Sheaf Theory W. D. Gillam

W. D. Gillam

Department of Mathematics, Brown University E-mail address: [email protected]

Contents

Introduction 7 Chapter I. Sheaves 9 1. Open sets 9 2. Presheaves 9 3. Representable presheaves 10 4. Generators 11 5. Sheaves 11 6. Stalks 13 7. Sheafification 15 8. Espaces étalé 16 9. Sieves 18 10. Properties of Sh(X) 20 11. Examples 22 12. Exercises 22 Chapter II. Direct and Inverse Images 23 1. Functoriality of Ouv(X) 23 2. Direct images 23 3. Skyscraper sheaves 23 4. Inverse images 24 5. Sheaf Hom 26 6. Constant sheaves 27 7. Representable case 28 8. Inverse images and espaces étalé 28 9. Faithfulness of X 7→ Sh(X) 29 Chapter III. Descent 31 1. Canonical topology 31 2. Open maps and variants 31 3. Descent for spaces 31 4. Descent for sheaves 31 Chapter IV. Cohomology 33 1. Abelian categories 33 2. Enough injectives 33 3. Homological algebra 33 4. Abelian Sheaves 33 3 4 CONTENTS

5. Ringed spaces 36 6. Sheaf Ext 37 7. Supports 38 8. Sheaf cohomology 39 9. Acyclic sheaves 39 10. Flasque sheaves 39 11. Godement resolution 42 12. Soft sheaves 42 13. Filtered direct limits 42 14. Vanishing theorems 46 15. Higher direct images 46 16. Leray spectral sequence 47 17. Group quotients 48 18. Local systems 48 19. Examples 48 20. Exercises 48

Chapter V. Base Change 51 1. Proper maps 51 2. Comparison map 55 3. Homotopy invariance 57

Chapter VI. Duality 59 1. Adjoints for immersions 59 2. Extension by zero 61 3. Proper maps - Second treatment 63

Chapter VII. Finite Spaces 65 1. Supports 65 2. Specialization 65 3. Coniveau 66 4. Equivalence 67 5. Stalk formulas 67 6. Hom formulas 67 7. Poincar´eduality 68 8. Examples 69 9. Exercises 74

Chapter VIII. Cechˇ Theory 75 1. Cechˇ resolution 75 2. Reﬁnement maps 77 3. Alternating variant 78 4. Cech cohomology for a ﬁxed cover 78 5. Basic properties 79 6. Cechˇ cohomology 79 7. Cechˇ acyclic sheaves 80 CONTENTS 5

8. Cechˇ presheaf 82 9. Comparison theorems 85 10. The unit interval 86 11. Simplicial objects 86 12. The standard simplex 87 13. Truncations and (co)skeleta 87 14. Simplicial homotopy 89 15. Simplicial sets 91 16. Standard simplicial sets 92 17. Geometric realization 93 18. The abelian setting 94 19. Simplicial rings 94 20. Simplicial sheaves 96 21. Weak equivalences 99 22. Properties of coskeleta 100 23. Cechˇ theory revisited 104 24. Exercises 107 Bibliography 109

Introduction

The aim of this document is to give a concise treatment of the “classical” theory of sheaves on topological spaces, with the following philosophical guidelines: (1) Appeal to standard facts about limits in the category of sets whenever possible, e.g. exactness of filtered direct limits. (2) When possible, formulate constructions and results about sheaves on a topological space so that make sense on a site or topos with little modification. (3) Appeal to general homological algebra whenever possible. (4) Follow Godement [G] closely as long as this does not conflict with the other guidelines. Regarding (1) and (2), my feeling is that the general philosophy of topos theory is that a topos is a category that behaves like the category of sets, so, for example, abelian group objects in a topos should behave like abelian groups, etc. Almost any theorem about sets that can be stated and proved using category theoretic language should also hold true in a topos. In particular, the category of sheaves on a topological space should be a lot like the category of sets. This philosophy is usually manifested by bootstrapping up from the category of presheaves, which is “very much” like the category of sets from the perspective of category theory. A related philosophical point is that, since our object of study is the category of sheaves on a topological space, we should try to phrase our constructions intrinsically using this category, rather than refering back to the topological space, its category of open sets, etc. The manifestation of this is the usage of the representable sheaf associated to an open set, rather than the open set itself. Regarding (3), my feeling is that basically every spectral sequence arises as a special case of the Grothendieck spectral sequence. One should never have to write down a double complex out of the blue and analyse what its horizontal and vertical cohomology mean, etc. Grothendieck already did that once and for all.

CHAPTER I

Sheaves

1. Open sets

For a topological space X, let Ouv(X) denote the category of open subsets of X, where the only morphisms are inclusions. The category Ouv(X) has arbitrary direct limits given by the formula [ lim (f : C → Ouv(X)) = f(C) −→ C∈C and inverse limits over any category with ﬁnitely many objects, given by the formula \ lim (f : C → Ouv(X)) = f(C). ←− C∈C In particular, ∅ ∈ Ouv(X) is initial (the direct limit of the empty functor, or the empty union) and X ∈ Ouv(X) is ﬁnal (the inverse limit of the empty functor).

2. Presheaves

A presheaf on a topological space X is a functor F : Ouv(X)op → Ens U 7→ F (U). Presheaves on X form a category PSh(X) where a morphism of presheaves is a natural transformation of functors. For U ∈ Ouv(X), elements of F (U) are called sections of F over U. For an inclusion of open sets V ⊆ U and s ∈ F (U) we often write s|V for the image of s under the morphism FV ⊆U : F (U) → F (V ). We refer to these morphisms as restriction morphisms. For U ∈ Ouv(X), the functor PSh(X) → Ens F 7→ F (U) is often denoted Γ(U, ). We sometimes write Γ(U, F ) in lieu of F (U), especially when U = X, in which case Γ(X, ) has a special name: the global sections functor. The category PSh(X), being deﬁned as a category of functors to Ens, inherites various properties from Ens. For example, PSh has all small direct and inverse limits. These limits are formed “objectwise.” For example, for a direct limit system 9 10 I. SHEAVES

i 7→ Fi in PSh(X), the direct limit presheaf F = lim Fi is given by −→

(2.1) F (U) = lim Fi(U). −→ In particular, notice that the section functors Γ(U, ): PSh(X) → Ens commute with all limits.

3. Representable presheaves

The prototypical presheaf is the representable presheaf hU associated to U ∈ Ouv(X), deﬁned by hU (V ) = HomOuv(X)(V,U). By abuse of notation, we often write U instead of hU . Formation of hU is functorial in U and deﬁnes a functor h : Ouv(X) → PSh(X) called the Yoneda functor.

Lemma 3.1. The Yoneda functor h is fully faithful and commutes with arbitrary inverse limits. There is a bijection

HomPSh(X)(hU , F ) = F (U) natural in U ∈ Ouv(X) and F ∈ PSh(X).

Proof. This is a straightforward variant of Yoneda’s Lemma. Remark 3.2. The Yoneda functor does not generally commute with direct limits, or even with ﬁnite direct sums. In particular, the presheaves hU ⊕ hV and hU∪V do not generally coincide. Indeed, for W ∈ Ouv(X), by deﬁnition of direct sums, we have a HomPSh(X)(hU ⊕ hV , hW ) = HomPSh(X)(hU , hW ) HomPSh(X)(hV , hW ) a = hW (U) hW (V ) a = HomOuv(X)(U, W ) HomOuv(X)(V,W ), while

HomPSh(X)(hU∪V , hW ) = HomOuv(X)(U ∪ V,W ). These two sets do not generally have the same cardinality. Now, we could remedy this particular failure, as some authors do, by using a variant of Ouv(X) where we consider not only open sets of X, but all maps f : Y → X which are local homeomorphisms. From our point of view, there is no reason to do this. We are only using Ouv(X) as a stepping stone to get to another category; we do not particularly care about the category Ouv(X) or the behaviour of its Yoneda functor. 5. SHEAVES 11

4. Generators

Recall that a set G of objects of a category C is called a generating set if, for any parallel C morphisms f, g : C → D, the following are equivalent: (1) f = g (2) 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” M M G → C

G∈G HomC(G,C) is an epimorphism for every C ∈ C. An object G of C is called a generator if {G} is a generating set. Dually, G is called a cogenerating set if G is a generating set for Cop. The paradigm example here is: any nonempty set is a generator of Ens.

Lemma 4.1. The representable presheaves {hU : U ∈ Ouv(X)} form a generating set for PSh(X).

Proof. Two morphisms f, g : F → G in PSh(X) are equal iﬀ f(U), g(U): F (U) → G (U) are equal for every U ∈ Ouv(X). On the other hand, f(U), g(U) are identiﬁed with

f∗, g∗ : HomPSh(X)(hU , F ) → HomPSh(X)(hU , G ) by (3.1).

5. Sheaves

Let U = {Ui : i ∈ I} be a family of open subsets of X with union U (a cover of U). A presheaf F is separated for U if the product of the restriction maps Y F (U) → F (Ui) i∈I is injective (that is, if a section of F is completely determined by its restrictions to the open sets in a cover). F has the gluing property (or descent property) with respect to U if, for any set of sections {si ∈ F (Ui): i ∈ I} with si|Uij = sj|Uij for all i, j ∈ I, there is a section s ∈ F (U) with s|Ui = si for all i ∈ I. F is separated (resp. has the gluing property) if it is separated (resp. has the gluing property) for every such family U. A presheaf is a sheaf if it is separated and it has the gluing property. 12 I. SHEAVES

We can sum this up by saying that F is a sheaf if and only if Y Y F (U) → F (Ui) ⇒ F (Uij)(5.1) i∈I (i,j)∈I×I is an equalizer diagram of sets for any U ∈ Ouv(X) and any cover U = {Ui : i ∈ I} of U. Remark 5.1. If F is a separated presheaf, then the set F (∅) is punctual (has one element) because the empty set is covered by the empty cover and the empty product in the category of sets is punctual.

The following criterion is often useful: Lemma 5.2. For F ∈ PSh(X), the following are equivalent: (1) F is a sheaf. (2) F commutes with ﬁltered inverse limits and has the sheaf property for covers of the form U = {U, V }. That is, (a) For any ﬁltered union U = ∪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 for this cover.

(2) =⇒ (1). Let U = {Ui : i ∈ I} be a family of open subsets of X with union U. To prove F is separated for U, suppose s, t ∈ F (U) have the same restriction to every Ui. By Zorn’s Lemma and the fact that F has the sheaf property for filtered unions, there is a maximal J ⊆ I such that s and t have the same restriction to V := ∪i∈J Ui. If J 6= I, then there is an i ∈ I \ J and, using the separation of F for the cover {Ui,V } we find that s and t agree on V ∪ Ui, contradicting maximality of J. The gluing property is proved similarly. Corollary 5.3. If X is a noetherian topological space, then F ∈ PSh(X) is a sheaf iff F (U ∪ V ) → F (U) × F (V ) ⇒ F (U ∩ V ) is an equalizer diagram for every U, V ∈ Ouv(X).

Proof. By definition, X is noetherian iff any filtered direct limit system in Ouv(X) is essentially constant, so any presheaf F trivially commutes with filtered inverse limits. 6. STALKS 13

A morphism of sheaves is deﬁned so that the category Sh(X) of sheaves on X forms a full subcategory of the category PSh(X).

Lemma 5.4. Every representable presheaf hU is a sheaf and

{hU ∈ Sh(X): U ∈ Ouv(X)} is a generating set for Sh(X).

Proof. The ﬁrst statement is straightforward, and the second is obvious from (4.1).

6. Stalks

The stalk of a presheaf F at a point x ∈ X is

x := lim (U). F −→ F U3x The direct limit is taken over open neighborhoods of x and restriction maps between them. Given a section s ∈ F (U) and a point x ∈ U we let sx ∈ Fx denote the image of s under the natural map

(U) → lim (V ) = x. F −→ F F V 3x Observe that a presheaf F is separated if and only if, for any object U of Top(X) and any s, t ∈ F (U), we have s = t if and only if sx = tx for all x ∈ U. A morphism of presheaves f : F → G induces a morphism fx : Fx → Gx on stalks for each x ∈ X by taking the direct limit of the morphisms f(U): F (U) → G (U) over all open neighborhoods U of x.

For example, the stalk hU,x of the representable sheaf hU at x is a punctual set if x ∈ U and empty otherwise. In contrast to the situation for presheaves, the stalks of a map f : F → G of sheaves are a good reﬂection of f: Lemma 6.1. For a morphism f : F → G in Sh(X), the following are equivalent: (1) f is a monomorphism in Sh(X). (2) f(U): F (U) → G (U) is a monomorphism in Ens for every U ∈ Ouv(X). (3) fx : Fx → Gx is a monomorphism in Ens for every x ∈ X. The following are equivalent: (4) f is an epimorphism in Sh(X). (5) fx : Fx → Gx is an epimorphism in Ens for every x ∈ X. The following are equivalent: (6) f is an isomorphism in Sh(X). (7) f(U): F (U) → G (U) is an isomorphism in Ens for every U ∈ Ouv(X). (8) fx : Fx → Gx is an isomorphism in Ens for every x ∈ X. 14 I. SHEAVES

Proof. (3) =⇒ (2) If f(U)(s) = f(U)(t), then fx(s(x)) = fx(t(x)) for every x ∈ U. Since the maps on stalks are monic, we have sx = tx for all x ∈ U, hence s = t because a sheaf is separated, so f(U) is monic. (2) =⇒ (3) A filtered direct limit of monic maps of sets is monic. (2)⇐⇒(1) By definition, f is monic in Sh(X) iff

f∗ : HomSh(X)(H , F ) → HomSh(X)(H , G ) is a monomorphism of sets for every H ∈ Sh(X). In any category, it suﬃces to check this for H running over a generating set, so by (4.1), we conclude that f is monic iﬀ

f∗ : HomSh(X)(hU , F ) → HomSh(X)(hU , G ) is monic for every U ∈ Ouv(X). But this map of sets is identiﬁed with the map f(U) by Yoneda (3.1).

(5) =⇒ (4) Suppose g1, g2 : G → H satisfy g1f = g2f. Then g1,xfx = g2,xfx for every x ∈ X, hence g1,x = g2,x for every x ∈ X because f is surjective on stalks. We conclude g1 = g2 using separation for H . (4) =⇒ (5) Since f is epic, by deﬁnition, the map

∗ f : HomSh(X)(G , H ) → HomSh(X)(F , H ) is a monomorphism of sets for every H ∈ Sh(X). If H = x∗A (c.f. (3) below) for A ∈ Ens, then this monomorphism is identiﬁed with

∗ fx : HomEns(Gx,A) → HomEns(Fx,A).

Since this is a monomorphism for every x, A, fx is an epimorphism in Ens for every x. (8) =⇒ (7) Consider a t ∈ G (U). Since the maps on stalks are epic, there are neighborhoods Ux of every x ∈ U and sections s(x) ∈ F (Ux) such that f(Ux)(s(x)) = t|Ux . After restricting to U ∩ Ux, we may assume every Ux is contained in U, so that the Ux form a cover of U. Since f commutes with restriction maps, we have

f(Ux ∩ Uy)(s(x)|Ux∩Uy ) = t|Ux∩Uy = f(Ux ∩ Uy)(sy|Ux∩Uy ).

Since each f(Ux ∩ Uy) is monic by ((3) =⇒ (2) above), the s(x) agree on overlaps, so they glue, by the sheaf property for F , to an s ∈ F (U). It then follows that f(U)(s) = t because this equality can be checked on each Ux by separation for G . (6)⇐⇒(7) Obvious. (7)⇐⇒(8) A map of sets is an isomorphism iﬀ it is both epic and monic, so this follows from the previous results. 7. SHEAFIFICATION 15

7. Sheaﬁﬁcation

The main goal of this section is to prove that Sh(X) is a reflexive subcategory of PSh(X), i.e. the inclusion functor Sh(X) → PSh(X) has a left adjoint, called sheafification. For a presheaf F , we will construct a sheaf F + (the sheafification of F ) and a morphism of presheaves F → F +, such that any morphism of presheaves F → G with G a sheaf will factor uniquely through F → F +. We set F +(U) equal to the set of all Y t = (t(x))x∈U ∈ Fx x∈U satisfying the following local coherence condition: There are a cover U = {Ui} of U and sections s(i) ∈ F (Ui) such that t(x) = s(i)x ∈ Fx for all x ∈ Ui. The presheaf F + has natural restriction maps given by restriction of functions and it is easy to check that F + is a sheaf. Indeed, we may actually glue functions together in the usual sense if they agree on overlaps. The construction is functorial because a map of presheaves f : F → G induces maps of stalks fx : Fx → Gx which we may use to define maps (pullback) between the sets of functions f +(U): F +(U) → G +(U) as above. The map f +(U) takes a function t satisfying the local coherence condition to another such function because if s(i) ∈ F (Ui) witness it for t then f(Ui)(s(i)) ∈ + G (Ui) witness it for f (U)(t). Notice that the map F → F + induces an isomorphism on every stalk. It then follows from (6.1) that if F is already a sheaf, then the natural map F → F + is an isomorphism. In particular, if F → G is a morphism of presheaves with G a sheaf, then we obtain the claimed factorization F → F + → G + ∼= G . This factorization is unique because one can check easily that a morphism of sheaves is determined by its values on stalks. The fundamental facts about sheafification are summed up in the following Theorem 7.1. The inclusion functor Sh(X) → PSh(X) commutes with arbitrary inverse limits and has a left adjoint: the sheafification functor PSh(X) → Sh(X) F 7→ F +. The adjunction morphism F + → F is an isomorphism for any sheaf F .

Proof. We proved the statements about adjointness in the discussion above. Saying that the forgetful functor commutes with inverse limits means that an inverse limit of sheaves (taken in the category of presheaves) is already a sheaf, which can be checked directly from the fact that the sheaf property is deﬁned in terms of a certain diagram being an equalizer diagram (a certain kind of inverse limit) and the fact that “inverse limits commute amongst themselves”. 16 I. SHEAVES

Corollary 7.2. The category Sh(X) has all small direct and inverse limits. The sheafification functor commutes with arbitrary direct limits and finite inverse limits.

Proof. We have already mentioned that Sh(X) has all inverse limits and that these coincide with the inverse limit taken in PSh(X). Using the adjointness properties of the sheafification functor, it follows from a formal exercise in category theory that the direct limit of a direct system of sheaves i 7→ Fi can be obtained by first taking the direct limit in the category of presheaves (c.f. (2.1)), then applying the sheafification functor to the result. Sheafification commutes with arbitrary direct limits simply because it is a right adjoint (formal category theory). To prove that it commutes with finite inverse limits, let f : C → PSh(X) be a functor with C a finite category (finitely many morphisms and finitely many objects). We must check that the natural map + lim (f : C → PSh(X)) → lim (f + : C → Sh(X)) ←− ←− is an isomorphism. By (6.1), it is enough to check this on stalks. On the other hand, stalks commute with sheafification, so this map is identified with the natural map lim lim f(C)(U) → lim lim f(C)(U), −→ ←− ←− −→ U3x C∈C C∈C U3x which is an isomorphism because finite inverse limits commute with filtered direct limits in the category of sets.

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

8. Espaces ´etal´e

Let F be a presheaf on a space X. We define a topological space Et F , called the espace étalé of F and a continuous map π : Et F → X as follows: As a set, we let a Et F := Fx x∈X be the disjoint union of the stalks of F . The set Et F is given the topology where a basic open subset is a set of the form

Us := {sx ∈ Fx ⊆ Et X : x ∈ U}, where U ∈ Ouv(X), s ∈ F (U). To show that these sets form the basic opens for a topology we should show that every sx ∈ Et F is in one of the Us (this is clear) and 8. ESPACES ETAL´ E´ 17 that an intersection Us ∩ Vt of such basic opens is again such a basic open. Indeed, the set W := {x ∈ U ∩ V : sx = tx} is open in U ∩ V (hence also in X) because equality at a stalk by definition implies equality on a neighborhood, and we have Us ∩ Vt = Wr, where r := s|W , or r := t|W (note that s, t have the same stalk everywhere in W , but they might not be equal in F (W ) because F might not be a sheaf). Define π : Et F → X so that π−1(x) = Fx ⊆ Et F for each x ∈ F . The map π is continuous because −1 [ π (U) = Vs V ⊆U,s∈F (V ) for any U ∈ Ouv(X). Given any U ∈ Ouv(X) and any s ∈ F (U), we define a map φ(U)(s): U → Et F by φ(U)(s)(x) := sx ∈ Fx ⊆ Et F . This map clearly sections π : Et F → X and is continuous because −1 φ(U)(s) (Vt) = {x ∈ U : x ∈ V and sx = tx}

= {x ∈ U ∩ V : sx = tx} is open in U as we argued above. Our maps

φ(U): F (U) → {s ∈ HomTop(U, Et F ): πs = IdU } are clearly compatible with restricting to smaller U, hence define a morphism φ from F to the sheaf of local sections of π : Et F → X (see (11.2)). Lemma 8.1. For any space X and presheaf F on X, the map φ from F to the sheaf of local sections of the espace étalé π : Et F → X identifies the latter sheaf with the sheafification of F . In particular, if F is a sheaf, then F is isomorphic to the sheaf of local sections of π : Et F → X.

Proof. By the stalk criterion (6.1) it suffices to prove that φ is an isomorphism on stalks. To see that φx is surjective, consider a local section f : U → Et F of π on a neighborhood U of x ∈ U. We must prove that f|V agrees with φ(V )(t): V → Et F for some t ∈ F (V ) for some neighborhood V of x in U. By definition of the stalk, we can find some neighborhood W of x in U and some r ∈ F (W ) such that rx = f(x) ∈ Fx. Since f is continuous, the set −1 V := f Wr := {y ∈ W ∩ U : ry = f(y)} is open in U and clearly contains x, hence V and t := r|V are as desired.

To see that φx is injective, suppose s, t ∈ F (U) for some neighborhood U of x and the functions φ(U)(s), φ(U)(t): U → Et X agree on a neighborhood of X. We have to show that sx = tx ∈ Fx. But this is obvious from the fact that φ(U)(s) and φ(U)(t) agree at x because

sx = φ(U)(s)(x) = φ(U)(t)(x) = tx. 18 I. SHEAVES

Evidently then, the theory of the espace étalégives another means of sheafifying a presheaf. It also gives a means of translating various sheaf-theoretic questions into questions of topology.

Remark 8.2. Given a map p : W → X, the espace étaléof the sheaf F of local sections of p is a topological space π : Et F → X over X that generally has little to do with the space W . One only knows that the sheaf of local sections of π is isomorphic to the sheaf of local sections of p. The map π is built solely from the sheaf of local sections of p, but one cannot hope to recover the map p from this sheaf. Another very special property of the espace étalé π is that the map [f] 7→ f(x) identifies germs of local sections of π near x with the fiber π−1(x). For a general space p : W → X over X, the map [f] 7→ f(x) from germs of local sections of p near x to the fiber p−1(x) is neither surjective nor injective.

The formation of the espace ´etal´eis functorial in F , so we may think of Et as a functor

Et : Sh(X) → Top/X.

(It is actually defined on PSh(X), but we saw above that the éspaceétaléof a presheaf is naturally isomorphic to the espace étaleof its sheafification.) Formation of the espace étaléis also functorial in X, in a sense that we will make precise in Section 8 once we discuss the functorial dependence of the category of sheaves Sh(X) on X.

