Sheaf Theory

W. D. Gillam

Department of Mathematics, Brown University

E-mail address : wgillam@math.brown.edu

Contents

Introduction 7

Chapter I. Sheaves 91. Open sets 92. Presheaves 93. Representable presheaves 104. Generators 115. Sheaves 116. Stalks 137. Sheafification 158. Espaces etale 169. Sieves 1810. Properties of Sh(X) 2011. Examples 2212. Exercises 22

Chapter II. Direct and Inverse Images 231. Functoriality of Ouv(X) 232. Direct images 233. Skyscraper sheaves 234. Inverse images 245. Sheaf Hom 266. Constant sheaves 277. Representable case 288. Inverse images and espaces etale 289. Faithfulness of X 7→ Sh(X) 29

Chapter III. Descent 311. Canonical topology 312. Open maps and variants 313. Descent for spaces 314. Descent for sheaves 31

Chapter IV. Cohomology 331. Abelian categories 332. Enough injectives 333. Homological algebra 334. Abelian Sheaves 33

3

4 CONTENTS

5. Ringed spaces 366. Sheaf Ext 377. Supports 388. Sheaf cohomology 399. Acyclic sheaves 3910. Flasque sheaves 3911. Godement resolution 4212. Soft sheaves 4213. Filtered direct limits 4214. Vanishing theorems 4615. Higher direct images 4616. Leray spectral sequence 4717. Group quotients 4818. Local systems 4819. Examples 4820. Exercises 48

Chapter V. Base Change 511. Proper maps 512. Comparison map 553. Homotopy invariance 57

Chapter VI. Duality 591. Adjoints for immersions 592. Extension by zero 613. Proper maps - Second treatment 63

Chapter VII. Finite Spaces 651. Supports 652. Specialization 653. Coniveau 664. Equivalence 675. Stalk formulas 676. Hom formulas 677. Poincare duality 688. Examples 699. Exercises 74

Chapter VIII. Cech Theory 751. Cech resolution 752. Refinement maps 773. Alternating variant 784. Cech cohomology for a fixed cover 785. Basic properties 796. Cech cohomology 797. Cech acyclic sheaves 80

CONTENTS 5

8. Cech presheaf 829. Comparison theorems 8510. The unit interval 8611. Simplicial objects 8612. The standard simplex 8713. Truncations and (co)skeleta 8714. Simplicial homotopy 8915. Simplicial sets 9116. Standard simplicial sets 9217. Geometric realization 9318. The abelian setting 9419. Simplicial rings 9420. Simplicial sheaves 9621. Weak equivalences 9922. Properties of coskeleta 10023. Cech theory revisited 10424. Exercises 107

Bibliography 109

Introduction

The aim of this document is to give a concise treatment of the “classical” theoryof sheaves on topological spaces, with the following philosophical guidelines:

(1) Appeal to standard facts about limits in the category of sets wheneverpossible, e.g. exactness of filtered direct limits.

(2) When possible, formulate constructions and results about sheaves on a topo-logical 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 theoryis 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. Almostany theorem about sets that can be stated and proved using category theoreticlanguage should also hold true in a topos. In particular, the category of sheaveson a topological space should be a lot like the category of sets. This philosophy isusually manifested by bootstrapping up from the category of presheaves, which is“very much” like the category of sets from the perspective of category theory. Arelated philosophical point is that, since our object of study is the category of sheaveson a topological space, we should try to phrase our constructions intrinsically usingthis category, rather than refering back to the topological space, its category of opensets, etc. The manifestation of this is the usage of the representable sheaf associatedto an open set, rather than the open set itself.

Regarding (3), my feeling is that basically every spectral sequence arises as aspecial case of the Grothendieck spectral sequence. One should never have to writedown a double complex out of the blue and analyse what its horizontal and verticalcohomology 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 ofX, where the only morphisms are inclusions. The category Ouv(X) has arbitrarydirect limits given by the formula

lim−→

(f : C→ Ouv(X)) =⋃C∈C

f(C)

and inverse limits over any category with finitely many objects, given by the formula

lim←−

(f : C→ Ouv(X)) =⋂C∈C

f(C).

In particular, ∅ ∈ Ouv(X) is initial (the direct limit of the empty functor, or theempty 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 anatural transformation of functors. For U ∈ Ouv(X), elements of F (U) are calledsections of F over U . For an inclusion of open sets V ⊆ U and s ∈ F (U) we oftenwrite s|V for the image of s under the morphism FV⊆U : F (U)→ F (V ). We referto 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), especiallywhen 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, inheritesvarious properties from Ens. For example, PSh has all small direct and inverselimits. 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 oftenwrite U instead of hU . Formation of hU is functorial in U and defines a functorh : Ouv(X)→ PSh(X) called the Yoneda functor.

Lemma 3.1. The Yoneda functor h is fully faithful and commutes with arbitraryinverse 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 donot generally coincide. Indeed, for W ∈ Ouv(X), by definition of direct sums, wehave

HomPSh(X)(hU ⊕ hV , hW ) = HomPSh(X)(hU , hW )∐

HomPSh(X)(hV , hW )

= hW (U)∐

hW (V )

= 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 remedythis particular failure, as some authors do, by using a variant of Ouv(X) wherewe consider not only open sets of X, but all maps f : Y → X which are localhomeomorphisms. From our point of view, there is no reason to do this. We are onlyusing Ouv(X) as a stepping stone to get to another category; we do not particularlycare 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, forany parallel C morphisms f, g : C → D, the following are equivalent:

(1) f = g(2) fh = 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”⊕G∈G

⊕HomC(G,C)

G→ C

is an epimorphism for every C ∈ C. An object G of C is called a generator if Gis a generating set. Dually, G is called a cogenerating set if G is a generating set forCop.

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 generat-ing 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 ofU). A presheaf F is separated for U if the product of the restriction maps

F (U)→∏i∈I

F (Ui)

is injective (that is, if a section of F is completely determined by its restrictionsto the open sets in a cover). F has the gluing property (or descent property) withrespect to U if, for any set of sections si ∈ F (Ui) : i ∈ I with si|Uij

= sj|Uijfor

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) forevery such family U . A presheaf is a sheaf if it is separated and it has the gluingproperty.

12 I. SHEAVES

We can sum this up by saying that F is a sheaf if and only if

F (U)→∏i∈I

F (Ui) ⇒∏

(i,j)∈I×I

F (Uij)(5.1)

is an equalizer diagram of sets for any U ∈ Ouv(X) and any cover U = Ui : i ∈ Iof U .

Remark 5.1. If F is a separated presheaf, then the set F (∅) is punctual (hasone element) because the empty set is covered by the empty cover and the emptyproduct 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 = ∪iUi 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 forthis cover.

(2) =⇒ (1). Let U = Ui : i ∈ I be a family of open subsets of X with unionU . To prove F is separated for U , suppose s, t ∈ F (U) have the same restriction toevery Ui. By Zorn’s Lemma and the fact that F has the sheaf property for filteredunions, there is a maximal J ⊆ I such that s and t have the same restriction toV := ∪i∈JUi. If J 6= I, then there is an i ∈ I \ J and, using the separation of F forthe cover Ui, V we find that s and t agree on V ∪Ui, contradicting maximality ofJ . The gluing property is proved similarly.

Corollary 5.3. If X is a noetherian topological space, then F ∈ PSh(X) isa 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 inOuv(X) is essentially constant, so any presheaf F trivially commutes with filteredinverse limits.

6. STALKS 13

A morphism of sheaves is defined so that the category Sh(X) of sheaves on Xforms 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−→U3x

F (U).

The direct limit is taken over open neighborhoods of x and restriction maps betweenthem. Given a section s ∈ F (U) and a point x ∈ U we let sx ∈ Fx denote theimage of s under the natural map

F (U)→ lim−→V 3x

F (V ) = Fx.

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 morphismof presheaves f : F → G induces a morphism fx : Fx → Gx on stalks for eachx ∈ X by taking the direct limit of the morphisms f(U) : F (U) → G (U) over allopen neighborhoods U of x.

For example, the stalk hU,x of the representable sheaf hU at x is a punctual setif x ∈ U and empty otherwise.

In contrast to the situation for presheaves, the stalks of a map f : F → G ofsheaves 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 everyx ∈ U . Since the maps on stalks are monic, we have sx = tx for all x ∈ U , hences = 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 tocheck this for H running over a generating set, so by (4.1), we conclude that f ismonic 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 mapf(U) by Yoneda (3.1).

(5) =⇒ (4) Suppose g1, g2 : G →H satisfy g1f = g2f . Then g1,xfx = g2,xfx forevery x ∈ X, hence g1,x = g2,x for every x ∈ X because f is surjective on stalks. Weconclude 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) forA ∈ Ens, then this monomorphism is identified with

f ∗x : HomEns(Gx, A) → HomEns(Fx, A).

Since this is a monomorphism for every x,A, fx is an epimorphism in Ens for everyx.

(8) =⇒ (7) Consider a t ∈ G (U). Since the maps on stalks are epic, there areneighborhoods 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 thatthe 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 thatf(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 thisfollows from the previous results.

7. SHEAFIFICATION 15

7. Sheafification

The main goal of this section is to prove that Sh(X) is a reflexive subcategoryof PSh(X), i.e. the inclusion functor Sh(X) → PSh(X) has a left adjoint, calledsheafification.

For a presheaf F , we will construct a sheaf F + (the sheafification of F ) and amorphism of presheaves F → F +, such that any morphism of presheaves F → Gwith G a sheaf will factor uniquely through F → F +. We set F +(U) equal to theset of all

t = (t(x))x∈U ∈∏x∈U

Fx

satisfying the following local coherence condition: There are a cover U = Ui of Uand 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 functionsand it is easy to check that F + is a sheaf. Indeed, we may actually glue functionstogether in the usual sense if they agree on overlaps. The construction is functorialbecause a map of presheaves f : F → G induces maps of stalks fx : Fx → Gx whichwe 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 conditionto 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 thenfollows from (6.1) that if F is already a sheaf, then the natural map F → F + is anisomorphism. 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 factorizationis unique because one can check easily that a morphism of sheaves is determined byits 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 arbi-trary 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 inverselimit of sheaves (taken in the category of presheaves) is already a sheaf, which canbe checked directly from the fact that the sheaf property is defined in terms of acertain diagram being an equalizer diagram (a certain kind of inverse limit) and thefact 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 inverselimits.

Proof. We have already mentioned that Sh(X) has all inverse limits and thatthese coincide with the inverse limit taken in PSh(X). Using the adjointness proper-ties of the sheafification functor, it follows from a formal exercise in category theorythat the direct limit of a direct system of sheaves i 7→ Fi can be obtained by firsttaking the direct limit in the category of presheaves (c.f. (2.1)), then applying thesheafification functor to the result.

Sheafification commutes with arbitrary direct limits simply because it is a rightadjoint (formal category theory). To prove that it commutes with finite inverselimits, let f : C → PSh(X) be a functor with C a finite category (finitely manymorphisms 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−→U3x

lim←−C∈C

f(C)(U) → lim←−C∈C

lim−→U3x

f(C)(U),

which is an isomorphism because finite inverse limits commute with filtered directlimits in the category of sets.

For example, the representable sheaf hX is the terminal object of Sh(X) (andPSh(X)). Indeed, hX(U) is punctual (terminal in Ens) for every U ∈ Ouv(X) sothere is clearly a unique map F → hX as there is a unique map F (U) → hX(U)for every U ∈ Ens.

8. Espaces etale

Let F be a presheaf on a space X. We define a topological space Et F , calledthe espace etale of F and a continuous map π : Et F → X as follows: As a set, welet

Et F :=∐x∈X

Fx

be the disjoint union of the stalks of F . The set Et F is given the topology wherea basic open subset is a set of the form

Us := sx ∈ Fx ⊆ EtX : x ∈ U,

where U ∈ Ouv(X), s ∈ F (U). To show that these sets form the basic opens for atopology we should show that every sx ∈ Et F is in one of the Us (this is clear) and

8. ESPACES ETALE 17

that an intersection Us ∩ Vt of such basic opens is again such a basic open. Indeed,the set

W := x ∈ U ∩ V : sx = txis open in U ∩ V (hence also in X) because equality at a stalk by definition impliesequality 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 equalin 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) =⋃

V⊆U,s∈F (V )

Vs

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 → Xand 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 = IdUare clearly compatible with restricting to smaller U , hence define a morphism φ fromF 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 thesheaf of local sections of the espace etale π : Et F → X identifies the latter sheafwith the sheafification of F . In particular, if F is a sheaf, then F is isomorphic tothe sheaf of local sections of π : Et F → X.

Proof. By the stalk criterion (6.1) it suffices to prove that φ is an isomorphismon stalks. To see that φx is surjective, consider a local section f : U → Et F of π on aneighborhood U of x ∈ U . We must prove that f |V agrees with φ(V )(t) : V → Et Ffor some t ∈ F (V ) for some neighborhood V of x in U . By definition of thestalk, we can find some neighborhood W of x in U and some r ∈ F (W ) such thatrx = f(x) ∈ Fx. Since f is continuous, the set

V := f−1Wr := 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 xand the functions φ(U)(s), φ(U)(t) : U → EtX agree on a neighborhood of X. Wehave to show that sx = tx ∈ Fx. But this is obvious from the fact that φ(U)(s) andφ(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 etale gives another means of sheafifyinga presheaf. It also gives a means of translating various sheaf-theoretic questions intoquestions of topology.

Remark 8.2. Given a map p : W → X, the espace etale of the sheaf F oflocal sections of p is a topological space π : Et F → X over X that generally haslittle to do with the space W . One only knows that the sheaf of local sections ofπ is isomorphic to the sheaf of local sections of p. The map π is built solely fromthe sheaf of local sections of p, but one cannot hope to recover the map p fromthis sheaf. Another very special property of the espace etale 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 ageneral space p : W → X over X, the map [f ] 7→ f(x) from germs of local sectionsof p near x to the fiber p−1(x) is neither surjective nor injective.

The formation of the espace etale is functorial in F , so we may think of Et as afunctor

Et : Sh(X) → Top/X.

(It is actually defined on PSh(X), but we saw above that the espace etale of apresheaf is naturally isomorphic to the espace etale of its sheafification.) Formationof the espace etale is also functorial in X, in a sense that we will make precisein Section 8 once we discuss the functorial dependence of the category of sheavesSh(X) on X.

9. Sieves

The following categorical constructions are used to define sheaves in more gen-eral settings. The constructions are not needed elsewhere in any serious way, thoughsome of the terminology will be natural in our study of Cech cohomology in Chap-ter VIII.

Given a category C, a sieve in C is a full subcategory U ⊆ C such that, for anyU ∈ 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 byinclusion. A set of objects G ⊆ C determines a sieve—namely the sieve consisting ofall 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 clearlycontained 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 categoryC/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 befactored 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) isnothing more than a family of open subsets of U closed under passage to smalleropen subsets. We declare a sieve U ∈ Crib(U) to be a covering sieve if U is a coverof U , and we let Cov(U) denote the set of covering sieves (it inherits an orderingfrom Crib(U)). A covering sieve U ∈ Cov(U) is nothing more than a cover of Uclosed under passage to smaller sets, and a generating set for U is a subset of G ⊂ Usuch that every V ∈ U is contained in some G ∈ G . This clearly implies that G is asubcover, but not every subcover is a generating set—G must contain all the largestopen 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 ×ZU ∈ 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 ′ ∈ PSh(X) by setting

F ′(U , U) := lim←−

∏Ui∈U

F (Ui) ⇒∏(i,j)

F (Ui ∩ Uj)

for a covering sieve

U = Ui : i ∈ I ∈ Cov(U).

Next note, for

U ′ = Ui : i ∈ I ′ ⊆ U ,

20 I. SHEAVES

with U ′ ∈ Cov(U) (i.e. for a Cov(U)op morphisms U → U ′), we have an obviousmap F ′(U , U)→ F ′(U ′, U) induced by the restriction map∏

i∈I

F (Ui)→∏i∈I′

F (Ui).

Set

F ′(U) := lim−→

U∈Cov(U)op

F ′(U , U).

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 ′(U)→F ′(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 ′ is separated.(2) For any separated F ∈ PSh(X), F ′ is a sheaf.(3) For any F ∈ PSh(X), there is a natural isomorphism (F ′)′ = 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 allowus to work more directly with Sh(X) as a category per se, without so much explicitreference 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 commutewith filtered direct limits), so by (6.1) we reduce to proving that epimorphisms inEns are closed under base change, which is easy.

Theorem 10.2. A morphism f : F → G in Sh(X) is an epimorphism iff, forevery 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 definitionof “epimorphism”, so the implication (⇐=) is obvious, and to establish ( =⇒ ), itremains only to show the following: suppose g : F → H is a Sh(X) morphismsuch that gπ1 = gπ2 : F ×G F → H , then g = hf for some h : G → H . Thefastest way to prove this is to use stalks: the theorem we are trying to prove isclear 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 thatgx = hxfx. Since H is a sheaf, we can view a section t ∈ H (U) as an elementt = (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) actuallysatisfies the local coherence condition (the desired equality g = hf will then followimmediately by checking on stalks). To check local coherence of h(U)(s) at a pointy ∈ U , we use surjectivity of fy to find an open neighborhood V of y in U and asection 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) = hzfz(tz) = hz(sz) = h(U)(s)(z)

for all z ∈ V .

In highfalutin language, the next result says that the “usual covers of X arecofinal 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 isan epimorphism iff there is a cover U = Ui : i ∈ I of X and a morphism∐

i∈I

hUi→ G .

