in
Y
Fn
[l],→[m],l≤n
as in the above discussion. We want to find a y ∈ (Coskn F )m+1 with dm+1
y = xi
i
for i ∈ [m + 1] \ {k}. Using the above description of sections y ∈ (Coskn F )m+1 , we
define y(φ : [n] ,→ [m + 1]) as follows: Since m > n, the image of φ must miss at
102
VIII. ČECH THEORY
least two elements of [m + 1] (this is the key point). In particular, it misses some
i ∈ [m + 1] with i 6= k, which is to say: φ can be factored as
ψ
[n]
/
[m]
si
/
[m + 1] .
:
φ
Declare y(φ) := xi (ψ). I claim this is independent of the choice of such a factorization
of φ. Indeed, suppose φ can be factored through both si , sj : [m] → [m + 1] (with
i < j, say). Then we have a commutative diagram
[m]
<
O
ψ
[n]
θ
η
si
/
sj−1
$
[m − 1]
"
si
[m + 1]
:
sj
[m]
of monomorphisms in ∆. Using the definition of the boundary maps in the coskeleton
and the fact that x is in the k-horn Fn+1 [k], we then compute
xi (ψ) = xi (sj−1 θ)
= (dm
j−1 )(θ)
= (dm
i xj )(θ)
= xj (si θ)
= xj (η).
It easily follows that y is actually a section of (Coskn F )m+1 . We now easily compute
(dm+1
y)(φ : [n] ,→ [m]) = y(si φ)
i
= xi (φ).
To prove the second statement, first note that Coskn F is certainly fibrant in
degrees m < n since fibrancy in degree m only depends on the degree m−1, m, m+1
parts, and F → Coskn F is an isomorphism in these degrees. By what we just
proved above, it remains only to prove fibrancy of Coskn F in degree n. Since
Fn = (Coskn F )n , and Fn−1 = (Coskn F )n−1 we have
Fn+1 [k] = (Coskn F )m+1 [k].
Given
x = (x1 , . . . , x̂k , . . . , xn+1 ) ∈ (Coskn F )m+1 [k] = Fn+1 [k],
22. PROPERTIES OF COSKELETA
103
we use the fact that F is fibrant to find y ∈ Fn+1 such that dn+1
= xi for i ∈
i
[n + 1] \ {k}. Define
Y
z∈
Fn
[n],→[n+1]
by
z(si : [n] ,→ [n + 1]) := dn+1
y ∈ Fn .
i
One easily checks that in fact z ∈ (Coskn F )n+1 and that dn+1
z = xi .
i
Lemma 22.3. For any based fibrant simplicial sheaf F , πm (Coskn F ) = 1 for
m ≥ n.
Proof. Suppose x ∈ Zm (Coskn F ) for m ≥ n. Define y ∈ (Coskn F )m+1 as
follows: Given φ : [n] ,→ [m + 1], choose a factorization
ψ
[n]
/
si
[m]
/
[m + 1]
:
φ
and declare
y(φ) :=
x(ψ), i = m + 1
1,
otherwise.
I claim that this is independent of the choice of factorization. The only issue is
when φ can be factored as si ψ for i < m + 1 and as sm+1 η, in which case we have a
commutative diagram
[m]
<
O
ψ
[n]
/
sj−1
θ
η
si
$
[m − 1]
"
si
[m + 1]
:
sm+1
[m]
of monomorphisms in ∆. But then we just compute
x(ψ) = x(sm θ)
= (dm
m x)(θ)
= 1
since x ∈ Zm (Coskn F ). This independence of choice of factorization implies that
Y
y∈
Fn
[n],→[m+1]
104
VIII. ČECH THEORY
is in fact in (Coskn F )m+1 . Now we compute, for i ∈ [m] that
(dm+1
y)(φ : [n] ,→ [m]) = y(si φ)
i
= 1
and
(dm+1
m+1 y)(φ : [n] ,→ [m]) = y(sm+1 φ)
= x(φ),
hence y witnesses x ∼ 1 in πm (F ).
