Homology of sheaves via Brown representability
aa r X i v : . [ m a t h . A T ] F e b HOMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY
FERNANDO SANCHO DE SALAS
Abstract.
We give an elementary construction of homology of sheaves from Brownrepresentability for the dual and see how its main properties are derived easily from theconstruction. Comparison with Poincar´e-Verdier duality and with homology of groupsare also developed.
Introduction
On an arbitrary topological space, cohomology is defined for any sheaf and studied withinthe framework of the theory of derived functors, whereas homology is usually defined onlyfor constant or locally constant coefficients and thus does not fit within such a framework.However, on finite topological spaces one may in fact define the homology and cohomologyof any sheaf and both of these constructions are developed within the framework of thetheory of derived functors. In other words, on a finite topological space it is possibleto consider the homology groups H i ( X, F ) with coefficients on any sheaf F , and in thiscase H i ( X, ) is the i -th (left-)derived functor of H ( X, ). This was already pointedout by Deheuvels ([3]) and a systematic treatment may be found in [13] (see also [2]). Ofcourse, on a finite topological space (more generally, on an Alexandroff space), the categoryShv( X ) of sheaves of abelian groups on X has enough projectives and this fails to be truefor general (and most common) topological spaces.Another context where one has an homology theory for sheaves is on locally compactand Haussdorff spaces, where one has Borel-Moore homology, developed in [1]; here wemean Borel-Moore homology with compact supports, which is the one with the expectedproperties of an homology theory (as singular homology). As it is mentioned in op. cit.,Bore-Moore homology is, more than a homology theory, a co-cohomology theory, and itsconstruction needs a detour through the theory of cosheaves.Another direction in homology theory is to construct homology of cosheaves, insteadof sheaves. The reason of this is that one expects homology to be covariant with respectto open inclusions, i.e., an open inclusion V ⊂ U must induce a morphism H i ( V, F ) → H i ( U, F ), hence cosheaves seem better adapted for this purpose. The problem here is thatthe category of cosheaves is not as good as that of sheaves, a main problem being the
Mathematics Subject Classification.
Key words and phrases. sheaves, homology, Brown representability.The author was supported by research project MTM2017-86042-P (MEC). existence of a cosheafification procedure of a precosheaf (the analog of the sheafification ofa presheaf) (see [11], [10]).The first aim of this paper is to give an elementary construction of an homology theoryof sheaves, under mild local hypothesis on the topological space, by using Brown repre-sentability for the dual, and which agrees with that of finite spaces and that of locallycompact Haussdorff ones. The second aim is to show that all the properties that onewould desire follow from the construction in a transparent way. Let us give some details.Let X be any topological space and π : X → {∗} the projection to a point; let us denoteby D ( X ) the (unbounded) derived category of the category of sheaves on X and D ( Z ) thederived category of abelian groups. Let us consider the inverse image functor π − : D ( Z ) → D ( X ) . This functor has a right adjoint, the derived global sections functor R Γ( X, ). This iscohomology. We shall use Brown representability for the dual to prove (Theorem 2.2)that π − has also a left adjoint, denoted by L ( X, ), under mild local hypothesis on X : we shall assume that X is locally connected and locally cohomologically trivial (seeDefinition 2.1); on locally compact Haussdorff spaces the latter condition can be weakenedto a locally cohomologically connected condition (see Remark 2.3). We shall define thehomology groups with coefficients on a sheaf F as H i ( X, F ) := H i L ( X, F ). It is importantto remark that one has to use the unbounded derived category (instead of D − ( X ), as onewould be tempted) in order to have Brown representability available, in the same way thatNeeman simplified Grothendieck duality via Brown representability ([7]) by consideringthe unbounded derived category of quasi-coherent sheaves.Once the homology groups H i ( X, F ) are defined, we show in an easy way its mainproperties: (a) Long exact sequence associated to an exact sequence of sheaves, (b) Dualitybetween homology and cohomology (Theorem 2.9), (c) Covariance with respect to conti-nuous maps X → Y (Proposition 2.9), (d) Mayer Vietoris sequences (Propositions 2.16and 2.17), (e) Vietoris theorem and homotopy invariance (Theorem 2.18 and Corollary2.19), (f) Excision (Proposition 2.26),(g) Universal coefficients formula (Theorem 2.27),(h) Cap product (Theorem 2.28), (i) K¨unneth formula (Theorem 2.34).In section 2.8 we make a comparison with Poincar´e-Verdier duality in a very general form.Let X be a topological space with a duality theory: an exact functor ω : D ( X ) → D ( Z )that admits a right adjoint ω ! (for example, X a locally compact Haussdorff space and ω = R Γ c ( X, ) the cohomology with compact support); let us denote D ωX := ω ! ( Z ) the“dualizing complex” and H iω ( X, F ) = H i ( ω ( F )). Then, there is a natural morphism(Proposition 2.37) H − i ( X, F L ⊗ D ωX ) → H i ( X, F )and we determine when this morphism is an isomorphism for any F (Theorem 2.38). Wealso characterize homological manifolds in section 2.8.1, as Bredon does on locally compactHaussdoff spaces ([1]) OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 3
Section 3 is devoted to the comparison between homology of sheaves a homology ofgroups, as it happens for cohomology of sheaves and cohomology of groups. This requiresthe introduction of a functor from sheaves to locally constant sheaves which is the homo-logical analog of the quasicoherator functor on schemes. This functor is again constructedfrom Brown representability.Finally, section 1 is devoted to the study of the underived version of homology, i.e, of thefunctor of cosections L(
X, F ) = H ( X, F ). It is nothing but the left adjoint of the inverseimage π − : Mod( Z ) → Shv( X ), where Mod( Z ) denotes the category of abelian groups.We shall see (Theorem 2.32) that there are enough L( X, )-acyclics and then L ( X, )may be viewed as a left derived functor of L( X, ). We also see in section 1.2 how thefunctor of cosections relates sheaves and cosheaves in a natural way.1. Cosections
Notations and conventions.
Let X be a topological space. By a sheaf on X wealways mean a sheaf of abelian groups. We shall denote by Shv( X ) the category of sheaveson X . If f : X ′ → X is a continuous map and F is a sheaf on X , f − F denotes the inverseimage sheaf by f . If j : U ֒ → X is an open subset, we also denote j − F = F | U and wedenote by j ! : Shv( U ) → Shv( X ) the extension by zero functor, which is left adjoint of j − .We shall denote F U = j ! F | U .Let X be a topological space, π X : X → {∗} the projection to a point, and Mod( Z ) thecategory of abelian groups. Proposition 1.1. If X is locally connected, the functor π − X : Mod( Z ) → Shv( X ) commutes with direct products.Proof. Let { G i } i ∈ I be a family of abelian groups. The natural morphism π − X ( Q i G i ) → Q i ( π − X G i ) is an isomorphism after taking sections on a connected open subset. Since X is locally connected, it is a stalkwise isomorphism. (cid:3) Theorem 1.2.
Let X be a locally connected space. The functor π − X : Mod( Z ) → Shv( X ) has a left adjoint L( X, ) : Shv( X ) → Mod( Z ) . Thus, for any sheaf F on X and any abelian group G one has Hom Z (L( X, F ) , G ) = Hom Shv( X ) ( F, π − X G ) . Proof.
Since π − X is exact and commutes with direct products, it has a left adjoint. (cid:3) Notation . As it is usual, the constant sheaf π − X G shall be denoted simply by G . Definition 1.3.