9. Sieves

The following categorical constructions are used to deﬁne sheaves in more general settings. The constructions are not needed elsewhere in any serious way, though some of the terminology will be natural in our study of Cechˇ cohomology in Chap- ter VIII. Given a category C, a sieve in C is a full subcategory U ⊆ C such that, for any U ∈ U and any V ∈ C,

HomC(V,U) 6= ∅ =⇒ V ∈ U.

Let Crib(C) denote the set of sieves in C. Crib(C) has a natural ordering by inclusion. A set of objects G ⊆ C determines a sieve—namely the sieve consisting of all C ∈ C which admit a morphism to some G ∈ G. For a sieve U and subset G ⊆ U, we say that G is a generating set for U if the sieve generated by G (which is clearly contained in U) is equal to U. For an object X ∈ C, we set Crib(X) := Crib(C/X). The trivial sieve is the sieve in Crib(X) generating by {IdX } (i.e. the entire category C/X). 9. SIEVES 19

Given a sieve U ∈ Crib(Y ), and C morphisms Y → Z, W → Z, let W ×Z U ∈ Crib(W ) be the sieve consisting of all V → W ﬁtting into a commutative diagram

V / U

W / Z for some (U → Y ) ∈ U. For an object Y ∈ C, a sieve U ∈ Crib(Y ), and a family of sieves V = {V(U) ∈ Crib(U): U ∈ U}, let V/U ∈ Crib(Y ) be the sieve consisting of all morphisms W → Y which can be factored W → V → U → Y with (U → Y ) ∈ U and (V → U) ∈ V(U). For a topological space X, and U ∈ Ouv(X), notice that U ∈ Crib(U) is nothing more than a family of open subsets of U closed under passage to smaller open subsets. We declare a sieve U ∈ Crib(U) to be a covering sieve if U is a cover of U, and we let Cov(U) denote the set of covering sieves (it inherits an ordering from Crib(U)). A covering sieve U ∈ Cov(U) is nothing more than a cover of U closed under passage to smaller sets, and a generating set for U is a subset of G ⊂ U such that every V ∈ U is contained in some G ∈ G . This clearly implies that G is a subcover, but not every subcover is a generating set—G must contain all the largest open sets in U. The covering sieves in Ouv(X) enjoy the following properties: (1) For all U ∈ Ouv(X), the trivial sieve is in Cov(U). (2) For all Ouv(X) morphisms Y → Z, W → Z and all U ∈ Cov(Y ), W ×Z U ∈ Cov(W ). (3) For all Y ∈ Ouv(X), all U ∈ Cov(Y ), and all families V = {V(U) ∈ Cov(U): U ∈ U}, V/U ∈ Cov(Y ). We remark that the sheafification of F ∈ PSh(X) can be constructed as follows. First define F 0 ∈ PSh(X) by setting   0 Y Y F (U,U) := lim  F (Ui) F (Ui ∩ Uj) ←− ⇒ Ui∈U (i,j) for a covering sieve

U = {Ui : i ∈ I} ∈ Cov(U). Next note, for 0 0 U = {Ui : i ∈ I } ⊆ U, 20 I. SHEAVES

with U 0 ∈ Cov(U) (i.e. for a Cov(U)op morphisms U → U 0), we have an obvious map F 0(U,U) → F 0(U 0,U) induced by the restriction map Y Y F (Ui) → F (Ui). i∈I i∈I0 Set 0(U) := lim 0(U,U). F −→ F U∈Cov(U)op Note that Cov(U)op is ﬁltered: any two covering sieves have a common reﬁnement. Note that, for an Ouv(X) morphism V → U, we have a natural morphism F 0(U) → F 0(V ) induced by the functor Cov(U) → Cov(V )

U 7→ V ×U U.

We leave it as an exercise to check that (1) For any F ∈ PSh(X), F 0 is separated. (2) For any separated F ∈ PSh(X), F 0 is a sheaf. (3) For any F ∈ PSh(X), there is a natural isomorphism (F 0)0 = F +. Notice that this construction avoids the use of stalks.

10. Properties of Sh(X)

In this section we establish some facts about the category Sh(X) that will allow us to work more directly with Sh(X) as a category per se, without so much explicit reference to the topological space X. Lemma 10.1. Epimorphisms in Sh(X) are closed under base change.

Proof. Base change commutes with stalks (finite inverse limits of sets commute with filtered direct limits), so by (6.1) we reduce to proving that epimorphisms in Ens are closed under base change, which is easy. Theorem 10.2. A morphism f : F → G in Sh(X) is an epimorphism iff, for every H ∈ Sh(X), the diagram

HomSh(X)(G , H ) → HomSh(X)(F , H ) ⇒ HomSh(X)(F ×G F , H ) is an equalizer diagram of sets.

Proof. Injectivity of the left map in the diagram for all H is the deﬁnition of “epimorphism”, so the implication (⇐=) is obvious, and to establish ( =⇒ ), it remains only to show the following: suppose g : F → H is a Sh(X) morphism such that gπ1 = gπ2 : F ×G F → H , then g = hf for some h : G → H . The fastest way to prove this is to use stalks: the theorem we are trying to prove is clear in Ens and stalks preserve epimorphism (6.1) and commute with base change, 10. PROPERTIES OF Sh(X) 21

so we at least have a (unique) map hx : Gx → Hx (for each x ∈ X) such that gx = hxfx. Since H is a sheaf, we can view a section t ∈ H (U) as an element Q t = (t(x)) ∈ x∈U Hx satisfying the local coherence condition (c.f. (7)). For U ∈ Ouv(X), and s ∈ G (U), it remains only to prove that h(U)(s)(x) := hx(sx) actually satisﬁes the local coherence condition (the desired equality g = hf will then follow immediately by checking on stalks). To check local coherence of h(U)(s) at a point y ∈ U, we use surjectivity of fy to ﬁnd an open neighborhood V of y in U and a section t ∈ F (V ) such that f(V )(t) = s|V (equivalently, fz(tz) = sz for all z ∈ V ). Then g(V )(t) ∈ H (V ) witnesses local consistency:

g(V )(t)z = gz(tz) = 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 are coﬁnal in the category of epimorphisms to the terminal object of Sh(X).”

Lemma 10.3. Let G ∈ Sh(X). The morphism G → hX to the terminal object is an epimorphism iﬀ there is a cover U = {Ui : i ∈ I} of X and a morphism a hUi → G . i∈I

Proof. ( =⇒ ). By (5.4), the hU form a generating set for Sh(X), so we can certainly ﬁnd an epimorphism of the form a a hUi → G

Ui∈Ouv(X) j∈Ji

(take Ji = HomSh(X)(hUi , G ) = G (Ui) and use the evaluation map, for example). If x ∈ X is any point, then since

(1) the composition of epimorphisms is an epimorphism, (2) coproducts (or any direct limits) commute with stalks, and (3) an epimorphism in Sh(X) induces an epimorphism on stalks (6.1), we have an epimorphism a a hUi,x → hX,x.

Ui∈Ouv(X) j∈Ji

Note hX,x is punctual, so this simply says there is some Ui(x) such that hUi(x),x 6= (i.e. x ∈ Ui(x), see (6)). The open sets Ui(x) hence clearly cover X and the coproduct of the hUi(x) clearly maps to G by ﬁrst mapping to the “big” coproduct above, which can be done in many ways. (⇐=). Check on stalks. 22 I. SHEAVES

11. Examples

For example, if X is a one point space, then PSh(X) is isomorphic to the arrow category of Ens via the map PSh(X) → Ens·→· F 7→ (F (X) → F (∅)), while Sh(X) is equivalent to Ens via the map Sh(X) → Ens F 7→ F (X), (c.f. Remark 5.1). Example 11.1. For any topological space X, the presheaf C (X, R) which 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 local in nature (can be checked on each open set in a cover). This sheaf contains the presheaf C b(X, R) of bounded functions. This latter presheaf is separated, but it is not generally a sheaf because being bounded is global in nature and one cannot glue together functions which are locally bounded into a function which is globally bounded. Indeed, any continuous map f : X → R is locally bounded. Example 11.2. Generalizing the previous example, for any topological spaces X,Y , the presheaf U 7→ HomTop(U, Y ) (with the obvious restriction maps) is a sheaf on X because continuity is local. Similarly, if π : W → X is a map of topological spaces the presheaf

U 7→ {s ∈ HomTop(U, W ): πs = IdU } of local sections of π is a sheaf on X. We will see in Section ?? that every sheaf is isomorphic to a sheaf of this form.

12. Exercises

Exercise 12.1. Let X be a topological space, F ∈ PSh(X), U a cover of X, V a reﬁnement of U. Show that, if F is a sheaf for V, then it is a sheaf for U. Exercise 12.2. A basis is a subset (full subcategory) B ⊆ Ouv(X) such that every U ∈ Ouv(X) is a union (direct sum) of open sets in B. Set PSh(B) = op HomCat(B , Ens). Say F ∈ PSh(B) is a sheaf if, for every U ∈ B and every cover {Ui ∈ B : i ∈ I} of U, the diagram Y Y F (U) → F (Ui) ⇒ F (Uij) i∈I (i,j)∈I×I is an equalizer diagram of sets. Let Sh(B) be the full subcategory of PSh(B) consisting of sheaves. Observe that the functor PSh(X) → PSh(B) induced by the inclusion Bop ⊆ Ouv(X)op takes Sh(X) into Sh(B). Show that Sh(X) → Sh(B) is an equivalence of categories. CHAPTER II

Direct and Inverse Images

So far we have only considered the category of (pre) sheaves on a ﬁxed topological space X. In this section, we study the functoriality of these categories in X.

1. Functoriality of Ouv(X)

First notice that Ouv(X) is contravariantly functorial in X. The functor Ouv : Espop → Cat X 7→ Ouv(X) is given on a morphism f : X → Y in Esp by Ouv(f): Ouv(Y ) → Ouv(X) U 7→ f −1(U). The functor Ouv(f) commutes with direct limits (unions) and ﬁnite inverse limits (intersections). Furthermore, Ouv(f) takes covers to covers. Though we do not wish to deﬁne this, Ouv(f) is a continuous morphism of sites.

2. Direct images

Let f : X → Y be a continuous map of topological spaces, F a presheaf on X. The direct image presheaf f∗F is the presheaf on Y given by op f∗F : Ouv(Y ) → Ens U 7→ F (f −1(U)).

The restriction maps for this presheaf f∗F (U) → f∗F (V ) are given by the restric- −1 −1 op tion maps F (f (U)) → F (f (V )) for F . In other words, f∗F : Ouv(Y ) → Ens is the composition of Ouv(f)op : Ouv(Y )op → Ouv(X)op and F : Ouv(X)op → Ens.

3. Skyscraper sheaves

For example, if x : {x} ,→ X is the inclusion of a point, and A is a set (regarded as a sheaf on {x} as in (11)), then x∗A is the sheaf on X given by A, x ∈ U U 7→ {0}, otherwise

23 24 II. DIRECT AND INVERSE IMAGES

A sheaf on X isomorphic to such a sheaf is called a skyscraper sheaf. The stalks of x∗A are given by A, y ∈ {x}− (x A) = ∗ y {0}, otherwise

If F is a sheaf, then so is f∗F because f∗F has the sheaf property with respect to a cover U if and only if F has it with respect to the cover f −1U, as is clear from the definitions. The construction f∗ is clearly functorial in F . Example 3.1. The following example shows why direct limits of sheaves do not typically commute with sections. Let X be any infinite set with the discrete topology and let ZX be the constant sheaf. The presheaf direct sum ⊕x∈X x∗Z of all stalks of Z is given by M U 7→ Z. x∈U This presheaf if clearly separated, but it does not have the gluing property for the cover of X by singleton sets. The sheafification ⊕x∈X x∗Z of this presheaf is given by Q U 7→ x∈U Z because these are the functions to the stalks which (trivially) satisfy the local coherence condition. If, on the other hand, X is given the cofinite topology (a noetherian topology), then this presheaf if already a sheaf.

4. Inverse images

For a continuous map f : X → Y , and a presheaf G on Y , then the inverse −1 image presheaf fpreG ∈ PSh(X) is defined by −1 op fpreG : Ouv(X) → Ens U 7→ lim (V ), −→ G V ⊇f(U) where the direct limit is taken over open subsets V of Y containing f(U) with respect to the restriction maps of G (this is a filtered subcategory of Ouv(Y )). Note V ⊇ f(U) iff U ⊂ f −1(V ), so the direct limit is over the category U ↓ Ouv(Y ), whose objects are pairs (V, g) where V ∈ Ouv(Y ) and g : U → Ouv(f)(V ) is morphism in Ouv(X). A morphism (V, g) → (V 0, g0) in U ↓ Ouv(Y ) is a morphism i : V → V 0 in Ouv(X) making the diagram U g g0

y Ouv(f)(i) % Ouv(f)(V ) / Ouv(f)(V 0) 4. INVERSE IMAGES 25

in Ouv(Y ) commute.1 For any U ∈ Ouv(X), we have f(f −1(U)) ⊆ U so a section s ∈ G (U) yields −1 −1 −1 a section fpre(s) ∈ (fpreG )(f U) given by the class of s in the direct limit system −1 −1 −1 deﬁning (fpreG )(f (U)). It is clear from the deﬁnition of fpreG that its stalks are given by −1 (fpreG )x = Gf(x).

The stalk of a presheaf F on a space X at a point x ∈ X is nothing but the inverse limit of F under the inclusion map x : {x} → X. −1 The functor fpre does not generally take sheaves to sheaves. We are led to define f −1 : Sh(Y ) → Sh(X) −1 + G 7→ (fpreG ) −1 by composing fpre and the sheafification functor: −1 −1 + f G := (fpreG ) . As for the presheaf inverse image, a section s ∈ G (U) determines a section f −1s ∈ −1 −1 −1 −1 −1 (f G )(f (U)), namely, the image of fpres ∈ (fpreG )(f (U)) under the sheafification morphism −1 −1 −1 −1 (fpreG )(f (U)) → (f G )(f (U)). Examining the construction of sheafification, we see that a section of (f −1G )(V ) can be viewed as an element Y s = (s(x)) ∈ Gf(x) x∈V satisfying the local coherence condition: For every x ∈ V , there is a neighborhood W of x in V , a neighborhood U of f(W ) in Y , and a section t ∈ G (U) such that 0 0 s(x ) = tf(x0) for every x ∈ W . −1 −1 The formation of fpre and f are both functorial in f in the sense that for f : X → Y and g : Y → Z the functors −1 −1 −1 fpregpre, (gf)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 −1 (gpreG )y = Gg(y) mentioned above.) −1 Theorem 4.1. The functor fpre : PSh(Y ) → PSh(X) is left adjoint to f∗ : −1 PSh(X) → PSh(Y ). The functor fpre commutes with arbitrary direct limits and finite inverse limits. The functor f∗ commutes with arbitrary inverse limits.

1Of course, the usage of the category U ↓ Ouv(Y ) is overly complex, but it generalizes well and allows us to link up with various general category theoretic techniques (comma categories, Kan extension). 26 II. DIRECT AND INVERSE IMAGES

Proof. The adjointness is a special case of the Kan Extension Theorem and is straightforward to check. The other statements are formal consequences of adjoint- −1 ness, except the fact that fpre commutes with finite inverse limits. This is proved by noting that the category of neighborhoods of any subspace is filtered, and filtered direct limits commute with finite inverse limits in Ens. −1 Corollary 4.2. The functor f : Sh(Y ) → Sh(X) is left adjoint to f∗ : Sh(X) → Sh(Y ). The functor f −1 commutes with arbitrary direct limits and finite inverse limits. The functor f∗ commutes with arbitrary inverse limits.

Proof. This follows formally from the theorem using the adjointness and exactness properties of the sheafification functor (7.1), (7.2). Corollary 4.3. Inverse images commute with sheafification: For f : X → Y and any presheaf F ∈ PSh(Y ) we have a natural isomorphism f −1(F +) = −1 + (fpreF ) in Sh(X).

Proof. This is a consequence of Yoneda’s Lemma and the adjointness results just established: For any G ∈ Sh(X) we have −1 + + HomSh(Y )(f (F ), G ) = HomSh(Y )(F , f∗G )

= HomPSh(Y )(F , f∗G ) −1 = HomPSh(X)(fpreF , G ) −1 + = HomPSh(X)((fpreF ) , G ). Remark 4.4. Suppose i : U,→ X is the inclusion of an open set and F is a sheaf on X. Then for any V ∈ Ouv(U), V is in Ouv(X) and is coﬁnal among −1 −1 open neighborhoods of i(V ) in X, hence ipreF (V ) = F (V ). Furthermore, ipreF is already a sheaf, so we have i−1F (V ) = F (V ). Evidently then, i−1F is nothing but the restriction of the functor F : Ouv(X)op → Ens to the full subcategory op op −1 Ouv(V ) ⊆ Ouv(X) . Accordingly, we often write F |U instead of i F .

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, g Hom(F |U , G |U ) agree when restricted

to Ui for every i, then for any V ∈ Ouv(U) and any s ∈ F (V ), we have f(V )(s)|Ui∩V =

g(V )(s)|Ui∩V for all i and {Ui ∩ V → V } is a cover, so f(V )(s) = g(V )(s) by separation for G . This holds for any s, V , so f = g, hence H om(F , G ) is separated.

Given maps fi ∈ Hom(F |Ui , G |Ui ) which agree on overlaps, deﬁne f : F |U → G |U as follows. For any V ∈ Ouv(U), and any s ∈ F (V ) consider the fi(Ui ∩

V )(s|Ui∩V ) ∈ G (Ui ∩ V ). Since fi(Ui ∩ Uj ∩ V ) = fj(Ui ∩ Uj ∩ V ), these agree on overlaps, so by the gluing property for G there is an f(V )(s) ∈ F (V ) restricting to

fi(Ui ∩ V )(s|Ui∩V ) in G (Ui ∩ V ) for every i. To show that the maps f(V ): F (V ) → G (V ) s 7→ f(V )(s)

actually deﬁne a map of presheaves F |U → G |U , it remains to show that F (V ) / G (V )

F (W ) / G (V ) commutes for any morphism W → V in Ouv(U). This follows from separation for G . In light of the lemma, we may view (5.1) as a functor H om( , ): Sh(X)op × Sh(X) → Sh(X) (F , G ) 7→ H om(F , G ). The sheaf H om(F , G ) is called sheaf hom from F to G . For a space X and an open set i : U,→ X, the formulae

H om(hX , F ) = F

H om(hU , F ) = i∗(F |U ) are clear from the Yoneda Lemma (3.1). For a continuous map f : X → Y , the −1 adjunction (f , f∗) “sheaﬁﬁes” to yield a natural isomorphism −1 f∗H omX (f F , G ) = H omY (F , f∗G ).

6. Constant sheaves

An extreme case of the formation of inverse images occurs when f is the map from X to a one point space Y . If we identify Sh(Y ) with Ens as in (11), then the −1 pre presheaf inverse image functor fpre is identiﬁed with the functor A 7→ AX , where pre −1 AX is the constant presheaf U 7→ A, and the inverse image functor f is the functor A 7→ AX , where op AX : Ouv(X) → Ens U 7→ Aπ0(U), 28 II. DIRECT AND INVERSE IMAGES

where π0(U) is the set of connected components of U. A sheaf isomorphic to some AX is called a constant sheaf.

7. Representable case

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

= hf −1(U) −1 are clear for any morphism f : X → Y and any U ∈ Ouv(Y ). The set (f hU )(V ) is punctual if f(V ) ⊆ U (equivalently V ⊆ f −1(U)) and empty otherwise. Similarly, for V ∈ Ouv(X), the sheaf f∗hV assigns the punctual set to U ∈ Ouv(Y ) if −1 f (U) ⊆ V , else (f∗hV )(U) is empty.

8. Inverse images and espaces ´etal´e

The formation of inverse images is compatible with the construction of the espace étalé(§8). Lemma 8.1. Let f : X → Y be a continuous map of topological spaces, F a presheaf on Y , π : Et F → Y its espace étalé. Then π1 : X ×Y Et F → X is −1 naturally isomorphic to the espace étaléof fpreF . Similarly, if F is a sheaf, π1 is naturally isomorphic to the espace étaléof f −1F .

Proof. Recall from Section ?? that, as a set, −1 a −1 a Et fpreF = (fpreF )x = Ff(x), x∈X x∈X

which we prefer to think of as the set of pairs (x, s) where x ∈ X and s ∈ Ff(x). On the other hand, X ×Y Et F is exactly the same set of pairs (the preimage of f(x) ∈ Y under π : Et F → Y is Ff(x) by construction of π). This identiﬁcation clearly respects the two maps to X, so it remains to prove that it is continuous. −1 A basic open set in Et fpreF is a set of the form −1 Ut = {(x, s) ∈ Et fpreF : x ∈ U, s = tx ∈ Ff(x)} −1 where U is an open subset of X and t ∈ (fpreF )(U). Such a t is represented by a V ∈ Ouv(Y ) with f(U) ⊆ V and a section r ∈ F (V ). For x ∈ U one then has tx = rf(x), so this open corresponds to the open subset U ×Y Vr under our identiﬁcation.

A basic open set of X ×Y Et F is given by U ×Y Vs where U ∈ Ouv(X), V ∈ Ouv(Y ), s ∈ F (V ) and Vs := {sy ∈ Fy ⊆ Et F : y ∈ V }. But we have −1 −1 U × V = f (V ) × V = f (V ) −1 Y s Y s fpres −1 under our identiﬁcation, so this basic open in X ×Y Et F is also open in Et fpreF . 9. FAITHFULNESS OF X 7→ Sh(X) 29

The statements about sheaves follow from the statements about presheaves because both inverse images and espaces étalécommute with sheafification.

9. Faithfulness of X 7→ Sh(X)

CHAPTER III

Descent

1. Canonical topology

canonical cover = universal eﬀective epimorphism

2. Open maps and variants

open maps, weakly open maps, universal w.o. maps, basic properties (composition, base change)

3. Descent for spaces

4. Descent for sheaves

CHAPTER IV

Cohomology

The purpose of this chapter is to give a concise, self-contained treatment of sheaf cohomology.

1. Abelian categories

2. Enough injectives

3. Homological algebra

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

4. Abelian Sheaves

For a topological space X, let Ab(X) denote the category of abelian group objects in Sh(X). An object F ∈ Ab(X) is often called a sheaf of abelian groups or just an abelian sheaf. Similarly, we let PAb(X) denote the category of abelian group objects in PSh(X). Note that PAb(X) is isomorphic to the category of functors from Ouv(X)op to Ab and that this identifies Ab(X) with the full subcategory consisting of those F : Ouv(X)op → Ab whose composition with the forgetful functor Ab → Ens is a sheaf. For F ∈ PAb(X) and any inclusion V ⊆ U of open subsets of X, the restriction map F (U) → F (V ) is a homomorphism of (abelian) groups. The basic theory of sheaves from Chapter ?? translates to the abelian setting in the same way that one can translate various statements about sets into statements about (abelian) groups. Lemma 4.1. The forgetful functor Ab(X) → PAb(X) has a left adjoint (sheafification) which commutes with arbitrary direct limits and finite inverse limits.