Lemma 22.4. Let F be a based fibrant simplicial sheaf. Suppose that the degree
n + 1 part πn+1 : Fn+1 → (Coskn F )n+1 of the adjunction morphism F → Coskn F
is an epimorphism for every n. Then F → 1 is a weak equivalence (i.e. πn (F ) = 1
for every n).
Proof. Given x ∈ Zn (F ) = Zn (Coskn F ), we know from the previous lemma
that πn (Coskn F ) = 1.4 So, since x ∼ 1 in Zn (Coskn F ), there is locally (in
fact, from the proof of the previous lemma, we don’t need “locally here”) some
z ∈ (Coskn F )n+1 such that
(1) dn+1
= 1 for i ∈ [n] and
i
n+1
(2) dn+1 y = x.
Note that we always give Coskn F the basepoint determined by the composition
1 → F → Coskn F .
Since Fn+1 → (Coskn−1 F )n is epic, we also know that, further locally, there is
some z ∈ Fn+1 with πn+1 z = y. But F → Coskn F is a map of simplicial objects,
so the diagrams
πn+1
Fn+1
dm+1
i
$
x
/
(Coskn F )n+1
dm+1
i
Fn = (Coskn F )n
commute for every i ∈ [n + 1], hence z witnesses x ∼ 1, so [x] = 1 in πn (F ).
23. Čech theory revisited
Given a sheaf F ∈ Ab(X) we used the Čech complex (1) of a cover {Ui : i ∈ I}
of X to produce a resolution of F , and hence a spectral sequence (5.1) converging
to the cohomology Hn (X, F ). This is putting the cart before the horse. Actually,
we know from the lemma before that that Coskn F is fibrant, so it makes sense to speak of
πn (Coskn F ).
4And
23. ČECH THEORY REVISITED
105
we will see in a moment that the Čech complexes are obtained by taking resolutions
“to the left” of ZX , then applying Hom( , F ), so we are really trying to compute
Hn (X, F ) = Extn (ZX , F )
(c.f. (6.1)) by resolving ZX . We cannot generally find a projective resolution of ZX
as there are not generally enough projectives in Ab(X), but to correctly compute
this Ext group, we only need a resolution G • → ZX where the G i are Hom( , F )
acyclic for this particular F . Even better, if we are willing to use a sequence
· · · G2• → G1• → G0•
of such resolutions (really a filtered category of resolutions), then we could get away
with the hypothesis:
Ext>0 (Gnm , F ) = 0
lim
−→
n→∞
The next thing to notice is that the failure of Čech cohomology to compute sheaf
cohomology in general has to do with the fact that, no matter how fine a cover
U = {Ui : i ∈ I}
of X we choose, we will have little control over the pairwise intersections Ui ∩ Uj
(and the triple intersections Ui ∩ Uj ∩ Uk , etc.). We saw an extreme example of this
in (8) where there was a finest possible cover U = {U0A , U0B } of a space Y , but
where many sheaves on Y had higher cohomology on X = U0A ∩ U0B .
“The solution” is to use hypercovers (21) keep track of not only a cover U = {Ui }
of X (i.e. a refinement of the trivial cover), but we also allow the possibility of taking
k
of each Ui ∩ Uj . Of course, once we do this, we will also want
a nontrivial cover Vi,j
k
to allow nontrivial covers of all new pairwise intersections Vi,j
∩ Uk , etc.
Given any G ∈ Sh(X), we can form its 0-coskeleton
Cosk0 G
∈ sSh(X),
which is just the simplicial sheaf with
(Cosk0 G )n = G n+1
(for reasons of notation, it is convenient to view G n+1 as G [n] =
morphism φ : [m] → [n],
Q
[n]
G ). For a ∆
(Cosk0 G )(φ) : G n+1 → G m+1
is given by (Cosk0 G )(φ)(s)(i) = s(φ(i)). The boundary map dni : G n+1 → G n is
given by omitting the ith entry of an n-tuple and the degeneracy map eni : G n → G n+1
repeats the ith entry.