The group L(
X, F ) shall be called cosections of F on X .For the rest section 1, the space is always assumed to be locally connected. FERNANDO SANCHO DE SALAS
Proposition 1.4 (Explicit computation) . Let I be the set of connected and non-emptyopen subsets of X . For each i ∈ I , let U i be the connected open subset corresponding to i .We say that i ≤ j if U i ⊇ U j . Then L( X, F ) = lim → i ∈ I F ( U i ) . For example, for any constant sheaf G : L( X, G ) = G ⊕ π ( X ) . Proof.
Let F be a sheaf. Since X is locally connected, for any open subset U , F ( U ) = Q j ∈ π ( U ) F ( U j ). In other words, F is uniquely determined by its value on connected opensubsets. Then, if G is a constant sheaf, a morphism F → G is equivalent to giving acollection of morphisms φ i : F ( U i ) → G ( U i ) which are compatible with restrictions. Since G ( U i ) = G and the restrictions morphisms G ( U i ) → G ( U j ) are the identity, we concludethat a morphism F → G is a collection of morphisms φ i : F ( U i ) → G such that F ( U i ) φ i / / $ $ ❍❍❍❍❍❍❍❍❍ GF ( U j ) φ j = = ③③③③③③③③③ is commutative for any i ≤ j . This is a morphism lim → i ∈ I F ( U i ) → G . (cid:3) Proposition 1.5.
From representability, it follows (1) L( X, ) is right exact: an epimorphism F → F ′ → induces an epimorphism L( X, F ) → L( X, F ′ ) → . (2) L( X, ) commutes with direct sums: L( X, ⊕ i F i ) = ⊕ i L( X, F i ) . More generally, L( X, ) commutes with direct limits. Proposition 1.6.
Let f : X → Y be a continuous map between locally connected spaces.For any sheaf F on Y one has a natural morphism f ∗ : L( X, f − F ) → L( Y, F ) . If g : Y → Z is another continuous map, then ( g ◦ f ) ∗ = g ∗ ◦ f ∗ . Moreover (id X ) ∗ = id .Proof. For any abelian group G one has a natural morphismHom Shv( Y ) ( F, G ) → Hom
Shv( X ) ( f − F, f − G ) = Hom Shv( X ) ( f − F, G )i.e., a morphism Hom Z (L( Y, F ) , G ) → Hom Z (L( X, f − F ) , G )and then the desired morphism f ∗ : L( X, f − F ) → L( Y, F ). The equality ( g ◦ f ) ∗ = g ∗ ◦ f ∗ follows from the construction and from the equality f − ◦ g − = ( g ◦ f ) − ; the equality(id X ) ∗ = id from the equality (id X ) − = id. (cid:3) OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 5
Proposition 1.7.
For any open subset j : U ֒ → X and any sheaf F ′ on U , one has L( U, F ′ ) = L( X, j ! F ′ ) . Hence, for any sheaf F on X one has L( U, F | U ) = L( X, F U ) and the morphism j ∗ : L( U, F | U ) → L( X, F ) is obtained by applying L( X, ) to the mor-phism F U → F .Proof. The functor j − : Shv( X ) → Shv( U ) is right adjoint of j ! : Shv( U ) → Shv( X ). Theequality π − U = j − ◦ π − X induces, by adjunction, an equality L( U, ) = L( X, ) ◦ j ! . Thestatement about j ∗ follows from its construction. (cid:3) Sheaves and cosheaves.Definition 1.8. ([1, Ch.V, Def. 1.1]) A precosheaf Q on X is a covariant functor from thecategory of open subsets on X to abelian groups. Thus, one has an abelian group Q ( U )for each open subset U , and a morphism of groups e V U : Q ( V ) → Q ( U ) for each V ⊆ U such that e UU = id for any U and e W U = e V U ◦ e W V for any W ⊆ V ⊆ U . A cosheaf is a precosheaf Q such that, for any open subset U and any open covering U = ∪ i U i , thesequence (with obvious morphisms) ⊕ i,j Q ( U ij ) → ⊕ i Q ( U i ) → Q ( U ) → U ij = U i ∩ U j . A morphism of cosheaves Q → Q is just a morphismof functors, i.e., a natural transformation. We shall denote by Coshv( X ) the category ofcosheaves on X .A sequence of cosheaves Q → Q → Q → exact if Q ( U ) → Q ( U ) →Q ( U ) → U . In this case, the sequence 0 → Hom( Q , Q ) → Hom( Q , Q ) → Hom( Q , Q ) is exact for any Q .The direct sum ⊕ i Q i of cosheaves is defined by ( ⊕ i Q i )( U ) = ⊕ i Q i ( U ); it is a cosheaf,and it is the categorical direct sum. Examples 1.9. (1) Any abelian group G defines a cosheaf G cos , named constant cosheaf,by G cos ( U ) = G ⊕ π ( U ) (notice that we assume that X is locally connected, and then the connected componentsare open subsets). For any cosheaf Q and any abelian group G one hasHom Coshv ( Q , G cos ) = Hom Z ( Q ( X ) , G ) . (2) Let j : U ֒ → X be an open subset. A cosheaf Q ′ on X induces a cosheaf Q ′| U on eachopen subset U , defined as Q ′| U ( V ) = Q ′ ( V ) FERNANDO SANCHO DE SALAS for any open subset V of U . This defines a functorCoshv( X ) → Coshv( U ) , Q ′
7→ Q ′| U that takes constant cosheaves into constant cosheaves (i.e., ( G cos ) | U = G cos ). Conversely,a cosheaf Q on U induces a cosheaf j ! Q on X , defined by( j ! Q )( V ) = Q ( U ∩ V ) . This defines a functor j ! : Coshv( U ) → Coshv( X )and one has an adjunctionHom Coshv( X ) ( j ! Q , Q ′ ) = Hom Coshv( U ) ( Q, Q ′| U ) . For any abelian group G , we shall denote G cos U := j ! ( G cos ). If V ⊆ U , one has a naturalmorphism G cos V → G cos U .(3) For any cosheaves Q , Q ′ , we can define a sheaf Hom( Q , Q ′ ) by:Hom( Q , Q ′ )( U ) = Hom( Q | U , Q ′| U )and obvious restriction morphisms. We leave the reader to check that it is in fact a cosheaf.If F is a sheaf on X , the covariant behaviour of L( U, F | U ) with respect to U shows thatthere is a cosheaf attached to F : Definition 1.10.
Let F be a sheaf on X . The associated cosheaf , denoted by cos( F ), isdefined by cos( F )( U ) := L( U, F | U )and for any V ⊆ U , the morphism cos( F )( V ) → cos( F )( U ) is i ∗ : L( V, F | V ) → L( U, F | U ),with i : V ֒ → U . By Proposition 1.7, cos( F )( U ) = L( X, F U ) and cos( F )( V ) → cos( F )( U )is obtained by applying L( X, ) to F V → F U . Remark 1.11. cos( F ) is indeed a cosheaf. For any open subset U and any open covering U = ∪ i U i , one has an exact sequence of sheaves (let us denote U ij = U i ∩ U j ) ⊕ i,j F U ij → ⊕ i F U i → F U → X, ), an exact sequence of abelian groups ⊕ i,j cos( F )( U ij ) → ⊕ i cos( F )( U i ) → cos( F )( U ) → . A morphism of sheaves F → F induces a morphism of cosheaves cos( F ) → cos( F ),thus we obtain a functor cos : Shv( X ) → Coshv( X ) . Proposition 1.12.