Proof. ( =⇒ ). By (5.4), the hU form a generating set for Sh(X), so we cancertainly find an epimorphism of the form∐

Ui∈Ouv(X)

∐j∈Ji

hUi→ G

(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 ∐Ui∈Ouv(X)

∐j∈Ji

hUi,x → hX,x.

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 ofthe hUi(x)

clearly maps to G by first mapping to the “big” coproduct above, whichcan 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 arrowcategory 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 as-sociates 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 localin nature (can be checked on each open set in a cover). This sheaf contains thepresheaf C b(X,R) of bounded functions. This latter presheaf is separated, but itis not generally a sheaf because being bounded is global in nature and one cannotglue together functions which are locally bounded into a function which is globallybounded. Indeed, any continuous map f : X → R is locally bounded.

Example 11.2. Generalizing the previous example, for any topological spacesX, Y , the presheaf U 7→ HomTop(U, Y ) (with the obvious restriction maps) is a sheafon X because continuity is local. Similarly, if π : W → X is a map of topologicalspaces the presheaf

U 7→ s ∈ HomTop(U,W ) : πs = IdUof local sections of π is a sheaf on X. We will see in Section ?? that every sheaf isisomorphic 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 thatevery U ∈ Ouv(X) is a union (direct sum) of open sets in B. Set PSh(B) =HomCat(B

op,Ens). Say F ∈ PSh(B) is a sheaf if, for every U ∈ B and every coverUi ∈ B : i ∈ I of U , the diagram

F (U)→∏i∈I

F (Ui) ⇒∏

(i,j)∈I×I

F (Uij)

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 theinclusion 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 topologicalspace 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 notwish 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 restric-tion 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 (regardedas a sheaf on x as in (11)), then x∗A is the sheaf on X given by

U 7→A, x ∈ U0, otherwise

23

24 II. DIRECT AND INVERSE IMAGES

A sheaf on X isomorphic to such a sheaf is called a skyscraper sheaf. The stalks ofx∗A are given by

(x∗A)y =

A, y ∈ x−0, otherwise

If F is a sheaf, then so is f∗F because f∗F has the sheaf property with respectto a cover U if and only if F has it with respect to the cover f−1U , as is clear fromthe definitions. The construction f∗ is clearly functorial in F .

Example 3.1. The following example shows why direct limits of sheaves donot typically commute with sections. Let X be any infinite set with the discretetopology and let ZX be the constant sheaf. The presheaf direct sum ⊕x∈Xx∗Z of allstalks of Z is given by

U 7→⊕x∈U

Z.

This presheaf if clearly separated, but it does not have the gluing property for thecover of X by singleton sets. The sheafification ⊕x∈Xx∗Z of this presheaf is givenby

U 7→∏

x∈U Z

because these are the functions to the stalks which (trivially) satisfy the local coher-ence condition. If, on the other hand, X is given the cofinite topology (a noetheriantopology), 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 inverseimage presheaf f−1

preG ∈ PSh(X) is defined by

f−1preG : Ouv(X)op → Ens

U 7→ lim−→

V⊇f(U)

G (V ),

where the direct limit is taken over open subsets V of Y containing f(U) withrespect to the restriction maps of G (this is a filtered subcategory of Ouv(Y )). NoteV ⊇ 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 ) ismorphism in Ouv(X). A morphism (V, g)→ (V ′, g′) in U ↓ Ouv(Y ) is a morphismi : V → V ′ in Ouv(X) making the diagram

Ug

yy

g′

%%Ouv(f)(V )

Ouv(f)(i)// Ouv(f)(V ′)

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) yieldsa section f−1

pre(s) ∈ (f−1preG )(f−1U) given by the class of s in the direct limit system

defining (f−1preG )(f−1(U)). It is clear from the definition of f−1

preG that its stalks aregiven by

(f−1preG )x = Gf(x).

The stalk of a presheaf F on a space X at a point x ∈ X is nothing but theinverse limit of F under the inclusion map x : x → X.

The functor f−1pre does not generally take sheaves to sheaves. We are led to define

f−1 : Sh(Y ) → Sh(X)

G 7→ (f−1preG )+

by composing f−1pre and the sheafification functor:

f−1G := (f−1preG )+.

As for the presheaf inverse image, a section s ∈ G (U) determines a section f−1s ∈(f−1G )(f−1(U)), namely, the image of f−1

pres ∈ (f−1preG )(f−1(U)) under the sheafifica-

tion morphism(f−1

preG )(f−1(U))→ (f−1G )(f−1(U)).

Examining the construction of sheafification, we see that a section of (f−1G )(V ) canbe viewed as an element

s = (s(x)) ∈∏x∈V

Gf(x)

satisfying the local coherence condition: For every x ∈ V , there is a neighborhoodW of x in V , a neighborhood U of f(W ) in Y , and a section t ∈ G (U) such thats(x′) = tf(x′) for every x′ ∈ W .

The formation of f−1pre and f−1 are both functorial in f in the sense that for

f : X → Y and g : Y → Z the functors

f−1preg

−1pre, (gf)−1

pre : PSh(Z) → PSh(X)

are naturally isomorphic, as are the functors

f−1g−1, (gf)−1 : PSh(Z) → PSh(X).

(In particular, the case where X = y is a point of Y gives the stalk formula(g−1

preG )y = Gg(y) mentioned above.)

Theorem 4.1. The functor f−1pre : PSh(Y ) → PSh(X) is left adjoint to f∗ :

PSh(X) → PSh(Y ). The functor f−1pre 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 andallows us to link up with various general category theoretic techniques (comma categories, Kanextension).

26 II. DIRECT AND INVERSE IMAGES

Proof. The adjointness is a special case of the Kan Extension Theorem and isstraightforward to check. The other statements are formal consequences of adjoint-ness, except the fact that f−1

pre commutes with finite inverse limits. This is proved bynoting that the category of neighborhoods of any subspace is filtered, and filtereddirect 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 finiteinverse limits. The functor f∗ commutes with arbitrary inverse limits.

Proof. This follows formally from the theorem using the adjointness and ex-actness 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 +) =(f−1

preF )+ in Sh(X).

Proof. This is a consequence of Yoneda’s Lemma and the adjointness resultsjust established: For any G ∈ Sh(X) we have

HomSh(Y )(f−1(F +),G ) = HomSh(Y )(F

+, f∗G )

= HomPSh(Y )(F , f∗G )

= HomPSh(X)(f−1preF ,G )

= HomPSh(X)((f−1preF )+,G ).

Remark 4.4. Suppose i : U → X is the inclusion of an open set and F is asheaf on X. Then for any V ∈ Ouv(U), V is in Ouv(X) and is cofinal amongopen neighborhoods of i(V ) in X, hence i−1

preF (V ) = F (V ). Furthermore, i−1preF

is already a sheaf, so we have i−1F (V ) = F (V ). Evidently then, i−1F is nothingbut the restriction of the functor F : Ouv(X)op → Ens to the full subcategoryOuv(V )op ⊆ Ouv(X)op. Accordingly, we often write F |U instead of i−1F .

5. Sheaf Hom

Consider the functor

H om( , ) : PSh(X)op ×PSh(X) → PSh(X)(5.1)

(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, gHom(F |U ,G |U) agree when restrictedto 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 sepa-ration 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 onoverlaps, so by the gluing property for G there is an f(V )(s) ∈ F (V ) restricting tofi(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 )

// G (V )

F (W ) // G (V )

commutes for any morphism W → V in Ouv(U). This follows from separation forG .

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 , theadjunction (f−1, f∗) “sheafifies” to yield a natural isomorphism

f∗H omX(f−1F ,G ) = H omY (F , f∗G ).

6. Constant sheaves

An extreme case of the formation of inverse images occurs when f is the mapfrom X to a one point space Y . If we identify Sh(Y ) with Ens as in (11), then thepresheaf inverse image functor f−1

pre is identified with the functor A 7→ ApreX , where

ApreX 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 someAX is called a constant sheaf.

7. Representable case

It is instructive to describe f∗F and f−1F in the case where F = hU is arepresentable sheaf. The equalities

f−1hU = f−1prehU

= hf−1(U)

are clear for any morphism f : X → Y and any U ∈ Ouv(Y ). The set (f−1hU)(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 ) iff−1(U) ⊆ V , else (f∗hV )(U) is empty.

8. Inverse images and espaces etale

The formation of inverse images is compatible with the construction of the espaceetale (§8).

Lemma 8.1. Let f : X → Y be a continuous map of topological spaces, F apresheaf on Y , π : Et F → Y its espace etale. Then π1 : X ×Y Et F → X isnaturally isomorphic to the espace etale of f−1

preF . Similarly, if F is a sheaf, π1 is

naturally isomorphic to the espace etale of f−1F .

Proof. Recall from Section ?? that, as a set,

Et f−1preF =

∐x∈X

(f−1preF )x =

∐x∈X

Ff(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 off(x) ∈ Y under π : Et F → Y is Ff(x) by construction of π). This identificationclearly respects the two maps to X, so it remains to prove that it is continuous.

A basic open set in Et f−1preF is a set of the form

Ut = (x, s) ∈ Et f−1preF : x ∈ U, s = tx ∈ Ff(x)

where U is an open subset of X and t ∈ (f−1preF )(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 thenhas tx = rf(x), so this open corresponds to the open subset U ×Y Vr under ouridentification.

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

U ×Y Vs = f−1(V )×Y Vs = f−1(V )f−1pres

under our identification, so this basic open in X ×Y Et F is also open in Et f−1preF .

9. FAITHFULNESS OF X 7→ Sh(X) 29

The statements about sheaves follow from the statements about presheaves be-cause both inverse images and espaces etale 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 (composi-tion, 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 sheafcohomology.

1. Abelian categories

2. Enough injectives

3. Homological algebra

basic theory of delta functors and derived functors, Grothendieck spectral se-quence, spectral sequence from complexes

4. Abelian Sheaves

For a topological space X, let Ab(X) denote the category of abelian groupobjects in Sh(X). An object F ∈ Ab(X) is often called a sheaf of abelian groups orjust an abelian sheaf. Similarly, we let PAb(X) denote the category of abelian groupobjects in PSh(X). Note that PAb(X) is isomorphic to the category of functorsfrom Ouv(X)op to Ab and that this identifies Ab(X) with the full subcategoryconsisting of those F : Ouv(X)op → Ab whose composition with the forgetfulfunctor Ab→ Ens is a sheaf. For F ∈ PAb(X) and any inclusion V ⊆ U of opensubsets 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 inthe same way that one can translate various statements about sets into statementsabout (abelian) groups.

Lemma 4.1. The forgetful functor Ab(X)→ PAb(X) has a left adjoint (sheafi-fication) which commutes with arbitrary direct limits and finite inverse limits.

Proof. The analogous statements are true of Sh(X)→ PSh(X) and preserva-tion of finite inverse limits implies that sheafification takes (abelian) group objectsto (abelian) group objects.

The categories PAb(X) and Ab(X) clearly carry the structure of additive cate-gories. 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 lim-its are formed by taking the inverse limit in PAb(X) and noting that it is alreadyin Ab(X). Indeed, Ab → Ens commutes with inverse limits, so this is obviousfrom the corresponding statement about inverse limits in Sh(X). However, to formdirect limits in Ab(X), one must sheafify (i.e. apply the sheafification functor of thelemma to) the direct limit taken in PAb(X).

Explicitly, the zero object 0 ∈ PAb(X) is the constant presheaf U 7→ 0. Notethat this is also a sheaf and that it is also the zero object in Ab(X). For anymorphism 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 similarstatement holds in Ab(X) because the natural map there is obtained by sheafifyingthis natural map. We have proved:

Proposition 4.2. Ab(X) and PAb(X) are abelian categories with all smalldirect and inverse limits.

Observe that Z is a generator for Ab (c.f. §??), and that this is obvious oncategory theoretic grounds from the fact that a one element set is a generator forEns 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 ofbootstrapping 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→ ⊕preF Z,

where ⊕preF Z ∈ PAb(X) is given by

⊕preF Z : Ouv(X)op → Ab

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→ (⊕preF Z)+,

Proof. Follows from the lemma and the adjointness property of sheafification(7.1).

4. ABELIAN SHEAVES 35

Theorem 4.5. ⊕prehU

Z : U ∈ Ouv(X) (resp. ⊕hUZ : U ∈ Ouv(X)) is agenerating set for PAb(X) (resp. Ab(X)). PAb(X) and Ab(X) have all smalldirect and inverse limits. Finite inverse limits and filtered direct limits commute inboth PAb(X) and Ab(X).

Proof. The first statement is purely formal; let us prove it for Ab(X) forconcreteness. For a sheaf F , we have

HomAb(X)(⊕hUZ,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)(⊕hUZ,F )→ HomAb(X)(⊕hUZ,G ).

The existence of limits was discussed above. The last statement is certainly true ofPAb(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 filtereddirect limits and inverse limits). Hence this is also true of Ab(X) by adjointnessand exactness properties of sheafification.

Corollary 4.6. Every F ∈ PAb(X) (resp. Ab(X)) admits a monomorphisminto an injective object of PAb(X) (resp. PAb(X)).

Proof. This follows formally from the theorem as in Grothendieck’s Tohokupaper.

The section functors

PAb(X) → Ab

F 7→ Γ(U,F )

commute with all direct and inverse limits, as these are formed “objectwise” inPAb(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 thesection functors for sheaves commute with inverse limits), but they do not generallycommute 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 coincidewith finite direct products. In other words, Γ(U, ) is a left exact morphism ofabelian categories—its right derived functors will be considered in more generalityin §??.

36 IV. COHOMOLOGY

The forgetful functor Ab(X) → PAb(X) is similarly a left exact morphism ofabelian categories. Its nth right derived functor is denoted F 7→ H n(F ) and iscalled 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 ofabelian categories f−1 : Ab(Y ) → Ab(X) and a left exact morphism of abeliancategories f∗ : Ab(X) → Ab(Y ). The right derived functors Rn f∗ are called thehigher 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) wecan consider the category Mod(OX) of modules (“module objects”) over OX . Con-cretely, OX ∈ An(X) is a functor OX : Ouv(X)op → An whose composition withthe 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 anOX(U) module structure making the restriction maps F (U) → F (V ) linear overOX(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 abeliancategory 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) iscalled 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 overthe stalk OX,x of OX .

The following direct proof that Mod(X) has enough injectives is sometimesuseful, especially if we need to get our hands on a supply of easily understoodinjectives.

Lemma 5.1. Let x : x → X be the inclusion of a point of a topological spaceX with a sheaf of rings OX . Let I be an injective OX,x module. Then x∗I is aninjective OX module.

Proof. We must show F 7→ HomX(F , x∗I). Note OX,x = x−1OX , so by theadjointness (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 in-jectives.

6. SHEAF EXT 37

Proof. Let F ∈ Mod(X). Since Mod(A) has enough injectives for any ringA, we can choose, for each x ∈ X, a monomorphism Fx → I(x) from the stalk ofF 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 ofall these maps is a map F →

∏x∈X x∗I(x) which is easily seen to be monic. Note∏

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)) inheritsan abelian group object structure from that of G . We may view sheaf Hom as afunctor

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 “internalHom” 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 , ) is left exact. Its right derivedfunctors are denoted

G 7→ E xtp(F ,G ).

Lemma 6.1. Let X be a ringed space, F ∈Mod(X). There is an isomorphismof 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 homo-morphism ·s ∈ Hom(OX ,F ) given by

(·s)(U) : OX(U) → F (U)

f 7→ f · s|Uyields an isomorphism Γ(X,F ) = Hom(X,F ). The result now follows since bothδ functors are effaceable, as Mod(X) has enough injectives.

The proof of the following lemma will require several constructions that we havenot 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 isa natural first quadrant spectral sequence

Ep,q2 = 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 aspecial case of the Grothendieck spectral sequence as long as we can prove thatH om(F ,G ) is Γ-acyclic for any injective sheaf G ∈Mod(X). Since flasque sheavesare Γ-acyclic by (10.4), below, it suffices to prove that H om(F ,G ) is flasque, sowhat we must show is that, for any open set ι : U → X and any homomorphismf : ι−1F → ι−1G , there is a homomorphism F : F → G in Mod(X) with ι−1F = 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−1F // _

ι!ι−1G

// G

FF

55

in Mod(X), which completes as indicated by injectivity of G since the verticalarrow is monic. Since ι−1ι! = Id, the vertical and second horizontal arrows becomeisomorphisms 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), theset 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 sheavesis the closure of the union of their supports.

A family Φ of closed subsets of a topological space X is called a family of supportsif 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 allclosed subsets of X is a family of supports. Another example is the family Φ of allclosed quasi-compact subsets of X (if X is Hausdorff, then the adjective “closed” isredundant, but this is not true for a general X). For a subspace Z of X, we couldtake Φ to be the family of all closed subsets of X contained in Z. We could also takeΦ to be the family of all closed, paracompact subsets of X. We say that a family ofsupports 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 paracom-pact,

then we say Φ is paracompactifying.

Recall that a space X is called paracompact iff X is Hausdorff and every opencover 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 ∈ ΦΦY := A ∈ Φ : A ⊆ Y

are both both families of supports on Y . We emphasize that, in general, these arevery different, so we caution the reader to take care in keeping track of which familyof supports we are using.