Proof. The analogous statements are true of Sh(X) → PSh(X) and preserva- tion of finite inverse limits implies that sheafification takes (abelian) group objects to (abelian) group objects. The categories PAb(X) and Ab(X) clearly carry the structure of additive categories. Kernels and cokernels (in fact all limits) are formed “objectwise” in PAb(X) just as they were in PSh(X); in particular, PAb(X) has all small direct and inverse 33 34 IV. COHOMOLOGY

limits. Consequently, Ab(X) also has all small direct and inverse limits: inverse limits are formed by taking the inverse limit in PAb(X) and noting that it is already in Ab(X). Indeed, Ab → Ens commutes with inverse limits, so this is obvious from the corresponding statement about inverse limits in Sh(X). However, to form direct limits in Ab(X), one must sheafify (i.e. apply the sheafification functor of the lemma to) the direct limit taken in PAb(X). Explicitly, the zero object 0 ∈ PAb(X) is the constant presheaf U 7→ {0}. Note that this is also a sheaf and that it is also the zero object in Ab(X). For any morphism f : F → G in PAb(X), the natural map Cok Ker f → Ker Cok f is an isomorphism because Cok Ker f(U) → Ker Cok f(U) is an isomorphism for each U ∈ Ouv(X)(Ab is an abelian category!). A similar statement holds in Ab(X) because the natural map there is obtained by sheafifying this natural map. We have proved: Proposition 4.2. Ab(X) and PAb(X) are abelian categories with all small direct and inverse limits.

Observe that Z is a generator for Ab (c.f. §??), and that this is obvious on category theoretic grounds from the fact that a one element set is a generator for Ens and the free abelian group functor is left adjoint to the forgetful functor Ab → Ens. These observations are easily promoted to Ab(X), by the usual method of bootstrapping through PAb(X). Lemma 4.3. The forgetful functor PAb(X) → PSh(X) has a left adjoint, the “free abelian group” functor PSh(X) → PAb(X) pre F 7→ ⊕F Z, pre where ⊕F Z ∈ PAb(X) is given by pre op ⊕F Z : Ouv(X) → 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) pre + F 7→ (⊕F Z) , Proof. Follows from the lemma and the adjointness property of sheaﬁﬁcation (7.1). 4. ABELIAN SHEAVES 35

4.5 {⊕pre : U ∈ Ouv(X)} (resp. {⊕ : U ∈ Ouv(X)}) is a Theorem . hU Z hU Z generating set for PAb(X) (resp. Ab(X)). PAb(X) and Ab(X) have all small direct and inverse limits. Finite inverse limits and ﬁltered direct limits commute in both PAb(X) and Ab(X).

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

HomAb(X)(⊕hU Z, F ) = HomSh(X)(hU , F ) = F (U). This is natural in F , so for parallel arrows f, g : F → G , the maps f(U), g(U): F (U) → G (U) (which are equal for all U iﬀ f = g) are identiﬁed with

f∗, g∗ : HomAb(X)(⊕hU Z, F ) → HomAb(X)(⊕hU Z, G ). The existence of limits was discussed above. The last statement is certainly true of PAb(X) since these limits are formed objectwise and this statement is true of Ab (because it is true of Ens and Ab → Ens is conservative and commutes with filtered direct limits and inverse limits). Hence this is also true of Ab(X) by adjointness and exactness properties of sheafification. Corollary 4.6. Every F ∈ PAb(X) (resp. Ab(X)) admits a monomorphism into an injective object of PAb(X) (resp. PAb(X)).

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

The section functors PAb(X) → Ab F 7→ Γ(U, F ) commute with all direct and inverse limits, as these are formed “objectwise” in PAb(X) just as they were in PSh(X). The section functors Ab(X) → Ab F 7→ Γ(U, F ) commute with inverse limits (note Ab → Ens commutes with inverse limits, and the section functors for sheaves commute with inverse limits), but they do not generally commute with direct limits. That is, Γ(U, F ) preserves kernels, but not cokernels. Note that Γ(U, ) does, however, preserve ﬁnite direct sums since these coincide with ﬁnite direct products. In other words, Γ(U, ) is a left exact morphism of abelian categories—its right derived functors will be considered in more generality in §??. 36 IV. COHOMOLOGY

The forgetful functor Ab(X) → PAb(X) is similarly a left exact morphism of abelian categories. Its nth right derived functor is denoted F 7→ H n(F ) and is called the nth cohomology presheaf of F . It will be studied further in Chapter ??. A morphism of topological spaces f : X → Y induces an exact morphism of abelian categories f −1 : Ab(Y ) → Ab(X) and a left exact morphism of abelian n categories f∗ : Ab(X) → Ab(Y ). The right derived functors R f∗ are called the higher direct images of f. These will be studied further in Section 15.

5. Ringed spaces

Let An(X) be the category of ring objects of Sh(X). For OX ∈ An(X) we can consider the category Mod(OX ) of modules (“module objects”) over OX . Con- op cretely, OX ∈ An(X) is a functor OX : Ouv(X) → An whose composition with the forgetful functor An → Ens is a sheaf. From this point of view, F ∈ Mod(OX ) is a functor F : Ouv(X)op → Ab (whose composition with Ab → Ens is a sheaf) such that, for each U ∈ Ouv(X), the abelian group F (U) is equipped with an OX (U) module structure making the restriction maps F (U) → F (V ) linear over OX (U) → OX (V ). That is, for s ∈ F (U) and t ∈ OX (U), we require

(t · s)|V = t|V · s|V .

Note that Mod(ZX ) is isomorphic to Ab(X). The results of §4 on Ab(X) carry over, mutatis mutandis, to Mod(OX ). In particular, Mod(OX ) is an abelian category with all small direct and inverse limits, a set of generators, etc. A pair (X, OX ) consisting of a topological space X and a sheaf of rings OX ∈ An(X) is called a ringed space. If OX is clear from context, we will simply write X for (X, OX ) and Mod(X) for Mod(OX ). We will write “X module” instead of “OX module,” etc. Note that the stalk Fx of an OX module has a natural module structure over the stalk OX,x of OX . The following direct proof that Mod(X) has enough injectives is sometimes useful, especially if we need to get our hands on a supply of easily understood injectives. Lemma 5.1. Let x : {x} ,→ X be the inclusion of a point of a topological space X with a sheaf of rings OX . Let I be an injective OX,x module. Then x∗I is an injective OX module.

−1 Proof. We must show F 7→ HomX (F , x∗I). Note OX,x = x OX , so by the −1 adjointness (x , x∗),

HomX (F , x∗I) = HomOX,x (Fx,I),

which is exact because the stalk functor is exact and because HomOX,x ( ,I) is exact by deﬁnition of injective. Theorem 5.2. For any ringed space X, the category Mod(X) has enough injectives. 6. SHEAF EXT 37

Proof. Let F ∈ Mod(X). Since Mod(A) has enough injectives for any ring A, we can choose, for each x ∈ X, a monomorphism Fx ,→ I(x) from the stalk of F into an injective OX,x module. The adjunction morphism for the adjoint functors −1 (x , x∗) furnishes a natural morphism F → x∗I(x) of X modules. The product of Q all these maps is a map F → x∗I(x) which is easily seen to be monic. Note Q x∈X x∈X x∗I(x) is a product of injectives and is hence injective.

6. Sheaf Ext

For G ∈ Ab(X) and F ∈ Sh(X), observe that H om(F , G ) (c.f. (5)) inherits an abelian group object structure from that of G . We may view sheaf Hom as a functor H om( , ): Sh(X)op × Ab(X) → Ab(X) (F , G ) 7→ H om(F , G ).

Similarly, if X is equipped with a sheaf of rings OX , then we have an “internal Hom” or “sheaf Hom” functor H om( , ): Mod(X)op × Mod(X) → Mod(X) (F , G ) 7→ H om(F , G ).

For any ﬁxed F ∈ ModX , the functor H om(F , ) is left exact. Its right derived functors are denoted G 7→ E xtp(F , G ). Lemma 6.1. Let X be a ringed space, F ∈ Mod(X). There is an isomorphism of δ functors from Mod(X) to Mod(Γ(X, OX )) : • • H (X, F ) = Ext (OX , F )

Proof. It is straightforward to check that mapping s ∈ Γ(X, F ) to the homomorphism ·s ∈ Hom(OX , F ) given by

(·s)(U): OX (U) → F (U)

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

The proof of the following lemma will require several constructions that we have not yet discussed, but in any case it is natural to state it now. Lemma 6.2. Local-to-Global Ext Sequence. For F , G ∈ Mod(X), there is a natural ﬁrst quadrant spectral sequence p,q p q p+q E2 = H (X, E xt (F , G )) =⇒ Ext (F , G ). 38 IV. COHOMOLOGY

Proof. Since Γ(X, H om(F , G )) = Hom(F , G ), this will be obtained as a special case of the Grothendieck spectral sequence as long as we can prove that H om(F , G ) is Γ-acyclic for any injective sheaf G ∈ Mod(X). Since flasque sheaves are Γ-acyclic by (10.4), below, it suffices to prove that H om(F , G ) is flasque, so what we must show is that, for any open set ι : U,→ X and any homomorphism f : ι−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

−1 −1 ι!ι F / ι!ι G / G _ 5

F F in Mod(X), which completes as indicated by injectivity of G since the vertical −1 arrow is monic. Since ι ι! = Id, the vertical and second horizontal arrows become −1 isomorphisms upon applying ι so this F is as desired.

7. Supports

Let F be an abelian presheaf on a topological space X, U an open subset of X, s ∈ F (U) a section. Let Supp s := {x ∈ U : s(x) 6= 0 ∈ Fx} be the support of s. The set Supp s is closed in U, hence locally closed in X. For any s, t ∈ F (U), the set {x ∈ U : s(x) = t(x)} is an open subset of U. The closed set − Supp F := {x ∈ X : Fx 6= 0} of X is called the support of F . The support of any direct sum or product of sheaves is the closure of the union of their supports. A family Φ of closed subsets of a topological space X is called a family of supports if it satisfies the conditions: (1) If A ∈ Φ and B is a closed subset of A, then B ∈ Φ. (2) If A, B ∈ Φ then A ∪ B ∈ Φ. The first axiom in particular implies that ∅ ∈ Φ. For example, the family Φ of all closed subsets of X is a family of supports. Another example is the family Φ of all closed quasi-compact subsets of X (if X is Hausdorff, then the adjective “closed” is redundant, but this is not true for a general X). For a subspace Z of X, we could take Φ to be the family of all closed subsets of X contained in Z. We could also take Φ to be the family of all closed, paracompact subsets of X. We say that a family of supports Φ has neighborhoods if, it satisfies the additional condition (3) Every A ∈ Φ has a neighborhood B with B− ∈ Φ. If, furthermore, Φ satisfies 10. FLASQUE SHEAVES 39

(4) Every A ∈ Φ has a neighborhood B such that B− ∈ Φ and B− is paracompact, then we say Φ is paracompactifying. Recall that a space X is called paracompact iff X is Hausdorff and every open cover of X has a locally finite (open) refinement. For a subspace Y ⊆ X, and a family of supports Φ on X, the families Φ ∩ Y := {A ∩ Y : A ∈ Φ} ΦY := {A ∈ Φ: A ⊆ Y } are both both families of supports on Y . We emphasize that, in general, these are very different, so we caution the reader to take care in keeping track of which family of supports we are using. If F is an abelian sheaf on X and i : U,→ X is an open subset of X, then the X −1 global sections with support in Φ of the abelian sheaf F |U = i!i F on X are given

by ΓΦ(X, F ) = ΓΦU (U, F ). This is immediate from the deﬁnitions.

8. Sheaf cohomology

For any abelian presheaf F on X, and any family of supports Φ on X,

ΓΦ(X, F ) := {s ∈ F (X) : Supp s ∈ Φ} is a subgroup of Γ(X, F ) (it contains 0 because ∅ ∈ Φ, and it is closed under addition because Φ is closed under ﬁnite unions and passage to smaller closed sets) called the group of global sections of F with support in Φ. It is easy to see that F 7→ ΓΦ(X, F ) is a left exact functor Ab(X) → Ab. The corresponding right derived functors are i denoted HΦ(X, F ) and called the cohomology of F with support in Φ. In case Φ is the family of supports consisting of all closed subsets contained in a closed subspace i i Y ⊆ X,HΦ(X, F ) is usually denoted HY (X, F ) and is called the cohomology of F with support in Y .

Unlike the usual global section functor, ΓΦ does not generally commute with inﬁnite inverse limits, or even with inﬁnite products.

9. Acyclic sheaves

Let X be a topological space, Φ a family of supports on X. A sheaf F ∈ Ab(X) n is called ΓΦ-acyclic iﬀ HΦ(X, F ) = 0 for n > 0. (PUT SOMETHING HERE TO THE EFFECT THAT COHOMOLOGY CAN BE COMPUTED USING GAMMA ACYCLIC RESOLUTIONS)

10. Flasque sheaves

A sheaf F on a topological space X is called ﬂasque iﬀ the restriction map F (U) → F (V ) is surjective for every inclusion V ⊆ U of open subsets of X. 40 IV. COHOMOLOGY

Lemma 10.1. The direct image of a ﬂasque sheaf under any morphism of topological spaces is again ﬂasque.

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

Lemma 10.2. (Gluing sections) Suppose A = {Ai : i ∈ I} is a locally ﬁnite family of subsets of X with union A and we have sections si of an abelian sheaf F

over Ai for each i ∈ I which agree on overlaps (si|Aij = sj|Aij for all i, j ∈ I). Then

there is a unique section s of F over A such that s|Ai = si for each i ∈ I. Q Proof. Certainly there is a unique map s : A → x∈A Fx with Y s(x) = si(x) ∈ Fx ⊂ Fa a∈A

for every i ∈ I with x ∈ Ai. The only question is whether s satisfies the local coherence condition. To see this, fix a ∈ A and use local finiteness to find an open neighborhood U of a which meets only Ai1 ...Ain . The sections sij satisfy the local coherence condition at a, so there are open neighborhoods Uij of a and sections tij ∈ F (Uij ) such that tij (x) = sij (x) = s(x) for all x ∈ Uij ∩ Aij . Let V be the intersection of U and the Uij so V is an open neighborhood of a since there are only

finitely many ij. The set W where the restrictions tij |V all agree is an open subset of V containing a. Let t ∈ F (W ) be the common value. Then t witnesses the local coherence of s at a. Theorem 10.3. Let X be a topological space, Φ a family of supports on X. The class of flasque sheaves satisfies the following properties: (1) For any short exact sequence of sheaves 0 → F 0 → F → F 00 → 0 0 00 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 0 → F → F 00 → 0 on X with F 0 and F flasque, F 00 is flasque. (3) A direct summand of a flasque sheaf is flasque. (4) Any sheaf admits a monomorphism into a flasque sheaf. If Φ is paracompactifying, then the class of Φ soft sheaves satisfies the same properties.

Proof. The fact that summands of flasque or Φ soft sheaves are again flasque or Φ-soft follows from the fact that (F ⊕ G )(U) = F (U) ⊕ G (U) for any open U ⊆ X. Any sheaf embeds in a flasque and Φ-soft sheaf by taking the sheaf of discontinuous sections (see ). 10. FLASQUE SHEAVES 41

Now assume 0 → F 0 → F → F 00 → 0 is a short exact sequence of sheaves on X with F 0 flasque. We will first show that Γ(X, F ) → Γ(X, F 00) is surjective. Let s ∈ Γ(X, F 00). Consider the set A of all pairs (U, t) where U is an open subset of X, and t ∈ F (U) maps to s|U under 00 0 0 0 0 F (U) → F (U). Order A by declaring (U, t) ≤ (U , t ) iff U ⊆ U and t |U = t. By the sheaf properties for F and F 0, any chain in A has an upper bound, so by Zorn’s Lemma, A has a maximal element (U, t). If U = X we’re done. If not, there is some x ∈ X \ U. Since F → F 00 is surjective on stalks, we can find a neighborhood 0 0 00 V of x in X and a section t ∈ F (V ) such that t 7→ s|V under F (V ) → F (V ). 0 Note t|U∩V − t |U∩V ∈ F (U ∩ V ) actually lies in F (U ∩ V ) because both t|U∩V and 0 00 0 t |U∩V map to s|U∩V under F (U ∩ V ) → F (U ∩ V ). Since F is flasque, there 0 0 is a w ∈ F (V ) restricting to t|U∩V − t |U∩V on U ∩ V . It follows that t ∈ F (U) and t0 + w ∈ F (V ) agree on U ∩ V , hence glue to a section t˜ ∈ F (U ∪ V ). The section t˜ maps to s|U∪V because this can be checked on U and V by separation for F 00 and adding w ∈ F 0(V ) to t0 doesn’t change its image in F 00(V ). We conclude that (U ∪ V, t˜) contradicts maximality of (U, t). 00 00 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 have support in Φ, so we wish to find a t0 ∈ F 0(X) such that t − t0 has support in Φ— subtracting off t0 from t doesn’t change its image in F 00(X), so this will complete 0 0 the proof. Note t|X\Supp s ∈ F (X \ Supp s) is actually in F (X \ Supp s). Take t 0 0 0 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 and a Φ-soft sheaf is acyclic for any paracompactifying Φ. Any injective sheaf is ﬂasque.

Proof. Follows from the theorem by standard homological algebra. • Lemma 10.5. Let F → G be a resolution of F ∈ Ab(X) by ΓΦ-acyclic sheaves. Then for all p ≥ 0 we have p • p H (ΓΦG ) = HΦ(X, F ).

Proof. By standard homological algebra, there is a spectral sequence with p,q p q E2 = HΦ(X, G ) p+q q converging to HΦ (X, F ). If every G 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 is paracompactifying). Lemma 10.6. (Flasque is local) Let F be a sheaf on X, U an open cover of X. Then F is flasque if and only if F |U is a flasque sheaf on U for every U ∈ U. 42 IV. COHOMOLOGY

Proof. The implication =⇒ is obvious since (F |U )(V ) = F (V ). For the other implication, suppose s ∈ F (V ) and we want to show there is a lift of s to F (X). Consider the set of pairs (U, t) where U is a neighborhood of V and t ∈ F (V ) restricts to s on V . Argue as in the proof of the above theorem, using Zorn’s Lemma. Lemma 10.7. A ﬂasque sheaf is Φ soft for any paracompactifying Φ.

Proof. Follows immediately from (1.7).

11. Godement resolution

Given a sheaf F on a space X, let C 0(F ) be the sheaf of discontinuous sections Y U 7→ Fx, x∈U with the obvious restriction maps. There is a natural morphism of sheaves ε : F → 0 0 C (F ) taking t ∈ F (U) to the map (tx)x∈U ∈ C (F )(U). This map is injective 0 0 because if tx = tx for all x ∈ U then t = t because F is separated. Taking C 1(F ) := C 0(Cok ε) and continuing yields a resolution ε : F → C •(F ) called the Godement resolution or (by Godement) canonical resolution of F . It is clear that from the formula for sections that, for any sheaf F and any n ≥ 0, the sheaf C n(F ) is ﬂasque. A morphism of sheaves f : F → G induces a natural morphism of canonical resolutions. Theorem 11.1. For a sheaf F on a topological space X, and any family of p • ∼ p supports Φ, there is a canonical isomorphism H (ΓΦG (F )) = HΦ(X, F ).

Proof. The Godement resolution is, in particular, a ﬂasque resolution, and a ﬂasque resolution is ΓΦ acyclic, so this follows from standard homological algebra.

12. Soft sheaves

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

13. Filtered direct limits

The formation of direct limits of sheaves on a space X, which involves sheafification, can often be difficult to understand. However, for filtered direct limits, the situation simplifies greatly when X is compact or noetherian. In this section, we 13. FILTERED DIRECT LIMITS 43

begin by showing that global sections commute with filtered direct limits on a compact or noetherian space. We use these results to prove that appropriate classes of Γ-acyclic sheaves are also preserved under filtered direct limits on such spaces. Com- bining this with some general nonsense, we find that sheaf cohomology commutes with filtered direct limits on a compact or noetherian space.

Theorem 13.1. Let X be a topological space, c 7→ Fc a direct limit system of sheaves on X indexed by a ﬁltered category C. If X is quasi-compact, then the natural map

lim Γ(X, Fc) → Γ(X, lim Fc)(13.1) −→ −→ is injective. If, furthermore, X is Hausdorﬀ (i.e. compact), or noetherian, then it is an isomorphism.

Proof. Recall that the direct limit sheaf F := lim Fc is given by the sheafifi- −→ cation of the presheaf direct limit F pre, whose sections are given by pre F (U) = lim Fc(U). −→ The map (13.1) is the map on global sections induced by the sheafification map F pre → F . The injectivity when X is quasi-compact is not hard: Recall from the usual construction of filtered direct limits that an element [s] ∈ lim Fc(X) is represented −→ by an index c ∈ C and an element s ∈ Fc(X). Two such pairs [s] = (c, s), [t] = (d, t) are equivalent iff there are C-morphisms f : c → e and g : d → e so that

F (f)(s) = F (g)(t) ∈ Fe(X). Unwinding the sheafification construction, we see that [s] and [t] map to the same element of Γ(X, F ) iff there is a cover {Ui : i ∈ I} of X, a function e : I → C, and C-morphisms fi : c → e(i), gi : d → e(i) for each each i ∈ I so that

(13.2) F (fi)(s|Ui) = F (g)(t|Ui) ∈ Fe(i)(Ui). If X is quasi-compact, we can assume that I is finite, hence, since C is filtered, we can find a single index e ∈ C and maps hi : e(i) → e so that hifi = hjfj and higi = hjgj for every i, j ∈ I. Call the respective common maps f : c → e and g : d → e. From the equality (13.2) we see that

F (f)(s)|Ui = F (f)(s|Ui)

= F (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 on each pairwise intersection Uij = Ui ∩ Uj: For every x ∈ Uij, there are: an index c(i, j, x) ∈ C, C morphisms f = f(i, j, x): c(i) → c(i, j, x), g = g(i, j, x): c(j) → c(i, j, x), and a neighborhood W = W (i, j, x) of x in Uij such that

(13.3) F (f)(si|W ) = F (g)(sj|W ) ∈ Fc(i,j,x)(W ). The reader can spell out the details of when two such collections of data represent the same section—we will implicitly replace one such data set with an equivalent one in what follows. First, since X is compact, we can assume I is finite, hence, since C is filtered, we can assume that c := c(i) is independent of i ∈ I. (Choose an index c so that there are C morphisms fi : c(i) → C for each i ∈ I and replace si by F (fi)(si).) Now, by 1 a standard topological argument we can find a cover {Vi : i ∈ I} (same indexing set I!) of X such that V i ⊆ Ui for all i ∈ I. Now, using compactness of V ij and the fact that C is filtered, we can find an index c(i, j) ∈ C, a neighborhood W (i, j) of V ij in Uij, and a map f(i, j): c → c(i, j) so that c(i, j) (resp. f(i, j), f(i, j), W (i, j)) can serve as c(i, j, x) (resp. f(i, j, x), g(i, j, x), W (i, j, x)) in the compatibility condition for each x ∈ V ij. In fact, since there are only finitely many pairs (i, j), we can (using C filtered) even arrange that d := c(i, j) and the map f := f(i, j): c → d are independent of the pair (i, j). By the same kind of calculation we made in the injectivity argument, the compatibility condition (13.3) now implies that

F (f)(si)|Vij = F (f)(sj)|Vij.