The functor cos satisfies: (1)
It commutes with restrictions: for any open subset U and any sheaf F , one has: cos( F ) | U = cos( F | U ) . OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 7 (2)
It commutes with j ! . For any open subset j : U ֒ → X and any sheaf F on U onehas: cos( j ! F ) = j ! cos( F ) . (3) It takes constant sheaves into constant cosheaves: cos( G ) = G cos , for any abeliangroup G . Combining with (2) , one has cos( G U ) = G cos U for any abelian group G and any open subset U . (4) It commutes with direct sums (more generally, with direct limits): cos( ⊕ i F i ) = ⊕ i cos( F i ) . (5) It is right exact: if F ′ → F → F ′′ → is an exact sequence of sheaves, then cos( F ′ ) → cos( F ) → cos( F ′′ ) → is an exact sequence of cosheaves.Proof. (1) For any open subset V of U ,[cos( F ) | U ]( V ) = cos( F )( V ) = L( V, F | V ) = L( V, ( F | U ) | V ) = cos( F | U )( V ) . (2) For any open subset V of X , one has, on the one hand( j ! cos F ))( V ) = cos( F )( U ∩ V ) = L( U ∩ V, F | U ∩ V ) , and, on the other hand (let us denote j ′ : U ∩ V ֒ → V )cos( j ! F )( V ) = L( V, ( j ! F ) | V ) = L( V, ( j ′ ) ! F | U ∩ V ) 1 .
7= L( U ∩ V, F | U ∩ V ) . (3) For any open subset U one hascos( G )( U ) = L( U, G ) = G ⊕ π ( U ) = G cos ( U ) . (4) cos( ⊕ i F i )( U ) = L( U, ( ⊕ i F i ) | U ) = L( U, ⊕ i F i | U ) = ⊕ i L( U, F i | U ) = ⊕ i cos( F i )( U ) =( ⊕ i cos( F i ))( U ) . Analogous proof for the direct limit.(5) Since F L( U, F | U ) is right exact, one concludes. (cid:3) Definition 1.13.
Since the functor cos : Shv( X ) → Coshv( X ) is right exact and commuteswith direct sums, it has a right adjointshf : Coshv( X ) → Shv( X ) . Thus, for any sheaf F and cosheaf Q one hasHom Coshv (cos( F ) , Q ) = Hom Shv ( F, shf( Q )) . The explicit description of shf( Q ) is given by the following: Proposition 1.14.
Let Q be a cosheaf. For any open subset U , one has shf( Q )( U ) = Hom Coshv( X ) ( Z cos U , Q ) = Hom Coshv( U ) ( Z cos , Q | U ) and for any V ⊆ U , the morphism shf( Q )( U ) → shf( Q )( V ) FERNANDO SANCHO DE SALAS is obtained by taking
Hom( , Q ) in the natural morphism Z cos V → Z cos U (or it is the naturalrestriction morphism Hom( Z cos , Q | U ) → Hom( Z cos , Q | V ) ). In other words (see Examples1.9, (3) ), shf( Q ) = Hom( Z cos , Q ) . Proof.
One has shf( Q )( U ) = Hom( Z U , shf( Q )) = Hom(cos( Z U ) , Q ). One concludes be-cause cos( Z U ) = Z cos U , by (3) of Proposition 1.12. The statement about V ⊆ U is immedi-ate. (cid:3) Remark 1.15.
An alternative construction of shf( Q ) can be done in an analogous wayto that of cos( F ) in the following way. We leave the reader to check the details. Thefunctor Mod( Z ) → Coshv( X ) , G G cos , has a right adjoint, because it is right exact andcommutes with direct sums. Thus, for any cosheaf Q there is an abelian group Γ( X, Q )(the group of sections of Q on X ) and an isomorphismHom( G, Γ( X, Q )) = Hom( G cos , Q )and it may be explicitely computed by the formulaΓ( X, Q ) = lim ← i ∈ I Q ( U i )where I is the set of connected and non empty open subsets of X and i ≤ j if U i ⊆ U j . Now,for any open subsets U ⊆ V , the natural morphism Hom( G cos , Q | V ) → Hom( G cos , Q | U )induces a morphism Γ( V, Q | V ) → Γ( U, Q | U ) . The assignation U Γ( U, Q | U ) is the sheaf shf( Q ) defined above; in other words,shf( Q )( U ) = Γ( U, Q | U ) . Proposition 1.16.
The functor shf satisfies (1)
It commutes with restrictions: shf( Q ) | U = shf( Q | U ) . (2) It takes constant cosheaves into constant sheaves: shf( G cos ) = G. (3) It is left exact: If → Q ′ → Q → Q is an exact sequence of cosheaves (i.e., it isexact when applied to each U ), then → shf( Q ′ ) → shf( Q ) → shf( Q ) is an exactsequence of sheaves.Proof. (1) follows from adjunction and (2) of Proposition 1.12.(2) For any open subset U one has:shf( G cos )( U ) = Hom Coshv( U ) ( Z cos , G cos ) = Hom Z ( Z cos ( U ) , G ) = Y i ∈ π ( U ) G = G ( U ) . (3) follows from adjunction. (cid:3) Remark 1.17.
The functors cos : Shv( X ) → Coshv( X ), shf : Coshv( X ) → Shv( X ) donot yield an equivalence between the categories of sheaves and cosheaves. However, they OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 9 produce an equivalence between the categories of locally constant sheaves and locally con-stant cosheaves. Indeed, for any locally constant sheaf L (resp. any locally constant cosheaf L ) the natural morphism L → shf(cos( L )) (resp., the natural morphism cos(shf( L )) → L )is an isomorphism because it is so after restricting to any open subset U such that L | U isconstant (resp. such that L | U is constant).Finally, to emphasize the symmetry between the functors cos and shf, we leave thereader to check the following: Remark 1.18.
Let j : U ֒ → X be an open subset. For any sheaf F and cosheaf Q onehas: Hom Z (cos( F )( U ) , G ) = Hom Shv( X ) ( F, j ∗ G )Hom Z ( G, shf( Q )( U )) = Hom Coshv( X ) ( j ! G cos , Q ) . PROBLEM: Let G be an abelian group and let us denote H ( X, G ) = G π ( X ) . We wantto derive this functor (by the left). We can take two directions. One is to think H ( X, G )as G cos ( X ), hence as the functor Q 7→ Q ( X ) on cosheaves applied to the constant cosheaf;then, we are lead to derive the functor Q 7→ Q ( X ). This direction finds serious difficultiesrelated to the pathologies of the category of cosheaves. The main difficulty is that there isnot a cosheafication procedure of a precosheaf (in an analogous way to the sheafication ofa presheaf), and then, there is not a good definition of the kernel (cosheaf) of a morphismof cosheaves; thus the category of cosheaves is not abelian. To overcome this difficulty oneis lead to work with cosheaves of abelian pro-groups (see [11], [10]), and things becomemore difficult and technical. To our knowledge, there is still not a theory of homology ofcosheaves avaialable.A second direction is to think H ( X, G ) as the cosections of the constant sheaf, i.e.L(
X, G ). This leads to derive the functor on sheaves F L( X, F ). The problem here isthat the category Shv( X ) has not enough projectives. If X is a finite topological space(more generally, an Alexandroff space), the category Shv( X ) has enough projectives andthe left derived functor L ( X, ) of L( X, ) is left adjoint of π − X : D (Ab) → D ( X ); thus,the simplest idea is to define (for a general topological space) L ( X, ) as the left adjointof π − X . The existence of this adjoint needs some additional local hypothesis on X , but doinclude, besides Alexandroff spaces, the “most common” topological spaces .2. Homology
Notations.