If F is an abelian sheaf on X and i : U → X is an open subset of X, then theglobal sections with support in Φ of the abelian sheaf F |XU = i!i

−1F on X are givenby 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 ∈ Φis a subgroup of Γ(X,F ) (it contains 0 because ∅ ∈ Φ, and it is closed under additionbecause Φ is closed under finite unions and passage to smaller closed sets) called thegroup 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 aredenoted Hi

Φ(X,F ) and called the cohomology of F with support in Φ. In case Φ isthe family of supports consisting of all closed subsets contained in a closed subspaceY ⊆ X, Hi

Φ(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 withinfinite 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 Hn

Φ(X,F ) = 0 for n > 0.

(PUT SOMETHING HERE TO THE EFFECT THAT COHOMOLOGY CANBE COMPUTED USING GAMMA ACYCLIC RESOLUTIONS)

10. Flasque sheaves

A sheaf F on a topological space X is called flasque iff the restriction mapF (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 topo-logical spaces is again flasque.

Proof. This is clear from the formula (f∗F )(U) = F (f−1U) for sections of thedirect image sheaf.

Lemma 10.2. (Gluing sections) Suppose A = Ai : i ∈ I is a locally finitefamily of subsets of X with union A and we have sections si of an abelian sheaf Fover Ai for each i ∈ I which agree on overlaps (si|Aij

= sj|Aijfor all i, j ∈ I). Then

there is a unique section s of F over A such that s|Ai= si for each i ∈ I.

Proof. Certainly there is a unique map s : A→∏

x∈A Fx with

s(x) = si(x) ∈ Fx ⊂∏a∈A

Fa

for every i ∈ I with x ∈ Ai. The only question is whether s satisfies the localcoherence condition. To see this, fix a ∈ A and use local finiteness to find an openneighborhood U of a which meets only Ai1 . . . Ain . The sections sij satisfy the localcoherence condition at a, so there are open neighborhoods Uij of a and sectionstij ∈ F (Uij) such that tij(x) = sij(x) = s(x) for all x ∈ Uij ∩ Aij . Let V be theintersection of U and the Uij so V is an open neighborhood of a since there are onlyfinitely many ij. The set W where the restrictions tij |V all agree is an open subsetof V containing a. Let t ∈ F (W ) be the common value. Then t witnesses the localcoherence of s at a.

Theorem 10.3. Let X be a topological space, Φ a family of supports on X. Theclass of flasque sheaves satisfies the following properties:

(1) For any short exact sequence of sheaves

0→ F ′ → F → F ′′ → 0

on X with F ′ flasque, the map ΓΦ(X,F )→ ΓΦ(X,F ′′) is surjective.(2) For any short exact sequence of sheaves

0→ F ′ → F → F ′′ → 0

on X with F ′ and F flasque, F ′′ 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 proper-ties.

Proof. The fact that summands of flasque or Φ soft sheaves are again flasque orΦ-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 discontinuoussections (see ).

10. FLASQUE SHEAVES 41

Now assume

0→ F ′ → F → F ′′ → 0

is a short exact sequence of sheaves on X with F ′ flasque. We will first show thatΓ(X,F ) → Γ(X,F ′′) is surjective. Let s ∈ Γ(X,F ′′). Consider the set A of allpairs (U, t) where U is an open subset of X, and t ∈ F (U) maps to s|U underF (U)→ F ′′(U). Order A by declaring (U, t) ≤ (U ′, t′) iff U ⊆ U ′ and t′|U = t. Bythe sheaf properties for F and F ′, any chain in A has an upper bound, so by Zorn’sLemma, A has a maximal element (U, t). If U = X we’re done. If not, there issome x ∈ X \U . Since F → F ′′ is surjective on stalks, we can find a neighborhoodV of x in X and a section t′ ∈ F (V ) such that t′ 7→ s|V under F (V ) → F ′′(V ).Note t|U∩V − t′|U∩V ∈ F (U ∩ V ) actually lies in F (U ∩ V ) because both t|U∩V andt′|U∩V map to s|U∩V under F (U ∩ V ) → F ′′(U ∩ V ). Since F ′ is flasque, thereis a w ∈ F ′(V ) restricting to t|U∩V − t′|U∩V on U ∩ V . It follows that t ∈ F (U)and t′ + w ∈ F (V ) agree on U ∩ V , hence glue to a section t ∈ F (U ∪ V ). Thesection t maps to s|U∪V because this can be checked on U and V by separation forF ′′ and adding w ∈ F ′(V ) to t′ doesn’t change its image in F ′′(V ). We concludethat (U ∪ V, t) contradicts maximality of (U, t).

Now we show that ΓΦ(X,F )→ ΓΦ(X,F ′′) is also surjective. Let s ∈ ΓΦ(X,F ′′).By what we just proved, there is some t ∈ F (X) mapping to s, but t may not havesupport in Φ, so we wish to find a t′ ∈ F ′(X) such that t − t′ has support in Φ—subtracting off t′ from t doesn’t change its image in F ′′(X), so this will completethe proof. Note t|X\Supp s ∈ F (X \ Supp s) is actually in F ′(X \ Supp s). Take t′

to be any lift of this to F ′(X) (such a t′ exists since F ′ is flasque).

Corollary 10.4. A flasque sheaf is ΓΦ-acyclic for any family of supports Φ anda 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(ΓΦG •) = HpΦ(X,F ).

Proof. By standard homological algebra, there is a spectral sequence with

Ep,q2 = Hp

Φ(X,G q)

converging to Hp+qΦ (X,F ). If every G q is acyclic, this degenerates to yield the

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 isparacompactifying).

Lemma 10.6. (Flasque is local) Let F be a sheaf on X, U an open cover ofX. 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 theother implication, suppose s ∈ F (V ) and we want to show there is a lift of sto F (X). Consider the set of pairs (U, t) where U is a neighborhood of V andt ∈ F (V ) restricts to s on V . Argue as in the proof of the above theorem, usingZorn’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

U 7→∏x∈U

Fx,

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 injectivebecause if tx = t′x for all x ∈ U then t = t′ because F is separated. TakingC 1(F ) := C 0(Cok ε) and continuing yields a resolution ε : F → C •(F ) calledthe Godement resolution or (by Godement) canonical resolution of F . It is clearthat from the formula for sections that, for any sheaf F and any n ≥ 0, the sheafC n(F ) is flasque. A morphism of sheaves f : F → G induces a natural morphismof canonical resolutions.

Theorem 11.1. For a sheaf F on a topological space X, and any family ofsupports Φ, there is a canonical isomorphism Hp(ΓΦG •(F )) ∼= Hp

Φ(X,F ).

Proof. The Godement resolution is, in particular, a flasque resolution, and aflasque 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−1F )

is surjective for every closed subset i : Y → X. More generally, if Φ is a family ofsupports on X, then F is called Φ-soft iff, for every closed embedding i : Y → Xwith Y ∈ Φ, the natural map F (X)→ Γ(Y, i−1F ) is surjective.

13. Filtered direct limits

The formation of direct limits of sheaves on a space X, which involves sheafifi-cation, can often be difficult to understand. However, for filtered direct limits, thesituation 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 com-pact or noetherian space. We use these results to prove that appropriate classes ofΓ-acyclic sheaves are also preserved under filtered direct limits on such spaces. Com-bining this with some general nonsense, we find that sheaf cohomology commuteswith 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 ofsheaves on X indexed by a filtered category C. If X is quasi-compact, then thenatural map

lim−→

Γ(X,Fc) → Γ(X, lim−→

Fc)(13.1)

is injective. If, furthermore, X is Hausdorff (i.e. compact), or noetherian, then itis 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 mapF pre → F .

The injectivity when X is quasi-compact is not hard: Recall from the usualconstruction 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 sameelement of Γ(X,F ) iff there is a cover Ui : i ∈ I of X, a function e : I → C, andC-morphisms fi : c→ e(i), gi : d→ e(i) for each each i ∈ I so that

F (fi)(s|Ui) = F (g)(t|Ui) ∈ Fe(i)(Ui).(13.2)

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 hifi = hjfj andhigi = hjgj for every i, j ∈ I. Call the respective common maps f : c → e andg : d→ e. From the equality (13.2) we see that

F (f)(s)|Ui = F (f)(s|Ui)= F (hifi)(s|Ui)= F (hi)(F (fi)(s|Ui))= F (hi)(F (gi)(t|Ui))= F (higi)(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 oneach pairwise intersection Uij = Ui ∩ Uj: For every x ∈ Uij, there are: an indexc(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

F (f)(si|W ) = F (g)(sj|W ) ∈ Fc(i,j,x)(W ).(13.3)

The reader can spell out the details of when two such collections of data representthe same section—we will implicitly replace one such data set with an equivalentone in what follows.

First, since X is compact, we can assume I is finite, hence, since C is filtered, wecan assume that c := c(i) is independent of i ∈ I. (Choose an index c so that thereare C morphisms fi : c(i)→ C for each i ∈ I and replace si by F (fi)(si).) Now, bya standard topological argument1 we can find a cover Vi : i ∈ I (same indexing setI!) of X such that V i ⊆ Ui for all i ∈ I. Now, using compactness of V ij and the factthat C is filtered, we can find an index c(i, j) ∈ C, a neighborhood W (i, j) of V ij inUij, 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)) canserve as c(i, j, x) (resp. f(i, j, x), g(i, j, x), W (i, j, x)) in the compatibility conditionfor 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 → dare independent of the pair (i, j). By the same kind of calculation we made in theinjectivity 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 thiscase, since an open subset of a noetherian space is quasi-compact, the intersectionsUij 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 com-patibility condition for each x ∈ Uij. The rest of the argument is identical (replaceU with V everywhere).

Corollary 13.2. Let X be a noetherian topological space, c 7→ Fc a filtereddirect 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 easyexercise 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 thisfollows from the previous Theorem.

Alternatively, one can just directly prove that the direct limit presheaf is a sheafby using Corollary 5.3 and the fact that a filtered direct limit of equalizer diagramsof sets is an equalizer diagram. (Filtered direct limits commute with finite inverselimits of sets.)

Corollary 13.3. A filtered direct limit of flasque sheaves on a noetherian topo-logical space is flasque. A filtered direct limit of soft sheaves on a compact topologicalspace is soft.

Proof. Since a filtered direct limit of surjective maps (of sets) is again sur-jective, the first statement is immediate from Corollary 13.2, and the second isimmediate from Theorem 13.1 (note Z is compact when X is compact).

Theorem 13.4. Sheaf cohomology commutes with filtered direct limits on a com-pact or noetherian space X. That is: Let c 7→ Fc be a filtered direct limit systemof sheaves of abelian groups on a compact or noetherian space X. Then the naturalmap

lim−→

Hp(X,Fc) → Hp(X, lim−→

Fc)(13.4)

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 sidesof (13.4) are natural in (c 7→ Fc) and may viewed as functors A → Ab. Thesefunctors agree when p = 0 by Theorem 13.1, so, by general nonsense, they will agreefor all p once we prove that both sides are effaceable δ-functors A → Ab. To seethat both sides are δ-functors, first note that the formation of filtered direct limitsof abelian groups

lim−→

: Hom(C,Ab) → Ab

is an exact functor (this is an easy exercise and is well-known). It follows easily fromthis 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 thatcohomology H• is a δ-functor.

It remains only to prove that both sides are effaceable. For any F ∈ Ab(X), wecan find an injection F → F ′ from F into a flasque sheaf, functorial in F (use thefirst step of the Godement resolution, for example), so we can find an injection fromany limit system (c 7→ Fc) into a limit system (c 7→ F ′

c) where each F ′c 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 afiltered direct limit of flasques is flasque (Corollary 13.3). In the compact case, theright side vanishes because a flasque sheaf is soft (Lemma ??), a direct limit of softsheaves is soft (Corollary 13.3), and a soft sheaf on a (para)compact space has nohigher cohomology.

14. Vanishing theorems

Recall that a closed subset Z of a topological space X is called irreducible iffit cannot be written as a union of two proper closed subsets. The dimension of atopological space X (denoted dimX) is the maximal length n of a strictly increasingchain

∅ 6= Z0 ( Z1 ( · · · ( Zn

of non-empty irreducible closed subsets of X. We set dimX = ∞ if X containsarbitrarily long such chains.

Theorem 14.1. (Grothendieck’s Vanishing) Let X be a noetherian topolog-ical space of dimension n, Φ a family of supports on X. Then Hp

Φ(X,F ) = 0 forevery 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 globalsection functor is exact.

Proof. It is enough to prove that, for any surjection f : F → G of sheaves onX, the map Γ(X,F ) → Γ(X,G ) is surjective. Let s ∈ Γ(X,G ). Since F → G issurjective, 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 theopen set

U i := Ui \ (∪i−1j=1Uj)

and restricting sections, we may even assume that the Ui are pairwise disjoint. Butthen the si automatically agree on overlaps, so by the sheaf property on F they liftto a section t ∈ Γ(X,F ) and f(X)(t) = s by the sheaf property on G , since this istrue on each Ui.

15. Higher direct images

Let f : X → Y be a morphism of topological spaces. The direct image functorf∗ : 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, wecan 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 nat-urally 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 toprove 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 bythe above formula are zero when p > 0 and F is injective. This is clear from thefact that F |f−1(U) is also injective in Ab(f−1(U)) by (??), hence has no highercohomology.

16. Leray spectral sequence

Theorem 16.1. (Leray) Let f : X → Y be a continuous map of topologicalspaces, F a sheaf on X. There is a first quadrant spectral sequence

Ep,q2 = 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 (whichare Γ(Y, ) acyclic by (10.4)), so this spectral sequence is a special case of theGrothendieck spectral sequence.

One consequence of the Leray spectral sequence is the following celebrated the-orem.

Corollary 16.2. (Vietoris-Begle) Let f : X → Y be a proper map withacyclic fibers:

Hp(f−1(y),Z) =

Z, p = 00, p > 0.

Then Hp(Y,F )→ Hp(X, f−1F ) 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 anyconstant sheaf. We will show that the Leray spectral sequence (??) degenerates by

48 IV. COHOMOLOGY

showing that f∗f−1F = F and Rp f∗f

−1F = 0 for p > 0. This follows from theformula

(Rp f∗ZX)y = Hp(f−1(y), (f−1F )|f−1(y))

of (15.1) and the fact that (f−1F )|f−1(y) is just the constant sheaf on f−1(y) asso-ciated to the abelian group Fy, as is clear from functoriality of inverse images andcommutativity of the cartesian diagram

f−1(y) //

X

f

y // Y.

17. Group quotients

18. Local systems

For an abelian group A, an A-local system or an A-locally constant sheaf on aspace X is a sheaf F ∈ Ab(X), which, locally on X, is isomorphic to the constantsheaf AX . The Leray spectral sequence is most easily understood when f : X → Yis a fiber bundle with fiber F (i.e. Y can be covered with open sets Ui so thatf−1(Ui) is isomorphic, over Ui, to Ui × F ). In this case, the sheaf Rp f∗ZX will bean Hp(F,Z)-local system on Y .

19. Examples

20. Exercises

The exercises below are intended to show that one cannot prove (1.7) withoutsome assumptions along the lines of (2) or (1).

Exercise 20.1. Construct an example where f : X → Y is a closed embeddingof 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 skewdiagonal (−x, x) : x ∈ R in the subspace topology it inherits from Y . Let F bethe constant sheaf associated to the two element set 0, 1. Show that X has thediscrete topology, hence

Γ(X, f−1F ) =∏x∈X

Fx

= 0, 1R.Show that the characteristic function of the irrational numbers f ∈ 0, 1R is not inthe 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 thefamily of supports Φ consisting of compact subsets of X is paracompactifying. Co-homology with support in Φ in this situation is usually called compactly supportedcohomology.

Exercise 20.4. Let A be an abelian group, X a topological space, F a subsheafof AX . Show that for any a ∈ A, the set

Ua := x ∈ X : a ∈ Fx ⊆ AX,x = Ais open inX. Now suppose that A is a ring and I is an ideal sheaf of AX (a subobjectof AX in the category of AX-modules). Show that, for a finitely generated ideal Iof A, the set

UI := x ∈ X : I ⊆ Ixis open in X.

Exercise 20.5. Let F be a subsheaf of ZX for a quasi-compact space X. Showthat there are only finitely many n ∈ Z>0 such that Un (notation from the previousexercise) is nonempty. For x ∈ X, let n(x) be the smallest non-negative integersuch 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 fors sufficiently large. Then Zs := Vs \ Vs−1 is locally closed subspace of X. := Showthat the sheaves Fi := (F |Vi)X define a finite filtration of F with Fs/Fs−1

∼= ZXZs.

Exercise 20.6. Let X be an infinite set with the cofinite topology, and let x ∈ Xbe a point. Show directly that HomX(x∗Z,ZX) = 0. Show that x!ZX = 0 (c.f. The-orem 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 topologicalspace X is the smallest family of subsets of X containing the open subsets, andclosed under finite intersections and complements (hence also containing the closedsubsets and closed under finite unions). If X is noetherian, show that a union oftwo locally closed subsets is locally closed, and hence the complement of a locallyclosed subset is locally closed. Conclude that the locally closed subsets coincidewith the constructible subsets. On the other hand, if X = R, show by example thatthe 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 flasquein general.

Exercise 20.9. Let i : U → X be an open set. Show that the sheaf ⊕hUZ onX 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 compactspace is locally closed iff it is locally compact. Let α be the smallest ordinal withinfinitely many limit ordinals less than it. Show that the space α+1 (with the ordertopology) embeds in R (as does any countable ordinal with its order topology), butis not locally compact. One can arrange that Fx 6= 0 if and only if x is in the imageof this embedding.