Hence the local sections F (f)(si)|Vi ∈ Fd(Vi) glue to a global section s ∈ Fd(X). The image of s in lim Fc(X) clearly maps to our [s] under (13.1). −→ The argument in the case where X is noetherian is similar, but easier. In this case, since an open subset of a noetherian space is quasi-compact, the intersections Uij are quasi-compact. Using this quasi-compactness and the fact that C is ﬁltered, we ﬁnd an index c(i, j) and a map f(i, j): c → c(i, j) so that c(i, j) (resp. f(i, j), f(i, j), Uij) can serve as c(i, j, x) (resp. f(i, j, x), g(i, j, x), W (i, j, x)) in the compatibility condition for each x ∈ Uij. The rest of the argument is identical (replace U with V everywhere).

Corollary 13.2. Let X be a noetherian topological space, c 7→ Fc a ﬁltered direct limit system of sheaves on X. The direct limit sheaf F := lim Fc coincides −→ with the direct limit presheaf:

F (U) = lim Fc(U) −→ for every open subset U ⊆ X.

1This uses in an essential way the fact that X is Hausdorﬀ. The statement we want is an easy exercise in our case, but it is also a very special case of a general “Shrinking Lemma” (c.f. [Eng, 1.5.18]). 13. FILTERED DIRECT LIMITS 45

Proof. An open subset of a noetherian topological space is noetherian, so this follows from the previous Theorem. Alternatively, one can just directly prove that the direct limit presheaf is a sheaf by using Corollary 5.3 and the fact that a filtered direct limit of equalizer diagrams of sets is an equalizer diagram. (Filtered direct limits commute with finite inverse limits of sets.) Corollary 13.3. A filtered direct limit of flasque sheaves on a noetherian topological space is flasque. A filtered direct limit of soft sheaves on a compact topological space is soft.

Proof. Since a filtered direct limit of surjective maps (of sets) is again surjective, the first statement is immediate from Corollary 13.2, and the second is immediate from Theorem 13.1 (note Z is compact when X is compact). Theorem 13.4. Sheaf cohomology commutes with filtered direct limits on a compact or noetherian space X. That is: Let c 7→ Fc be a filtered direct limit system of sheaves of abelian groups on a compact or noetherian space X. Then the natural map p p lim H (X, Fc) → H (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 sides of (13.4) are natural in (c 7→ Fc) and may viewed as functors A → Ab. These functors agree when p = 0 by Theorem 13.1, so, by general nonsense, they will agree for all p once we prove that both sides are effaceable δ-functors A → Ab. To see that both sides are δ-functors, first note that the formation of filtered direct limits of abelian groups lim : Hom(C, Ab) → Ab −→ is an exact functor (this is an easy exercise and is well-known). It follows easily from this fact and the fact that sheafification is exact that lim : Hom(C, Ab(X)) → Ab(X) −→ is also exact. Now the fact that both sides are δ-functors follows from the fact that cohomology H• is a δ-functor. It remains only to prove that both sides are effaceable. For any F ∈ Ab(X), we can find an injection F ,→ F 0 from F into a flasque sheaf, functorial in F (use the first step of the Godement resolution, for example), so we can find an injection from 0 0 any limit system (c 7→ Fc) into a limit system (c 7→ Fc) where each Fc is flasque. It now remains only to prove that, for p > 0, both sides of (13.4) vanish when the Fc are flasque. The left side vanishes (with no hypotheses on X) because a flasque sheaf 46 IV. COHOMOLOGY

has no higher cohomology. In the noetherian case, the right side vanishes because a filtered direct limit of flasques is flasque (Corollary 13.3). In the compact case, the right side vanishes because a flasque sheaf is soft (Lemma ??), a direct limit of soft sheaves is soft (Corollary 13.3), and a soft sheaf on a (para)compact space has no higher cohomology.

14. Vanishing theorems

Recall that a closed subset Z of a topological space X is called irreducible iﬀ it cannot be written as a union of two proper closed subsets. The dimension of a topological space X (denoted dim X) is the maximal length n of a strictly increasing chain ∅= 6 Z0 ( Z1 ( ··· ( Zn of non-empty irreducible closed subsets of X. We set dim X = ∞ if X contains arbitrarily long such chains. Theorem 14.1. (Grothendieck’s Vanishing) Let X be a noetherian topolog- p ical space of dimension n, Φ a family of supports on X. Then HΦ(X, F ) = 0 for every p > n and every abelian sheaf F on X.

Proof. The following cohomology vanishing theorem sometimes comes in handy. Theorem 14.2. Let X be a zero dimensional, second countable topological space, F an abelian sheaf on X. Then Hp(X, F ) = 0 for every p > 0. Indeed, the global section functor is exact.

Proof. It is enough to prove that, for any surjection f : F → G of sheaves on X, the map Γ(X, F ) → Γ(X, G ) is surjective. Let s ∈ Γ(X, G ). Since F → G is surjective, there are a cover U = {Ui : i ∈ I} and sections ti with f(Ui)(ti) = s|Ui for every i ∈ I. Since X is zero dimensional and second countable, we may assume, after possibly passing to a reﬁnement and restricting sections that U = {U1,U2,... } is countable and that every Ui is open-and-closed in X. By replacing Ui with the open set i−1 U i := Ui \ (∪j=1Uj)

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

15. Higher direct images

Let f : X → Y be a morphism of topological spaces. The direct image functor f∗ : Ab(X) → Ab(Y ) is easily seen to be left exact (this left exactness is inherited 16. LERAY SPECTRAL SEQUENCE 47

from left exactness of the section functors). Since Ab(X) has enough injectives, we p can form the right derived functors R f∗ : Ab(X) → Ab(Y ). p Lemma 15.1. For a sheaf F ∈ Ab(X), the sheaf R f∗F ∈ Ab(Y ) on Y is nat- p −1 urally isomorphic to the sheaf associated to the presheaf U 7→ H (f (U), F |f −1(U)). p In particular, the stalks of R f∗F are given by p p −1 (R f∗F )y = lim H (f (U), F |f −1(U)), −→ where the ﬁltered direct limit system is over neighborhoods U of y in Y .

Proof. Certainly this is true for p = 0, so by general nonsense it suffices to prove that the indicated sheafifications form an effaceable δ-functor from Ab(X) to Ab(Y ). To see that they form a δ-functor, first observe that before sheafifying, the presheaves defined by the above formula certain form a δ-functor Ab(X) → PAb(Y ). But sheafification is exact, so the sheafified versions also form a δ-functor. To see that it is effaceable, it will suffice to show that the presheaves defined by the above formula are zero when p > 0 and F is injective. This is clear from the −1 fact that F |f −1(U) is also injective in Ab(f (U)) by (??), hence has no higher cohomology.

16. Leray spectral sequence

Theorem 16.1. (Leray) Let f : X → Y be a continuous map of topological spaces, F a sheaf on X. There is a ﬁrst quadrant spectral sequence p,q p q p+q E2 = H (Y, R f∗F ) =⇒ H (X, F ).

Proof. We have Γ(Y, f∗F ) = Γ(X, F ) for any sheaf F on X, and by (10.1) f∗ takes injective sheaves on X (which are ﬂasque) to ﬂasque sheaves on Y (which are Γ(Y, ) acyclic by (10.4)), so this spectral sequence is a special case of the Grothendieck spectral sequence.

One consequence of the Leray spectral sequence is the following celebrated theorem. Corollary 16.2. (Vietoris-Begle) Let f : X → Y be a proper map with acyclic ﬁbers: , p = 0 Hp(f −1(y), ) = Z Z 0, 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 any constant sheaf. We will show that the Leray spectral sequence (??) degenerates by 48 IV. COHOMOLOGY

−1 p −1 showing that f∗f F = F and R f∗f F = 0 for p > 0. This follows from the formula p p −1 −1 (R f∗ZX )y = H (f (y), (f F )|f −1(y)) −1 −1 of (15.1) and the fact that (f F )|f −1(y) is just the constant sheaf on f (y) associated to the abelian group Fy, as is clear from functoriality of inverse images and commutativity 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 a space X is a sheaf F ∈ Ab(X), which, locally on X, is isomorphic to the constant sheaf AX . The Leray spectral sequence is most easily understood when f : X → Y is a ﬁber bundle with ﬁber F (i.e. Y can be covered with open sets Ui so that −1 p f (Ui) is isomorphic, over Ui, to Ui × F ). In this case, the sheaf R f∗ZX will be p an H (F, Z)-local system on Y .

19. Examples

20. Exercises

The exercises below are intended to show that one cannot prove (1.7) without some assumptions along the lines of (2) or (1). Exercise 20.1. Construct an example where f : X → Y is a closed embedding of ﬁnite topological spaces, but the natural monomorphism of (1.7) is not surjective. Exercise 20.2. Let S be the Sorgenfrey line (R with basic open sets [a, b) for a < b), let Y = S × S, and let f : X → Y be the inclusion of the skew diagonal {(−x, x): x ∈ R} in the subspace topology it inherits from Y . Let F be the constant sheaf associated to the two element set {0, 1}. Show that X has the discrete topology, hence −1 Y Γ(X, f F ) = Fx x∈X = {0, 1}R. Show that the characteristic function of the irrational numbers f ∈ {0, 1}R is not in the image of the natural map of (1.7). Hint: Baire category theorem. 20. EXERCISES 49

Exercise 20.3. Let X be a locally compact Hausdorﬀ space. Show that the family of supports Φ consisting of compact subsets of X is paracompactifying. Co- homology with support in Φ in this situation is usually called compactly supported cohomology. Exercise 20.4. Let A be an abelian group, X a topological space, F a subsheaf of AX . Show that for any a ∈ A, the set

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

Exercise 20.5. Let F be a subsheaf of ZX for a quasi-compact space X. Show that there are only finitely many n ∈ Z>0 such that Un (notation from the previous exercise) is nonempty. For x ∈ X, let n(x) be the smallest non-negative integer such that n(x) ∈ Fx. Show that the set V := {x ∈ X : n(x) > 0} is open in X. Show that the sets Vs := {x ∈ V : n(x) ≤ s} are open in V and that Vs = V for s sufficiently large. Then Zs := Vs \ Vs−1 is locally closed subspace of X. := Show X ∼ X that the sheaves Fi := (F |Vi ) define a finite filtration of F with Fs/Fs−1 = ZZs . Exercise 20.6. Let X be an infinite set with the cofinite topology, and let x ∈ X ! be a point. Show directly that HomX (x∗Z, ZX ) = 0. Show that x ZX = 0 (c.f. 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 topological space X is the smallest family of subsets of X containing the open subsets, and closed under finite intersections and complements (hence also containing the closed subsets and closed under finite unions). If X is noetherian, show that a union of two locally closed subsets is locally closed, and hence the complement of a locally closed subset is locally closed. Conclude that the locally closed subsets coincide with the constructible subsets. On the other hand, if X = R, show by example that the union of a closed subset and a locally closed subset need not be locally closed. Hint: Take a sequence of positive real numbers

x1 > x2 > x3 > ··· with xi → 0 as i → ∞. Let

A := [x1, x2) ∪ [x3, x4) ∪ · · · and let B = {0}. Note that A is locally closed, but A ∪ B is not. Exercise 20.8. Show that a direct limit of ﬂasque sheaves need not be ﬂasque in general.

Exercise 20.9. Let i : U,→ X be an open set. Show that the sheaf ⊕hU Z on X considered in Section 4 coincides with the sheaf i!ZU . 50 IV. COHOMOLOGY

Exercise 20.10. Give an example of a sheaf F on R such that the set {x ∈ R : Fx 6= 0} is not locally closed in R. Hint: Recall that a subspace of a locally compact space is locally closed iff it is locally compact. Let α be the smallest ordinal with infinitely many limit ordinals less than it. Show that the space α+1 (with the order topology) embeds in R (as does any countable ordinal with its order topology), but is not locally compact. One can arrange that Fx 6= 0 if and only if x is in the image of this embedding. Exercise 20.11. If f : F → G is a surjective morphism of flasque sheaves then the induced maps f p : C p(F ) → C p(G ) are surjective for every p ≥ 0. Hint: Induct on p using 10.3. Exercise 20.12. Supp F ⊆ Supp C p(F ) for every p. Exercise 20.13. For any nonnegative integer n, the functor Ab(X) → Ab(X) given by F 7→ C n(F ) is exact. Hint. Check on stalks for p = 0 and induct using the 3x3 Lemma. Exercise 20.14. Construct an example of a topological space X, a basis B ⊆ Ouv(X) and a sheaf F ∈ Ab(X) such that F (U) → F (V ) is surjective for every B morphism V,→ U, but F is not flasque. Exercise 20.15. Show that the Sorgenfrey line S is zero dimensional, but not second countable. Is there a sheaf of abelian groups on S (or S × S) with nonvanishing higher cohomology? c.f. (14.2). CHAPTER V

Base Change

1. Proper maps

Some of the most topologically intricate arguments in sheaf theory occur in the study of the “compatibility” of direct and inverse images with base change—the subject of this section and the next.

Definition 1.1. Let X be a topological space, Y ⊆ X a subspace, {Ai : i ∈ I} a family of subsets of X. We say that Y is relatively Hausdorff in X iff any two distinct points of Y have disjoint neighborhoods in X. We say that {Ai} is locally finite iff any point x ∈ X has a neighborhood U in X such that {i ∈ I : U ∩ Ai 6= ∅} is finite. We say that X is paracompact iff every open cover of X has a locally finite open refinement. Definition 1.2. We say that a map f : X → Y of topological spaces is proper iff it is closed and each fiber f −1(y) is compact and relatively Hausdorff in X.

The condition that the ﬁbers of f be relatively Hausdorﬀ is equivalent to saying that the diagonal X ⊆ X ×Y X is closed (i.e. f is separated). It is easy to see that this condition is closed under base change. The property of being a closed map is 2 certainly not preserved preserved under base change. For example, π1 : R → R is not closed even though it is a base change of the closed map from R to a point. Never-the-less, we have: Lemma 1.3. Proper maps of topological spaces are closed under base change and composition. Being proper is local on the base.

Proof. Everything is straightforward except perhaps the statement that the base change π1 : Z ×Y X → Z of a proper map f : X → Y along an arbitrary map g : Z → Y is a closed map. To see this, suppose W ⊆ Z ×Y X is closed, and z∈ / π1(W ). We want to see that z has a neighborhood disjoint from π1(W ), so that −1 π1(W ) is closed. Since z∈ / π1(W ), we know that, for any x ∈ f (g(z)), (z, x) ∈/ W . Since W is closed we can find a basic open neighborhood Ux ×Y Vx of (z, x) disjoint −1 from W . Since f (g(z)) is compact, we only need finitely many such Ux ×Y Vx to −1 cover π1 (z), so we can intersect the finitely many Ux that we need to find a single open neighborhood U of z ∈ Z and a neighborhood V of f −1(g(z)) in X so that −1 U ×Y V is a neighborhood of π1 (z) in Z ×Y X. The set T := Y \ f(X \ V ) is open in Y since f is closed, and T contains g(z) because V contains f −1(g(z)), so g−1(T ) is an open neighborhood of z in Z. I claim that U ∩ g−1(T ) is disjoint from 51 52 V. BASE CHANGE

0 −1 0 −1 0 π1(W ). Indeed, if z ∈ U ∩ g (T ), then g(z ) ∈ T , so f (g(z )) ⊆ V , hence no 0 point (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 iff f −1(K) is compact for every compact subspace K ⊆ Y . If f : X → Y and g : Y → Z are maps of locally compact Hausdorff spaces such that gf and g are proper, then f is proper.

Proof. (=⇒) Suppose {Ui : i ∈ I} is a family of open subsets of X covering Z := f −1(K). Since f is proper, each fiber of f is compact, so for each x ∈ K, there −1 is a finite subset Ix ⊆ I such that Ux := ∪i∈Ix Ui covers f (x). Since the proper map f is closed, Vx := Y \ f(Z \ Ux) is an open neighborhood of x in Y , and it is −1 clear that f (y) ⊆ Ux for every y ∈ Vx ∩ K. Since K is compact, we can find a finite subset S ⊆ K so that {Vx : x ∈ S} covers K. Then J := ∪x∈SIx is a finite subset of I and such that the Uj cover Z. For (⇐=), the only issue is to prove that f is closed. Suppose Z ⊆ X is closed and y∈ / f(Z), but y is in the closure of f(Z). Choose a neighborhood W of y in Y with compact closure W . Then y is still in the closure of W ∩ f(Z). Since f −1(W ) is compact and Z is closed, Z ∩ f −1(W ) is compact, so K := f(Z ∩ f −1(W )) is also compact, hence closed. But this K is a closed subset of Y containing W ∩f(Z) with y∈ / K, which contradicts y being in the closure of W ∩ f(Z). For the final statement, note that by the first part of the lemma, it suffices to prove that f −1(K) is compact for any compact K ⊆ Y . Certainly g(K) is compact in Z, so (gf)−1(g(K)) is compact because gf is proper. But f −1(K) is a closed −1 subspace of (gf) (g(K)), so it is also compact.

The following technical lemmas are the main topological input for the proper base change theorem: Lemma 1.5. Let X be a topological space, Y ⊆ X a compact, relatively Hausdorﬀ subspace, {Ui : i ∈ I} a family of opens in X covering Y . Then there is a ﬁnite family {Vj : j ∈ J} of opens in X whose union V contains Y and a map φ : J → I such that (V ∩ V j) ⊆ Uφ(j) for all j ∈ J.

Proof. Fix y ∈ Y . Choose φ(y) ∈ I so that y ∈ Uφ(y). By the relatively Haus- 0 dorﬀ assumption, we can choose, for each y ∈ Y \Uφ(y), disjoint open neighborhoods 0 Sy,y0 ⊆ Uφ(y) and Wy,y0 of y and y respectively. By compactness of Y , there is a ﬁnite subset Iy ⊆ Y \ Uφ(y) so that 0 0 Ty := Uφ(y) ∪ ∪y ∈Iy Wy,y

0 is an open neighborhood of Y in X. Set Sy := ∩y∈Iy Sy,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∈J Sj contains Y . Set T := ∩jTj. Then it is easy to see that {Vj := Sj ∩ T } is as desired. 1. PROPER MAPS 53

Lemma 1.6. Let X be a paracompact topological space, {Ui : i ∈ I} an open cover of X. Then there exists a locally ﬁnite open cover {Vj : j ∈ J} of X and a map φ : J → I so that V j ⊆ Uφ(j).

Proof. Fix x ∈ X and choose µ(x) ∈ I so that x ∈ Uµ(x). For each y ∈ X\Uµ(x), choose disjoint open neighborhoods Sx,y ⊆ Uµ(x) and Wx,y of x and y respectively. By paracompactness of X, we can ﬁnd a locally ﬁnite family of opens {Tk : k ∈ K} and a map ψ : K → X \ Uµ(x) so that Tk ⊆ Wx,ψ(k) and so that the Tk and Uµ(x) cover X. Find an open neighborhood Rx ⊆ Uµ(x) so that

K(x) := {k ∈ K : Tk ∩ Rx 6= ∅}

is finite and let Sx := ∩k∈K(x)Sx,ψ(k). Then it is easy to see that Sx ⊆ Uµ(x). Choose a locally finite refinement {Vj : j ∈ J} of the cover {Sx : x ∈ X}, so we have a map ν : J → X with Vj ⊆ Sν(j). Then this {Vj} is as desired if we set φ := µν. Lemma 1.7. Suppose X is a topological space and Y ⊆ X is a subspace. Then the natural map

lim (U) → Γ(Y, |Y ) −→ F F U⊇Y is monic (the direct limit is over open neighborhoods U of Y in X). Suppose furthermore that at least one of the following assumptions holds: (1) Y is compact and relatively Hausdorﬀ in X. (2) There is a paracompact neighborhood U of Y in X so that Y is closed in U. Then the natural map above is an isomorphism.

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

Proof. To prove the first statement, suppose U, V ⊇ Y are open neighborhoods and s ∈ F (U) and t ∈ F (V ) agree in Γ(Y, F |Y ). Then sy = ty ∈ Fy for any y ∈ Y , so we can find a neighborhood Uy of y in X (contained in U ∩ V ) with s|Uy = t|Uy. Then W := ∪y∈Y Uy is a neighborhood of y in X on which s and t agree, hence they are equal in the (filtered) direct limit. For the second statement it remains only to prove surjectivity. By definition of the inverse image F |Y , a section s ∈ Γ(Y, F |Y ) can be viewed as an element Y s = (s(y)) ∈ Fy y∈Y

so that there is a family of opens {Ui : i ∈ I} in X with Y ⊆ U := ∪Ui and sections i i s ∈ F (Ui) so that s(y) = sy for all y ∈ Ui ∩ Y . We can always pass to a smaller i neighborhood of Y in U and restrict our cover Ui and our sections s , so under either assumption, we can assume that {Ui} is a locally ﬁnite cover of U. When the second assumption holds, we can (and will) assume without loss of generality that X itself is paracompact and Y is closed in X. Our goal is to produce (after 54 V. BASE CHANGE

possibly shrinking U to a smaller neighborhood of Y in X) a section t ∈ F (U) so that ty = s(y) for all y ∈ Y . I next claim that we can further assume that there is a locally finite open cover {Vj : j ∈ J} of U and a map φ : J → I with (V j ∩ U) ⊆ Uφ(j). To see this under the first assumption, apply Lemma 1.5 and then make the replacements U 7→ U ∩ V , Ui 7→ Ui ∩ V , Vj 7→ Vj ∩ U, etc. to assume V = U. (We can even assume that J is finite in this case.) To see this under the second assumption, apply Lemma 1.6 to the cover of X given by the Ui and X \ Y .

Now let {Tx : x ∈ U} be neighborhoods in U witnessing the local ﬁniteness of {Ui} and {Vj}, so the sets

I(x) := {i ∈ I : Tx ∩ Ui 6= ∅} 0 I (x) := {i ∈ I : x ∈ Ui}

J(x) := {j ∈ J : x ∈ Vj} are all finite and φ(J(x)) ⊆ I0(x) ⊆ I(x). For any x ∈ U, the sections {si : i ∈ I0(x)} all have the same stalk (namely s(x)) at x, so, since I0(x) is finite, we can find a neighborhood 00 Wx ⊆ ∩i∈I0(x)Ui ∩ Tx i 0 00 of x in U so that the sections {s : i ∈ I (x)} all agree on Wx . Call this common 0 00 section t (x) ∈ F (Wx ). Set 0 00 Wx := Wx ∩ ∩j∈J(x)Vj 0 Wx := Wx ∩ ∩j∈J(x):φ(j)∈I(x)\I0(x)(X \ V j) 0 = Wx \ ∪j∈J(x):φ(j)∈I(x)\I0(x)V j

so that Wx is a neighborhood of x in U because

0 φ(j) ∈ I(x) \ I (x) =⇒ x∈ / Uφ(j)

=⇒ x∈ / V j.

0 Set t(x) := t (x)|Wx.

I claim that t(x1)|(Wx1 ∩ Wx2 ) = t(x2)|(Wx1 ∩ Wx2 ) for all x1, x2 ∈ U so that the t(x) glue to a global section over U (which will then clearly map to s as desired).