For this section we shall use an extensive use of basic results on the(unbounded) derived category of sheaves on a topological space, which may be found in[14]. Throughout the paper an injective or flat complex means a K -injective or K -flatcomplex in the sense of [14].For any topological space X , D ( X ) denotes the (unbounded) derived category of sheaveson X . If X is a point, we obtain the derived category of abelian groups, denoted by D ( Z ).For any F ∈ D ( X ) and any open subset U , R Γ( U, F ) denotes the left derived functor of sections on U . For any complexes of sheaves F, F ′ on X , Hom • ( F, F ′ ) denotes the complexof homomorphisms and R Hom • ( F, F ′ ) its left derived functor. We denote by Hom • ( F, F ′ )the complex of sheaves of homomorphisms and R Hom • ( F, F ′ ) its left derived functor. Wedenote F L ⊗ F ′ the derived tensor product of F and F ′ . One has natural isomorphisms R Hom • ( Z U , F ) = R Γ( U, F ) R Γ( U, R Hom • ( F, F ′ )) = R Hom • ( F | U , F ′| U ) R Hom • ( F L ⊗ F ′ , F ′′ ) = R Hom • ( F, R Hom • ( F ′ , F ′′ )) . For any complex of sheaves F , we denote by F ∨ its derived dual: F ∨ := R Hom • ( F, Z ).For any continuous map f : X → Y , one has the inverse image functor f − : D ( Y ) → D ( X )and the (derived) direct image R f ∗ : D ( X ) → D ( Y ), which is right adjoint of f − . For anyopen subset j : U ֒ → X one has the functor j ! : D ( U ) → D ( X ), which is left adjoint of j − ,and we denote F U := j ! j − F . One has an exact triangle F U → F → F X − U where F X − U := i ∗ i − F , with i : X − U ֒ → X .For any closed subset Y of X , R Γ Y ( X, F ) denotes de right derived functor of globalsections supported on Y and R Γ Y F its sheafified version. One has R Γ Y ( X, F ) = R Hom • ( Z Y , F ) , R Γ Y F = R Hom • ( Z Y , F )A perfect complex is a complex E which is locally isomorphic to a bounded complex0 → G p → · · · → G q → G i is a finite and free Z -module. In other words, foreach point x ∈ X , there is a neighbourhood U of X and an isomorphism E | U ≃ π − U E ,where E is a finite complex of finite and free Z -modules. A complex E of abelian groupsis perfect if and only if is bounded (i.e. has bounded cohomology) and H i ( E ) is finitelygenerated for any i . Finally, we shall use that an abelian group G is zero if an only ifHom Z ( G, Q / Z ) = 0 and a complex E of abelian groups is 0 (in the derived category) if andonly if E ∨ = 0 (see [1, Ch.V, Prop. 13.7]); in particular, a morphism E → E in D ( Z ) isan isomorphism if and only if its dual E ∨ → E ∨ is so.2.2. Homology.Definition 2.1.
We say that X is cohomologically trivial if H i ( X, G ) = 0 for any i > G . Any homotopically trival space is cohomologically trivial. Anyintegral scheme is cohomologically trivial.Let D ( Z ) be the derived category of abelian groups and D ( X ) the derived category ofsheaves on X (notice: we are considering the unbounded derived category, i.e., unboundedcomplexes). The category D ( X ) has direct products, but it is not the direct product of OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 11 complexes. Given a family of complexes K i , we shall denote Q i K i the direct productcomplex and Q Di K i the direct product in D ( X ). If K i → I i are injective resolutions, then Y Di K i := Y i I i . This pathology does not occur in D ( Z ); that is, for any collection G i of objects in D ( Z ),one has that Q i G i = Q Di G i . Theorem 2.2.
Assume that X is locally connected and locally cohomologically trivial. Thefunctor π − X : D ( Z ) → D ( X ) has a left adjoint L ( X, ) : D ( X ) → D ( Z ) . Thus, for any F ∈ D ( X ) and any G ∈ D ( Z ) one has Hom D ( Z ) ( L ( X, F ) , G ) = Hom D ( X ) ( F, π − X G ) . Proof.
Let us denote π = π X . By Brown representability for the dual (see [8],[9], [5])it suffices to see that π − : D ( Z ) → D ( X ) commutes with direct products, i.e.: For anycollection { G i } i ∈ I , G i ∈ D ( Z ), the natural morphism π − ( Y i G i ) → Y Di π − G i is an isomorphism (in D ( X )). Since any object G ∈ D ( Z ) is isomorphic to Q n H n ( G )[ − n ],we may assume that G i is an abelian group (in some degree). Now, if π − G i → I i is aninjective resolution of π − G i , we have to prove that the morphism of complexes Y i π − G i → Y i I i is a quasi-isomorphism. For any connected and cohomologically trivial open subset U Y i Γ( U, π − G i ) = Γ( U, Y i π − G i ) → Γ( U, Y i I i ) = Y i Γ( U, I i )is a quasi-isomorphism because Γ( U, π − G i ) → Γ( U, I i ) is so. (cid:3) Remark 2.3.
On locally compact Haussdorff spaces the locally cohomological trivial con-dition can be weakened by a cohomologically locally connected condition (see [1])). Thiscondition means that for each point x ∈ X and neighbourhood U of x , there is a neigh-bourhood V ⊂ U of x such that H n ( U, Z ) → H n ( V, Z ) is null for any n >
0. Then,by universal coefficients theorem, one has that H n ( U, G ) → H n ( V, G ) is also null for anyabelian group G . Hence, for any family of abelian groups G i one has thatlim → x ∈ U Y i H n ( U, G i ) = 0for any n >
0, and then (under locally connected hypothesis) one concludes that Q i π − G i → Q i I i is a quasi-isomorphism, and Theorem 2.2 still holds. Remark 2.4.
For the rest of the paper we shall assume that the topological spaces involvedsatisfy that π − has a left adjoint, i.e., that L ( , ) exists, since, for most of the results, thelocal hypothesis of Theorem 2.2 or Remark 2.3 will not play any role. When they do, itwill be explicitly stated. Definition 2.5.
For any complex K , we shall denote H i ( K ) := H − i ( K ). For any F ∈ D ( X ) we shall denote H i ( X, F ) := H i [ L ( X, F )]and call it the i -th homology group with coefficients in F . Remark 2.6.
Since π − X is exact, L ( X, ) is exact too (see [8]). Thus, for any exacttriangle F → F ′ → F ′′ one has an exact triangle L ( X, F ) → L ( X, F ′ ) → L ( X, F ′′ )and then a long exact sequence of homology groups · · · → H i ( X, F ) → H i ( X, F ′ ) → H i ( X, F ′′ ) → · · · . Moreover, L ( X, ) commutes with direct sums. Notation.
For any G ∈ D ( Z ), we shall often denote G := π − X G the complex of constantsheaves on X . Remark 2.7.
For any F ∈ D ( X ) and any G ∈ D ( Z ) one has: R Hom • ( L ( X, F ) , G ) = R Hom • ( F, G ) . Proof.