Exercise 20.11. If f : F → G is a surjective morphism of flasque sheaves thenthe induced maps fp : C p(F )→ C p(G ) are surjective for every p ≥ 0. Hint: Inducton 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 usingthe 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 everyB morphism V → U , but F is not flasque.

Exercise 20.15. Show that the Sorgenfrey line S is zero dimensional, but notsecond countable. Is there a sheaf of abelian groups on S (or S × S) with non-vanishing 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 thestudy of the “compatibility” of direct and inverse images with base change—thesubject of this section and the next.

Definition 1.1. Let X be a topological space, Y ⊆ X a subspace, Ai : i ∈ Ia family of subsets of X. We say that Y is relatively Hausdorff in X iff any twodistinct points of Y have disjoint neighborhoods in X. We say that Ai is locallyfinite 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 finiteopen refinement.

Definition 1.2. We say that a map f : X → Y of topological spaces is properiff 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 sayingthat the diagonal X ⊆ X ×Y X is closed (i.e. f is separated). It is easy to see thatthis condition is closed under base change. The property of being a closed map iscertainly not preserved preserved under base change. For example, π1 : R2 → Ris 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 andcomposition. Being proper is local on the base.

Proof. Everything is straightforward except perhaps the statement that thebase change π1 : Z ×Y X → Z of a proper map f : X → Y along an arbitrarymap g : Z → Y is a closed map. To see this, suppose W ⊆ Z ×Y X is closed, andz /∈ π1(W ). We want to see that z has a neighborhood disjoint from π1(W ), so thatπ1(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) disjointfrom W . Since f−1(g(z)) is compact, we only need finitely many such Ux ×Y Vx tocover π−1

1 (z), so we can intersect the finitely many Ux that we need to find a singleopen neighborhood U of z ∈ Z and a neighborhood V of f−1(g(z)) in X so thatU ×Y V is a neighborhood of π−1

1 (z) in Z ×Y X. The set T := Y \ f(X \ V ) isopen in Y since f is closed, and T contains g(z) because V contains f−1(g(z)), sog−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′ ∈ U ∩ g−1(T ), then g(z′) ∈ T , so f−1(g(z′)) ⊆ V , hence nopoint (z′, 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 ifff−1(K) is compact for every compact subspace K ⊆ Y . If f : X → Y and g : Y → Zare maps of locally compact Hausdorff spaces such that gf and g are proper, then fis proper.

Proof. (=⇒) Suppose Ui : i ∈ I is a family of open subsets of X coveringZ := f−1(K). Since f is proper, each fiber of f is compact, so for each x ∈ K, thereis a finite subset Ix ⊆ I such that Ux := ∪i∈IxUi covers f−1(x). Since the propermap f is closed, Vx := Y \ f(Z \ Ux) is an open neighborhood of x in Y , and it isclear that f−1(y) ⊆ Ux for every y ∈ Vx ∩ K. Since K is compact, we can find afinite subset S ⊆ K so that Vx : x ∈ S covers K. Then J := ∪x∈SIx is a finitesubset of I and such that the Uj cover Z.

For (⇐=), the only issue is to prove that f is closed. Suppose Z ⊆ X is closedand y /∈ f(Z), but y is in the closure of f(Z). Choose a neighborhood W of y in Ywith 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 alsocompact, hence closed. But this K is a closed subset of Y containing W ∩f(Z) withy /∈ 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 toprove that f−1(K) is compact for any compact K ⊆ Y . Certainly g(K) is compactin Z, so (gf)−1(g(K)) is compact because gf is proper. But f−1(K) is a closedsubspace of (gf)−1(g(K)), so it is also compact.

The following technical lemmas are the main topological input for the properbase change theorem:

Lemma 1.5. Let X be a topological space, Y ⊆ X a compact, relatively Hausdorffsubspace, Ui : i ∈ I a family of opens in X covering Y . Then there is a finitefamily Vj : j ∈ J of opens in X whose union V contains Y and a map φ : J → Isuch 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 Haus-dorff assumption, we can choose, for each y′ ∈ Y \Uφ(y), disjoint open neighborhoodsSy,y′ ⊆ Uφ(y) and Wy,y′ of y and y′ respectively. By compactness of Y , there is afinite subset Iy ⊆ Y \ Uφ(y) so that

Ty := Uφ(y) ∪(∪y′∈IyWy,y′

)is an open neighborhood of Y in X. Set Sy := ∩y∈IySy,y′ . Then it is easy to see

that (Sy ∩ Ty) ⊆ Uφ(y). Again using compactness of Y , find a finite J ⊆ Y so that∪j∈JSj contains Y . Set T := ∩jTj. Then it is easy to see that Vj := Sj ∩ T is asdesired.

1. PROPER MAPS 53

Lemma 1.6. Let X be a paracompact topological space, Ui : i ∈ I an opencover of X. Then there exists a locally finite open cover Vj : j ∈ J of X and amap 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 ∈ Kand 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 Sx ⊆ Uµ(x). Choosea locally finite refinement Vj : j ∈ J of the cover Sx : x ∈ X, so we have a mapν : 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. Thenthe natural map

lim−→U⊇Y

F (U) → Γ(Y,F |Y )

is monic (the direct limit is over open neighborhoods U of Y in X). Suppose fur-thermore 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.

Remark 1.8. The “natural map” is Γ of the sheafification morphism f−1preF →

f−1F .

Proof. To prove the first statement, suppose U, V ⊇ Y are open neighborhoodsand 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∈YUy is a neighborhood of y in X on which s and t agree, hence theyare equal in the (filtered) direct limit.

For the second statement it remains only to prove surjectivity. By definition ofthe inverse image F |Y , a section s ∈ Γ(Y,F |Y ) can be viewed as an element

s = (s(y)) ∈∏y∈Y

Fy

so that there is a family of opens Ui : i ∈ I in X with Y ⊆ U := ∪Ui and sectionssi ∈ 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 undereither assumption, we can assume that Ui is a locally finite cover of U . Whenthe second assumption holds, we can (and will) assume without loss of generalitythat 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) sothat ty = s(y) for all y ∈ Y .

I next claim that we can further assume that there is a locally finite open coverVj : j ∈ J of U and a map φ : J → I with (V j ∩U) ⊆ Uφ(j). To see this under thefirst 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 isfinite in this case.) To see this under the second assumption, apply Lemma 1.6 tothe cover of X given by the Ui and X \ Y .

Now let Tx : x ∈ U be neighborhoods in U witnessing the local finiteness ofUi and Vj, so the sets

I(x) := i ∈ I : Tx ∩ Ui 6= ∅I ′(x) := i ∈ I : x ∈ UiJ(x) := j ∈ J : x ∈ Vj

are all finite and φ(J(x)) ⊆ I ′(x) ⊆ I(x). For any x ∈ U , the sections si : i ∈ I ′(x)all have the same stalk (namely s(x)) at x, so, since I ′(x) is finite, we can find aneighborhood

W ′′x ⊆

(∩i∈I′(x)Ui

)∩ Tx

of x in U so that the sections si : i ∈ I ′(x) all agree on W ′′x . Call this common

section t′(x) ∈ F (W ′′x ). Set

W ′x := W ′′

x ∩(∩j∈J(x)Vj

)Wx := W ′

x ∩(∩j∈J(x):φ(j)∈I(x)\I′(x)(X \ V j)

)= W ′

x \(∪j∈J(x):φ(j)∈I(x)\I′(x)V j

)so that Wx is a neighborhood of x in U because

φ(j) ∈ I(x) \ I ′(x) =⇒ x /∈ Uφ(j)

=⇒ x /∈ V j.

Set t(x) := t′(x)|Wx.

I claim that t(x1)|(Wx1 ∩Wx2) = t(x2)|(Wx1 ∩Wx2) for all x1, x2 ∈ U so that thet(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 , inparticular u ∈ Tx1 . Since u ∈ Wx2 there is j2 ∈ J(x2) so that u ∈ Vj2 . Then φ(j2) isin I(x1) because

u ∈ Tx1 ∩ Vj2 ⊆ Tx1 ∩ Uφ(j2),

but φ(j2) must actually be in I ′(x1) ⊆ I(x1), because otherwise u wouldn’t be in

Wx1 since it is in V j2 . Since φ(j2) ∈ I ′(x1) ∩ I ′(x2), we have t(x1)y = sφ(j2)y = 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

X ′

f ′

g // X

f

Y ′h // Y

of topological spaces. For any sheaf F on X, there is a natural map

h−1f∗F → (f ′)∗g−1F(2.1)

of sheaves on Y ′, defined as follows. First of all, by the adjointness (h−1, h∗), it isequivalent to give a map

f∗F → h∗(f′)∗g

−1F(2.2)

of sheaves on Y . For any open set U ⊆ Y , we have (f∗F )(U) = F (f−1U) and

(h∗(f′)∗g

−1preF )(U) = (g−1

preF )((f ′)−1h−1U)

= lim−→

V⊇g((f ′)−1h−1(U))

F (V ).

But commutativity implies

g((f ′)−1h−1U) = g(g−1f−1U)

⊆ f−1(U),

so f−1(U) is one of the V ’s in the direct limit system, hence we have a natural map

(f∗F )(U) → (h∗(f′)∗g

−1preF )(U)

given by the structure map to the direct limit. This is clearly natural in U since therestriction maps are induced by the restriction maps for F on both sides, hence weobtain a map f∗F → h∗(f

′)∗g−1preF . Composing with h∗(f

′)∗ of the sheafification

map g−1preF → g−1F yields the map (2.2).

There are variants for pushforward with proper support, as follows: If the dia-gram is cartesian, then there is a natural map

h−1f!F → (f ′)!g−1F(2.3)

defined in the same way. The key point is that, if s ∈ F (U) has proper support Zover Y , then g−1s ∈ F (g−1(U)) has proper support over Y ′ because

Supp(g−1s) = g−1 Supp s = g−1Z,

56 V. BASE CHANGE

and we have a cartesian diagram

g−1Z //

Z

X ′ //

X

Y ′ // Y

where Z → Y is proper, hence g−1Z → Y ′ is also proper (Lemma 1.3).

One also has the map (2.3) when the spaces X,X ′, Y, Y ′ in the diagram arelocally compact (and Hausdorff) and g and h are proper. In this case the point isthat f ′|g−1Z : g−1Z → Y ′ is proper when f |Z : Z → Y is proper. To see this,first note that g|g−1Z → Z is proper (since it is a base change of g), hence thecomposition (f |Z)(g|g−1Z) is proper (Lemma 1.3) and we have

h(f ′|g−1Z) = (f |Z)(g|g−1Z)

by commutativity, hence f ′|g−1Z is proper by the 2-out-of-3 property of proper mapsfrom Lemma 1.4.

An important special case of the map (2.5) occurs when h : Y ′ → Y is theinclusion of a point y ∈ Y and the diagram is cartesian so that X ′ = Xy is the fiberof f over y ∈ Y . In this case, h−1 is the stalk functor F 7→ Fy and f ′ is the functorof 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 (Theo-rem ??), but it actually turns out that the more general base change theorem reducesto 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−→U

F (f−1(U)) → Γ(Xy,F |Xy),

where the direct limit runs over all neighborhoods U of y in Y . But the fact that fis closed implies that the f−1(U) are cofinal in the set of all neighborhoods of thefiber Xy, so the map in question is the same as the map

lim−→V

F (V ) → Γ(Xy,F |Xy),

where V runs over all neighborhoods of Xy in X. Since the fiber Xy is compact andrelatively 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

X ′

f ′

g // X

f

Y ′h // Y

is a cartesian diagram of topological spaces with f proper. Then the natural map

h−1f∗F → (f ′)∗g−1F(2.5)

is an isomorphism for every sheaf F on X.

Proof. We can check this on stalks at any y′ ∈ Y ′. But f ′ is also proper(Lemma 1.3) and f and f ′ have the same fibers, so we reduce formally to thesituation of the previous theorem.

3. Homotopy invariance

Let I := [0, 1] denote the unit interval. For a continuous map H : X×I → Y andt ∈ 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 mapH : X × I → Y such that H0 = f and H1 = g. Being homotopic is an equivalencerelation 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. Forany 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 induceisomorphisms on cohomology by the “2-out-of-3” property of isomorphisms, the factthat π1ιi = Id : X → X, and the fact that π1 : X × I → X induces isomorphismson all constant sheaf cohomology by Vietoris-Begle (use the fact that the interval isacyclic—see the computations in (??)). The result is now immediate since Hι0 = fand 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 topolog-ical 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 cannotbe the case:

Example 3.2. Let S1 ⊆ R2 be the unit circle and let f : S1 → I be the pro-jection from S1 onto the interval I = [−1, 1] so that f is a double cover branchedover ±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 cohomol-ogy on a space with at most two points. The Leray spectral sequence for f thereforeyields an isomorphism

H1(S1,Z) = H1(I, f∗ZS1).

Note that this group is Z by the computations in (??). To get a feel for the sheafF := f∗ZS1 , note that

F ([−1, 1)) = ZF ((−1, 1]) = ZF ((−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 leftadjoint f−1 : Ab(Y ) → Ab(X) to f∗ : Ab(X) → Ab(Y ). It would also bedesirable, 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 adjointf ! to f∗ would imply that f∗ commutes with all direct limits, which it generally doesnot (even when Y is a point). In the general case then, one is forced to look for theright adjoint f ! only in an appropriate derived category sense. However, in the casewhere f is a closed embedding (so f∗ does preserve direct limits) we will constructthe desired right adjoint f ! in this section by elementary means.

In fact, it is natural to work not only with closed embeddings, but with immer-sions. Recall that f : X → Y is an immersion iff f can be factored f = ji as anopen embedding i followed by a closed embedding j. In other words, f is an immer-sion iff it is an isomorphism onto a subset A ⊆ X which can be written A = Z ∩ Uwhere 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 neighborhoodUx of x in X such that A ∩ Ux is closed in Ux.

Proof. The implication =⇒ is clear. If A satisfies the second condition, thenA will be closed in U := ∪x∈AUx because being closed is local, hence A = U ∩Z forsome 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 thelemma and discussion above.

Proposition 1.2. An open embedding is an immersion, and a closed embeddingis an immersion. Immersions are the smallest class of morphisms containing openembeddings and closed embeddings and closed under composition. Immersions areclosed under base change, and the property of being an immersion is local on thecodomain (base).

59

60 VI. DUALITY

In a noetherian topological space, locally closed subsets are the same as con-structible 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 fullyfaithful.

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 anyU ∈ Ouv(X), the fact that the f(U) respect the restriction maps implies thatthe 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 restrictionmaps for the inclusions (V ∩ U) → V and (V ∩ U) → U are the identity forboth presheaves j∗F , j∗G . In other words, for U ∈ Ouv(Z), we can define amap f(U) : F (U) → G (U) unambiguously by choosing any U ′ ∈ Ouv(X) withU = U ′∩Z and setting f(U) := f(U ′). The maps f(U) clearly respect the restrictionmaps since this is true for the maps f(U ′) and any inclusion of opens in Z lifts toan 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 directand inverse limits and is fully faithful. Its essential image consists of thoseF ∈ 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 ∩ Uon X (with the obvious restriction maps inherited from those of F ). Thefunctor j! : Ab(X) → Ab(Z) is right adjoint to j∗ and the adjunctionmorphism 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 satisfiesSupp s ⊆ Z ∩ U since the stalks of j∗F at any x ∈ X \ Z are zero, so the finalstatement 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

−1G ⊆ G , hence we have

HomAb(X)(j∗F ,G ) = HomAb(X)(j∗F , j∗j−1G ).

But by Lemma 1.3, j∗ is fully faithful, so

HomAb(X)(j∗F , j∗j−1G ) = HomAb(Z)(F , f−1G )

2. EXTENSION BY ZERO 61

and putting the two together yields the desired adjunction. The functor j∗ commuteswith 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. Fora F ∈ Ab(U), let i!F be the sheaf associated to the presheaf ipre

! F defined by

V 7→

F (V ), V ⊆ U0, otherwise.

The functor i! : Ab(U) → Ab(X) is left adjoint to i−1 (i.e. the restriction functorF 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 adjoint-ness property of sheafification.

Remark 1.6. Unravelling the local coherence condition in the sheafificationconstruction (7), we see that the sheaf i!F can be explicitly described as follows: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(x′) = 0for all x′ ∈ W ∩ U .

(2) For every x ∈ V ∩U , there is a neighborhood W of x in V ∩U and a sectiont ∈ F (W ) such that tx′ = s(x′) for x′ ∈ W .

2. Extension by zero

Extension by zero will allow us to construct a kind of “wrong way” map whichis the basis for Poincare 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

ipre! F : V 7→

F (V ), V ⊆ U0, 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 closedin V , so there is a natural morphism ipre

! F → i!F . To see that this induces anisomorphism (ipre

! F )+ → i!F , it suffices to check that it is an isomorphism onstalks. Certainly if x ∈ U ⊆ X, then (ipre

! F )x → (i!F )x is an isomorphism becauseboth sheaves coincide on all sufficiently small neighborhoods of x (namely thosecontained in U). It remains only to show that (i!F )x = 0 for x ∈ X \ U . Indeed, ifV is any neighborhood of x and s ∈ (i!F )(V ) any section, then s restricts to zeroon 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 setι : U → X.