`
If G = i hUi for a cover {Ui }, then since finite products commute with coproducts in Sh(X) and the Yoneda functor h preserves inverse limits (intersections in
106
VIII. ČECH THEORY
Ouv(X)), Cosk0 G is the “expected” simplicial object:
a
(Cosk0 F )n =
hUi0 ∩···∩Uin
(i0 ,...,in )
To obtain the Čech complex constructed in (1) for an F ∈ Ab(X), we simply apply
the sheaf Hom functor
H om( , F )
to Cosk0 G and note that the result is a cosimplicial object in Ab(X). In this case
we have
H om(Cosk0 G , F )n = H om((Cosk0 G )n , F )
a
= H om(
hUi0 ∩···∩Uin , F )
(i0 ,...,in )
=
Y
H om(hUi0 ∩···∩Uin , F )
(i0 ,...,in )
=
Y
FUi0 ∩···∩Uin .
(i0 ,...,in )
To this cosimplicial object, one can associate a chain complex supported in nonnegative degrees, which is just the Čech resolution of F . Of course, there are varying
ways to make this complex (the unnormalized chain complex, normalized chain
complex, complement of the degeneracies in the normalized complex, etc.—all are
homotopy equivalent) and these correspond to the variations on the Čech complex
(e.g. the alternating Čech complex).
The fact that F → H om(Cosk0 G , F ) is a resolution of F (i.e. a quasiisomorphism of complexes) boils down to the following simple observations:
(1) It suffices to check this on each Ui , since the Ui form a cover.
(2) The coskeleton is constructed via finite inverse limits, hence it certainly
commutes with restriction to Ui , as does H om.
(3) The restriction G |Ui → hX |Ui has an obvious section obtained from the
structure map of hUi = hX |Ui into the coproduct G .
(4) For any object C in any category (with finite inverse limits, say) if the map
to the terminal object 1 has a section, then Cosk0 C → 1 is a homotopy
equivalence (14.1). Hence G |Ui → hUi is a homotopy equivalence in sSh(Ui ).
(5) Functors preserve homotopy equivalences, so F |Ui → H om(Cosk0 G |Ui , F |Ui )
is also a homotopy equivalence in csAb(X).
(6) The Dold-Kan theorem identifies csAb(X) with the category of nonnegatively graded cochain complexes in Ab(X) and this identification takes
homotopies in the sense of simplicial objects to homotopies in the sense
of chain complexes, hence F |Ui → H om(Cosk0 G |Ui , F |Ui ) is a homotopy
equivalence.
24. EXERCISES
107
24. Exercises
Many of the constructions of this section involve factoring ΓΦ as H 0 followed
Φ
by Ȟ (X, ), under the assumption that Φ has neighborhoods (then applying the
Grothendieck spectral sequence). On the other hand, we can always factor ΓΦ as
HΦ0 followed by the usual global section functor Γ : PAb(X) → Ab, even without
the neighborhoods assumption. However, it is not so clear whether we can apply
the Grothendieck spectral sequence.
Exercise 24.1. The class of flasque presheaves satisfies the following properties:
(1) For any short exact sequence
0 → F 0 → F → F 00 → 0
in PAb(X) with F 0 and F flasque, F 00 is flasque.
(2) A direct summand of a flasque presheaf is flasque.
(3) For any F ∈ PAb(X), there is a flasque G ∈ PAb(X) and a monomorphism F ,→ G .
Exercise 24.2. For any family of supports Φ on a topological space X, show
that the functor HΦ0 : Ab(X) → PAb(X) takes flasque sheaves to separated flasque
presheaves. If X is notherian, show that HΦ0 takes flasque sheaves to flasque sheaves.
•
Exercise 24.3. Is the δ-functor Ȟ (U, ) from PAb(X) → Ab (c.f. ) acyclic
•
for the class of flasque presheaves? Assume Φ has neighborhoods, so that ȞΦ (X, )
is also a δ-functor. Is it acyclic for flasque presheaves?
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[DP]
[GZ]
[Qui]
[Hov]
[Mil]
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109