Let ǫ : F → π − X L ( X, F ) be the unit map. The composition R Hom • ( L ( X, F ) , G ) → R Hom • ( π − X L ( X, F ) , π − X G ) ǫ ∗ → R Hom • ( F, π − X G )is an isomorphism, since it is so after taking H i . (cid:3) Proposition 2.8.
For any sheaf F on X one has: (a) H i ( X, F ) = 0 for any i < . (b) H ( X, F ) = L(
X, F ) . (c) H i ( X, ) commutes with direct sums. (d) L ( X, ) maps D ≤ n ( X ) into D ≤ n ( Z ) where D ≤ n stands for complexes with H i = 0 for i > n .Proof. (a) It suffices to see that Hom Z ( H i ( X, F ) , Q / Z ) = 0 for i <
0. NowHom Z ( H i ( X, F ) , Q / Z ) = H i Hom • ( L ( X, F ) , Q / Z ) = H i R Hom • ( F, Q / Z ) 2 .
7= Ext i ( F, Q / Z ) , which is 0 for i < OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 13 (b) Since H i ( X, F ) = 0 for i >
0, for any abelian group G one hasHom Z ( H ( X, F ) , G ) = H R Hom • ( L ( X, F ) , G ) = H R Hom • ( F, G )= Hom
Shv ( F, G ) = Hom Z (L( X, F ) , G ) . (c) This is immediate, since L ( X, ) commutes with direct sums.(d) It is an easy consequence of Remark 2.7 and is left to the reader. We shall not makeuse of it. (cid:3) Theorem 2.9 (Duality between homology and cohomology) . For any F ∈ D ( X ) and any G ∈ D ( Z ) one has: R Hom • ( L ( X, F ) , G ) = R Γ( X, R Hom • ( F, G )) , In particular, R Hom • ( L ( X, Z ) , G ) = R Γ( X, G ) and L ( X, F ) ∨ = R Γ( X, F ∨ ) . where ( ) ∨ stands for the derived dual R Hom • ( , Z ) . Then one has the (split) exactsequence → Ext Z ( H i − ( X, F ) , Q / Z ) → H i ( X, F ∨ ) → Hom Z ( H i ( X, F ) , Z ) → . Proof.
In fact, R Hom • ( L ( X, F ) , G ) = R Hom • ( F, G ) = Γ( X, R Hom • ( F, G )) . If F = Z , onobtains R Hom • ( L ( X, Z ) , G ) = R Γ( X, R Hom • ( Z , G )) = R Γ( X, G ). The last formula isjust to take G = Z . (cid:3) Corollary 2.10. X is cohomologically trivial if and only if H i ( X, Z ) = 0 for any i > .Proof. X is cohomologically trivial if and only if R Γ( X, G ) = Γ(
X, G ) for any abeliangroup G . Since R Γ( X, G ) = R Hom • ( L ( X, Z ) , G ), this is equivalent to say that L ( X, Z ) =L( X, Z ), i.e., H i ( X, Z ) = 0 for any i > (cid:3) Proposition 2.11.
Let j : U ֒ → X be an open subset, Y ֒ → X a closed subset and G ∈ D ( Z ) . (1) For any F ∈ D ( U ) one has L ( U, F ) = L ( X, j ! F ) . In particular, for any F ∈ D ( X ) one has L ( U, F | U ) = L ( X, F U ) . (2) For any F ∈ D ( X ) one has R Hom • ( L ( Y, F | Y ) , G ) = R Hom • ( F, G Y ) . (3) Let S = Y ∩ U . For any F ∈ D ( X ) one has R Hom • ( L ( S, F | S ) , G ) = R Hom • ( F U , G Y ) . Proof. (1) The functor j ! : D ( U ) → D ( X ) is left adjoint of j − . Since π − U = j − ◦ π − X , weconclude that L ( U, ) = L ( X, ) ◦ j ! .(2) R Hom • ( L ( Y, F | Y ) , G ) = R Hom • ( F | Y , G ) = R Hom • ( F, i ∗ G ).(3) By (2) applied to S ֒ → U , one has R Hom • ( L ( S, F | S ) , G ) = R Hom • ( F | U , ( G | U ) S ).Since ( G | U ) S = ( G Y ) | U , we obtain R Hom • ( L ( S, F | S ) , G ) = R Hom • ( F | U , ( G Y ) | U ) = R Hom • ( F U , G Y ) . (cid:3) Definition 2.12.
Let F be a complex of sheaves on X . We denote by L( X, F ) the complexof abelian groups · · · → L( X, F n ) → L( X, F n +1 ) → · · · and one has a natural morphism of complexes F → π − X L( X, F ), hence a morphism in D ( Z ) L ( X, F ) → L( X, F ) . An object F ∈ D ( X ) is called L -acyclic if the morphism L ( X, F ) → L( X, F ) is an iso-morphism in D ( Z ). For example, a sheaf F is L-acyclic iff H i ( X, F ) = 0 for any i >
Proposition 2.13.
An open subset U is cohomologically trivial if and only if Z U is L -acyclic.Proof. Since H i ( X, Z U ) 2 . H i ( U, Z ), the result follows from Corollary 2.10. (cid:3) Proposition 2.14.
Let f : X → Y a continuous map between locally connected spaces.For any F ∈ D ( Y ) one has a natural morphism f ∗ : L ( X, f − F ) → L ( Y, F ) hence a morphism f ∗ : H i ( X, f − F ) → H i ( Y, F ) . For any continuous maps X f → Y g → Z one has: ( g ◦ f ) ∗ = g ∗ ◦ f ∗ . For any X , Id X ∗ = Id .Proof. The inverse image morphism Hom D ( Y ) ( F, G ) → Hom D ( X ) ( f − F, G ) gives, by Yoneda,the desired morphism. The equality ( g ◦ f ) − = f − ◦ g − yields the equality ( g ◦ f ) ∗ = g ∗ ◦ f ∗ and the equality Id − X = Id yields that Id X ∗ = Id. (cid:3) Remark 2.15.
Let F ∈ D ( Y ) and ǫ F : F → π − Y L ( Y, F ) the unit morphism. The diagram f − F f − ( ǫ F ) / / ǫ f − F ' ' ◆◆◆◆◆◆◆◆◆◆◆ π − X L ( Y, F ) π − X L ( X, f − F ) π − X ( f ∗ ) ♠♠♠♠♠♠♠♠♠♠♠♠ is commutative. OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 15
Proposition 2.16 (Mayer-Vietoris) . Let X = U ∪ V , with U, V open subsets. For any F ∈ D ( X ) one has an exact triangle: L ( U ∩ V, F | U ∩ V ) → L ( U, F | U ) ⊕ L ( V, F | V ) → L ( X, F ) Proof.
Applying L ( X, ) to the exact triangle F U ∩ V → F U ⊕ F V → F , one concludes byProposition 2.11. (cid:3) Proposition 2.17 (Mayer-Vietoris2) . Let X = Y ∪ Z , with Y, Z closed subsets. For any F ∈ D ( X ) one has a long exact sequence · · · → H i ( Y ∩ Z, F | Y ∩ Z ) → H i ( Y, F | Y ) ⊕ H i ( Z, F | Z ) → H i ( X, F ) → · · · Proof.