Proof. If 0→ F ′ → F → F ′′ → 0 is exact on U , then one easily checks that

0→ ι!F′ → ι!F → ι!F

′′ → 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 denotedFX). 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 closedinterval [0, 1] and F = AU is a constant sheaf on U , then i∗F is just the constantsheaf AX , which has stalk A at every point of X, while the stalk of i!F is zero at0, 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

−1F → j!j−1F . 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 itis still closed in W and this is a well-defined map

(i!i−1F )(W )→ (j!j

−1F )(W )

which clearly respects restriction.

Theorem 2.3. (Mayer-Vietoris Sequences) Let i : U → X be the inclusionof an open set, j : Z → X its closed complement. There is a short exact sequence

0→ i!i−1F → F → j∗j

−1F → 0

in Ab(X) natural in F ∈ Ab(X). Let k : V → X be another open set and letl : U ∪ V → X and m : U ∩ V → X be the inclusions. There is a short exactsequence

0→ m!m−1F → i!i

−1F ⊕ k!k−1F → l!l

−1F → 0.

Proof. Check exactness on stalks by using the previous lemma.

Theorem 2.4. (Duality for an open embedding) Let i : U → X be theinclusion 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−1i! = Id.The functor i−1 : Ab(X)→ Ab(U) takes injectives to injectives.

Proof. Checking the adjointness is a straightforward exercise and the equalityi−1i! = Id is clear from the definitions. The final statement then follows formally:Suppose F ′ ∈ Ab(X) is injective and

0→ ι−1F ′ → F → F ′′ → 0

3. PROPER MAPS - SECOND TREATMENT 63

is an exact sequence in Ab(U). Applying ι! and (2.2), we obtain a diagram of solidarrows with exact row

0 // ι!ι−1F ′ _

// ι!F

szz

// ι!F ′′ // 0

F ′

in Ab(X), which completes as indicated by injectivity of F ′ ∈ Ab(X). Applying thefunctor ι−1 and using i−1i! = Id, the vertical arrow above becomes an isomorphismand ι−1s 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 somerather technical lemmas. It turns out that the stalk formula behaves very well forproper maps, which we now introduce.

Definition 3.1. A subspace Z ⊆ X is called relative Hausdorff is any two pointsof Z have disjoint neighborhoods in X. A map of topological spaces f : X → Y iscalled proper if f is closed (the image of any closed set is closed) and the fibers off 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 propertiesof proper maps are summed up in the following theorem.

Theorem 3.2. The composition of proper maps is proper. Proper maps arepreserved 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. Iff : X → Y is proper and y ∈ Y , then the neighborhoods f−1(U) of f−1(y) (as Uranges over all neighborhoods of y ∈ Y ) are cofinal in the set of neighborhoods off−1(y) in X.

Proof. We first prove that proper maps are closed under base change. Letg : Z → Y be arbitrary, and let Z ×Y X be the fibered product. To check thatπ1 : Z×Y X → Z is closed, it is enough to check that π1(A×Y B) is closed wheneverA ⊆ Z and B ⊆ X are closed, since closed sets of this form form a basis for allclosed sets. On the other hand, we have π1(A ×Y B) = A ∩ g−1(f [B]), which is

closed when f is closed. Similarly, π(1z) = 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 Uand 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 thatbeing closed is local; the “compact relative Hausdorff fibers” condition is clearlylocal on the codomain.

64 VI. DUALITY

The last statement is immediate from the fact that f is closed: if V is an openneighborhood of f−1(y) in X, then f [X \ V ] is closed in X since f is closed, soY \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 compactiff 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 compactHausdorff spaces such that gf is proper, then f is proper. It follows that any mapbetween compact Hausdorff spaces is proper.

Proof. By the lemma, it suffices to prove that f−1(K) is compact for anycompact K ⊆ Y . Certainly g[K] is compact in Z, so (gf)−1(g[K]) is compactbecause gf is proper. But f−1(K) is a closed subspace of (gf)−1(g[K]), so it is alsocompact.

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 theassumption (1) and use (3.2) to note that the f−1(U) are cofinal in the neighborhoodsof f−1(y).

CHAPTER VII

Finite Spaces

This section is devoted to sheaf theory on finite, typically non-Hausdorff, topolog-ical spaces. We begin with some preliminaries on the topology of a finite topologicalspace X.

1. Supports

For any family of supports Φ on X, there is a closed subspace Y such that Φconsists of all closed subsets of Y . Indeed, just take Y to be the union of every closedset in Φ: then Y is in Φ because Φ is closed under finite unions and every closedsubset of Y is in Φ because Φ is closed under passage to smaller closed subsets. Asusual, 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 ≤ yiff x ∈ y−. We are only interesting in making statements about Sh(X), so therewould 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 par-ticularly 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 ofs for some s ∈ S. A subset of X is open iff it is closed under passage to largerelements.

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 isopen in Z1 and Z2 := Z1 \ U2 is hence closed in X. Continuing in this manner, weobtain 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 ofX 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 subspacesin 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 irreducibleclosed subspaces ∅ ( x0− ( · · · ( xn−.

3. Coniveau

Let X be any ringed space, not necessarily finite, and let F ∈Mod(X). For anyspecialization x ≤ y, we have a natural map fx,y : Fx → Fy which is OX,x → OY,ylinear. It is nothing but the direct limit of the maps F (U)→ Fy over neighborhoodsU of x, all of which contain y since x ∈ y−. We view fx,y as an OX,x linear mapby 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,zfx,y,

so we may in fact view the stalk functor as a functor

Ab(X)→ HomCat((X,≤),Ab).

The coniveau complex is the complex

0→ F →∏x0

x0∗Fx0 →∏x0<x1

x0∗Fx1 →∏

x0<x1<x2

x0∗Fx2 → · · ·

in Mod(X), where the products are over strictly increasing chains in (X,≤) andwhere the boundary maps are given by

(dns)(x0, . . . , xn+1) := x1∗fx1,xn+1s(x1, . . . , xn+1)

+n∑i=1

(−1)ix0∗fx0,xn+1s(x0, . . . , xi, . . . , xn+1)

+(−1)n+1x0∗fx0,xns(x0, . . . , xn).

By abuse of notation, we will often simply write

(dns)(x0, . . . , xn+1) :=n∑i=1

(−1)is(x0, . . . , xi, . . . , xn+1)|xn+1 .

Observe that this complex “depends on X” even when Sh(X) does not. Forexample, the coniveau complex computed on X is not the same as the coniveaucomplex computed on its sobrification Xsob even though Sh(X) = Sh(Xsob). It isgenerally more natural to work with the coniveau complex computed on Xsob, wherethe 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 ≤ yiff Uy ⊆ Ux. When x ≤ y, the restriction map F (Ux)→ F (Uy) may be viewed as amap fx,y : Fx → Fy (we will usually denote this s 7→ s|y). The restriction maps fx,ycompletely determine the sheaf F , as we will now show. Note that U = ∪x∈UUx forany U ∈ Ouv(X) and

Ux ∩ Uy =⋃z≥x,y

Uz,

so, by using the sheaf property for F , we can recover F (U) as the equalizer

F (U) = lim←−

∏x∈U

Fx ⇒∏

(x,y)∈U×U

∏z≥x,y

Fz

,(4.1)

where the two maps are (sx) 7→ (fx,z(sx)) and (sx) 7→ (fy,z(sx)). There is anequivalence 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 factsheaf) whose sections are defined by the formula (4.1).

5. Stalk formulas

The formulae

(x∗A)(y) =

A, y ≤ x0, otherwise

(ix!A)y =

A, x ≤ y0, otherwise

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 gxfx,y ifx ≤ y (or zero otherwise).

68 VII. FINITE SPACES

There is a kind of duality for finite spaces obtained by inverting the specializationordering (replacing (X,≤) with the opposite category). This exchanges x∗ andix!( ), etc.

Lemma 6.1. Let X be a finite ringed space, x a point of X. If I is an injectiveOX,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 ex-actness 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

0→ F →⊕x0

x0∗Fx0 →⊕x0<x1

x0∗Fx1 →⊕

x0<···<xn

x0∗Fxn → 0

by finite direct sums/products of skyscraper sheaves. There is a “dual” resolution bydirect 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 con-tractible complex. The key point is that we can commute the finite direct productsin the definition of the coniveau complex with stalks (finite inverse limits commutewith filtered direct limits). Therefore, by (3), this stalk complex is given by

0→ Fz →⊕z≤x0

Fx0 →⊕

z≤x0<x1

Fx1 → · · · →⊕

z≤x0<···<xn

Fxn → 0.

Define

cn :⊕

z≤x0<···<xn<xn+1

Fxn+1 →⊕

z≤x0<···<xn

Fxn

by

(cns)(x0, . . . , xn) :=

s(z, x0, . . . , xn), z < x0

0, otherwise

with the convention that c−1 : ⊕xFx → Fz sends s to s(z). One can check that

dnz cn + cn+1dn+1z = Id .

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. Poincare duality

We now prove an instance of the Poincare Duality theorem. Let X be a finitetopological 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 derivedcategory 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 resolutionof 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

Homk(⊕

x0<···<xn

Fxn , V )→ · · · → Homk(⊕x0

Fx0 , V ),

sitting in the interval [−n, 0]. On the other hand, by the Hom formulae (6), this isnaturally isomorphic to the complex⊕

x0<···<xn

Homk(F , xn∗V )→ · · · →⊕x0

Homk(F , x0∗V ),

which is nothing but the complex

Hom•k(F , ωX ⊗ V ),

where ωX is the complex ⊕x0<···<xn

xn∗k → · · · →⊕x0

x0∗k

(sitting in the interval [−n, 0]). Furthermore, ωX ⊗ V is a complex of injective kmodules, so this is also

R Hom•k(F , ωX ⊗ V ).

This discussion is easily extended to complexes F •, V • by applying the coniveau res-olution termwise, and making the same arguments with the resulting double/triplecomplexes.

8. Examples

In this section we work out some sheaf cohomology groups on finite topologicalspaces. Finite (non-Hausdorff) topological spaces are a rich source of examples inthe theory of sheaves.

Our main examples will involve the following spaces. The Sierpinski space Sis the unique topology on 1, 2 where 1 is closed, but 2 is not. The spaceX = 1A, 2, 1B has three points with basic open sets

U1A := 1A, 2U2 := 2U1B := 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, 1BU0B := 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 setF (∅) = 0) because S has only two interesting open sets S, 2 and neither of thesecan be covered nontrivially. A sheaf F ∈ Ab(S) is nothing but the data of therestriction map F (S) → F (2), which is also the data of the specialization mapF1 → F2 (note 0 < 1 is the specialization ordering of S), so Ab(S) is nothing butthe arrow category Ab•→• of Ab. The global section functor is identified with thestalk functor F 7→ F1, hence it is exact. However, sections Γ1 with support in theclosed 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 exactsequences in Ab, and the sequence of kernels of such a morphism need not form anexact 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 // F1//

F1 ⊕F2

π2

// F2

// 0

0 // F2// F2

// 0 // 0,

that

Hp1(S,F ) =

Ker(F1 → F2), p = 0Cok(F1 → F2), p = 10, p > 1

Notice that, furthermore, the forgetful functor Ab(S) → PAb(S) is exact, soΓ1 is not even exact as a functor Γ1 : PAb(X)→ Ab. Even worse, there is no hopeof “making it exact” (computing its derived functors) by using Cech cohomology.Indeed, the trivial cover S refines every other cover, and, since we may computewith alternating Cech cohomology, the Cech cohomology groups in this case aresimply

Hp

1(S,F ) =

Γ1(S,F ), p = 00, 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

0A

''

0B

ww1A

1B

~~2

and

Y

!!U0A

!!

U0B

X

!!U1A

!!

U1B

U2

∅

respectively. The sheafification functor

PAb(Y ) → Ab(Y )

is given by:

F (Y )

##F (U0A)

##

F (U0B)

F (X)

##F (U1A)

##

F (U1B)

F (U2)

F (∅)

7→

F (U0A)×F+(X) F (U0B)

ww ''F (U1A)

''

F (U1B)

wwF (U0A)×F (U2) F (U0B)

ww ''F (U1A)

''

F (U1B)

wwF (U2)

0

Note the “double fiber product” formula for global sections F +(Y ). The fact thatsheafification preserves stalks is manifest in the fact that F (U)→ F +(U) is alwaysan isomorphism for

U ∈ U0A, U0B, U1A, U1B, U2as 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 canuse ordered Cech cohomology with U to compute the absolute Cech cohomology ofany F ∈ PAb(Y ). The ordered Cech complex C(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 thepresheaf

Z1

1

Z

(1,0)

Z

(1,1)Z⊕ Z

(1,0)

(1,0)

Z

1

Z

1Z

0

this Cech complex is (1 −10 −1

): Z2 → Z2,

which has no homology at all. On the other hand, F + (or even F sep) is just theconstant sheaf ZY and the sheafification map F → F + is an isomorphism on everyopen set except that F (X)→ F +(X) is given by (1, 0) : Z⊕Z. The monomorphism

H0(Y,F ) → Γ(Y,F +)

of Lemma 8.4 is simply 0 → Z. The “reason” this is not surjective is because thetwo sections 1 ∈ F (U0A) and 1 ∈ F (U0B) “should” glue to form a global sectionsince their difference in F (X) is zero when restricted to F (U1A) and F (U1B) andU1A and U1B cover X.

Higher Cech cohomology of sheaves on Y is also poorly behaved. We will nowconstruct a sheaf F ′ ∈ Ab(Y ) where the natural monomorphism

H2(Y,F ′)→ H2(Y,F ′)

of Theorem 5.1 is given by 0 → Z. Of course, we already know H2(Y,F ′) vanishes

for any F ′ ∈ PAb(Y ′) as the Cech complex is supported in degrees 0, 1, so wesimply seek an F ′ ∈ Ab(Y ) with nonvanishing H2. Given that Y has noetheriandimension two, it is not terribly surprising that we can find such a sheaf.

8. EXAMPLES 73

Our sheaf F ′ is part of the short exact sequence

0→ F ′ → F → F ′′ → 0(8.1)

in Ab(Y ) given below.

0

Z=

=

Z=

=

##0

0

Z

=

Z

=

Z

∆ ""

Z

∆||0

// Z=

=

// Z⊕ Zπ1

||

π2

""0

0

Z

=

Z

=

Z

##

Z

Z Z 0

Note that exactness in Ab(Y ) can be checked on stalks, which is to say it canbe checked on U0A, U0B, U1A, U1B, and U2. To compute the cohomology groups ofF ′′, we consider the SES of sheaves

0→ F ′′ → G → G ′′ → 0(8.2)

given by the diagram below.

Z=

~~

=

Z⊕ Z=

=

##

Z⊕ Zπ1

~~

π2

Z

∆

Z

∆~~

Z⊕ Z

= ##

Z⊕ Z

=

Z

Z

~~Z⊕ Z

π1

~~

π2

// Z⊕ Zπ1

π2

##

// 0

~~ Z

Z

~~

Z

##

Z

0

0

~~0 0 0

Here the maps on global sections

F ′′(Y ) = Z→ G (Y ) = Z⊕ Z→ G ′′(Y ) = Z⊕ Z

74 VII. FINITE SPACES

are given by ∆ and ∆π2, respectively. The sheaves G and G ′′ are flasque, hencehave no higher cohomology, so by examining the LES in cohomology, we compute:

Hp(Y,F ′′) =

Z p = 0, 10 p > 1

The sheaf F in the appearing in the sequence (8.1) is also flasque, so we can nowtake the LES in cohomology there to show that

Hp(X ′,F ′) =

Z p = 20 otherwise

as claimed.

Note that the exact sequence

0→ F ′ → F → G → G ′′ → 0

obtained by concatenating the exact sequences (8.1) and (8.2) (which we have usedto work out the cohomology of F ′) is nothing but the coniveau resolution of F ′

constructed in (6.2).

9. Exercises

Exercise 9.1. Let Y ⊆ X be a closed subspace of a finite space X. Write downa formula for ΓY (X,F ) in the langauge of (4) (i.e. making reference only to thestalks of F and the maps between them arising from specializations). Specializeyour formula to the case where F is a skyscraper.

Exercise 9.2. Extend the above Poincare Duality to cohomology with supportin a closed subspace Y ⊆ X (c.f. (1)).

Exercise 9.3. A ringed space (X,OX) is specialization flat if the natural ringmap OX,x → OX,y is flat for any specialization x ≤ y. Show that every schemeis specialization flat, but show that every ring map arises in this manner for someappropriate (X,OX).

CHAPTER VIII

Cech Theory

Cech cohomology is named for the Cech mathematician Eduard Cech (1893-1960), after whom the Stone-Cech compactification is also named.

1. Cech resolution

Let X be a topological space, F ∈ PAb(X). It will be important to developa theory of Cech cohomology for presheaves as well as sheaves. For an open seti : U → X, we set FU = i∗i

−1preF . Note that FU is given by

V 7→ F (U ∩ V ),

and that this is a sheaf if F is a sheaf (i−1pre = i−1 in this situation). Furthermore,

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 presheaveson X induced by the restriction maps of F . Note that the stalk of this map is anisomorphism 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) ∈ Ip+1, we use the following notation:

U(i0, . . . , ip) := Ui0 ∩ · · · ∩ UipF (i0, . . . , ip) := FU(i0,...,ip).

The convention

F () := F(1.1)

is convenient. For j ∈ 0, . . . , p the natural inclusion U(i0, . . . , ip)→ U(i0, . . . , ij, . . . , ip)of open sets induces a PAb(X) morphism

F (i0, . . . , ij, . . . , ip) → F (i0, . . . , ip)

s 7→ s|U(i0,...,ip).