We can check this on stalks at a typical point u ∈ Wx1 ∩ Wx2 . Since u ∈ Wx1 , in particular u ∈ Tx1 . Since u ∈ Wx2 there is j2 ∈ J(x2) so that u ∈ Vj2 . Then φ(j2) is in I(x1) because

u ∈ Tx1 ∩ Vj2 ⊆ Tx1 ∩ Uφ(j2), 0 but φ(j2) must actually be in I (x1) ⊆ I(x1), because otherwise u wouldn’t be in 0 0 φ(j2) Wx1 since it is in V j2 . Since φ(j2) ∈ I (x1) ∩ I (x2), we have t(x1)y = sy = 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

g X0 / X

f 0 f h Y 0 / Y of topological spaces. For any sheaf F on X, there is a natural map

−1 0 −1 (2.1) h f∗F → (f )∗g F

0 −1 of sheaves on Y , deﬁned as follows. First of all, by the adjointness (h , h∗), it is equivalent to give a map

0 −1 (2.2) f∗F → h∗(f )∗g F

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

0 −1 −1 0 −1 −1 (h∗(f )∗gpreF )(U) = (gpreF )((f ) h U) = lim (V ). −→ F V ⊇g((f 0)−1h−1(U))

But commutativity implies

g((f 0)−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

0 −1 (f∗F )(U) → (h∗(f )∗gpreF )(U) given by the structure map to the direct limit. This is clearly natural in U since the restriction maps are induced by the restriction maps for F on both sides, hence we 0 −1 0 obtain a map f∗F → h∗(f )∗gpreF . Composing with h∗(f )∗ of the sheaﬁﬁcation −1 −1 map gpreF → g F yields the map (2.2). There are variants for pushforward with proper support, as follows: If the diagram is cartesian, then there is a natural map

−1 0 −1 (2.3) h f!F → (f )!g F deﬁned in the same way. The key point is that, if s ∈ F (U) has proper support Z over Y , then g−1s ∈ F (g−1(U)) has proper support over Y 0 because

Supp(g−1s) = g−1 Supp s = g−1Z, 56 V. BASE CHANGE and we have a cartesian diagram

g−1Z / Z

X0 / X

Y 0 / Y where Z → Y is proper, hence g−1Z → Y 0 is also proper (Lemma 1.3). One also has the map (2.3) when the spaces X,X0,Y,Y 0 in the diagram are locally compact (and Hausdorff) and g and h are proper. In this case the point is that f 0|g−1Z : g−1Z → Y 0 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 the composition (f|Z)(g|g−1Z) is proper (Lemma 1.3) and we have h(f 0|g−1Z) = (f|Z)(g|g−1Z) by commutativity, hence f 0|g−1Z is proper by the 2-out-of-3 property of proper maps from Lemma 1.4. An important special case of the map (2.5) occurs when h : Y 0 → Y is the 0 inclusion of a point y ∈ Y and the diagram is cartesian so that X = Xy is the fiber −1 0 of f over y ∈ Y . In this case, h is the stalk functor F 7→ Fy and f is the functor of global sections over Xy, so (2.5) can be viewed as a map

(2.4) (f∗F )y → Γ(Xy, F |Xy).

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 reduces to this special case rather easily. Theorem 2.1. Suppose f : X → Y is a proper map. Then the natural map (2.4) is an isomorphism for every y ∈ Y and every sheaf F on X. For any F ∈ Ab(X), i i the natural map (R f∗F )x → H (Xy, F |Xy ) is an isomorphism for all i.

Proof. The natural map in question is the natural map −1 lim (f (U)) → Γ(Xy, |Xy), −→ F F U where the direct limit runs over all neighborhoods U of y in Y . But the fact that f is closed implies that the f −1(U) are coﬁnal in the set of all neighborhoods of the ﬁber Xy, so the map in question is the same as the map

lim (V ) → Γ(Xy, |Xy), −→ F F V where V runs over all neighborhoods of Xy in X. Since the ﬁber Xy is compact and relatively Hausdorﬀ in X, this map is an isomorphism by Lemma 1.7(1). 3. HOMOTOPY INVARIANCE 57

The second statement follows from the ﬁrst by general “universal δ-functors” nonsense. Theorem 2.2. Suppose g X0 / X

f 0 f h Y 0 / Y is a cartesian diagram of topological spaces with f proper. Then the natural map −1 0 −1 (2.5) h f∗F → (f )∗g F is an isomorphism for every sheaf F on X.

Proof. We can check this on stalks at any y0 ∈ Y 0. But f 0 is also proper (Lemma 1.3) and f and f 0 have the same ﬁbers, so we reduce formally to the situation of the previous theorem.

3. Homotopy invariance

Let I := [0, 1] denote the unit interval. For a continuous map H : X ×I → Y and t ∈ I, let ιt : X,→ X × I denote the map x 7→ (x, t) and set Ht := Hιt : X → Y . Maps f, g : X → Y are called homotopic if there is a homotopy f to g: a map H : X × I → Y such that H0 = f and H1 = g. Being homotopic is an equivalence relation on maps preserved by compositions. Sheaf cohomology enjoys the following homotopy invariance property: Theorem 3.1. Let f, g : X → Y be homotopic maps of topological spaces. For any abelian group A, the maps f ∗, g∗ :Hp(Y,A) → Hp(X,A) coincide.

Proof. Let H : X × I → Y be a homotopy from f to g. The maps ι0, ι1 induce isomorphisms on cohomology by the “2-out-of-3” property of isomorphisms, the fact that π1ιi = Id : X → X, and the fact that π1 : X × I → X induces isomorphisms on all constant sheaf cohomology by Vietoris-Begle (use the fact that the interval is acyclic—see the computations in (??)). The result is now immediate since Hι0 = f and Hι1 = g.

Warning: Do not read too much into the “homotopy invariance” property above. Many people falsely believe that a sheaf of abelian groups on a contractible topological space can have no higher cohomology groups (this is true for constant sheaves). In fact, the very existence of the Leray spectral sequence ensures that this cannot be the case:

Example 3.2. Let 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 branched over {±1} ⊆ I. Certainly f is a proper map by (3.4), so we easily conclude that 58 V. BASE CHANGE

>0 R f∗ = 0 by the stalk formula (3.5) and the fact that no sheaf has higher cohomology on a space with at most two points. The Leray spectral sequence for f therefore yields an isomorphism 1 1 1 H (S , Z) = H (I, f∗ZS1 ). Note that this group is Z by the computations in (??). To get a feel for the sheaf F := f∗ZS1 , note that F ([−1, 1)) = Z F ((−1, 1]) = Z 2 F ((−1, 1)) = Z and both restriction maps F ([−1, 1)), F ((−1, 1]) → F ((−1, 1)) are the diagonal ∆ : Z → Z2. CHAPTER VI

Duality

1. Adjoints for immersions

Let f : X → Y be a morphism of topological spaces. We always have a left −1 adjoint f : Ab(Y ) → Ab(X) to f∗ : Ab(X) → Ab(Y ). It would also be desirable, for many reasons, to have a right adjoint f ! : Ab(Y ) → Ab(X) to f∗. In general, we know this is impossible because the existence of a right adjoint ! f to f∗ would imply that f∗ commutes with all direct limits, which it generally does not (even when Y is a point). In the general case then, one is forced to look for the right adjoint f ! only in an appropriate derived category sense. However, in the case where f is a closed embedding (so f∗ does preserve direct limits) we will construct the desired right adjoint f ! in this section by elementary means. In fact, it is natural to work not only with closed embeddings, but with immersions. Recall that f : X → Y is an immersion iﬀ f can be factored f = ji as an open embedding i followed by a closed embedding j. In other words, f is an immersion iﬀ it is an isomorphism onto a subset A ⊆ X which can be written A = Z ∩ U where Z is closed in X and U is open in X. Such a subset A is called locally closed, and an immersion is hence sometimes called a locally closed embedding. The term “locally closed” is used because of the following:

Lemma 1.1. A ⊆ X is locally closed iﬀ, for each x ∈ A there is a neighborhood Ux of x in X such that A ∩ Ux is closed in Ux.

Proof. The implication =⇒ is clear. If A satisﬁes the second condition, then A will be closed in U := ∪x∈AUx because being closed is local, hence A = U ∩ Z for some Z closed in X, i.e. A is locally closed in our original sense.

Here are the basic facts about immersions, all of which follow easily from the lemma and discussion above.

Proposition 1.2. An open embedding is an immersion, and a closed embedding is an immersion. Immersions are the smallest class of morphisms containing open embeddings and closed embeddings and closed under composition. Immersions are closed under base change, and the property of being an immersion is local on the codomain (base). 59 60 VI. DUALITY

In a noetherian topological space, locally closed subsets are the same as constructible subsets, but this is not true in general (Exercise 20.7).

Lemma 1.3. Let j : Z,→ X be the inclusion of any subspace. Then j∗ : PAb(Z) → PAb(X) is fully faithful, hence j∗ : Ab(Z) → Ab(X) is also fully faithful.

Proof. We want to show that for any F , G ∈ PAb(Z), the map

j∗ : HomPAb(Z)(F , G ) → HomPAb(X)(j∗F , j∗G ) is bijective. Given any morphism PAb(X) morphism f : j∗F → j∗G and any U ∈ Ouv(X), the fact that the f(U) respect the restriction maps implies that the morphism f(U): F (U ∩ Z) → G (U ∩ Z) depends only on U ∩ Z: Indeed, if V ∩ Z = U ∩ Z, then both are also equal to (V ∩ U) ∩ Z and the restriction maps for the inclusions (V ∩ U) ,→ V and (V ∩ U) ,→ U are the identity for both presheaves j∗F , j∗G . In other words, for U ∈ Ouv(Z), we can deﬁne a map f(U): F (U) → G (U) unambiguously by choosing any U 0 ∈ Ouv(X) with U = U 0 ∩Z and setting f(U) := f(U 0). The maps f(U) clearly respect the restriction maps since this is true for the maps f(U 0) and any inclusion of opens in Z lifts to an inclusion of opens in X. The function f 7→ f hence deﬁnes an inverse to our j∗ above. Theorem 1.4. Let j : Z,→ X be a closed subspace of a topological space X.

(1) The direct image functor j∗ : Ab(Z) → Ab(X) commutes with all direct and inverse limits and is fully faithful. Its essential image consists of those F ∈ Ab(X) where Supp F ⊆ Z. (2) For F ∈ Ab(X), deﬁne j!F ∈ Ab(Z) by

U 7→ ΓY ∩U (U, F ) = {s ∈ F (U) : Supp s ⊆ Z ∩ U} on X (with the obvious restriction maps inherited from those of F ). The ! functor j : Ab(X) → Ab(Z) is right adjoint to j∗ and the adjunction ! morphism F → j j∗F is an isomorphism for any F ∈ Ab(Z).

Proof. First of all, any section s of j∗F over U ∈ Ouv(X) trivially satisﬁes Supp s ⊆ Z ∩ U since the stalks of j∗F at any x ∈ X \ Z are zero, so the ﬁnal statement of the theorem is clear. Also, if f : j∗F → G is any Ab(X) morphism, U ∈ Ouv(X), s ∈ (j∗F )(U), then obviously Supp f(U)(s) ⊆ Supp s ⊆ Z ∩ U, −1 so f factors uniquely through the subsheaf j∗j G ⊆ G , hence we have −1 HomAb(X)(j∗F , G ) = HomAb(X)(j∗F , j∗j G ).

But by Lemma 1.3, j∗ is fully faithful, so −1 −1 HomAb(X)(j∗F , j∗j G ) = HomAb(Z)(F , f G ) 2. EXTENSION BY ZERO 61

and putting the two together yields the desired adjunction. The functor j∗ commutes ! −1 with all limits because it is both a left adjoint (to j ) and a right adjoint (to j ). Theorem 1.5. Let i : U,→ X be an open subspace of a topological space X. For pre a F ∈ Ab(U), let i!F be the sheaf associated to the presheaf i! F deﬁned by (V ),V ⊆ U V 7→ F {0}, otherwise. −1 The functor i! : Ab(U) → Ab(X) is left adjoint to i (i.e. the restriction functor F 7→ F |U).

pre −1 Proof. The functor i! : PAb(U) → PAb(X) is clearly left adjoint to i : PAb(X) → PAb(U), so the desired adjointness follows formally from the adjointness property of sheafification. Remark 1.6. Unravelling the local coherence condition in the sheafification construction (7), we see that the sheaf i!F can be explicitly described as follows: Q A section s ∈ (i!F )(V ) is an element s = (s(x)) ∈ x∈U∩V Fx satisfying the two conditions: (1) For every x ∈ V \U there is a neighborhood W of x in V such that s(x0) = 0 for all x0 ∈ W ∩ U. (2) For every x ∈ V ∩U, there is a neighborhood W of x in V ∩U and a section 0 0 t ∈ F (W ) such that tx0 = s(x ) for x ∈ W .

2. Extension by zero

Extension by zero will allow us to construct a kind of “wrong way” map which is the basis for Poincar´eand 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 (V ),V ⊆ U ipre : V 7→ F ! F 0, otherwise is given by (i!F )(V ) = {s ∈ F (V ∩ U) : Supp s is closed in V }. The stalk of i!F at x ∈ X is Fx if x ∈ U and zero otherwise.

Proof. First note that i!F is a sheaf because being closed is a local property. If V ⊆ U, then U ∩ V = V and the support of any s ∈ F (V ) is certainly closed pre in V , so there is a natural morphism i! F → i!F . To see that this induces an pre + isomorphism (i! F ) → i!F , it suﬃces to check that it is an isomorphism on pre stalks. Certainly if x ∈ U ⊆ X, then (i! F )x → (i!F )x is an isomorphism because both sheaves coincide on all suﬃciently small neighborhoods of x (namely those contained in U). It remains only to show that (i!F )x = 0 for x ∈ X \ U. Indeed, if V is any neighborhood of x and s ∈ (i!F )(V ) any section, then s restricts to zero on the neighborhood V \ Supp s of x and is hence zero in the stalk. 62 VI. DUALITY

Corollary 2.2. The functor ι! : Ab(U) → Ab(X) is exact for any open set ι : U,→ X.

Proof. If 0 → F 0 → F → F 00 → 0 is exact on U, then one easily checks that 0 00 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 denoted X F ). It is manifestly a subsheaf of i∗F , though the two do not generally coincide. For example, if i : U → X is the inclusion of the open interval (0, 1) in the closed

interval [0, 1] and F = AU is a constant sheaf on U, then i∗F is just the constant sheaf AX , which has stalk A at every point of X, while the stalk of i!F is zero at 0, 1 ∈ X. Whenever i : U,→ X and j : V,→ X are open sets with U ⊆ V and F ∈ −1 −1 Ab(X), we have a natural morphism i!i F → j!j F . On W ∈ Ouv(X), this map sends a section s ∈ F (W ∩ U) with Supp s closed in W (hence also in W ∩ V ) to the section of F (W ∩ V ) obtained by gluing s and 0 ∈ F ((W ∩ V ) \ Supp s). Note that the support of this glued section is the same as the support of s, hence it is still closed in W and this is a well-deﬁned map −1 −1 (i!i F )(W ) → (j!j F )(W ) which clearly respects restriction. Theorem 2.3. (Mayer-Vietoris Sequences) Let i : U,→ X be the inclusion of an open set, j : Z,→ X its closed complement. There is a short exact sequence −1 −1 0 → i!i F → F → j∗j F → 0 in Ab(X) natural in F ∈ Ab(X). Let k : V,→ X be another open set and let l : U ∪ V,→ X and m : U ∩ V,→ X be the inclusions. There is a short exact sequence −1 −1 −1 −1 0 → m!m F → i!i F ⊕ k!k F → l!l F → 0.

Proof. Check exactness on stalks by using the previous lemma. Theorem 2.4. (Duality for an open embedding) Let i : U,→ X be the inclusion of an open subset of a topological space X. The functor i−1 : Ab(X) → −1 Ab(U) is right adjoint to the exact functor i! : Ab(U) → Ab(X) and i i! = Id. The functor i−1 : Ab(X) → Ab(U) takes injectives to injectives.

Proof. Checking the adjointness is a straightforward exercise and the equality −1 i i! = Id is clear from the deﬁnitions. The ﬁnal statement then follows formally: Suppose F 0 ∈ Ab(X) is injective and 0 → ι−1F 0 → F → F 00 → 0 3. PROPER MAPS - SECOND TREATMENT 63

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

−1 0 00 0 / ι!ι F / ι!F / ι!F / 0 _

s z F 0 in Ab(X), which completes as indicated by injectivity of F 0 ∈ Ab(X). Applying the −1 −1 functor ι and using i i! = Id, the vertical arrow above becomes an isomorphism −1 and ι s provides a splitting of the original sequence.

3. Proper maps - Second treatment

To better understand the stalk formula in (15.1), we will need to prove some rather technical lemmas. It turns out that the stalk formula behaves very well for proper maps, which we now introduce. Definition 3.1. A subspace Z ⊆ X is called relative Hausdorff is any two points of Z have disjoint neighborhoods in X. A map of topological spaces f : X → Y is called proper if f is closed (the image of any closed set is closed) and the fibers of f are compact and relative Hausdorff in X.

For example, the inclusion of a closed subspace Z,→ X is always a proper map. A space X is proper over a point iff X is compact Hausdorff. The basic properties of proper maps are summed up in the following theorem. Theorem 3.2. The composition of proper maps is proper. Proper maps are preserved under arbitrary base change. A map f : X → Y is proper iff for some −1 (equivalently any) cover {Ui} of Y , the maps f : f (Ui) → Ui are proper. If f : X → Y is proper and y ∈ Y , then the neighborhoods f −1(U) of f −1(y) (as U ranges over all neighborhoods of y ∈ Y ) are cofinal in the set of neighborhoods of f −1(y) in X.

Proof. We first prove that proper maps are closed under base change. Let g : Z → Y be arbitrary, and let Z ×Y X be the fibered product. To check that π1 : Z ×Y X → Z is closed, it is enough to check that π1(A×Y B) is closed whenever A ⊆ Z and B ⊆ X are closed, since closed sets of this form form a basis for all −1 closed sets. On the other hand, we have π1(A ×Y B) = A ∩ g (f[B]), which is ( −1 −1 closed when f is closed. Similarly, π1z) = {z} × f (g(z)) is compact if f (g(z)) is −1 compact and if p, q ∈ f (g(z)) have disjoint neighborhoods U, V in X, then Z ×Y U and Z ×Y V are disjoint neighborhoods of (z, p) and (z, q) in Z ×Y X. The fact that being proper is local on the codomain follows from the fact that being closed is local; the “compact relative Hausdorff fibers” condition is clearly local on the codomain. 64 VI. DUALITY

The last statement is immediate from the fact that f is closed: if V is an open neighborhood of f −1(y) in X, then f[X \ V ] is closed in X since f is closed, so −1 Y \f[X\V ] is an open neighborhood of y in Y and we have f (Y \f[X\V ]) ⊆ V . Lemma 3.3. A map f : X → Y of locally compact Hausdorﬀ spaces is compact iﬀ f −1(K) is compact for every compact subset K ⊆ Y .

Proof. Exercise ??. Corollary 3.4. If f : X → Y and g : Y → Z are maps of locally compact Hausdorﬀ spaces such that gf is proper, then f is proper. It follows that any map between compact Hausdorﬀ spaces is proper.

Proof. By the lemma, it suﬃces to prove that f −1(K) is compact for any compact K ⊆ Y . Certainly g[K] is compact in Z, so (gf)−1(g[K]) is compact because gf is proper. But f −1(K) is a closed subspace of (gf)−1(g[K]), so it is also compact. Lemma 3.5. Let f : X → Y be a proper map, F a sheaf on X. For every y ∈ Y , p p −1 the natural map (R f∗F )y → H (f (y), F |f −1(y)) is an isomorphism.

Proof. Apply (1.7) to the inclusion of the subspace f −1(y) ⊆ X using the assumption (1) and use (3.2) to note that the f −1(U) are coﬁnal in the neighborhoods −1 of f (y). CHAPTER VII

Finite Spaces

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

1. Supports

For any family of supports Φ on X, there is a closed subspace Y such that Φ consists of all closed subsets of Y . Indeed, just take Y to be the union of every closed set in Φ: then Y is in Φ because Φ is closed under ﬁnite unions and every closed subset of Y is in Φ because Φ is closed under passage to smaller closed subsets. As usual, we will write ΓY (X, F ) instead of ΓΦ(X, F ).

2. Specialization

The points of any topological space X are quasi-ordered by specialization: x ≤ y iff x ∈ {y}−. We are only interesting in making statements about Sh(X), so there would be no harm in replacing X with its sobrification, thereby ensuring that ≤ is, in fact, a partial order (equivalently: X is T0). The specialization ordering is particularly important when X is finite, as it completely determines the topology of X. A set is closed iff it is closed under passage to smaller elements (“specializations”): this is because any point in the closure of a finite set S must be in the closure of {s} for some s ∈ S. A subset of X is open iff it is closed under passage to larger elements.

In particular, the set U1 consisting of ≤ maximal points of (ﬁnite) X is open. Let Z1 be its closed complement. Similarly, the set U2 of ≤ maximal points of Z1 is open in Z1 and Z2 := Z1 \ U2 is hence closed in X. Continuing in this manner, we obtain a strictly decreasing sequence

X = Z0 ) Z1 ) ··· ) Zn ) ∅ of closed subspaces of X. Lemma 2.1. Let X be a sober noetherian space. The noetherian dimension n of X is equal to the length of the longest chain x0 < x1 < ··· < xn in (X, ≤) (n = ∞ iﬀ there are arbitrarily long such chains).

Proof. Suppose ∅ ( Z0 ( ··· ( Zn is a chain of irreducible closed subspaces in X. Since X is sober, each Zi has a generic point xi. We have x0 < x1 < ··· < xn 65 66 VII. FINITE SPACES

in (X, ≤). Conversely, given such a chain in (X, ≤), we get a chain of irreducible − − closed subspaces ∅ ( {x0} ( ··· ( {xn} .

3. Coniveau

Let X be any ringed space, not necessarily ﬁnite, and let F ∈ Mod(X). For any specialization x ≤ y, we have a natural map fx,y : Fx → Fy which is OX,x → OY,y linear. It is nothing but the direct limit of the maps F (U) → Fy over neighborhoods − U of x, all of which contain y since x ∈ {y} . We view fx,y as an OX,x linear map by regarding Fy as an OX,x module by restriction of sclars along OX,x → OY,y. Applying x∗, we obtain a Mod(X) morphism

x∗fx,y : x∗Fx → x∗Fy. If we have a sequence of specializations x ≤ y ≤ z, then it is clear that

fx,z = fy,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 Y Y Y 0 → F → x0∗Fx0 → x0∗Fx1 → x0∗Fx2 → · · · x0 x0

n (d s)(x0, . . . , xn+1) := x1∗fx1,xn+1 s(x1, . . . , xn+1) n X i + (−1) x0∗fx0,xn+1 s(x0,..., xˆi, . . . , xn+1) i=1 n+1 +(−1) x0∗fx0,xn s(x0, . . . , xn). By abuse of notation, we will often simply write

n n X i (d s)(x0, . . . , xn+1) := (−1) s(x0,..., xˆi, . . . , xn+1)|xn+1 . i=1

Observe that this complex “depends on X” even when Sh(X) does not. For example, the coniveau complex computed on X is not the same as the coniveau complex computed on its sobriﬁcation Xsob even though Sh(X) = Sh(Xsob). It is generally more natural to work with the coniveau complex computed on Xsob, where the specialization ordering better reﬂects the topology of X. 6. HOM FORMULAS 67

4. Equivalence

When X is ﬁnite, every x ∈ X has a smallest neighborhood Ux and Fx = F (Ux). We will write ix : Ux → X for the inclusion of this open set throughout. Note x ≤ y iﬀ Uy ⊆ Ux. When x ≤ y, the restriction map F (Ux) → F (Uy) may be viewed as a map fx,y : Fx → Fy (we will usually denote this s 7→ s|y). The restriction maps fx,y completely determine the sheaf F , as we will now show. Note that U = ∪x∈U Ux for any U ∈ Ouv(X) and [ Ux ∩ Uy = Uz, z≥x,y so, by using the sheaf property for F , we can recover F (U) as the equalizer   Y Y Y (4.1) F (U) = lim  Fx Fz , ←− ⇒ x∈U (x,y)∈U×U z≥x,y

where the two maps are (sx) 7→ (fx,z(sx)) and (sx) 7→ (fy,z(sx)). There is an equivalence of categories

Sh(X) = HomCat((X, ≤), Ens)

F 7→ (x 7→ Fx) natural in X, whose inverse is given by taking a functor f to the presheaf (in fact sheaf) whose sections are deﬁned by the formula (4.1).