For simplicity, we shall use the notation L ( C, F ) := L ( C, F | C ) for any closed subset C and any F ∈ D ( X ). For any G ∈ D ( Z ) we have an exact sequence0 → G → G Y ⊕ G Z → G Y ∩ Z → . Applying R Hom • ( F, ) and Proposition 2.11 we obtain an exact triangle( ∗ ) R Hom • ( L ( X, F ) , G ) → R Hom • ( L ( Y, F ) ⊕ L ( Z, F ) , G ) → R Hom • ( L ( Y ∩ Z, F ) , G )hence morphisms L ( X, F )[ − → L ( Y ∩ Z, F | Y ∩ Z ) → L ( Y, F | Y ) ⊕ L ( Z, F | Z ) → L ( X, F )and then a sequence · · · → H i +1 ( X, F ) → H i ( Y ∩ Z, F | Y ∩ Z ) → H i ( Y, F | Y ) ⊕ H i ( Z, F | Z ) → H i ( X, F ) → · · · which is exact because, taking Hom( , Q / Z ), one obtains the long exact sequence associ-ated to the triangle ( ∗ ) (for G = Q / Z ). (cid:3) Vietoris and homotopy invariance.Theorem 2.18 (See [1, Ch.V, Thm. 6.1]) . Let f : X → Y be a proper, separated andsurjective morphism. Assume that f − ( y ) is connected and cohomologically trivial for any y ∈ Y . Then, for any F ∈ D ( Y ) , the morphism f ∗ : L ( X, f − F ) → L ( Y, F ) is an isomorphism.Proof. Recall that f ∗ is defined, by Yoneda, from the inverse image morphismHom D ( Y ) ( F, G ) → Hom D ( X ) ( f − F, G ) = Hom D ( Y ) ( F, R f ∗ f − G ) . Since G → R f ∗ f − G is an isomorphism by the hypothesis and base change theorem (see[4]), we are done. (cid:3) Corollary 2.19 (See [1, Ch. V, Cor. 6.13]) . If f, g : X → Y are homotopic, then, for any G ∈ D ( Z ) , the morphisms f ∗ , g ∗ : L ( X, G ) → L ( Y, G ) coincide. Consequently, if X and Y are homotopic, then L ( X, G ) ≃ L ( Y, G ) for any G ∈ D ( Z ) .Proof. By hypothesis, there exists a continuous map H : X × [0 , → Y such that H = f , H = g . It suffices to see that the morphism H t ∗ : L ( X, G ) → L ( Y, G ) does not dependon t . Let i t : X → X × [0 , i t ( x ) = ( x, t ). Since H t = H ◦ i t , it suffices to see that i t ∗ : L ( X, G ) → L ( X × [0 , , G ) does not depend on t . Let π : X × [0 , → X be theprojection. Since i t ∗ ◦ π ∗ = id and π ∗ is an isomorphism by Theorem 2.18, we concludethat i t ∗ is the inverse of π ∗ , and then it does not depend on t . (cid:3) Local homology and Excision.Definition 2.20.
Let Y be a closed subset of X . Let us denote by L Y ( X, F ) the cokernelof the map L( X − Y, F | X − Y ) → L( X, F ) . Since L( X − Y, F | X − Y ) = L( X, F X − Y ), it follows from the exact sequence0 → F X − Y → F → F Y → Y ( X, F ) = L(
X, F Y ) and thenHom Z (L Y ( X, F ) , G ) = Hom Shv ( F Y , G ) = (cid:26) Hom
Shv ( F, Γ Y G )Γ Y ( X, Hom(
F, G )) . Definition 2.21.
For any F ∈ D ( X ), we define L Y ( X, F ) := L ( X, F Y ) and one has R Hom • ( L Y ( X, F ) , G ) = R Hom • ( F Y , G ) = (cid:26) R Hom • ( F, R Γ Y G ) R Γ Y ( X, R Hom • ( F, G )) . We shall denote H Yi ( X, F ) := H i [ L Y ( X, F )] . Remark 2.22.
The usual notation for this local homology would be H i ( X, U ; F ); we haveprefered to use a notation “dual” to that of local cohomology. Remark 2.23.
Applying L ( X, ) to the exact triangle F X − Y → F → F Y one obtains theexact triangle L ( X − Y, F | X − Y ) = L ( X, F X − Y ) → L ( X, F ) → L ( X, F Y ) = L Y ( X, F )and then a long exact sequence · · · → H i ( X − Y, F | X − Y ) → H i ( X, F ) → H Yi ( X, F ) → · · · OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 17
Proposition 2.24.
For any closed subsets
Y, Z of X and any F ∈ D ( X ) one has an exacttriangle L Y ∪ Z ( X, F ) → L Y ( X, F ) ⊕ L Z ( X, F ) → L Y ∩ Z ( X, F ) and then a long exact sequence · · · → H Y ∪ Zi ( X, F ) → H Yi ( X, F ) ⊕ H Zi ( X, F ) → H Y ∩ Zi ( X, F ) → · · · Proof.
Apply L ( X, ) to the exact triangle F Y ∪ Z → F Y ⊕ F Z → F Y ∩ Z . (cid:3) Remark 2.25.
For any open subset j : U ֒ → X and any F ∈ D ( U ) one has: L Y ∩ U ( U, F ) = L Y ( X, j ! F ) . Proof. L Y ∩ U ( U, F ) = L ( U, F Y ∩ U ) 2 . L ( X, j ! F Y ∩ U ) = L ( X, ( j ! F ) Y ) = L Y ( X, j ! F ). (cid:3) Proposition 2.26 (Excision) . Let U be an open subset of X containing a closed subset Y of X . The natural morphism L Y ( U, F | U ) → L Y ( X, F ) is an isomorphism.Proof. Since L Y ( U, F | U ) 2 . L Y ( X, F U ), applying L Y ( X, ) to the exact triangle F U → F → F X − U , it suffices to see that L Y ( X, F X − U ) = 0. But L Y ( X, F X − U ) = L ( X, ( F X − U ) Y ) =0 because Y ∩ ( X − U ) is empty. (cid:3) Universal coefficients formula.Theorem 2.27 (See [1, Ch.V, 3.10]) . For any F ∈ D ( X ) and any G ∈ D ( Z ) , there is anatural isomorphism L ( X, F L ⊗ G ) ∼ → L ( X, F ) L ⊗ G In particular, for any abelian group G one has a (split) exact sequence → H i ( X, F ) ⊗ G → H i ( X, F L ⊗ G ) → Tor ( H i − ( X, F ) , G ) → . Proof.
Let us first define the morphism. Let π : X → {∗} be the projection to a point.The unit map F → π − L ( X, F ) induces a morphism F L ⊗ π − G → π − L ( X, F ) L ⊗ π − G = π − ( L ( X, F ) L ⊗ G ), hence a morphism L ( X, F L ⊗ π − G ) → L ( X, F ) L ⊗ G . In order to seethat it is an isomorphism, we see the isomorphism after applying Hom D ( Z ) ( , A ) for any A ∈ D ( Z ). Now,Hom D ( Z ) ( L ( X, F ) L ⊗ G, A ) = Hom D ( Z ) ( L ( X, F ) , R Hom • ( G, A ))= Hom D ( X ) ( F, π − R Hom • ( G, A )) and Hom D ( Z ) ( L ( X, F L ⊗ π − G ) , A ) = Hom D ( X ) ( F L ⊗ π − G, π − A )= Hom D ( X ) ( F, R Hom • ( π − G, π − A )and we conclude if we prove that π − R Hom • ( G, A ) → R Hom • ( π − G, π − A )is an isomorphism. If G is a free Z -module, then the result follows because π − com-mutes with direct products. If G is a Z -module, we conclude by putting G as a coker-nel of a monomorphism between free Z -modules. Since any G ∈ D ( Z ) is isomorphic to ⊕ n H n ( G )[ − n ], the result follows. (cid:3) Cap product. (See [1, Ch.V, Sec.10])
Theorem 2.28.