For each nonnegative integer p, define

C p(U ,F ) :=∏

(i0,...,ip)∈Ip+1

F (i0, i1, . . . , ip).

75

76 VIII. CECH THEORY

The presheaves C p(U ,F ) ∈ PAb(X) can be assembled into a complex C •(U ,F )whose boundary maps

dp : C p(U ,F ) → C p+1(U ,F )

are given by

(dps)(i0, . . . , ip+1) :=

p+1∑j=0

(−1)js(i0, . . . , ij, . . . , ip+1)|U(i0,...,ip+1).(1.2)

There is a natural augmentation morphism

d−1 : F → C 0(U ,F )

given by setting s(i0) := s|U(i0) (the usual formula using convention (1.1)). The

abelian presheaves C p(U ,F ) are called the Cech presheaves associated to the presheafF and the covering sieve U .

Theorem 1.1. If U is a cover of X, then d−1 : F → C •(U ,F ) is a resolutionfor any F ∈ Ab(X).

Proof. We have already noted that, since F is a sheaf, each F (i0, . . . , ip) is

a sheaf, and hence C p(U ,F ) (being a product of such sheaves) is also a sheaf. Ittherefore suffices to check exactness of the augmented complex F → C •(U ,F )when restricted to each of the open sets Ui, since the Ui form a cover.1 In fact, wewill show that

(F → C •(U ,F ))|Ui

is null-homotopic. Define a contraction map

cp :∏i0,...,ip

F (i0, . . . , ip)|Ui→

∏i0,...,ip−1

F (i0, . . . , ip−1)|Ui

by

(cps)(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 : C 0(U ,F )|Ui→ F |Ui

simply sends s to s(i).

Now we just check that the cp form a retraction for C •(U ,F )x. That is,

dp−1cp + cp+1dp = 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

(dp−1cps)(i0, . . . , ip) =

p−1∑j=0

(cps)(i0, . . . , ij, . . . , ip)

=

p−1∑j=0

(−1)js(i, i0, . . . , ij, . . . , ip)

and

(cp+1dps)(i0, . . . , ip) = (dps)(i, i0, . . . , ip)

= s(i0, . . . , ip) +

p−1∑j=0

(−1)j+1s(i, i0, . . . , ij, . . . , ip).

The resolution of the theorem is called the Cech resolution of F ∈ Ab(X).

2. Refinement maps

Suppose that U = Ui : i ∈ I and V = Vj : j ∈ J are two families of opensets in X. Suppose r : J → I is a refinement in the sense that Vj ⊆ Ur(j) for allj ∈ J . Then we have a natural restriction map FU(r(j)) → FVj for every j, and infact a map of complexes

r∗ : C •(U ,F ) → C •(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

r∗1, r∗2 : C •(U ,F ) → C •(V ,F )

are homotopic.

Proof. The maps

hp : C p(U ,F ) → C p−1(V ,F )

given by

(hps)(j0, . . . , jp−1) :=

p−1∑k=0

(−1)ks(r1(j0), . . . , r1(jk), r2(jk), . . . , rp−1(jk))

provide a homotopy from r∗1 to r∗2.

78 VIII. CECH THEORY

3. Alternating variant

The alernating (or ordered) Cech complex is a “smaller” version of the Cechcomplex which is convenient for computations, and has theoretical implications (re-lations between covering dimension and cohomological dimension). Let Ui : i ∈ Ibe a family of open sets in X. Fix a total ordering ≤ of I. The sheaves

C p(U , <,F ) :=∏

i0<···<ip F (i0, . . . , ip)

form a complex C •(U , <,F ) whose boundary map is defined by the same formula(1.2) as the boundary map for C •(U ,F ). There is an obvious projection map

C •(U ,F ) → C •(U , <,F )(3.1)

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 Cech complex certainly provides aresolution of any F ∈ Ab(X). This can be proved directly (without the abovelemma) just as in the proof of (1.1) using the contraction operator

(cps)(i0, . . . , ip−1) :=

s(i, i0, . . . , ip−1), i < i00, otherwise.

4. Cech cohomology for a fixed cover

Finally we define Cech cohomology. Let U = Ui : i ∈ I be a cover of X. LetF ∈ PAb(X), let C•Φ(U ,F ) be the complex in Ab(X) obtained by applying ΓΦ tothe Cech complex C •(U ,F ).

Warning. The obvious inclusion

CpΦ(U ,F ) ⊆

∏(i0,...,ip)∈Ip+1

ΓΦ∩U(i0,...,ip)(U(i0, . . . , ip),F )

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, Cp

Φ(U ,F ) isthe subcomplex of Cp(U ,F ) consisting of those

s = (s(i0, . . . , ip)) ∈∏

(i0,...,ip)∈Ip+1

F (U(i0, . . . , ip))

such that there is a Y ∈ Φ such that

Supp s(i0, . . . , ip) ⊆ Y

for all (i0, . . . , ip) ∈ Ip+1.

The abelian groups

Hn

Φ(U ,F ) := Hn(C•Φ(U ,F ))

6. CECH COHOMOLOGY 79

are called the Cech cohomology2 groups of F with respect to the cover U .

5. Basic properties

We now prove some of the basic properties of the Cech cohomology groupsHn

Φ(U ,F ).

Theorem 5.1. For any F ∈ Ab(X), there is a first quadrant spectral sequence

Ep,q2 = Hp(Hq

Φ(C (U ,F )•)) =⇒ Hp+qΦ (X,F ).

In particular, there are natural maps Ep,02 = H

p

Φ(U ,F )→ HpΦ(X,F ) for every p.

Proof. Since the Cech complex is a resolution by (1.1), this is just a specialcase of the spectral sequence associated to a resolution of an object in an abeliancategory.

When each of the sheaves C p(U ,F ) is ΓΦ-acyclic, we say that U is a good coverfor 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

Hp

Φ(U ,F ) → HpΦ(X,F )

is an isomorphism for every p.

Proof. The spectral sequence of (5.1) degenerates.

6. Cech cohomology

In this section we introduce the (absolute) Cech cohomology groups, which haveno dependence on a choice of cover.

Set

Hp

Φ(X,F ) := lim−→U

Hp

Φ(U ,F ),

where the direct limit is over the the category of covers U = Ui : i ∈ I where thereis a unique morphism from U → V = Vj : j ∈ J iff there is a refinement mapr : J → I. The restriction maps are defined by taking the maps Hp(r∗) induced oncohomology and noting that there are independent of the choice of r.

An alternative means of calculating the Cech cohomology groups is by meansof covering sieves (c.f. (9)). Let U = Ui : i ∈ I be the set of all open subsets ofX (the trivial sieve on X ∈ Ouv(X)). By slight abuse of notation, say J ⊆ I is acovering 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. CECH THEORY

by reverse inclusion. The resulting set is filtered and we may compute the Cechcohomology as the cohomology of the direct limit complex:

Hp

Φ(X,F ) := Hn

(lim−→J

C•Φ(Uj : j ∈ J,F )

).

This is because filtered direct limits commute with cohomology and the set of cover-ing sieves is cofinal in the set of all covers, ordered by the existence of a refinement.

The abelian groups Hp

Φ(X,F ) are called the (absolute) Cech cohomology groupsof F . We will revisit these constructions are first proving some “classical” factsabout Cech cohomology.

7. Cech acyclic sheaves

As with usual sheaf cohomology, we will need a supply of presheaves with van-ishing higher Cech cohomology. Unfortunately, our supply of such presheaves willbe rather lame.

Theorem 7.1. Let U be an open cover of X, F ∈ PAb(X). The Cech coho-mology groups Hp

Φ(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 fromthe definition that ι∗ι

∗F is flasque for any inclusion ι : U → X. Furthermore, aproduct of flasque sheaves is flasque (since (

∏i Fi)(U) =

∏i Fi(U) and a product of

surjections is surjective). Hence the Cech resolution ε : F → C •(U ,F ) is a flasqueresolution of F , so

Hp

Φ(U ,F ) = Hp(ΓΦC •(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 Cech cohomology is, in some sense, effaceable.However, it turns out that Cech cohomology does not generally form a δ-functorfrom Ab(X) to Ab: a short exact sequence of sheaves does not generally inducea long exact sequence of Cech cohomology groups. This is the only obstruction toidentifying Cech cohomology with sheaf cohomology.

If we replace sheaves with presheaves, Cech cohomology is better behaved:

Theorem 7.2. For any cover U of X, the Cech cohomology groups H•(U , )

form an effaceable δ-functor from PAb(X) to Ab, hence Hn(U ,F ) = Rn H

0(U ,F )

for every F ∈ PAb(X).

7. CECH ACYCLIC SHEAVES 81

Proof. Suppose

F = 0→ F ′ → F → F ′′ → 0

is a short exact sequence in PAb(X). Then

0→ F ′(U)→ F (U)→ F ′′(U)→ 0

is exact for every U ∈ Ouv(X) (this is why it is important to use presheaves).Consequently,

0→ C•(U ,F ′)→ C•(U ,F )→ C•(U ,F ′′)→ 0

is a short exact sequence of complexes of abelian groups, since, in each degree it isjust a product of short exact sequences of the above form. The associated long exactcohomology sequence is natural in F hence the first claim is proved. To see thatH•(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) hasenough 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 Cech cohomology groups H

•Φ(X, ) form an effaceable δ-

functor from PAb(X) to Ab, hence Hn

Φ(U ,F ) = Rn H0

Φ(U ,F ) for every F ∈PAb(X).

Proof. Let U = Ui : i ∈ I be the set of all open subsets of X. By the sameargument as in the previous proof, we reduce to proving that

0→ lim−→I′

CpΦ(I ′,F ′)→ lim

−→I′

CpΦ(I ′,F )→ lim

−→I′

CpΦ(U ,F ′′)→ 0

(the filtered direct limit is over all I ′ ⊆ I such that Ui : i ∈ I ′ is a covering sieve,ordered by reverse inclusion) is exact for any exact sequence of presheaves

0→ F ′ → F → F ′′ → 0.

Exactness on the left and in the middle follows from exactness of filtered directlimits and left exactness of ΓΦ, so the issue is to prove surjectivity on the right. Asmentioned in (??), an element [(s, I ′)] on the right is represented by a pair (s, I ′)where I ′ indexes a covering seive and s is a set of sections

s(i0, . . . , ip) ∈ F ′′(U(i0, . . . , ip))

such that there is a Y ∈ Φ with

Supp s(i0, . . . , ip) ⊆ Y

for all (i0, . . . , ip) ∈ (I ′)p+1.

Since Φ has neighborhoods, we can find an open set V ∈ Ouv(X) with Y ⊆ Vand V − ∈ Φ. By choosing a common refinement of the cover indexed by I ′ andthe cover X \ Y, V we can an I ′′ ⊂ I ′ indexing a covering seive, such that, for

82 VIII. CECH THEORY

every i ∈ I ′′, either Ui ⊆ V or Ui ∩ Y = ∅. In particular, for (i0, . . . , ip) ∈ (I ′′)p+1,s(i0, . . . , ip) = 0 unless U(i0, . . . , ip) ⊆ V . The presheaf exactness implies that

F (U(i0, . . . , ip))→ F ′′(U(i0, . . . , ip))

is always surjective, so we can find, for each (i0, . . . , ip) ∈ (I ′′)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 ′′)p where U(i0, . . . , ip) is not contained in V , we obtaina preimage [(t, I ′′)] for [(s, I ′)] = [(s|I′′ , I ′′)]. Note that

Supp t(i0, . . . , ip) ⊆ V − ∈ Φ

for all (i0, . . . , ip) ∈ (I ′′)p+1.

The assumption that Φ has neighborhoods cannot be removed. See (??).

8. Cech presheaf

Let X be a topological space, Φ a family of supports on X, F ∈ Ab(X). Forn ≥ 0, let H n

Φ (F ) ∈ PAb(X) be the presheaf

H nΦ (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

H nΦ : Ab(X) → PAb(X)

F 7→ H nΦ (F )

coincides with the nth right derived functor of H 0Φ : Ab(X)→ PAb(X).

Proof. The functors F 7→H nΦ (F ) form an effaceable (by flasque sheaves, say)

δ-functor which manifestly has the right degree zero term.

Remark 8.2. The functor H 0Φ 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 valuedfunctions, then H 0

Φ is not even a sheaf.

The restriction maps for the presheaf H qΦ (F ) can be described concretely as

follows. Fix an injective resolution ε : F → I • of F . Then an inclusion of opensets 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

H qΦ (F )(U) = Hq(ΓΦ∩U(U,I •))

8. CECH PRESHEAF 83

and similarly with U replaced by V . The abelian presheaves H qΦ (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 separatedpresheaf) associated to the abelian presheaf H q

Φ (F ) is zero for all q > 0.

Proof. Fix a q > 0, an injective resolution F → I •, an open set U , and anelement s ∈H q

Φ (F )(U). Then s is represented by some q-cocycle s in the complexΓΦ∩U(U,I •), in other words by an element s ∈ I q(U) where the closure S of Supp sin X is in Φ. Since Φ has neighborhoods, there is a B ∈ Φ such that S is containedin the interior V of B. Since the complex I • is exact in positive degrees, the q-cocycle s|U∩V is a “coboundary on a cover.” That is, there is an open cover Ui ofU ∩ V and sections si ∈ I q−1(Ui) such that dq−1(Ui)(si) = s|U∩V for all i. Since Uiis contained in V , the closure in X of Supp si is contained in B, hence is in Φ, sosi ∈ ΓΦ∩Ui

(Ui,I q−1). The Ui together with the set U \ Supp s form an open coverof U , and taking the zero section of I q−1 on the latter open set we find that s is acoboundary on each set in this open cover, hence the restriction of s to every set inthis open cover is zero in cohomology.

Lemma 8.4. Let X be a topological space, Φ a family of supports. The naturalmap

H0

Φ(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 inducedby the natural map of presheaves F → F +, naturality of the the Cech complexassociated to a presheaf, and the fact that, for any G ∈ Ab(X), we have a naturalisomorphism

H0

Φ(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 keeptrack of supports by noting that the support of the section obtained by gluingis the unions of the supports of the sections being glued). Now let us prove thelemma. Making the natural identification just established, this map send a class

[(s, I ′)] ∈ H0

Φ(X,F ) to the corresponding class [(s+, I ′)] ∈ H0

Φ(X,F +). ViewingF + as the sheaf of maps to the stalks with the local coherence condition, it is clearthat [(s+, I ′)] is zero iff we can find a finer cover (hence a finer covering seive) I ′′ ⊆ I ′

on which (s, I ′) is zero.

Now let us try to prove surjectivity. We will ignore the supports for simplicity.A section sΓ(X,F

+) is an element s ∈∏

x∈X Fx satisfying the local coherencecondition (c.f. (??)), so we can find an I ′ ⊂ 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 whichis a subset of some Ui and restricting sections, we may assume I ′ indexes a coveringseive, so (I ′, s) ∈ C0(I ′,F ) is a class mapping to s ∈ Γ(X,F +) under the natural

84 VIII. CECH THEORY

map... but the difficulty is in showing that (I ′, s) is in the kernel of d0 of the Cechcomplex, and this is where we need separation of F . When F is separated, wesimply 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) = 0by separation.

We encountered an example where H0(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

Ep,q2 = H

p(U ,H q(F )) =⇒ Hp+q(X,F ).

If Φ is a family of supports with neighborhoods, then there is a similar spectralsequence

Ep,q2 = H

p

Φ(U ,H q(F )) =⇒ Hp+qΦ (X,F )

using the absolute Cech cohomology.

Proof. The functor Γ : Ab(X) → Ab is the composition of the forgetful

functor H 0 : Ab(X) → PAb(X) and the functor H0(U , ) : PAb(X) → Ab,

hence we have the Grothendieck spectral sequence

Ep,q2 = Rp H

0(U ,Rq H 0F ) =⇒ Hp+q(X,F ).

This spectral sequence is identified with the desired one by (8.1) and (7.2). Thesecond statement is proved similarly using (7.3).

Note that, for the existence of the Grothendieck spectral sequence we need to

know that H 0 takes injectives in Ab(X) to H0(U , ) acyclics (H

0

Φ(X, ) acyclicsfor the second statement); this follows from (7.1).

Theorem 8.6. For any topological space X and any abelian sheaf F on X, thenatural maps

Hp(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

0→ H1(X,F )→ H1(X,F )→ H0(X,H 1F )→ H

2(X,F )→ H2(X,F )

in the spectral sequence of (8.5), together with (8.4) and (8.3), which together implyH0(X,H 1F ) = 0.

9. COMPARISON THEOREMS 85

9. Comparison theorems

We conclude our treatment of classical Cech theory with two theorems aboutthe relationship between Cech cohomology and the usual sheaf cohomology.

Theorem 9.1. (Cartan’s Criterion) Let X, be a topological space, Φ a familyof 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.(3) H

p

Φ(U,F |U) = 0 for any p > 0 and any U ∈ U .

Then the natural map

Hp

Φ(X,F )→ HpΦ(X,F )

is an isomorphism for every p.

Proof. We first prove, by induction on p, that the natural map Hp

Φ(U,F |U)→Hp

Φ(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 cofinalin the family of all open covers of U , so we may compute the Cech cohomologygroups H

p(U,F |U) using only such covers. It then follows from assumption (3)

and the induction hypothesis that H qΦ (F )(U) = 0 for every U ∈ U and every

q with 1 ≤ q < p, so that Hi(U,H q

Φ (F )) = 0 for every such q because we cancompute Cech cohomology only using covers by open sets from U . Furthermore,

H0(U,H p(F )) = 0 by (8.3) and (8.4). Consider the spectral sequence of 8.5 on U .