5. Stalk formulas

The formulae A, y ≤ x (x A)(y) = ∗ 0, otherwise A, x ≤ y (i A) = x! y 0, 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 if x ≤ y (or zero otherwise). 68 VII. FINITE SPACES

There is a kind of duality for ﬁnite spaces obtained by inverting the specialization ordering (replacing (X, ≤) with the opposite category). This exchanges x∗ and ix!( ), etc. Lemma 6.1. Let X be a ﬁnite ringed space, x a point of X. If I is an injective OX,x module, then x∗I ∈ Mod(X) is injective. If P is a projective OX,x module, then ix!P ∈ Mod(X) is projective.

Proof. Both statements follow from the formulas for Hom above and the exactness of the stalk functor. Theorem 6.2. Let X be a ﬁnite ringed space of dimension n. For every F ∈ Mod(X), the coniveau complex of (3) is a resolution M M M 0 → F → x0∗Fx0 → x0∗Fx1 → x0∗Fxn → 0 x0 x0

direct sums of sheaves of the form ix!F x.

Proof. We will show the the stalk of this complex at a point z ∈ X is a contractible complex. The key point is that we can commute the finite direct products in the definition of the coniveau complex with stalks (finite inverse limits commute with filtered direct limits). Therefore, by (3), this stalk complex is given by M M M 0 → Fz → Fx0 → Fx1 → · · · → Fxn → 0.

z≤x0 z≤x0

z≤x0<···

with the convention that c−1 : ⊕xFx → Fz sends s to s(z). One can check that n n+1 dz cn + cn+1dz = 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. Poincar´eduality

We now prove an instance of the PoincaréDuality theorem. Let X be a finite topological space with the constant sheaf of rings k associated to a field k. Let Db(k) be the bounded derived category of Mod(k) and let Db(k) be the bounded derived category of k vector spaces. 8. EXAMPLES 69

◦ b b Theorem 7.1. There is a complex ωX ∈ D (k) and a D (k) isomorphism • • • • • ◦ • R Homk(R ΓF ,V ) = R Homk(F , ωX ⊗ V ) natural in F • ∈ Db(k) and V • ∈ Db(k).

Proof. By (6.1), the coniveau resolution F → C (F )• is an injective resolution of X (every vector space Fx is injective). Hence, we have an isomorphism R ΓF = ΓC (F )•, natural in F . For a k vector space V , we have • • • R Homk(R ΓF ,V ) = Homk(ΓC (F ) F ,V ). This latter complex looks like M M Homk( Fxn ,V ) → · · · → Homk( Fx0 ,V ), x0<···

x0<···

8. Examples

In this section we work out some sheaf cohomology groups on ﬁnite topological spaces. Finite (non-Hausdorﬀ) topological spaces are a rich source of examples in the theory of sheaves. Our main examples will involve the following spaces. The Sierpinski space S is the unique topology on {1, 2} where {1} is closed, but {2} is not. The space X = {1A, 2, 1B} has three points with basic open sets

U1A := {1A, 2}

U2 := {2}

U1B := {2, 1B}. 70 VII. FINITE SPACES

The space Y = {0A, 1A, 2, 1B, 0B} has basic open sets those of X, plus the sets

U0A := {0A, 1A, 2, 1B}

U0B := {1A, 2, 1B, 0B}.

Let us start with the Sierpinski space S. Here every presheaf F ∈ PAb(S) is a sheaf as long as F (∅) = 0 (the sheafification functor does nothing but set F (∅) = 0) because S has only two interesting open sets S, {2} and neither of these can be covered nontrivially. A sheaf F ∈ Ab(S) is nothing but the data of the restriction map F (S) → F ({2}), which is also the data of the specialization map F1 → F2 (note 0 < 1 is the specialization ordering of S), so Ab(S) is nothing but the arrow category Ab•→• of Ab. The global section functor is identified with the stalk functor F 7→ F1, hence it is exact. However, sections Γ1 with support in the closed set {1} is identified with the kernel functor

Γ1(S, F ) = Ker(F1 → F2).

This is not exact. Indeed, an exact sequence in Ab(S) is just a morphism of exact sequences in Ab, and the sequence of kernels of such a morphism need not form an exact sequence; they must be linked up with the cokernels as in the Snake Lemma. In fact it is straightforward to check, using the obvious ﬂasque resolution

0 / F1 / F1 ⊕ F2 / F2 / 0

π2 0 / F2 / F2 / 0 / 0, that  Ker(F1 → F2), p = 0 p  H1(S, F ) = Cok(F1 → F2), p = 1  0, p > 1

Notice that, furthermore, the forgetful functor Ab(S) → PAb(S) is exact, so Γ1 is not even exact as a functor Γ1 : PAb(X) → Ab. Even worse, there is no hope of “making it exact” (computing its derived functors) by using Cechˇ cohomology. Indeed, the trivial cover {S} reﬁnes every other cover, and, since we may compute with alternating Cechˇ cohomology, the Cechˇ cohomology groups in this case are simply p Γ (S, ), p = 0 Hˇ (S, ) = 1 F 1 F 0, p > 0

This shows the necessity of the “neighborhoods” assumption in (7.3). The specialization ordering (Y, ≤) and the category Ouv(Y )op look like 8. EXAMPLES 71

0A 0B and } ! U0A U0B

w ' ! } 1A 1B X

2 ~ } ! U1A U1B

! } U2

∅ respectively. The sheaﬁﬁcation functor PAb(Y ) → Ab(Y ) is given by:

F (Y ) F (U0A) ×F +(X) F (U0B)

{ # w ' F (U0A) F (U0B) F (U1A) F (U1B) 7→ # { ' w F (X) F (U0A) ×F (U2) F (U0B)

{ # w ' F (U1A) F (U1B) F (U1A) F (U1B)

# { ' w F (U2) F (U2)

F (∅) 0

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

U ∈ {U0A,U0B,U1A,U1B,U2} as these are the smallest neighborhoods of the ﬁve points of Y . 72 VII. FINITE SPACES

The open cover U = {U0A,U0B} of Y reﬁnes any other open cover, hence we can use ordered Cechˇ cohomology with U to compute the absolute Cechˇ cohomology of any F ∈ PAb(Y ). The ordered Cechˇ complex Cˇ(U, <, F ) is simply

F (U0A) ⊕ F (U0B) → F (X), where the map is the diﬀerence of the restriction maps. For example, if F is the presheaf Z 1 1 Z Z

(1,0) (1,1) Z ⊕ Z (1,0) (1,0)

Z Z

1 1 Z

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

Our sheaf F 0 is part of the short exact sequence

0 → F 0 → F → F 00 → 0(8.1) in Ab(Y ) given below.

0 Z Z = = = =

Ð { # 0 0 Z Z Z Z

= = ∆ ∆ Ð " | 0 / Z / Z ⊕ Z = = π1 π2

Ð | " 0 0 Z Z Z Z

= = Ð # { ZZ 0

Note that exactness in Ab(Y ) can be checked on stalks, which is to say it can be checked on U0A,U0B,U1A,U1B, and U2. To compute the cohomology groups of F 00, we consider the SES of sheaves

0 → F 00 → G → G 00 → 0(8.2) given by the diagram below.

Z Z ⊕ Z Z ⊕ Z = = = = π1 π2

~ { # ~ Z Z Z ⊕ Z Z ⊕ Z Z Z

∆ ∆ = = ~ # { ~ Z ⊕ Z / Z ⊕ Z / 0 π1 π2 π1 π2

~ { # ~ Z Z Z Z 0 0

0~ # 0{ 0 ~

Here the maps on global sections

00 00 F (Y ) = Z → G (Y ) = Z ⊕ Z → G (Y ) = Z ⊕ Z 74 VII. FINITE SPACES

00 are given by ∆ and ∆π2, respectively. The sheaves G and G are ﬂasque, hence have no higher cohomology, so by examining the LES in cohomology, we compute: p = 0, 1 Hp(Y, 00) = Z F 0 p > 1 The sheaf F in the appearing in the sequence (8.1) is also ﬂasque, so we can now take the LES in cohomology there to show that p = 2 Hp(X0, 0) = Z F 0 otherwise as claimed. Note that the exact sequence 0 → F 0 → F → G → G 00 → 0 obtained by concatenating the exact sequences (8.1) and (8.2) (which we have used to work out the cohomology of F 0) is nothing but the coniveau resolution of F 0 constructed in (6.2).

9. Exercises

Exercise 9.1. Let Y ⊆ X be a closed subspace of a ﬁnite space X. Write down a formula for ΓY (X, F ) in the langauge of (4) (i.e. making reference only to the stalks of F and the maps between them arising from specializations). Specialize your formula to the case where F is a skyscraper. Exercise 9.2. Extend the above Poincar´eDuality to cohomology with support in a closed subspace Y ⊆ X (c.f. (1)).

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

Cechˇ Theory

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

1. Cechˇ resolution

Let X be a topological space, F ∈ PAb(X). It will be important to develop a theory of Cechˇ cohomology for presheaves as well as sheaves. For an open set −1 i : U,→ X, we set FU = i∗ipreF . Note that FU is given by V 7→ F (U ∩ V ), −1 −1 and that this is a sheaf if F is a sheaf (ipre = i 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 presheaves on X induced by the restriction maps of F . Note that the stalk of this map is an isomorphism at any x ∈ V (both stalks are just Fx). Let

U = {Ui : i ∈ I}

be a family of open sets of X. For a ﬁnite subset {i0, . . . , ip} ⊆ I, or any vector p+1 (i0, . . . , ip) ∈ I , we use the following notation:

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

F (i0, . . . , ip) := FU(i0,...,ip). The convention (1.1) F () := F

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

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

s 7→ s|U(i0,...,ip).

For each nonnegative integer p, deﬁne ˇp Y C (U, F ) := F (i0, i1, . . . , ip). p+1 (i0,...,ip)∈I 75 76 VIII. CECHˇ THEORY

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

dp : Cˇp(U, F ) → Cˇp+1(U, F ) are given by

p+1 p X j ˆ (1.2) (d s)(i0, . . . , ip+1) := (−1) s(i0,..., ij, . . . , ip+1)|U(i0,...,ip+1). j=0 There is a natural augmentation morphism

d−1 : F → Cˇ0(U, F )

given by setting s(i0) := s|U(i0) (the usual formula using convention (1.1)). The abelian presheaves Cˇp(U, F ) are called the Cechˇ presheaves associated to the presheaf F and the covering sieve U. Theorem 1.1. If U is a cover of X, then d−1 : F → Cˇ•(U, F ) is a resolution for any F ∈ Ab(X).

Proof. We have already noted that, since F is a sheaf, each F (i0, . . . , ip) is a sheaf, and hence Cˇp(U, F ) (being a product of such sheaves) is also a sheaf. It therefore suﬃces to check exactness of the augmented complex F → Cˇ•(U, F ) 1 when restricted to each of the open sets Ui, since the Ui form a cover. In fact, we will show that ˇ• (F → C (U, F ))|Ui is null-homotopic. Deﬁne a contraction map Y Y cp : F (i0, . . . , ip)|Ui → F (i0, . . . , ip−1)|Ui i0,...,ip i0,...,ip−1 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 .

ˇ0 By our convention (1.1), c0 : C (U, F )|Ui → F |Ui simply sends s to s(i). ˇ• Now we just check that the cp form a retraction for C (U, F )x. That is,

p−1 p d cp + cp+1d = 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

p−1 p−1 X (d cps)(i0, . . . , ip) = (cps)(i0,...,ˆij, . . . , ip) j=0 p−1 X j = (−1) s(i, i0,...,ˆij, . . . , ip) j=0 and

p p (cp+1d s)(i0, . . . , ip) = (d s)(i, i0, . . . , ip) p−1 X j+1 = s(i0, . . . , ip) + (−1) s(i, i0,...,ˆij, . . . , ip). j=0

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

2. Reﬁnement maps

Suppose that U = {Ui : i ∈ I} and V = {Vj : j ∈ J} are two families of open sets in X. Suppose r : J → I is a reﬁnement in the sense that Vj ⊆ Ur(j) for all

j ∈ J. Then we have a natural restriction map FU(r(j)) → FVj for every j, and in fact a map of complexes

r∗ : Cˇ•(U, F ) → Cˇ•(V, F )

s 7→ s(r(j0), . . . , r(jp))|V (j0,...,jp).

Lemma 2.1. Suppose r1, r2 : J → I are two reﬁnements. Then the maps ∗ ∗ ˇ• ˇ• r1, r2 : C (U, F ) → C (V, F ) are homotopic.

Proof. The maps ˇp ˇp−1 hp : C (U, F ) → C (V, F ) given by

p−1 X k (hps)(j0, . . . , jp−1) := (−1) s(r1(j0), . . . , r1(jk), r2(jk), . . . , rp−1(jk)) k=0

∗ ∗ provide a homotopy from r1 to r2. 78 VIII. CECHˇ THEORY

3. Alternating variant

The alernating (or ordered) Cechˇ complex is a “smaller” version of the Cechˇ complex which is convenient for computations, and has theoretical implications (relations between covering dimension and cohomological dimension). Let {Ui : i ∈ I} be a family of open sets in X. Fix a total ordering ≤ of I. The sheaves ˇp(U, <, ) := Q (i , . . . , i ) C F i0<···

Proof. We will obtain this later from the simplicial point of view. For now, we just note that the alternating Cechˇ complex certainly provides a resolution of any F ∈ Ab(X). This can be proved directly (without the above lemma) just as in the proof of (1.1) using the contraction operator s(i, i , . . . , i ), i < i (c s)(i , . . . , i ) := 0 p−1 0 p 0 p−1 0, otherwise.

4. Cech cohomology for a ﬁxed cover ˇ Finally we deﬁne Cech cohomology. Let U = {Ui : i ∈ I} be a cover of X. Let ˇ• F ∈ PAb(X), let CΦ(U, F ) be the complex in Ab(X) obtained by applying ΓΦ to the Cechˇ complex Cˇ•(U, F ). Warning. The obvious inclusion ˇp Y CΦ(U, F ) ⊆ ΓΦ∩U(i0,...,ip)(U(i0, . . . , ip), F ) p+1 (i0,...,ip)∈I

is not generally an equality because ΓΦ does not generally commute with products. ˇp This formula is, however, valid if ΓΦ = Γ, or if I is ﬁnite. In general, CΦ(U, F ) is the subcomplex of Cˇp(U, F ) consisting of those Y s = (s(i0, . . . , ip)) ∈ F (U(i0, . . . , ip)) p+1 (i0,...,ip)∈I such that there is a Y ∈ Φ such that

Supp s(i0, . . . , ip) ⊆ Y p+1 for all (i0, . . . , ip) ∈ I . The abelian groups ˇ n n ˇ• HΦ(U, F ) := H (CΦ(U, F )) 6. CECHˇ COHOMOLOGY 79 are called the Cechˇ cohomology 2 groups of F with respect to the cover U.

5. Basic properties

We now prove some of the basic properties of the Cechˇ cohomology groups ˇ n HΦ(U, F ). Theorem 5.1. For any F ∈ Ab(X), there is a ﬁrst quadrant spectral sequence p,q p q ˇ • p+q E2 = H (HΦ(C (U, F ) )) =⇒ HΦ (X, F ). p,0 ˇ p p In particular, there are natural maps E2 = HΦ(U, F ) → HΦ(X, F ) for every p.

Proof. Since the Cechˇ complex is a resolution by (1.1), this is just a special case of the spectral sequence associated to a resolution of an object in an abelian category.

ˇp When each of the sheaves C (U, F ) is ΓΦ-acyclic, we say that U is a good cover for F . We say U is a good cover if it is a good cover for every abelian sheaf F . Corollary 5.2. If U is a good cover for F , then the natural map ˇ p p HΦ(U, F ) → HΦ(X, F ) is an isomorphism for every p.

Proof. The spectral sequence of (5.1) degenerates.

6. Cechˇ cohomology

In this section we introduce the (absolute) Cechˇ cohomology groups, which have no dependence on a choice of cover. Set p p Hˇ (X, ) := lim Hˇ (U, ), Φ F −→ Φ F U where the direct limit is over the the category of covers U = {Ui : i ∈ I} where there is a unique morphism from U → V = {Vj : j ∈ J} iff there is a refinement map r : J → I. The restriction maps are defined by taking the maps Hp(r∗) induced on cohomology and noting that there are independent of the choice of r. An alternative means of calculating the Cechˇ cohomology groups is by means of covering sieves (c.f. (9)). Let U = {Ui : i ∈ I} be the set of all open subsets of X (the trivial sieve on X ∈ Ouv(X)). By slight abuse of notation, say J ⊆ I is a covering sieve if {Uj : j ∈ J} is a covering seive. Order the set of covering sieves

2Our deﬁnition agrees with Godement’s [G, 5.2]. 80 VIII. CECHˇ THEORY

by reverse inclusion. The resulting set is filtered and we may compute the Cechˇ cohomology as the cohomology of the direct limit complex: ! ˇ p n ˇ• H (X, ) := H lim C ({Uj : j ∈ J}, ) . Φ F −→ Φ F J This is because filtered direct limits commute with cohomology and the set of covering sieves is cofinal in the set of all covers, ordered by the existence of a refinement. ˇ p ˇ The abelian groups HΦ(X, F ) are called the (absolute) Cech cohomology groups of F . We will revisit these constructions are first proving some “classical” facts about Cechˇ cohomology.

7. Cechˇ acyclic sheaves

As with usual sheaf cohomology, we will need a supply of presheaves with vanishing higher Cechˇ cohomology. Unfortunately, our supply of such presheaves will be rather lame. Theorem 7.1. Let U be an open cover of X, F ∈ PAb(X). The Cechˇ coho- p mology groups HΦ(U, F ) vanish for p > 0 under either of the following hypotheses: (1) F is ﬂasque and is in Ab(X) (2) F is Φ soft, Φ is paracompactifying, and F ∈ Ab(X)

Proof. We first prove the statement when F is flasque. Then it is clear from ∗ the definition that ι∗ι F is flasque for any inclusion ι : U,→ X. Furthermore, a Q Q product of flasque sheaves is flasque (since ( i Fi)(U) = i Fi(U) and a product of surjections is surjective). Hence the Cechˇ resolution ε : F → Cˇ•(U, F ) is a flasque resolution of F , so ˇ p p ˇ• HΦ(U, F ) = H (ΓΦC (U, F )) p = HΦ(X, F ) by 10.5, but this vanishes for p > 0 by 10.4 because F is flasque.

The above theorem says that Cechˇ cohomology is, in some sense, eﬀaceable. However, it turns out that Cechˇ cohomology does not generally form a δ-functor from Ab(X) to Ab: a short exact sequence of sheaves does not generally induce a long exact sequence of Cechˇ cohomology groups. This is the only obstruction to identifying Cechˇ cohomology with sheaf cohomology. If we replace sheaves with presheaves, Cechˇ cohomology is better behaved: • Theorem 7.2. For any cover U of X, the Cechˇ cohomology groups Hˇ (U, ) n 0 form an eﬀaceable δ-functor from PAb(X) to Ab, hence Hˇ (U, F ) = Rn Hˇ (U, F ) for every F ∈ PAb(X). 7. CECHˇ ACYCLIC SHEAVES 81

Proof. Suppose F = 0 → F 0 → F → F 00 → 0 is a short exact sequence in PAb(X). Then 0 → F 0(U) → F (U) → F 00(U) → 0 is exact for every U ∈ Ouv(X) (this is why it is important to use presheaves). Consequently, 0 → Cˇ•(U, F 0) → Cˇ•(U, F ) → Cˇ•(U, F 00) → 0 is a short exact sequence of complexes of abelian groups, since, in each degree it is just a product of short exact sequences of the above form. The associated long exact cohomology sequence is natural in F hence the first claim is proved. To see that • Hˇ (U, ) is effaceable it is enough to check that it is left exact and preserves finite direct sums (clear) because then it can be erased by injectives, which PAb(X) has enough of by (4.6).

With supports, we will need to be able to reﬁne covers. The best we can do is: Theorem 7.3. Let X, be a topological space, Φ a family of supports with neigh- ˇ ˇ • borhoods. The absolue Cech cohomology groups HΦ(X, ) form an eﬀaceable δ- ˇ n n ˇ 0 functor from PAb(X) to Ab, hence HΦ(U, F ) = R HΦ(U, F ) for every F ∈ PAb(X).

Proof. Let U = {Ui : i ∈ I} be the set of all open subsets of X. By the same argument as in the previous proof, we reduce to proving that 0 → lim Cˇp (I0, 0) → lim Cˇp (I0, ) → lim Cˇp (U, 00) → 0 −→ Φ F −→ Φ F −→ Φ F I0 I0 I0 0 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 0 → F → F 00 → 0. Exactness on the left and in the middle follows from exactness of ﬁltered direct limits and left exactness of ΓΦ, so the issue is to prove surjectivity on the right. As mentioned in (??), an element [(s, I0)] on the right is represented by a pair (s, I0) where I0 indexes a covering seive and s is a set of sections 00 s(i0, . . . , ip) ∈ F (U(i0, . . . , ip)) such that there is a Y ∈ Φ with

Supp s(i0, . . . , ip) ⊆ Y 0 p+1 for all (i0, . . . , ip) ∈ (I ) . Since Φ has neighborhoods, we can ﬁnd an open set V ∈ Ouv(X) with Y ⊆ V and V − ∈ Φ. By choosing a common reﬁnement of the cover indexed by I0 and the cover {X \ Y,V } we can an I00 ⊂ I0 indexing a covering seive, such that, for 82 VIII. CECHˇ THEORY

00 00 p+1 every i ∈ I , either Ui ⊆ V or Ui ∩ Y = ∅. In particular, for (i0, . . . , ip) ∈ (I ) , s(i0, . . . , ip) = 0 unless U(i0, . . . , ip) ⊆ V . The presheaf exactness implies that 00 F (U(i0, . . . , ip)) → F (U(i0, . . . , ip)) 00 p is always surjective, so we can ﬁnd, for each (i0, . . . , ip) ∈ (I ) with U(i0, . . . , ip) ⊆ V , a t(i0, . . . , ip) ∈ F (U(i0, . . . , ip)) mapping to s(i0, . . . , ip). By setting t(i0, . . . , ip) = 00 p 0 for those (i0, . . . , ip) ∈ (I ) where U(i0, . . . , ip) is not contained in V , we obtain 00 0 00 a preimage [(t, I )] for [(s, I )] = [(s|I00 ,I )]. Note that − Supp t(i0, . . . , ip) ⊆ V ∈ Φ 00 p+1 for all (i0, . . . , ip) ∈ (I ) . The assumption that Φ has neighborhoods cannot be removed. See (??).