For any
F, F ′ ∈ D ( X ) there is a morphism ∪ : L ( X, F ) L ⊗ R Γ( X, F ′ ) → L ( X, F L ⊗ F ′ ) and hence morphisms ∪ : H p ( X, F ) ⊗ H q ( X, F ′ ) → H p − q ( X, F L ⊗ F ′ ) . Proof.
The unit morphism F L ⊗ F ′ → π − X L ( X, F L ⊗ F ′ ) defines a morphismΦ : F ′ → R Hom • ( F, π − X L ( X, F L ⊗ F ′ )) . Taking global sections, one obtains a morphismΦ ∪ : R Γ( X, F ′ ) → R Hom • ( F, π − X L ( X, F L ⊗ F ′ )) = R Hom • ( L ( X, F ) , L ( X, F L ⊗ F ′ )) , which is equivalent, by adjunction, to a morphism ∪ : L ( X, F ) L ⊗ R Γ( X, F ′ ) → L ( X, F L ⊗ F ′ ) . (cid:3) Theorem 2.29 (See [1, Ch.V, Sec.10, (9)]) . Let f : X → Y be a continuous map. For any F, F ′ ∈ D ( Y ) the diagram L ( Y, F ) Ψ ∪ Y / / R Hom • ( R Γ( Y, F ′ ) , L ( Y, F L ⊗ F ′ )) L ( X, f − F ) f ∗ O O Ψ ∪ X / / R Hom • ( R Γ( X, f − F ′ ) , L ( X, f − ( F L ⊗ F ′ ))) f ∗∗ O O is commutative, where f ∗∗ is the morphism induced by the morphisms f ∗ : L ( X, f − ( F L ⊗ F ′ )) → L ( Y, F L ⊗ F ′ ) , f ∗ : R Γ( Y, F ′ ) → R Γ( X, f − F ′ ) . OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 19 and Ψ ∪ Y , Ψ ∪ X are the morphisms corresponding to ∪ Y , ∪ X by adjunction. Thus, one has f ∗ ( α ∪ f ∗ ( β )) = f ∗ ( α ) ∪ β for any α ∈ H p ( X, f − F ) , β ∈ H q ( Y, F ′ ) .Proof. The statement is a consequence of the functoriality of all the constructions. Let usgive the details. First notice that the morphism f ∗ : R Γ( Y, F ′ ) → R Γ( X, f − F ′ ) is nothingbut a particular case of the natural morphism R Hom • ( F, F ′ ) → R Hom • ( f − F, f − F ′ )(when F = Z ) that we also denote by f ∗ . Let us simplify the notations, by putting L ( F ) = L ( Y, F ), R Γ( F ′ ) = R Γ( Y, F ′ ) and so on.The morphism Φ Y : F → R Hom • ( F ′ , π − Y L ( F L ⊗ F ′ )), induces a morphism f − Φ Y : f − F → R Hom • ( f − F ′ , π − X L ( F L ⊗ F ′ )) and a commutative diagram R Γ( F ′ ) Φ ∪ Y / / f ∗ (cid:15) (cid:15) R Hom • ( F, π − Y L ( F L ⊗ F ′ )) f ∗ (cid:15) (cid:15) R Γ( f − F ′ ) Γ( f − Φ Y ) / / R Hom • ( f − F, π − X L ( F L ⊗ F ′ )) . On the other hand, the composition R Γ( f − F ′ ) Γ( f − Φ Y ) −→ R Hom • ( f − F, π − X L ( F L ⊗ F ′ )) f ∗ → R Hom • ( f − F, π − X L ( f − ( F L ⊗ F ′ )))is the morphism Φ ∪ X (see Remark 2.15).The statement now follows from the adjunctions between Φ ∪ Y and Ψ ∪ Y (and betweenΦ ∪ X and Ψ ∪ X ). (cid:3) K¨unneth formula.
Let X , X be two topological spaces and X × X the productspace. For any F ∈ D ( X ), F ∈ D ( X ) let us denote F ⊠ F := ( π − F ) L ⊗ ( π − F )where π i : X × X → X i is the natural projection. Proposition 2.30.
For any F ∈ D ( X ) , F ∈ D ( X ) there is a natural morphism L ( X × X , F ⊠ F ) → L ( X , F ) L ⊗ L ( X , F ) . Proof.
The unit morphisms F i → π − X i L ( X i , F i ) induce morphisms π − i F i → π − i π − X i L ( X i , F i ) = π − X × X L ( X i , F i )hence a morphism F ⊠ F → π − X × X L ( X , F ) L ⊗ π − X × X L ( X , F ) = π − X × X [ L ( X , F ) L ⊗ L ( X , F )] and then a morphism L ( X × X , F ⊠ F ) → L ( X , F ) L ⊗ L ( X , F ) . (cid:3) Our aim now is to show that the K¨unneth morphism is an isomorphism under localcohomological triviality assumptions (Theorem 2.34). For this, let us first construct astandard L-acyclic resolution of any complex of sheaves.2.7.1.
Standard resolution.
We assume that X is locally connected and locally cohomo-logically trivial. For any F ∈ D ( X ) we shall construct an L-acyclic complex C • F and aquasi-isomorphism C • F → F , which are functorial on F . The construction mimics that ofa flat resolution of F . Definition 2.31.