The natural map

Ep02 = H

p

Φ(U,F |U)→ HpΦ(U,F |U)

in this spectral sequence must be an isomorphism because we have shown thatEi,j

2 = 0 in this spectral sequence for every i, j with i+ j = p and j > 0.

Finally, we conclude the desired result by again considering the spectral se-quence of 8.5, using the first part of the proof and the assumption (3) to know thatHp(X,H q

Φ (F )) = 0 for every q > 0, so that the SES degenerates.

Theorem 9.2. Suppose Φ is a paracompactifying family of supports (§7) on atopological space X. Then for every p ≥ 0 and every F ∈ Ab(X), the natural map

Hp

Φ(X,F ) → HpΦ(X,F )

is an isomorphism.

Proof.

Corollary 9.3. For a paracompact topological space X, the natural map

Hp(X,F ) → Hp(X,F )

is an isomorphism for every p.

86 VIII. CECH THEORY

10. The unit interval

It is possible to compute the Cech cohomology groups Hp(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 ofroughly equal length where Ui intersects only Ui−1 and Ui+1 (and this intersection isconnected). (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 theCech 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:

d =

1 −1 0 0 · · · 0 00 1 −1 0 · · · 0 00 0 1 −1 · · · 0 0...

......

......

...0 0 0 0 · · · −1 00 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, Cech cohomology and the usual sheafcohomology coincide by Corollary 9.3, so we find

Hp(I,Z) =

Z, p = 00, p > 0.

(10.1)

11. Simplicial objects

For a nonnegative integer n, let [n] := 0, . . . , n. Let ∆ denote the categorywhose objects are the finite sets [0], [1], . . . and where a morphism [m] → [n] is anondecreasing 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”). Fori ∈ [n], let eni : [n + 1] → [n] be the unique nondecreasing map which is surjectiveand 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) viathe constant functor [n] 7→ C (which takes every ∆op morphism to IdC). Simplicialobjects in C form a category

sC := HomCat(∆op,C),

which evidently contains C as a full subcategory by identifying C with the constantsimplicial objects in sC. For C ∈ sC, it is customary to write Cn for C([n]) anddni : 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 mapseni 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

For a non-negative integer n, let ∆topn ⊆ Rn+1 denote the convex hull of the

standard basis vectors e0, . . . , en:

∆stdn := t0e0 + · · ·+ tnen ∈ Rn+1 : ti ∈ [0, 1],

∑ti = 1.

A morphism σ : [m] → [n] in the simplicial category ∆ induces a morphism oftopological spaces

∆top(σ) : ∆topm → ∆top

n

(t0e0 + · · ·+ tmem) 7→ (∑

i∈σ−1(0)

tie0 + · · ·+∑

i∈σ−1(n)

tien),

so we can regard ∆top as a cosimplicial topological space called the standard (topo-logical) simplex.

The algebraic analog of the standard simplex is the following: For a non-negativeinteger n, let An denote the ring

An := Z[t0, . . . , tn]/〈1− t0 − t1 − · · · − tn〉.

A ∆-morphism σ : [m]→ [n] induces a ring homomorphism

A(σ) : An → Am

ti 7→∑

j∈σ−1(i)

tj,

so that the An form a simplicial ring A, and hence the ∆algn := SpecAn form a

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 ∆ whoseobjects are [0], . . . , [n]. An n-truncated simplicial object in C is a functor ∆op

n → C.Let

snC := HomCat(∆opn ,C)

be the category of n-truncated simplicial objects in C. The inclusion ∆opn → ∆op

induces a restriction functor

trn : sC → snC

C 7→ C|∆opn,

which we call the n truncation.

88 VIII. CECH THEORY

Let ∆mon (resp. ∆monn ) 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 ∆epin similarly.

Assume C has finite inverse limits. By the Kan Extension Theorem, the trun-cation trn : sC→ snC admits a right adjoint

Coskn : snC → sC

called the n coskeleton functor. Specifically,

(CosknC)m := lim←−Ck : ([m]→ [k]) ∈ [m] ↓ ∆op

n

= lim←−Ck : ([k]→ [m]) ∈ ∆n ↓ [m]

= lim←−Ck : ([k]→ [m]) ∈ ∆mon

n ↓ [m].

The first equality is the usual “formula” for Kan extension; the second is a trans-lation of it. To be clear, we emphasize that the inverse limit is the inverse limit ofthe functor

(∆n ↓ [m])op → C

([k]→ [m]) 7→ Ck.

The third equality is by cofinality of ∆monn ↓ [m] in ∆n ↓ [m]: every object ([k]→ [m])

in ∆n ↓ [m] admits a map to an object ([l]→ [m]) where [l]→ [m] is monic.

In particular, notice that Cm = (CosknC)m for m ≤ n. Indeed, when m ≤ n,Id : [m]→ [m] is the terminal object of ∆n ↓ m.

The adjunction isomorphism

HomsnC(trnC,D) → HomsC(C,CosknD)

takes a map f : trnC → D to the map C → CosknD given in degree m by the mapCm → (CosknD)m obtained as the inverse limit of the maps

fkX(σ) : Xm → Yk

over all σ : [k]→ [m] in ∆n ↓ [m].

For C ∈ sC, the abuse of notation

CosknC := Coskn(trnC)

is standard.

Dually, if C has finite direct limits, then the Kan Extension Theorem impliesthat truncation trn admits a left adjoint

Skn : snC → sC

called the n skeleton functor, given on C ∈ snC by the formula

(SknC)m = lim−→Ck : ([m]→ [k]) ∈ [m] ↓ ∆n

= lim−→Ck : ([m]→ [k]) ∈ [m] ↓ ∆epi

n .

14. SIMPLICIAL HOMOTOPY 89

The structure map

(SknC)(σ) : (SknC)m → (SknC)l

induced by a ∆ morphism σ : [l] → [m] is the map on direct limits induced by thefollowing commutative diagram of functors:

([m] ↓ ∆n)op τ 7→σ∗τ //

(τ :[m]→[k]) 7→Ck %%

([l] ↓ ∆n)op

([l]→[k])7→CkyyC

Note that s0C is naturally isomorphic to C, so we can regard Sk0 and Cosk0 asfunctors C→ sC. With this understanding, Sk0C is the constant simplicial objectand Cosk0C is the simplicial object with (Cosk0C)n = C [n] (the n+ 1 fold productof C) whose boundary maps are the natural projections C [n] → C [n−1] “forgettingone coordinate” and whose degeneracy maps C [n] → C [n+1] are the diagonals givenby “repeating one coordinate”.

14. Simplicial homotopy

Let f, g : X → Y be sC morphisms. A homotopy h from f to g consists of a Cmorphism h(φ) : Xn → Yn, defined for each morphism φ : [n]→ [1] of ∆, satisfyingthe conditions:

(1) For any commutative triangle

[m]

φ

ψ // [n]

θ[1]

in ∆, the square

Xn

h(θ)//

X(ψ)

Yn

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 isa homotopy h from f to g, and k, l : D → E are sC morphisms with a homotopy h′

90 VIII. CECH THEORY

from k to l, then there is a homotopy from kf to lg, obtained by composing h andh′:

(φ : [n]→ [1]) 7→ (h′(φ)h(φ) : Cn → En).

We let ∼ be the equivalence relation generated by this relation and we say f ishomotopic to g if f ∼ g. In other words, f ∼ g means there is a finite string ofmorphisms f = f0, f1, . . . , fn = g such that, for every i, either there is a homotopyfrom 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 sCmorphism g : D → C such that fg ∼ 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 com-posing them. This is well defined because of the remarks about composition of homo-topies made above. There is an obvious forgetful functor sC→ HotsC. Evidentlyf : 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 mapf : C → 1 to the terminal object (empty product) admits section s (i.e. there is somemap 1→ C). Then

Cosk0 f : Cosk0C → 1

is a homotopy equivalence in sC with homotopy inverse

Cosk0 s : 1→ Cosk0C.

Proof. Certainly (Cosk0 f)(Cosk0 s) = Cosk0(fs) is the identity of Cosk0 1 = 1because Cosk0 is a functor and fs = Id1 by definition of “section”. We need to showthat

(Cosk0 s)(Cosk0 f) = Cosk0(sf)

is homotopic to the identity of Cosk0C. The map

(Cosk0(sf))n : Cn+1 → Cn+1

is given by

(Cosk0(sf))n(c)(i) = sfc(i)

(i = 0, . . . , n). We define a homotopy to the identity by associating to φ : [n]→ [1]the “straight line homotopy” map

h(φ) : Cn+1 → Cn+1

given by

h(φ)(c)(i) :=

sfc(i), φ(i) = 0c(i), φ(i) = 1.

15. SIMPLICIAL SETS 91

When φ is the constant map to 0 ∈ [1], obviously h(φ) = (Cosk0(sf))n and whenφ is the constant map to 1 ∈ [1] obviously h(φ) is the identity. To finish the proofthat h is the desired homotopy, we must show that, for any commutative square

[m]

φ

ψ // [n]

θ[1]

in ∆, the square

Cn+1h(θ)

//

ψ∗

Cn+1

ψ∗

Cm+1

h(φ)// Cm+1

commutes in C.

Here ψ∗ is short for (Cosk0C)(ψ). It is given by (ψ∗c)(i) = c(ψ(i)). This is asimple computation:

(ψ∗ h(θ))(c)(i) =

sfc(ψ(i)), θ(ψ(i)) = 0c(ψ(i)), θ(ψ(i)) = 1

=

sfc(ψ(i)), φ(i) = 0c(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 setX, we refer to the elements of Xn as simplicies of X of dimension n. Fix a simplexx ∈ 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 simplexX(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 easyto see that any simplex x ∈ Xm is a degeneracy of a unique non-degenerate simplexy: 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 familiarwith from topology:

Lemma 15.1. For a simplicial set X and a non-negative integer n, the image ofthe adjunction morphism SknX := Skn(trnX) → X consists of the simplices of X

92 VIII. CECH THEORY

which are degeneracies of some k simplex with k ≤ n. This adjunction morphism isan isomorphism iff all non-degenerate simplices of X have dimension ≤ n.

Proof. According to the general formulas of §13, an element [σ, x] of (SknX)mis 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′] from σto σ′ : [m] → [k′] in [m] ↓ ∆n and any x′ ∈ Xk′ we introduce the relation(σ′, x′) ∼ (σ,X(τ)(x′)). Two such pairs (x, σ) and (x′, σ′) represent the same el-ement of (SknX)m iff they are equivalent under the smallest equivalence relationcontaining the relations ∼. The adjunction morphism (SknX)m → Xm takes [σ, x]to X(σ)(x) ∈ Xm. The description of the image of the adjunction morphism is nowclear.

It remains only to prove that the adjunction morphism is injective when all non-degenerate simplices of X have dimension ≤ n. Suppose [x, σ], [x′, σ′] ∈ (SknX)mhave X(σ)(x) = X(σ′)(x′) =: y ∈ Xm and we wish to show that [x, σ] = [x′, σ]. Byhypothesis, 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]σ

~~

σ′

ρ

[k]

τ

[k′]

τ ′~~[l]

in ∆ as indicated (for example, by choosing sections of the surjections σ, σ′). Wehave 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 ′)(z) = x′, so(σ′, x′) ∼ (ρ, z). From transitivity of the equivalence relation induced by ∼, weconclude [σ, x] = [σ′, x′] ∈ (SknX)m.

16. Standard simplicial sets

There are several special simplicial sets that will be useful later. We collect themhere as a series of examples.

Example 16.1. For each n ∈ N we have the n-simplex ∆[n] ∈ sEns defined bysetting

∆[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], whichis 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−1f(j) = dj−1n−1f(i) for all 0 ≤ i < j ≤ n.

Example 16.3. For example, the simplicial n sphere Sn ∈ sEns is the simplicialset with exactly two non-degenerate simplices: one in dimension zero and one indimension n. Using Lemma 15.1 it is not hard to show that such a simplicial setexists 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 geometricrealization of X as follows. We start with the space

X :=∐n

∐Xn

∆topn ,

where ∆topn 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 mapof topological spaces ∆top(σ) : ∆top

m → ∆topn reflecting the cosimplicial structure of

∆top. For any x ∈ Xn we glue the component ∆topm (X(σ)(x)) of X to the component

∆topn (x) of X via the map

∆top(σ) : ∆topm (X(σ)(x)) → ∆top

n (x).

That is, we introduce the relation ∼ on X where P ∼ ∆top(σ)(P ) for all P ∈∆topm (X(σ)(x)). We let |X| be the space obtained from X by making all such glu-

ings. That is, we let |X| be the quotient of X (with the quotient topology) by theequivalence relation generated by the relations ∼ introduced above.

94 VIII. CECH 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 generatedspace (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, thefunctor || 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 thesame thing is chain complexes in A supported in non-negative degrees.

Given an A ∈ sA and an n ∈ N, we set

Nn(A) := ∩n−1i=0 Ker(din : An → An−1).

(Note that the intersection is over the kernels of all but the last boundary mapdnn : An → An−1. By convention, N0(A) := A0. It is easy to check that the lastboundary map dnn takes Nn(A) into Nn−1(A) and that dn−1

n−1dnn = 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≥0A,

which is in fact an equivalence of categories. Furthermore, this functor takes ho-motopies in the simplicial sense of §14 to homotopies in the usual sense of chaincomplexes. When A is the category of modules over a ring R, then the simplicialset underlying any simplicial R module M is fibrant and the homotopy groups ofthis simplicial set (with any choice of basepoint; the natural one being 0 ∈ M) areidentified 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 objectin An is called a simplicial ring. Given an A ∈ sAn, a module over A is a simplicialabelian group M such that each Mn is equipped with the structure of an An modulecompatibly 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 Kahler differentials ΩBn/An ∈Mod(Bn) fit together to form aB module ΩB/A.

19. SIMPLICIAL RINGS 95

For a simplicial ring A, we set

Nn(A) := ∩n−1i=0 Ker(din : An → An−1)

Zn(A) := ∩ni=0 Ker(din : An → An−1)

Hn(A) := Nn(A)/dn+1n+1Zn+1(A).

Note that Nn(A),Zn(A) ⊆ An are ideals of An. The subset dn+1n+1Zn+1(A) ⊆ An is

also an ideal because dn+1n+1 : An+1 → An has a section snn : An → An+1 so for a ∈ An

and b ∈ Zn+1(A) we have

adn+1n+1(b) = dn+1

n+1(snn(a))dn+1n+1(b)

= dn+1n+1(snn(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+1n+1Zn+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 geometricrealization §17 of the simplicial set underlying the simplicial ring A. The space |A| isof course pointed by 0 ∈ |A|. Since A is a ring object in simplicial sets and geometricrealization (viewed as a functor to K) preserves finite inverse limits, |A| is a ringobject in K. An element [f ] of πm(|A|) is represented by a map f : Im → A takingthe boundary of the cube Im to 0 ∈ |A| and an element [g] of πn(A) is representedby a map g : In → A taking the boundary of In to 0. Using the ring structure onA, we can define a map

fg : Im+n → |A|(x, y) 7→ f(x)g(y)

which also clearly takes the boundary of Im+n to 0 ∈ |A|, hence represents a class[fg] ∈ πm+n(|A|). The multiplication [f ][g] := [fg] is clearly well-defined; it isgraded commutative because the automorphism (x, y) 7→ (y, x) of Im+n acts by(−1)mn on the orientation of Im+n.

Similarly, the normalized chain complex N(A) of a simplicial ring A carries anatural differential graded ring structure. The multiplication maps

Nm(A)⊗ Nn(A) → Nm+n(A)

96 VIII. CECH THEORY

are defined using the “shuffle product” map

Am ⊗ An → Am+n

a⊗ b 7→∑(σ,τ)

Sign(σ, τ)A(σ)(a)A(τ)(b).

The sum here....

20. Simplicial sheaves

Let X be a topological space, Sh(X) the category of sheaves on X. The categorysSh(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 thatthis property holds for each s|Ui

. For example, we can say that f : F → G is anepimorphism in Sh(X) iff, for every s ∈ G , there exists (locally) a t ∈ F suchthat f(t) = s. Let 1 be the terminal object of Sh(X) (the sheaf represented by theterminal object X ∈ Ouv(X)).

Remark 20.1. The discussion to follow makes sense in an arbitrary topos. Inthis 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, . . . , xk, . . . , xn+1 ∈ Fn

such that dni xj = dnj−1xi for all i < j in [n] both distinct from k, there is (locally)

a y ∈ Xn+1 with dn+1i y = xi for all i 6= k in [n]. A simplicial sheaf is fibrant if it is

fibrant in every degree.

The definition is motivated as follows: Label the ith codimension one face ti = 0of the standard topological n+ 1 simplex

∆topn+1 := (t0, . . . , tn+1) ∈ Rn+1 : t0 + · · ·+ tn+1 = 1

with xi, so the kth face remains unlabelled; the labelled faces form the so calledk-horn ∆top

n+1[k] of the standard n+ 1 simplex (I would call it the closed star of thevertex ek opposite the kth codimension one face). The condition dni xj = dnj−1xi saysthat the labellings “agree” on the codimension two faces and the fibrancy conditionasserts the existence of a “map on the whole n+1 simplex” whose restriction to eachof the faces in the k-horn is the map indicated by the labelling. Note that we makeno demands on the restriction of this “filling map” to the kth face. Such a fillingmap always exists in topology because the inclusion of the k-horn ∆top

n+1[k] → ∆topn+1

admits a retract r, hence any map f from the k-horn to a topological space X canbe lifted to a map fr : ∆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 ∈ Imφ.