8. Cechˇ presheaf

Let X be a topological space, Φ a family of supports on X, F ∈ Ab(X). For n n ≥ 0, let HΦ (F ) ∈ PAb(X) be the presheaf n op HΦ (F ): Ouv(X) → Ab n U 7→ HΦ∩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 n HΦ : Ab(X) → PAb(X) n F 7→ HΦ (F ) th 0 coincides with the n right derived functor of HΦ : Ab(X) → PAb(X).

n Proof. The functors F 7→ HΦ (F ) form an effaceable (by flasque sheaves, say) δ-functor which manifestly has the right degree zero term. 0 Remark 8.2. The functor HΦ does not necessarily take a flasque sheaf to a sheaf. If X is an infinite discrete space, Φ is the family of compact (i.e. finite) subsets of X, and F is the flasque sheaf of continuous (i.e. arbitrary) Z valued 0 functions, then HΦ is not even a sheaf.

q The restriction maps for the presheaf HΦ (F ) can be described concretely as follows. Fix an injective resolution ε : F → I • of F . Then an inclusion of open sets V,→ U induces a morphism of complexes • • ΓΦ∩U (U, I ) → ΓΦ∩V (V, I ) and thus a morphism on the cohomology of these complexes. On the other hand, we have q q • HΦ (F )(U) = H (ΓΦ∩U (U, I )) 8. CECHˇ PRESHEAF 83

q and similarly with U replaced by V . The abelian presheaves HΦ (F ) are called the local cohomology presheaves of F with support in Φ. Lemma 8.3. Assume Φ has neighborhoods. Then the sheaf (or even the separated q presheaf) associated to the abelian presheaf HΦ (F ) is zero for all q > 0.

Proof. Fix a q > 0, an injective resolution F → I •, an open set U, and an q element s ∈ HΦ (F )(U). Then s is represented by some q-cocycles ˜ in the complex • q ΓΦ∩U (U, I ), in other words by an elements ˜ ∈ I (U) where the closure S of Supps ˜ in X is in Φ. Since Φ has neighborhoods, there is a B ∈ Φ such that S is contained in the interior V of B. Since the complex I • is exact in positive degrees, the q- cocycles ˜|U∩V is a “coboundary on a cover.” That is, there is an open cover {Ui} of q−1 q−1 U ∩ V and sections si ∈ I (Ui) such that d (Ui)(si) = s|U∩V for all i. Since Ui is contained in V , the closure in X of Supp si is contained in B, hence is in Φ, so q−1 si ∈ ΓΦ∩Ui (Ui, I ). The Ui together with the set U \ Supps ˜ form an open cover of U, and taking the zero section of I q−1 on the latter open set we ﬁnd thats ˜ is a coboundary on each set in this open cover, hence the restriction of s to every set in this open cover is zero in cohomology. Lemma 8.4. Let X be a topological space, Φ a family of supports. The natural map ˇ 0 + HΦ(X, F ) → ΓΦ(X, F ) is injective for any F ∈ PAb(X) and an isomorphism if F is separated.

Proof. Let {Ui : i ∈ I} be the open sets of X. The natural map is induced by the natural map of presheaves F → F +, naturality of the the Cechˇ complex associated to a presheaf, and the fact that, for any G ∈ Ab(X), we have a natural isomorphism ˇ 0 HΦ(X, G ) = ΓΦ given by sending s ∈ ΓΦ(X, G ) to the class of (s|Ui ,I) in s ∈ ΓΦ(X, G ) (this is injective by separation for G and surjective by gluing for G , taking care to keep track of supports by noting that the support of the section obtained by gluing is the unions of the supports of the sections being glued). Now let us prove the lemma. Making the natural identification just established, this map send a class 0 ˇ 0 + 0 ˇ 0 + [(s, I )] ∈ HΦ(X, F ) to the corresponding class [(s ,I )] ∈ HΦ(X, F ). Viewing F + as the sheaf of maps to the stalks with the local coherence condition, it is clear that [(s+,I0)] is zero iff we can find a finer cover (hence a finer covering seive) I00 ⊆ I0 on which (s, I0) is zero. Now let us try to prove surjectivity. We will ignore the supports for simplicity. + Q A section sΓ(X, F ) is an element s ∈ x∈X Fx satisfying the local coherence condition (c.f. (??)), so we can find an I0 ⊂ I indexing a cover and sections s(i) ∈ F (Ui) such that s(i)x = s(x) for all x ∈ Ui. Of course, by adding in every set which 0 is a subset of some Ui and restricting sections, we may assume I indexes a covering seive, so (I0, s) ∈ Cˇ0(I0, F ) is a class mapping to s ∈ Γ(X, F +) under the natural 84 VIII. CECHˇ THEORY

map... but the diﬃculty is in showing that (I0, s) is in the kernel of d0 of the Cechˇ complex, and this is where we need separation of F . When F is separated, we simply note that the stalk of s(i)|U(i,j) − s(j)|U(i,j) is zero at every point x ∈ U(i, j) (because it is just s(x) − s(x) by local coherence), hence s(i)|U(i,j) − s(j)|U(i,j) = 0 by separation.

0 We encountered an example where Hˇ (X, F ) → Γ(X, F +) is not surjective in (8).

Theorem 8.5. Let X be a topological space, F ∈ Ab(X), U a cover of X. There is a ﬁrst quadrant spectral sequence

p,q ˇ p q p+q E2 = H (U, H (F )) =⇒ H (X, F ). If Φ is a family of supports with neighborhoods, then there is a similar spectral sequence

p,q ˇ p q p+q E2 = HΦ(U, H (F )) =⇒ HΦ (X, F ) using the absolute Cechˇ cohomology.

Proof. The functor Γ : Ab(X) → Ab is the composition of the forgetful 0 functor H 0 : Ab(X) → PAb(X) and the functor Hˇ (U, ): PAb(X) → Ab, hence we have the Grothendieck spectral sequence

p,q p ˇ 0 q 0 p+q E2 = R H (U, R H F ) =⇒ H (X, F ). This spectral sequence is identiﬁed with the desired one by (8.1) and (7.2). The second statement is proved similarly using (7.3). Note that, for the existence of the Grothendieck spectral sequence we need to 0 ˇ 0 ˇ 0 know that H takes injectives in Ab(X) to H (U, ) acyclics (HΦ(X, ) acyclics for the second statement); this follows from (7.1). Theorem 8.6. For any topological space X and any abelian sheaf F on X, the natural maps p Hˇ (X, F ) → Hp(X, F ) are isomorphisms for p = 0, 1 and injective for p = 2.

Proof. This follows from the exact sequence of low order terms

1 2 0 → Hˇ (X, F ) → H1(X, F ) → H0(X, H 1F ) → Hˇ (X, F ) → H2(X, F ) in the spectral sequence of (8.5), together with (8.4) and (8.3), which together imply 0 1 H (X, H F ) = 0. 9. COMPARISON THEOREMS 85

9. Comparison theorems

We conclude our treatment of classical Cechˇ theory with two theorems about the relationship between Cechˇ cohomology and the usual sheaf cohomology. Theorem 9.1. (Cartan’s Criterion) Let X, be a topological space, Φ a family of supports, F ∈ Ab(X). Suppose U is an open cover of X satisfying: (1) If U, V ∈ U, then U ∩ V ∈ U. (2) U contains arbitrarily small neighborhoods of any x ∈ X. ˇ p (3) HΦ(U, F |U ) = 0 for any p > 0 and any U ∈ U. Then the natural map ˇ p p HΦ(X, F ) → HΦ(X, F ) is an isomorphism for every p. ˇ p Proof. We first prove, by induction on p, that the natural map HΦ(U, F |U ) → p HΦ(U, F |U ) is an isomorphism for all p, and all U ∈ U. The case p = 0 is trivial. Suppose we know this for all q with 0 ≤ q < p. Because of our assumptions (1),(2) about U, we note that, for any U ∈ U, the covers of U by open sets from U are cofinal in the family of all open covers of U, so we may compute the Cechˇ cohomology ˇ p groups H (U, F |U ) using only such covers. It then follows from assumption (3) q and the induction hypothesis that HΦ (F )(U) = 0 for every U ∈ U and every ˇ i q q with 1 ≤ q < p, so that H (U, HΦ (F )) = 0 for every such q because we can compute Cechˇ cohomology only using covers by open sets from U. Furthermore, 0 Hˇ (U, H p(F )) = 0 by (8.3) and (8.4). Consider the spectral sequence of 8.5 on U. The natural map p0 ˇ p p E2 = HΦ(U, F |U ) → HΦ(U, F |U ) in this spectral sequence must be an isomorphism because we have shown that i,j E2 = 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 sequence of 8.5, using the first part of the proof and the assumption (3) to know that p q H (X, HΦ (F )) = 0 for every q > 0, so that the SES degenerates. Theorem 9.2. Suppose Φ is a paracompactifying family of supports (§7) on a topological space X. Then for every p ≥ 0 and every F ∈ Ab(X), the natural map ˇ p p HΦ(X, F ) → HΦ(X, F ) is an isomorphism.

Proof. Corollary 9.3. For a paracompact topological space X, the natural map p Hˇ (X, F ) → Hp(X, F ) is an isomorphism for every p. 86 VIII. CECHˇ THEORY

10. The unit interval

p It is possible to compute the Cechˇ cohomology groups Hˇ (X, F ) directly from the definitions for reasonably simple topological spaces X and sheaves F . For example, the interval I = [0, 1] can be covered by connected open subinvervals U1,...,Un of roughly equal length where Ui intersects only Ui−1 and Ui+1 (and this intersection is connected). (Note U1 only meets U2 and Un only meets Un−1 and we assume n ≥ 2.) These sort of covers are cofinal in all covers of the interval. Global sections of the Cechˇ complex for this cover and the constant sheaf Z give the complex d : Zn → Zn−1 (in degrees 0, 1) where the boundary map is given by the matrix: 1 −1 0 0 ··· 0 0  0 1 −1 0 ··· 0 0    0 0 1 −1 ··· 0 0  d = ......  ......    0 0 0 0 · · · −1 0  0 0 0 0 ··· 1 −1 This map has cohomology only in degree zero, given by the diagonal ∆ : Z ,→ Zn. Since the interval is certainly paracompact, Cechˇ cohomology and the usual sheaf cohomology coincide by Corollary 9.3, so we find , p = 0 (10.1) Hp(I, ) = Z Z 0, p > 0.

11. Simplicial objects

For a nonnegative integer n, let [n] := {0, . . . , n}. Let ∆ denote the category whose objects are the ﬁnite sets [0], [1],... and where a morphism [m] → [n] is a n nondecreasing function. For each i ∈ [n], let si :[n − 1] → [n] be the unique ∆ n morphism whose image is [n] \{i} (the pneumonic naming si is for “skip i”). For n i ∈ [n], let ei :[n + 1] → [n] be the unique nondecreasing map which is surjective and hits i ∈ [n] twice. Let C be a category. A simplicial object in C is a functor ∆op → C. For example, any object C ∈ C determines a constant simplicial object (also denoted C ∈ sC) via op the constant functor [n] 7→ C (which takes every ∆ morphism to IdC ). Simplicial objects in C form a category op sC := HomCat(∆ , C), which evidently contains C as a full subcategory by identifying C with the constant simplicial objects in sC. For C ∈ sC, it is customary to write Cn for C([n]) and n n n di : Cn → Cn−1 for C(si ). The maps di are called the boundary maps of C. n n Similarly, we will write ei : Cn → Cn+1 as abuse of notation for C(ei ). The maps n ei 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

top n+1 For a non-negative integer n, let ∆n ⊆ R denote the convex hull of the standard basis vectors e0, . . . , en:

std n+1 X ∆n := {t0e0 + ··· + tnen ∈ R : ti ∈ [0, 1], ti = 1}. A morphism σ :[m] → [n] in the simplicial category ∆ induces a morphism of topological spaces top top top ∆ (σ) : ∆m → ∆n X X (t0e0 + ··· + tmem) 7→ ( tie0 + ··· + tien), i∈σ−1(0) i∈σ−1(n) so we can regard ∆top as a cosimplicial topological space called the standard (topological) simplex. The algebraic analog of the standard simplex is the following: For a non-negative integer n, let An denote the ring

An := Z[t0, . . . , tn]/h1 − t0 − t1 − · · · − tni. A ∆-morphism σ :[m] → [n] induces a ring homomorphism

A(σ): An → Am X ti 7→ tj, j∈σ−1(i)

alg so that the An form a simplicial ring A, and hence the ∆n := Spec An 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 ∆ whose op objects are [0],..., [n]. An n-truncated simplicial object in C is a functor ∆n → C. Let op snC := HomCat(∆n , C) op op be the category of n-truncated simplicial objects in C. The inclusion ∆n ,→ ∆ induces a restriction functor

trn : sC → snC

op C 7→ C|∆n , which we call the n truncation. 88 VIII. CECHˇ THEORY

mon mon Let ∆ (resp. ∆n ) denote the subcategory of ∆ (resp. ∆n) with the same objects, but where a morphism [m] → [k] is also required to be monic. Deﬁne ∆epi epi and ∆n similarly. Assume C has ﬁnite inverse limits. By the Kan Extension Theorem, the truncation trn : sC → snC admits a right adjoint

Coskn : snC → sC called the n coskeleton functor. Speciﬁcally, op (Coskn C)m := lim {Ck : ([m] → [k]) ∈ [m] ↓ ∆n } ←−

= lim {Ck : ([k] → [m]) ∈ ∆n ↓ [m]} ←− mon = lim {Ck : ([k] → [m]) ∈ ∆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 of the functor op (∆n ↓ [m]) → C

([k] → [m]) 7→ Ck. mon The third equality is by coﬁnality of ∆n ↓ [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 = (Coskn C)m for m ≤ n. Indeed, when m ≤ n, Id : [m] → [m] is the terminal object of ∆n ↓ m. The adjunction isomorphism

HomsnC(trn C,D) → HomsC(C, Coskn D) takes a map f : trn C → D to the map C → Coskn D given in degree m by the map Cm → (Coskn D)m obtained as the inverse limit of the maps

fkX(σ): Xm → Yk over all σ :[k] → [m] in ∆n ↓ [m]. For C ∈ sC, the abuse of notation

Coskn C := Coskn(trn C) is standard. Dually, if C has ﬁnite direct limits, then the Kan Extension Theorem implies that truncation trn admits a left adjoint

Skn : snC → sC called the n skeleton functor, given on C ∈ snC by the formula

(Skn C)m = lim {Ck : ([m] → [k]) ∈ [m] ↓ ∆n} −→ epi = lim {Ck : ([m] → [k]) ∈ [m] ↓ ∆n }. −→ 14. SIMPLICIAL HOMOTOPY 89

The structure map

(Skn C)(σ) : (Skn C)m → (Skn C)l induced by a ∆ morphism σ :[l] → [m] is the map on direct limits induced by the following commutative diagram of functors:

∗ op τ7→σ τ op ([m] ↓ ∆n) / ([l] ↓ ∆n)

(τ:[m]→[k])7→Ck ([l]→[k])7→Ck % C y

Note that s0C is naturally isomorphic to C, so we can regard Sk0 and Cosk0 as functors C → sC. With this understanding, Sk0 C is the constant simplicial object [n] and Cosk0 C is the simplicial object with (Cosk0 C)n = C (the n + 1 fold product of C) whose boundary maps are the natural projections C[n] → C[n−1] “forgetting one coordinate” and whose degeneracy maps C[n] → C[n+1] are the diagonals given by “repeating one coordinate”.

14. Simplicial homotopy

Let f, g : X → Y be sC morphisms. A homotopy h from f to g consists of a C morphism h(φ): Xn → Yn, deﬁned for each morphism φ :[n] → [1] of ∆, satisfying the conditions: (1) For any commutative triangle

ψ [m] / [n]

φ θ [1] in ∆, the square h(θ) Xn / Yn

X(ψ) Y (ψ)

h(φ) Xm / Ym commutes in C. (2) For any n, if φ :[n] → [1] is the constant map to 0 ∈ [1], then h(φ) = fn and if φ is the constant map to 1 ∈ [1], then h(φ) = gn. The relation consisting of pairs

(f, g) ∈ HomsC(C,D) × HomsC(C,D) deﬁned by “there is a homotopy from f to g” is not generally an equivalence relation. It is clearly reﬂexive, but not generally symmetric or transitive. However, if there is a homotopy h from f to g, and k, l : D → E are sC morphisms with a homotopy h0 90 VIII. CECHˇ THEORY from k to l, then there is a homotopy from kf to lg, obtained by composing h and h0:

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

We let ∼ be the equivalence relation generated by this relation and we say f is homotopic to g if f ∼ g. In other words, f ∼ g means there is a ﬁnite string of morphisms f = f0, f1, . . . , fn = g such that, for every i, either there is a homotopy from fi to fi+1 or there is a homotopy from fi+1 to fi. An sC morphism f : C → D is a homotopy equivalence if there is an sC morphism g : D → C such that 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 composing them. This is well defined because of the remarks about composition of homotopies made above. There is an obvious forgetful functor sC → HotsC. Evidently f : C → D is a homotopy equivalence iff its image in HotsC is an isomorphism. Lemma 14.1. Suppose C has products and C ∈ C is such that the unique map f : C → 1 to the terminal object (empty product) admits section s (i.e. there is some map 1 → C). Then

Cosk0 f : Cosk0 C → 1 is a homotopy equivalence in sC with homotopy inverse

Cosk0 s : 1 → Cosk0 C.

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

(Cosk0 s)(Cosk0 f) = Cosk0(sf) is homotopic to the identity of Cosk0 C. The map n+1 n+1 (Cosk0(sf))n : C → C 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 sfc(i), φ(i) = 0 h(φ)(c)(i) := c(i), φ(i) = 1. 15. SIMPLICIAL SETS 91

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

ψ [m] / [n]

φ θ [1] in ∆, the square h(θ) Cn+1 / Cn+1

ψ∗ ψ∗ h(φ) Cm+1 / Cm+1 commutes in C. ∗ ∗ Here ψ is short for (Cosk0 C)(ψ). It is given by (ψ c)(i) = c(ψ(i)). This is a simple computation: sfc(ψ(i)), θ(ψ(i)) = 0 (ψ∗ ◦ h(θ))(c)(i) = c(ψ(i)), θ(ψ(i)) = 1 sfc(ψ(i)), φ(i) = 0 = c(ψ(i)), φ(i) = 1 sf(ψ∗c)(i), φ(i) = 0 = (ψ∗c)(i), φ(i) = 1 = (h(φ) ◦ ψ∗)(c)(i).

15. Simplicial sets

In this section we study the category sEns of simplicial sets. For a simplicial set X, we refer to the elements of Xn as simplicies of X of dimension n. Fix a simplex x ∈ Xn. For any injective ∆ morphism σ :[k] → [n], the simplex X(σ)(x) ∈ Xk is called a face of x. For any surjective ∆ morphism σ :[k] → [n], the simplex X(σ)(x) ∈ Xk is called a degeneracy of x. Recall that the degeneracy map X(σ) corresponding to a ∆ surjection σ always has a retract, so in particular it is injective. The simplex x is called non-degenerate if it is a degeneracy only of itself. It is easy to see that any simplex x ∈ Xm is a degeneracy of a unique non-degenerate simplex y: just choose l minimal such that there is a ∆ surjection σ :[m] → [l] and a y ∈ Xl with X(σ)(y) = x. The n skeleton functor of §13 behaves like the one the reader may be familiar with from topology: Lemma 15.1. For a simplicial set X and a non-negative integer n, the image of the adjunction morphism Skn X := Skn(trn X) → X consists of the simplices of X 92 VIII. CECHˇ THEORY

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

Proof. According to the general formulas of §13, an element [σ, x] of (Skn X)m is represented by a pair (σ, x) consisting of a surjective ∆ morphism σ :[m] → [k], 0 with k ≤ n, and an element x ∈ Xk. Given any map τ :[k] → [k ] from σ 0 0 0 to σ :[m] → [k ] in [m] ↓ ∆n and any x ∈ Xk0 we introduce the relation (σ0, x0) ∼ (σ, X(τ)(x0)). Two such pairs (x, σ) and (x0, σ0) represent the same element of (Skn X)m iﬀ they are equivalent under the smallest equivalence relation containing the relations ∼. The adjunction morphism (Skn X)m → Xm takes [σ, x] to X(σ)(x) ∈ Xm. The description of the image of the adjunction morphism is now clear. It remains only to prove that the adjunction morphism is injective when all non- 0 0 degenerate simplices of X have dimension ≤ n. Suppose [x, σ], [x , σ ] ∈ (Skn X)m 0 0 0 have X(σ)(x) = X(σ )(x ) =: y ∈ Xm and we wish to show that [x, σ] = [x , σ]. By hypothesis, we can write y = X(ρ)(z) for some surjective ∆ morphism ρ :[m] → [l] with l ≤ n and some z ∈ Xl. We can complete the diagram [m] σ σ0 ~ [k] ρ [k0]

τ τ 0 ~ [l] in ∆ as indicated (for example, by choosing sections of the surjections σ, σ0). We have X(τ)(z) = x because this is true after applying the monic map of sets X(σ) (both sides are then y), so (σ, x) ∼ (ρ, z). Similarly we have X(τ 0)(z) = x0, so (σ0, x0) ∼ (ρ, z). From transitivity of the equivalence relation induced by ∼, we 0 0 conclude [σ, x] = [σ , x ] ∈ (Skn X)m.

16. Standard simplicial sets

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

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

∆[n]m := {(i0, . . . , im) : 0 ≤ i0 ≤ · · · ≤ im ≤ n}. The boundary maps for ∆[n] are given by j dm : ∆[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 j sm : ∆[n]m → ∆[n]m+1

(i0, . . . , im) 7→ (i0, . . . , ij−1, ij, ij, ij+1, . . . , im). It is easy to check that for any X ∈ sEns, the map

HomsEns(∆[n],X) 7→ Xn

f 7→ fn((0, 1, . . . , n)) is bijective. Example 16.2. For each n ∈ N we also have the boundary ∂∆[n] of ∆[n], which is the sub-simplicial set of ∆[n] generated by

(1, 2, . . . , n), (0, 2, . . . , n),..., (0, 1, . . . , n − 1) ∈ ∆[n]n−1. The map

HomsEns(∂∆[n],X) → HomEns([n],Xn−1)

f 7→ (i 7→ fn−1(0, . . . , i − 1, i + 1, . . . , n)) is a bijection onto the set of f ∈ HomEns([n],Xn−1) satisfying i j−1 dn−1f(j) = dn−1f(i) for all 0 ≤ i < j ≤ n. Example 16.3. For example, the simplicial n sphere Sn ∈ sEns is the simplicial set with exactly two non-degenerate simplices: one in dimension zero and one in dimension n. Using Lemma 15.1 it is not hard to show that such a simplicial set exists and is characterized up to isomorphism (unique isomorphism unless n = 0) by this property.

17. Geometric realization

Given a simplicial set X, we deﬁne a topological space |X| called the geometric realization of X as follows. We start with the space a a top X := ∆n , n Xn top top where ∆n is the standard topological n simplex of §12. For clarity, we write ∆n (x) for the component of the coproduct indexed by an n simplex x ∈ Xn. Given any ∆ morphism σ :[m] → [n], we have the map of sets X(σ): Xn → Xm and the map top top top of topological spaces ∆ (σ) : ∆m → ∆n reﬂecting the cosimplicial structure of top top ∆ . For any x ∈ Xn we glue the component ∆m (X(σ)(x)) of X to the component top ∆n (x) of X via the map top top top ∆ (σ) : ∆m (X(σ)(x)) → ∆n (x). That is, we introduce the relation ∼ on X where P ∼ ∆top(σ)(P ) for all P ∈ top ∆m (X(σ)(x)). We let |X| be the space obtained from X by making all such glu- ings. That is, we let |X| be the quotient of X (with the quotient topology) by the equivalence relation generated by the relations ∼ introduced above. 94 VIII. CECHˇ THEORY

The construction is clearly functorial in X, hence defines a functor || : sEns → top Top. In fact, the standard n simplex ∆n is compact, so X is a compactly generated space (a set is open iff its intersection with each compact set is open), hence so is |X| since it is a quotient of X. We can therefore view || as a functor || : sEns → K, where K is the category of compactly generated topological spaces. As such, the functor || preserves finite inverse limits [Hov, 3.2.4]. For more on geometric realization, see [Mil]

18. The abelian setting

Here we brieﬂy review the theory of simplicial objects in an abelian category A. The basic theorems of Dold-Kan-Puppe assert that simplicial objects in A are the same thing is chain complexes in A supported in non-negative degrees. Given an A ∈ sA and an n ∈ N, we set n−1 i Nn(A) := ∩i=0 Ker(dn : An → An−1). (Note that the intersection is over the kernels of all but the last boundary map n dn : An → An−1. By convention, N0(A) := A0. It is easy to check that the last n n−1 n boundary map dn takes Nn(A) into Nn−1(A) and that dn−1dn = 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 homotopies in the simplicial sense of §14 to homotopies in the usual sense of chain complexes. When A is the category of modules over a ring R, then the simplicial set underlying any simplicial R module M is ﬁbrant and the homotopy groups of this simplicial set (with any choice of basepoint; the natural one being 0 ∈ M) are identiﬁed with the homology of the normalized chain complex Nn(A).

19. Simplicial rings

Let An denote the category of commutative rings with unit. A simplicial object in An is called a simplicial ring. Given an A ∈ sAn, a module over A is a simplicial abelian group M such that each Mn is equipped with the structure of an An module compatibly with the simplicial structures: that is, for each ∆-morphism σ :[m] → [n], the map of abelian groups M(σ): Mn → Mm should be A(σ): An → Am linear: M(σ)(a · m) = A(σ) · M(σ)(m)

for all a ∈ An, m ∈ Mn. For example, if A → B is a morphism of simplicial rings, then the modules of Kählerdifferentials ΩBn/An ∈ Mod(Bn) fit together to form a B module ΩB/A. 19. SIMPLICIAL RINGS 95

For a simplicial ring A, we set n−1 i Nn(A) := ∩i=0 Ker(dn : An → An−1) n i Zn(A) := ∩i=0 Ker(dn : An → An−1) n+1 Hn(A) := Nn(A)/dn+1Zn+1(A). n+1 Note that Nn(A), Zn(A) ⊆ An are ideals of An. The subset dn+1Zn+1(A) ⊆ An is n+1 n also an ideal because dn+1 : An+1 → An has a section sn : An → An+1 so for a ∈ An and b ∈ Zn+1(A) we have n+1 n+1 n n+1 adn+1(b) = dn+1(sn(a))dn+1(b) n+1 n = dn+1(sn(a)b) n and sn(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. n+1 If An is a noetherian ring, then the An modules Zn(A), dn+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 geometric realization §17 of the simplicial set underlying the simplicial ring A. The space |A| is of course pointed by 0 ∈ |A|. Since A is a ring object in simplicial sets and geometric realization (viewed as a functor to K) preserves finite inverse limits, |A| is a ring m object in K. An element [f] of πm(|A|) is represented by a map f : I → A taking m the boundary of the cube I to 0 ∈ |A| and an element [g] of πn(A) is represented by a map g : In → A taking the boundary of In to 0. Using the ring structure on A, 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 is graded 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 a natural differential graded ring structure. The multiplication maps

Nm(A) ⊗ Nn(A) → Nm+n(A) 96 VIII. CECHˇ THEORY

are deﬁned using the “shuﬄe product” map

Am ⊗ An → Am+n X a ⊗ b 7→ Sign(σ, τ)A(σ)(a)A(τ)(b). (σ,τ) The sum here....

20. Simplicial sheaves

Let X be a topological space, Sh(X) the category of sheaves on X. The category sSh(X) is called the category of simplicial sheaves on X. For F ∈ Sh(X), we will write s ∈ F to mean s ∈ F (U) for some U ∈ Ouv(X), and we say that some property of s holds “locally” if there is a cover {Ui} so that this property holds for each s|Ui . For example, we can say that f : F → G is an epimorphism in Sh(X) iﬀ, for every s ∈ G , there exists (locally) a t ∈ F such that f(t) = s. Let 1 be the terminal object of Sh(X) (the sheaf represented by the terminal object X ∈ Ouv(X)). Remark 20.1. The discussion to follow makes sense in an arbitrary topos. In this more general setting, s ∈ F means s ∈ Hom(G , F ) for some G and “locally” means “after precomposing with an epimorphism H → G .”

A simplicial sheaf F ∈ sSh(X) is called ﬁbrant in degree n if, for every k ∈ [n+1] and every

x0,..., xˆk, . . . , xn+1 ∈ Fn n n such that di xj = dj−1xi for all i < j in [n] both distinct from k, there is (locally) n+1 a y ∈ Xn+1 with di y = xi for all i 6= k in [n]. A simplicial sheaf is fibrant if it is fibrant in every degree. th The definition is motivated as follows: Label the i codimension one face {ti = 0} of the standard topological n + 1 simplex

top n+1 ∆n+1 := {(t0, . . . , tn+1) ∈ R : t0 + ··· + tn+1 = 1} th with xi, so the k face remains unlabelled; the labelled faces form the so called top k-horn ∆n+1[k] of the standard n + 1 simplex (I would call it the closed star of the th n n vertex ek opposite the k codimension one face). The condition di xj = dj−1xi says that the labellings “agree” on the codimension two faces and the fibrancy condition asserts the existence of a “map on the whole n+1 simplex” whose restriction to each of the faces in the k-horn is the map indicated by the labelling. Note that we make no demands on the restriction of this “filling map” to the kth face. Such a filling top top map always exists in topology because the inclusion of the k-horn ∆n+1[k] ,→ ∆n+1 admits a retract r, hence any map f from the k-horn to a topological space X can top be lifted to a map fr : ∆n+1 → X. 20. SIMPLICIAL SHEAVES 97

We can abstractly deﬁne the k-horn of a simplicial sheaf F to be

Fn+1[k] := lim {Fm : φ :[m] ,→ [n + 1], m ≤ n, k ∈ Im φ}. ←−

Then a section of Fn+1[k] is nothing but a tuple x0,..., xˆk, . . . , xn ∈ Fn as above. There is an obvious restriction map Fn+1 → Fn+1[k]; the fibrancy condition says that this should be surjective. A map from the terminal object 1 (the terminal object of sSh(X) is the terminal object 1 of Sh(X) regarded as a constant simplicial sheaf) to a simplicial sheaf F will be called a basepoint of F . To give such a map is the same thing as giving a map 1 → F0, for then the map 1 → Fn must be the composition of 1 → F0 and any degeneracy map F0 → Fn, since this degeneracy map is an isomorphism in the constant simplicial object 1. We will systematically use “1” as abuse of notation for the map 1 → F . The existence of a basepoint implies, in particular, that each map Fn → 1 is an epimorphism (since it has a section). The category of based or pointed simplicial sheaves is the category of simplicial sheaves under 1. Let 1 → F be a based fibrant simplicial sheaf, n a nonnegative integer. Let n ZnF be the subsheaf of Fn consisting of those x ∈ Fn with di = 1 for all i ∈ [n]. The sheaf ZnF is called the sheaf of n cycles in F . Define a relation ∼ on ZnF by declaring x ∼ y iff, locally, there is some z ∈ Fn+1 such that

n+1 (1) di z = 1 for i ∈ [n + 1] \{n, n + 1} and n+1 n+1 (2) dn = x, dn+1z = y.

I claim ∼ is an equivalence relation.

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

n+1 (1) di z = 1 for i ∈ [n − 1] and n+1 (2) di z = x for i ∈ {n, n + 1}.

First check that

(1,..., 1, x) ∈ Fn+1[n],

n+1 so by ﬁbrancy in degree n we can ﬁnd, locally, a y ∈ Fn+1 such that di y = 1 for n+1 n+1 i ∈ [n − 1] and dn+1y = x (we know nothing about dn 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

n+2 (1) di w = 1 for i ∈ [n − 1] and n+2 n+2 (2) di w = di w = y for i ∈ {n, n + 1}. 98 VIII. CECHˇ THEORY

n+2 I claim z := dn+2w ∈ Fn+1 is as desired. Indeed, for i ∈ [n − 1], we compute n+1 n+1 n+2 di z = di dn+2w n+1 n+2 = dn+1di w n+1 = dn+11 = 1, and for i ∈ {n, n + 1}, we compute n+1 n+1 n+2 di z = di dn+2w n+1 n+2 = dn+1di w n+1 = dn+1y = x.

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

n+1 (1) di z = 1 for i ∈ [n − 1] and n+1 n+1 (2) dn z = x, dn+1z = y.

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

n+1 (1) di r = 1 for i ∈ [n − 1] and n+1 n+1 (2) dn r = dn+1r = y.

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

(1,..., 1, z, r) ∈ Fn+2[n + 2],

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

n+2 (1) di u = 1 for i ∈ [n − 1] and n+2 n+2 (2) dn z = z, dn+1u = r.

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

n+1 (1) di w = 1 for i ∈ [n − 1] and n+1 n+1 (2) dn w = y, dn+1w = x.

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

n+1 (1) di u = 1 for i ∈ [n − 1] and n+1 n+1 (2) dn u = y, dn+1u = x and since y ∼ z there is locally some v ∈ Fn+1 with

n+1 (1) di v = 1 for i ∈ [n − 1] and n+1 n+1 (2) dn v = y, dn+1v = z. 21. WEAK EQUIVALENCES 99

n+1 n+1 From the equality dn u = dn v it follows that

(1,..., 1, u, v) ∈ Fn+2[n + 2].

By ﬁbrancy, there is locally some w ∈ Fn+2 with n+2 (1) di w = 1 for i ∈ [n − 1] and n+2 n+2 (2) dn w = u, dn+1w = v n+2 and one checks easily that dn+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 can see easily from the deﬁnition that

1 ! d0 / π0(F ) = lim F1 / F0 . −→ 1 d1 Theorem 20.2. Let f, g : F → G be two maps of based ﬁbrant simplicial sheaves which are simplicially homotopic. Then

πn(f) = πn(g): πn(F ) → πn(G ) for every n.

Proof. Exercise.

21. Weak equivalences

A morphism f : F → G of based ﬁbrant simplicial sheaves induces a morphism ZnF → 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 every n. A morphism of (unbased) fibrant simplicial sheaves f : F → G is called a weak equivalence if, for every U ∈ Ouv(X), every basepoint 1 → F |U , and every n, 3 the map πn(f|U ) is an isomorphism. It is necessary to look at all locally defined basepoints in order to have a notion of weak equivalence that is local (even the existence of a global basepoint is a serious restriction, so otherwise too many maps would vacuously be weak equivalences).

Lemma 21.1. A morphism of fibrant simplicial sheaves f : F → G is a weak equivalence iff fx : Fx → Gx is a weak equivalence of fibrant simplicial sets (bijection on homotopy sets for every basepoint) for every point x ∈ X.

Proof. f is a weak equivalence iff πn(f|U ): F |U → G |U is an isomorphism for every basepoint 1 → F |U . This can be checked on stalks by (6.1). Formation of homotopy sheaves commutes with stalks since the construction of πn(F |U ) can be phrased in terms of finite inverse limits (to construct 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 the map to the terminal object F → 1 is a weak equivalence. It is clear that πn(1) = 1 for all n, so F is a hypercover iﬀ πn(F |U ) = 1 for all n for all U ∈ Ouv(X) and for all basepoints 1 → F |U . The category HCov(X) of hypercovers of X is the full subcategory of HotsSh(X) consisting of hypercovers. Note, by (20.2), that the property of being a hypercover is invariant under homotopy equivalences.

22. Properties of coskeleta

The coskeleta functors play well with ﬁbrancy and homotopy sheaves. In order to streamline the proofs in this section, we begin with some notation for coskeleta. Like any inverse limit, (Coskn C)m sits inside the corresponding product, so we may view a section of (“map to”) (Coskn C)m as a section Y s = (s(φ)) ∈ Ck φ:[k],→[m],k≤n satisfying the condition: whenever φ :[k] ,→ [m] can be factored in ∆ as

ψ θ [k] / [l] / [m] =

φ with l ≤ n, we have s(φ) = C(ψ)(s(θ)). Of course, if m ≤ n, then such a section s is uniquely determined by s(Id : [m] ,→ [m]) ∈ Cm and conversely any such s(Id : [m] ,→ [m]) determines a unique section of (Coskn C)m by setting s(φ :[k] ,→ [m]) := C(φ)(s(Id : [m] ,→ [m])) and we see that Cm = (Coskn C)m as we already noted in (11). Evidently then, we should concentrate on the case m ≥ n. Here, s is determined by the s(φ :[n] ,→ [m]) since any map [k] → [m] with k ≤ n factors through such a 22. PROPERTIES OF COSKELETA 101

φ. The condition that Y s = (s(φ :[n] ,→ [m])) ∈ Cn φ:[n],→[m]

actually determines a section of (Coskn C)m is the following: For any two monic ∆ morphisms φ, ψ :[n] ,→ [m], deﬁne an integer k by k + 1 := | Im φ ∩ Im ψ|. If k = −1, then no condition is imposed. Otherwise, there is a (unique) commutative diagram [n] ? θ φ

[k] [m] >

η ψ [n] of ∆ monomorphisms and the requirement is: C(θ)(s(φ)) = C(η)(s(ψ)).

Lemma 22.1. For any C ∈ sC, and any 0 ≤ m ≤ n, the adjunction morphism

Coskm C → Coskn(Coskm C) is an isomorphism.

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

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

Proof. Suppose m > n and consider

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

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

m+1 as in the above discussion. We want to ﬁnd a y ∈ (Coskn F )m+1 with di y = xi for i ∈ [m + 1] \{k}. Using the above description of sections y ∈ (Coskn F )m+1, we deﬁne y(φ :[n] ,→ [m + 1]) as follows: Since m > n, the image of φ must miss at 102 VIII. CECHˇ THEORY

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

ψ si [n] / [m] / [m + 1] . :

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

[m] < O ψ si sj−1

θ $ [n] / [m − 1] [m + 1] : sj s η i " [m]

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

xi(ψ) = xi(sj−1θ) m = (dj−1)(θ) m = (di xj)(θ)

= xj(siθ)

= xj(η).

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

m+1 (di y)(φ :[n] ,→ [m]) = y(siφ)

= xi(φ).

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

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

x = (x1,..., xˆk, . . . , xn+1) ∈ (Coskn F )m+1[k] = Fn+1[k], 22. PROPERTIES OF COSKELETA 103

n+1 we use the fact that F is fibrant to find y ∈ Fn+1 such that di = xi for i ∈ [n + 1] \{k}. Define Y z ∈ Fn [n],→[n+1] by n+1 z(si :[n] ,→ [n + 1]) := di y ∈ Fn. n+1 One easily checks that in fact z ∈ (Coskn F )n+1 and that di z = xi.

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

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

ψ si [n] / [m] / [m + 1] :

φ and declare x(ψ), i = m + 1 y(φ) := 1, otherwise. I claim that this is independent of the choice of factorization. The only issue is when φ can be factored as siψ for i < m + 1 and as sm+1η, in which case we have a commutative diagram [m] < O ψ si sj−1

θ $ [n] / [m − 1] [m + 1] : sm+1 s η i " [m] of monomorphisms in ∆. But then we just compute

x(ψ) = x(smθ) m = (dmx)(θ) = 1

since x ∈ Zm(Coskn F ). This independence of choice of factorization implies that Y y ∈ Fn [n],→[m+1] 104 VIII. CECHˇ THEORY

is in fact in (Coskn F )m+1. Now we compute, for i ∈ [m] that m+1 (di y)(φ :[n] ,→ [m]) = y(siφ) = 1 and m+1 (dm+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 degree n+1 part πn+1 : Fn+1 → (Coskn F )n+1 of the adjunction morphism F → Coskn F is an epimorphism for every n. Then F → 1 is a weak equivalence (i.e. πn(F ) = 1 for every n).

Proof. Given x ∈ Zn(F ) = Zn(Coskn F ), we know from the previous lemma 4 that πn(Coskn F ) = 1. So, since x ∼ 1 in Zn(Coskn F ), there is locally (in fact, from the proof of the previous lemma, we don’t need “locally here”) some z ∈ (Coskn F )n+1 such that n+1 (1) di = 1 for i ∈ [n] and n+1 (2) dn+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 is some z ∈ Fn+1 with πn+1z = y. But F → Coskn F is a map of simplicial objects, so the diagrams

πn+1 Fn+1 / (Coskn F )n+1

m+1 m+1 di di $ x Fn = (Coskn F )n

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

23. Cechˇ theory revisited ˇ Given a sheaf F ∈ Ab(X) we used the Cech complex (1) of a cover {Ui : i ∈ I} of X to produce a resolution of F , and hence a spectral sequence (5.1) converging to the cohomology Hn(X, F ). This is putting the cart before the horse. Actually,

4 And we know from the lemma before that that Coskn F is ﬁbrant, so it makes sense to speak of πn(Coskn F ). 23. CECHˇ THEORY REVISITED 105 we will see in a moment that the Cechˇ complexes are obtained by taking resolutions “to the left” of ZX , then applying Hom( , F ), so we are really trying to compute n n H (X, F ) = Ext (ZX , F )

(c.f. (6.1)) by resolving ZX . We cannot generally ﬁnd a projective resolution of ZX as there are not generally enough projectives in Ab(X), but to correctly compute • i this Ext group, we only need a resolution G → ZX where the G are Hom( , F ) acyclic for this particular F . Even better, if we are willing to use a sequence • • • ··· G2 → G1 → G0 of such resolutions (really a ﬁltered category of resolutions), then we could get away with the hypothesis: lim Ext>0( m, ) = 0 −→ Gn F n→∞

The next thing to notice is that the failure of Cechˇ cohomology to compute sheaf cohomology in general has to do with the fact that, no matter how ﬁne a cover

U = {Ui : i ∈ I} of X we choose, we will have little control over the pairwise intersections Ui ∩ Uj (and the triple intersections Ui ∩ Uj ∩ Uk, etc.). We saw an extreme example of this in (8) where there was a ﬁnest possible cover U = {U0A,U0B} of a space Y , but where many sheaves on Y had higher cohomology on X = U0A ∩ U0B.

“The solution” is to use hypercovers (21) keep track of not only a cover U = {Ui} of X (i.e. a reﬁnement of the trivial cover), but we also allow the possibility of taking k a nontrivial cover Vi,j of each Ui ∩ Uj. Of course, once we do this, we will also want k to allow nontrivial covers of all new pairwise intersections Vi,j ∩ Uk, etc. Given any G ∈ Sh(X), we can form its 0-coskeleton

Cosk0 G ∈ sSh(X), which is just the simplicial sheaf with

n+1 (Cosk0 G )n = G n+1 [n] Q (for reasons of notation, it is convenient to view G as G = [n] G ). For a ∆ morphism φ :[m] → [n],

n+1 m+1 (Cosk0 G )(φ): G → G n n+1 n is given by (Cosk0 G )(φ)(s)(i) = s(φ(i)). The boundary map di : G → G is th n n n+1 given by omitting the i entry of an n-tuple and the degeneracy map ei : G → G repeats the ith entry. ` If G = i hUi for a cover {Ui}, then since ﬁnite products commute with coproducts in Sh(X) and the Yoneda functor h preserves inverse limits (intersections in 106 VIII. CECHˇ THEORY

Ouv(X)), Cosk0 G is the “expected” simplicial object: a (Cosk ) = h 0 F n Ui0 ∩···∩Uin (i0,...,in)

To obtain the Cechˇ complex constructed in (1) for an F ∈ Ab(X), we simply apply the sheaf Hom functor H om( , F ) to Cosk0 G and note that the result is a cosimplicial object in Ab(X). In this case we have

H om(Cosk0 G , F )n = H om((Cosk0 G )n, F ) a = om( h , ) H Ui0 ∩···∩Uin F (i0,...,in) Y = om(h , ) H Ui0 ∩···∩Uin F (i0,...,in) Y = . FUi0 ∩···∩Uin (i0,...,in) To this cosimplicial object, one can associate a chain complex supported in nonnegative degrees, which is just the Cechˇ resolution of F . Of course, there are varying ways to make this complex (the unnormalized chain complex, normalized chain complex, complement of the degeneracies in the normalized complex, etc.—all are homotopy equivalent) and these correspond to the variations on the Cechˇ complex (e.g. the alternating Cechˇ 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 |Ui has an obvious section obtained from the

structure map of hUi = hX |Ui into the coproduct G . (4) For any object C in any category (with ﬁnite inverse limits, say) if the map to the terminal object 1 has a section, then Cosk0 C → 1 is a homotopy

equivalence (14.1). Hence G |Ui → hUi is a homotopy equivalence in sSh(Ui).

(5) Functors preserve homotopy equivalences, so F |Ui → H om(Cosk0 G |Ui , F |Ui ) is also a homotopy equivalence in csAb(X). (6) The Dold-Kan theorem identiﬁes csAb(X) with the category of nonneg- atively graded cochain complexes in Ab(X) and this identiﬁcation takes homotopies in the sense of simplicial objects to homotopies in the sense

of chain complexes, hence F |Ui → H om(Cosk0 G |Ui , F |Ui ) is a homotopy equivalence. 24. EXERCISES 107

24. Exercises

0 Many of the constructions of this section involve factoring ΓΦ as H followed Φ by Hˇ (X, ), under the assumption that Φ has neighborhoods (then applying the Grothendieck spectral sequence). On the other hand, we can always factor ΓΦ as 0 HΦ followed by the usual global section functor Γ : PAb(X) → Ab, even without the neighborhoods assumption. However, it is not so clear whether we can apply the Grothendieck spectral sequence. Exercise 24.1. The class of flasque presheaves satisfies the following properties: (1) For any short exact sequence 0 → F 0 → F → F 00 → 0 in PAb(X) with F 0 and F flasque, F 00 is flasque. (2) A direct summand of a flasque presheaf is flasque. (3) For any F ∈ PAb(X), there is a flasque G ∈ PAb(X) and a monomorphism F ,→ G . Exercise 24.2. For any family of supports Φ on a topological space X, show 0 that the functor HΦ : Ab(X) → PAb(X) takes flasque sheaves to separated flasque 0 presheaves. If X is notherian, show that HΦ takes flasque sheaves to flasque sheaves. • Exercise 24.3. Is the δ-functor Hˇ (U, ) from PAb(X) → Ab (c.f. ) acyclic ˇ • for the class of flasque presheaves? Assume Φ has neighborhoods, so that HΦ(X, ) is also a δ-functor. Is it acyclic for flasque presheaves?

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