Then a section of Fn+1[k] is nothing but a tuple x0, . . . , xk, . . . , xn ∈ Fn as above.There is an obvious restriction map Fn+1 → Fn+1[k]; the fibrancy condition saysthat this should be surjective.

A map from the terminal object 1 (the terminal object of sSh(X) is the terminalobject 1 of Sh(X) regarded as a constant simplicial sheaf) to a simplicial sheaf Fwill be called a basepoint of F . To give such a map is the same thing as giving amap 1 → F0, for then the map 1 → Fn must be the composition of 1 → F0 andany degeneracy map F0 → Fn, since this degeneracy map is an isomorphism in theconstant simplicial object 1. We will systematically use “1” as abuse of notationfor the map 1 → F . The existence of a basepoint implies, in particular, that eachmap Fn → 1 is an epimorphism (since it has a section). The category of based orpointed simplicial sheaves is the category of simplicial sheaves under 1.

Let 1 → F be a based fibrant simplicial sheaf, n a nonnegative integer. LetZnF be the subsheaf of Fn consisting of those x ∈ Fn with dni = 1 for all i ∈ [n].The sheaf ZnF is called the sheaf of n cycles in F . Define a relation ∼ on ZnFby declaring x ∼ y iff, locally, there is some z ∈ Fn+1 such that

(1) dn+1i z = 1 for i ∈ [n+ 1] \ n, n+ 1 and

(2) dn+1n = x, dn+1

n+1z = y.

I claim ∼ is an equivalence relation.

Reflexive: (This is the most difficult property to establish.) Given x ∈ ZnFn, weseek, locally, a z ∈ Fn+1 such that

(1) dn+1i z = 1 for i ∈ [n− 1] and

(2) dn+1i z = x for i ∈ n, n+ 1.

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+1i y = 1 for

i ∈ [n− 1] and dn+1n+1y = x (we know nothing about dn+1

n y). Next check that

(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+2i w = 1 for i ∈ [n− 1] and

(2) dn+2i w = dn+2

i w = y for i ∈ n, n+ 1.

98 VIII. CECH THEORY

I claim z := dn+2n+2w ∈ Fn+1 is as desired. Indeed, for i ∈ [n− 1], we compute

dn+1i z = dn+1

i dn+2n+2w

= dn+1n+1d

n+2i w

= dn+1n+11

= 1,

and for i ∈ n, n+ 1, we compute

dn+1i z = dn+1

i dn+2n+2w

= dn+1n+1d

n+2i w

= dn+1n+1y

= x.

Symmetric: Suppose x, y ∈ ZnF are such that (locally), there is some z ∈ Fn+1

with

(1) dn+1i z = 1 for i ∈ [n− 1] and

(2) dn+1n z = x, dn+1

n+1z = y.

By reflexivity, further locally, there is r ∈ Fn+1 such that

(1) dn+1i r = 1 for i ∈ [n− 1] and

(2) dn+1n r = dn+1

n+1r = y.

Since dn+1r = dn+1n z it follows that

(1, . . . ,1, z, r) ∈ Fn+2[n+ 2],

so there is, further locally, some u ∈ Fn+2 with

(1) dn+2i u = 1 for i ∈ [n− 1] and

(2) dn+2n z = z, dn+2

n+1u = r.

Now we check, just as above, that w := dn+2n+2u satisfies

(1) dn+1i w = 1 for i ∈ [n− 1] and

(2) dn+1n w = y, dn+1

n+1w = x.

Transitive: Suppose x, y, z ∈ ZnF have x ∼ y and y ∼ z. We know ∼ is symmetricso, since y ∼ x, locally there is some u ∈ Fn+1 with

(1) dn+1i u = 1 for i ∈ [n− 1] and

(2) dn+1n u = y, dn+1

n+1u = x

and since y ∼ z there is locally some v ∈ Fn+1 with

(1) dn+1i v = 1 for i ∈ [n− 1] and

(2) dn+1n v = y, dn+1

n+1v = z.

21. WEAK EQUIVALENCES 99

From the equality dn+1n u = dn+1

n v it follows that

(1, . . . ,1, u, v) ∈ Fn+2[n+ 2].

By fibrancy, there is locally some w ∈ Fn+2 with

(1) dn+2i w = 1 for i ∈ [n− 1] and

(2) dn+2n w = u, dn+2

n+1w = v

and one checks easily that dn+2n+2w ∈ Fn+1 witnesses x ∼ z.

Because of the judicious insertions of the word “locally,” the quotient

πn(F ) := (ZnF )/ ∼

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 cansee 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 sheaveswhich 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 morphismZnF → ZnG 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 everyn.

A morphism of (unbased) fibrant simplicial sheaves f : F → G is called a weakequivalence 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 definedbasepoints in order to have a notion of weak equivalence that is local (even theexistence of a global basepoint is a serious restriction, so otherwise too many mapswould vacuously be weak equivalences).

3Note here that f |U : F |U → G |U can be viewed as a map of based fibrant simplicial sheaves on Ufor a unique choice of basepoint on G |U , namely the one given by the composition 1→ F |U → GU .

100 VIII. CECH THEORY

Lemma 21.1. A morphism of fibrant simplicial sheaves f : F → G is a weakequivalence iff fx : Fx → Gx is a weak equivalence of fibrant simplicial sets (bijectionon 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 forevery basepoint 1 → F |U . This can be checked on stalks by (6.1). Formation ofhomotopy sheaves commutes with stalks since the construction of πn(F |U) can bephrased in terms of finite inverse limits (to construct ZnF |U) and finite direct limits(to take the quotient (ZnF |U)/ ∼) and these limits commute with stalks.

A fibrant simplicial sheaf F ∈ sSh(X) is called acyclic (or a hypercover) if themap to the terminal object F → 1 is a weak equivalence. It is clear that πn(1) = 1for all n, so F is a hypercover iff πn(F |U) = 1 for all n for all U ∈ Ouv(X) andfor all basepoints 1→ F |U .

The category HCov(X) of hypercovers ofX is the full subcategory of HotsSh(X)consisting of hypercovers. Note, by (20.2), that the property of being a hypercoveris invariant under homotopy equivalences.

22. Properties of coskeleta

The coskeleta functors play well with fibrancy and homotopy sheaves. In orderto streamline the proofs in this section, we begin with some notation for coskeleta.Like any inverse limit, (CosknC)m sits inside the corresponding product, so we mayview a section of (“map to”) (CosknC)m as a section

s = (s(φ)) ∈∏

φ:[k]→[m],k≤n

Ck

satisfying the condition: whenever φ : [k] → [m] can be factored in ∆ as

[k]ψ //

φ

==[l]

θ // [m]

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 sectionof (CosknC)m by setting

s(φ : [k] → [m]) := C(φ)(s(Id : [m] → [m]))

and we see that Cm = (CosknC)m as we already noted in (11).

Evidently then, we should concentrate on the case m ≥ n. Here, s is determinedby 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

s = (s(φ : [n] → [m])) ∈∏

φ:[n]→[m]

Cn

actually determines a section of (CosknC)m is the following: For any two monic ∆morphisms φ, ψ : [n] → [m], define an integer k by

k + 1 := | Imφ ∩ Imψ|.

If k = −1, then no condition is imposed. Otherwise, there is a (unique) commutativediagram

[n]φ

[k]

θ??

η

[m]

[n]

ψ

>>

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

CoskmC → Coskn(CoskmC)

is an isomorphism.

Proof. Follows directly from the definitions and the fact that inverse limitscommute amongst themselves.

Lemma 22.2. For any F ∈ sSh(X), Coskn F is fibrant in degrees > n. If Fis fibrant, then Coskn F is fibrant.

Proof. Suppose m > n and consider

x = (x1, . . . , xk, . . . , xm+1) ∈ (Coskn F )m+1[k].

Note each xi is in (Coskn F )m and we view the latter as being contained in∏[l]→[m],l≤n

Fn

as in the above discussion. We want to find a y ∈ (Coskn F )m+1 with dm+1i y = xi

for i ∈ [m+ 1] \ k. Using the above description of sections y ∈ (Coskn F )m+1, wedefine y(φ : [n] → [m + 1]) as follows: Since m > n, the image of φ must miss at

102 VIII. CECH THEORY

least two elements of [m + 1] (this is the key point). In particular, it misses somei ∈ [m+ 1] with i 6= k, which is to say: φ can be factored as

[n]ψ //

φ

::[m]

si // [m+ 1] .

Declare y(φ) := xi(ψ). I claim this is independent of the choice of such a factorizationof φ. Indeed, suppose φ can be factored through both si, sj : [m] → [m + 1] (withi < j, say). Then we have a commutative diagram

[m]

si

$$[n]

ψ

<<

θ //

η

""

[m− 1]

sj−1

OO

si

[m+ 1]

[m]

sj

::

of monomorphisms in ∆. Using the definition of the boundary maps in the coskeletonand the fact that x is in the k-horn Fn+1[k], we then compute

xi(ψ) = xi(sj−1θ)

= (dmj−1)(θ)

= (dmi xj)(θ)

= xj(siθ)

= xj(η).

It easily follows that y is actually a section of (Coskn F )m+1. We now easily compute

(dm+1i y)(φ : [n] → [m]) = y(siφ)

= xi(φ).

To prove the second statement, first note that Coskn F is certainly fibrant indegrees m < n since fibrancy in degree m only depends on the degree m−1,m,m+1parts, and F → Coskn F is an isomorphism in these degrees. By what we justproved above, it remains only to prove fibrancy of Coskn F in degree n. SinceFn = (Coskn F )n, and Fn−1 = (Coskn F )n−1 we have

Fn+1[k] = (Coskn F )m+1[k].

Given

x = (x1, . . . , xk, . . . , 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+1i = xi for i ∈

[n+ 1] \ k. Define

z ∈∏

[n]→[n+1]

Fn

by

z(si : [n] → [n+ 1]) := dn+1i y ∈ Fn.

One easily checks that in fact z ∈ (Coskn F )n+1 and that dn+1i z = xi.

Lemma 22.3. For any based fibrant simplicial sheaf F , πm(Coskn F ) = 1 form ≥ n.

Proof. Suppose x ∈ Zm(Coskn F ) for m ≥ n. Define y ∈ (Coskn F )m+1 asfollows: Given φ : [n] → [m+ 1], choose a factorization

[n]ψ //

φ

::[m]

si // [m+ 1]

and declare

y(φ) :=

x(ψ), i = m+ 11, otherwise.

I claim that this is independent of the choice of factorization. The only issue iswhen φ can be factored as siψ for i < m+ 1 and as sm+1η, in which case we have acommutative diagram

[m]

si

$$[n]

ψ

<<

θ //

η

""

[m− 1]

sj−1

OO

si

[m+ 1]

[m]

sm+1

::

of monomorphisms in ∆. But then we just compute

x(ψ) = x(smθ)

= (dmmx)(θ)

= 1

since x ∈ Zm(Coskn F ). This independence of choice of factorization implies that

y ∈∏

[n]→[m+1]

Fn

104 VIII. CECH THEORY

is in fact in (Coskn F )m+1. Now we compute, for i ∈ [m] that

(dm+1i y)(φ : [n] → [m]) = y(siφ)

= 1

and

(dm+1m+1y)(φ : [n] → [m]) = y(sm+1φ)

= x(φ),

hence y witnesses x ∼ 1 in πm(F ).

Lemma 22.4. Let F be a based fibrant simplicial sheaf. Suppose that the degreen+1 part πn+1 : Fn+1 → (Coskn F )n+1 of the adjunction morphism F → Coskn Fis an epimorphism for every n. Then F → 1 is a weak equivalence (i.e. πn(F ) = 1for every n).

Proof. Given x ∈ Zn(F ) = Zn(Coskn F ), we know from the previous lemmathat πn(Coskn F ) = 1.4 So, since x ∼ 1 in Zn(Coskn F ), there is locally (infact, from the proof of the previous lemma, we don’t need “locally here”) somez ∈ (Coskn F )n+1 such that

(1) dn+1i = 1 for i ∈ [n] and

(2) dn+1n+1y = 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 issome z ∈ Fn+1 with πn+1z = y. But F → Coskn F is a map of simplicial objects,so the diagrams

Fn+1

dm+1i

$$

πn+1 // (Coskn F )n+1

dm+1i

xxFn = (Coskn F )n

commute for every i ∈ [n+ 1], hence z witnesses x ∼ 1, so [x] = 1 in πn(F ).

23. Cech theory revisited

Given a sheaf F ∈ Ab(X) we used the Cech complex (1) of a cover Ui : i ∈ Iof X to produce a resolution of F , and hence a spectral sequence (5.1) convergingto the cohomology Hn(X,F ). This is putting the cart before the horse. Actually,

4And we know from the lemma before that that Coskn F is fibrant, so it makes sense to speak ofπn(Coskn F ).

23. CECH THEORY REVISITED 105

we will see in a moment that the Cech 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 ZXas there are not generally enough projectives in Ab(X), but to correctly computethis 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

· · ·G •2 → G •1 → G •0

of such resolutions (really a filtered category of resolutions), then we could get awaywith the hypothesis:

lim−→n→∞

Ext>0(Gmn ,F ) = 0

The next thing to notice is that the failure of Cech cohomology to compute sheafcohomology 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 thisin (8) where there was a finest possible cover U = U0A, U0B of a space Y , butwhere 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 = Uiof X (i.e. a refinement of the trivial cover), but we also allow the possibility of takinga nontrivial cover V k

i,j of each Ui ∩Uj. Of course, once we do this, we will also want

to allow nontrivial covers of all new pairwise intersections V ki,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] =∏

[n] G ). For a ∆

morphism φ : [m]→ [n],

(Cosk0 G )(φ) : G n+1 → Gm+1

is given by (Cosk0 G )(φ)(s)(i) = s(φ(i)). The boundary map dni : G n+1 → G n isgiven 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 hUifor a cover Ui, then since finite products commute with coprod-

ucts in Sh(X) and the Yoneda functor h preserves inverse limits (intersections in

106 VIII. CECH THEORY

Ouv(X)), Cosk0 G is the “expected” simplicial object:

(Cosk0 F )n =∐

(i0,...,in)

hUi0∩···∩Uin

To obtain the Cech complex constructed in (1) for an F ∈ Ab(X), we simply applythe sheaf Hom functor

H om( ,F )

to Cosk0 G and note that the result is a cosimplicial object in Ab(X). In this casewe have

H om(Cosk0 G ,F )n = H om((Cosk0 G )n,F )

= H om(∐

(i0,...,in)

hUi0∩···∩Uin

,F )

=∏

(i0,...,in)

H om(hUi0∩···∩Uin

,F )

=∏

(i0,...,in)

FUi0∩···∩Uin

.

To this cosimplicial object, one can associate a chain complex supported in nonneg-ative degrees, which is just the Cech resolution of F . Of course, there are varyingways to make this complex (the unnormalized chain complex, normalized chaincomplex, complement of the degeneracies in the normalized complex, etc.—all arehomotopy equivalent) and these correspond to the variations on the Cech complex(e.g. the alternating Cech complex).

The fact that F → H om(Cosk0 G ,F ) is a resolution of F (i.e. a quasi-isomorphism 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 |Uihas 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 Cosk0C → 1 is a homotopyequivalence (14.1). Hence G |Ui

→ hUiis 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 nonneg-

atively graded cochain complexes in Ab(X) and this identification takeshomotopies in the sense of simplicial objects to homotopies in the senseof chain complexes, hence F |Ui

→ H om(Cosk0 G |Ui,F |Ui

) is a homotopyequivalence.

24. EXERCISES 107

24. Exercises

Many of the constructions of this section involve factoring ΓΦ as H 0 followed

by HΦ

(X, ), under the assumption that Φ has neighborhoods (then applying theGrothendieck spectral sequence). On the other hand, we can always factor ΓΦ asH 0

Φ followed by the usual global section functor Γ : PAb(X)→ Ab, even withoutthe neighborhoods assumption. However, it is not so clear whether we can applythe Grothendieck spectral sequence.

Exercise 24.1. The class of flasque presheaves satisfies the following properties:

(1) For any short exact sequence

0→ F ′ → F → F ′′ → 0

in PAb(X) with F ′ and F flasque, F ′′ 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 monomor-

phism F → G .

Exercise 24.2. For any family of supports Φ on a topological space X, showthat the functor H 0

Φ : Ab(X)→ PAb(X) takes flasque sheaves to separated flasquepresheaves. If X is notherian, show that H 0

Φ takes flasque sheaves to flasque sheaves.

Exercise 24.3. Is the δ-functor H•(U , ) from PAb(X) → Ab (c.f. ) acyclic

for the class of flasque presheaves? Assume Φ has neighborhoods, so that H•Φ(X, )

is also a δ-functor. Is it acyclic for flasque presheaves?

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Fourier 11 (1961), 201-312.[GZ] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer- Verlag,

1966.[Qui] D. Quillen, Rational homotopy theory, Ann. Math. 90 (1969), 205-295.[Hov] M. Hovey, Model Categories[Mil] J. Milnor, The geometric realization of a semi-simplicial complex, Ann. Math. 65 (1957),

357-362.

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