Let F be a sheaf on X . We define C F := M U Z ⊕ F ( U ) U where U runs over the set of connected and cohomologically trivial open subsets of X .The equality Hom Shv( X ) ( Z U , F ) = F ( U ) induces a natural morphism C F → F which is an epimorphism, because the set of cohomologically trivial open subsets is a basisof X . Moreover, C F is L-acyclic by Proposition 2.13. Now, to construct a resolution of F by L-acyclics one proceeds in the standard way:Let F be the kernel of C F → F . We define C F := C F and F the kernel of C F → F .Recurrently, one defines C i F := C F i and F i +1 the kernel of C F i → F i . One obtains acomplex C • F = · · · → C i F → · · · → C F → C F which is a resolution of F by L-acyclic sheaves, and then C • F is L -acyclic.If F is a bounded above complex of sheaves, we define C • F as the simple complexassociated to the bicomplex C p F q . One has a quasi-isomorphism C • F → F and C • F isL-acyclic.Finally, for a general (unbounded) complex F , let us consider the truncations τ ≤ n F andthe morphisms of complexes φ n : τ ≤ n F → τ ≤ n +1 F . We define C • F := lim → n C • ( τ ≤ n F ). Onehas a quasi-isomorphism C • F → F . From the exact sequence0 → ⊕ n C • ( τ ≤ n F ) − φ n −→ ⊕ n C • ( τ ≤ n F ) → C • F → X, ) commutes with direct limits and direct sums)0 → ⊕ n L( X, C • ( τ ≤ n F )) → ⊕ n L( X, C • ( τ ≤ n F )) → L( X, C • F ) → OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 21 and an exact triangle ⊕ n L ( X, C • ( τ ≤ n F )) → ⊕ n L ( X, C • ( τ ≤ n F )) → L ( X, C • F )and a commutative diagram of exact triangles ⊕ n L( X, C • ( τ ≤ n F )) / / ⊕ n L( X, C • ( τ ≤ n F )) / / L( X, C • F ) ⊕ n L ( X, C • ( τ ≤ n F )) / / ≀ O O ⊕ n L ( X, C • ( τ ≤ n F )) / / ≀ O O L ( X, C • F ) O O and we conclude that C • F is L-acyclic. Thus, we have proved: Theorem 2.32. If X is locally cohomologically trivial, the category of complexes of sheaveson X has enough L -acyclics. More precisely, for each complex of sheaves F there are an L -acyclic complex C • F and a quasi-isomorphism C • F → F which are functorial in F .Hence L ( X, F ) ≃ L( X, C • F ) . Remark 2.33. C • F is also a flat resolution of F . Theorem 2.34 (See [1, Ch. V, Thm 13.1]) . Assume that X , X are locally cohomologicallytrivial topological spaces. If U × U is cohomologically trivial for any cohomological trivialopen subsets U , U of X , X (it would suffice this condition on a basis of cohomologicaltrivial open subsets of X and X ), then for any F ∈ D ( X ) , F ∈ D ( X ) the K¨unnethmorphism L ( X × X , F ⊠ F ) → L ( X , F ) L ⊗ L ( X , F ) is an isomorphism. Hence one has an exact sequence → M p + q = n H p ( F ) ⊗ H q ( F ) → H n ( F ⊠ F ) → M p + q = n − Tor ( H p ( F ) , H q ( F )) → , where we have used the abbreviated notation H i ( F ) := H i ( X, F ) .Proof. For any open subsets U , U of X , X , one has Z U ⊠ Z U = Z U × U ; hence, byhypothesis, Z U ⊠ Z U is L-acyclic if U , U are cohomologically trivial, and (assuming U i connected) L( X × X , Z U ⊠ Z U ) = Z = L( X , Z U ) ⊗ L( X , Z U ) . (1)Let C • F i be the standard resolution of F i (see 2.7.1). Recall that C • F i = lim → C • ( τ ≤ n F i )and C j ( τ ≤ n F i ) is a direct sum of sheaves of the form Z U i , with U i cohomologically trivial.Then π − i C • F i is a flat resolution of π − i F i and then F ⊠ F = ( π − C • F ) ⊗ ( π − C • F ) = lim → [ π − C • ( τ ≤ n F ) ⊗ π − C • ( τ ≤ n F )] . Since π − C • ( τ ≤ n F ) ⊗ π − C • ( τ ≤ n F ) is a bounded above complex of sheaves that are directsums of sheaves of the form Z U ⊠ Z U , we conclude that it is L-acyclic, and then ( π − C • F ) ⊗ ( π − C • F ) is also L-acyclic. Then L ( X × X , F ⊠ F ) = L( X × X , ( π − C • F ) ⊗ ( π − C • F )) . Since L( X × X , ) commutes with direct limits and direct sums, the equality (1) yieldsan isomorphismL( X × X , ( π − C • F ) ⊗ ( π − C • F )) = L( X , C • F ) ⊗ L( X , C • F )and we conclude because L( X i , C • F i ) is a flat complex of abelian groups and L ( X i , F i ) ≃ L( X i , C • F i ).2.7.2. Comparison with Borel-Moore homology.
Let X be a locally compact Haussdorffspace. Assume that X is locally connected and locally cohomologically trivial. For anysheaf F , let us denote H ci ( X, F ) its Borel-Moore homology groups with compact support.All the following properties of Borel-Moore homology may be found in [1]. If U is aconnected and cohomologically trivial open subset, then H ci ( X, Z U ) = 0 for i > H c ( X, Z U ) = Z . Since Borel-Moore homology commutes with direct sums, we obtain that H ci ( X, C F ) = 0 for i > H c ( X, C F ) = H ( X, C F ) = L( X, C F ). Then H ci ( X, F )may be computed with the standard resolution C • F , i.e., H ci ( X, F ) ≃ H i [ H c ( X, C • F )] 2 . ≃ H i ( X, F ). (cid:3) Comparison with Poincar´e-Verdier duality.
Let us consider two types of topo-logical spaces where one has a Poincar´e-Verdier duality theory:(A) X is Haussdorff, locally compact and locally of finite dimension. Let π : X → {∗} the projection to a point. Then (see [14]):The functor of cohomology with compact support R Γ c ( X, ) : D ( X ) → D ( Z )has a right adjoint π ! : D + ( Z ) → D + ( X ) . We shall denote D X = π ! Z , the dualizing complex. For any open subset U , one has( D X ) | U = D U . For any G ∈ D ( Z ) there is a morphism π − G L ⊗ D X → π ! G. Indeed, applying R Hom • ( , G ) to the unit morphism R Γ c ( X, D X ) → Z , gives a morphism G → R Hom • ( R Γ c ( X, D X ) , G ) = R Hom • ( D X , π ! G ) = R π ∗ R Hom • ( D X , π ! G )and then a morphism π − G → R Hom • ( D X , π ! G ), i.e., a morphism π − G L ⊗ D X → π ! G .(B) X is a finite topological space. The functor R Γ( X, ) : D ( X ) → D ( Z ) OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 23 has a right adjoint (see [6], or [13]), that we also denote by π ! (and then D X = π ! Z ). Animportant difference with (A) is that, for an open subset U , ( D X ) | U needs not coincide with D U . As in (A), one has a natural morphism π − G L ⊗ D X → π ! G (by the same arguments).In order to treat simultaneously both cases let us consider the following definition. Definition 2.35. A duality theory on X is an exact functor ω : D ( X ) → D ( Z ) that has aright adjoint ω ! : D ( Z ) → D ( X ). We shall denote D ωX = ω ! ( Z ). For each G ∈ D ( Z ), onehas a natural morphism π − G L ⊗ D ωX → ω ! G. (1)For each F ∈ D ( X ), we shall denote H iω ( X, F ) := H i ( ω ( F )) . Remark 2.36.
If we take ω = L ( X, ), then ω ! = π − , the morphism (1) is the identityand H iω ( X, F ) = H − i ( X, F ).If j : U ֒ → X is an open subset of X , we shall denote ω | U = ω ◦ j ! , whose right adjointis j − ◦ ω ! . Then D ω | U U = ( D ωX ) | U . In case (A), ω = R Γ c ( X, ) and ω | U = R Γ c ( U, ).However, in case (B), ω = R Γ( X, ) but ω | U needs not agree with R Γ( U, ). Thus, incase (B) we have two different duality theories on U : R Γ( U, ) and R Γ( X, ) | U . Proposition 2.37.
Let ( X, ω ) be a topological space with a duality theory. For any F ∈ D ( X ) there is a natural morphism Φ F : ω ( F L ⊗ D ωX ) → L ( X, F ) hence a morphism H − iω ( X, F L ⊗ D ωX ) → H i ( X, F ) . Proof.
The unit morphism ǫ : F → π − L ( X, F ) induces a morphism F L ⊗ D ωX ǫ ⊗ → π − L ( X, F ) L ⊗ D ωX (1) → ω ! L ( X, F )and then a morphism ω ( F L ⊗ D ωX ) → L ( X, F ) . (cid:3) The characterization of those topological spaces (
X, ω ) for which H − iω ( X, F L ⊗ D ωX ) → H i ( X, F ) is an isomorphism for any F is given in the following: Theorem 2.38 (See [1, Ch.V, Thm. 9.2]) . Let ( X, ω ) be a topological space with a dualitytheory. The following conditions are equivalent (1) H − iω ( X, F L ⊗ D ωX ) → H i ( X, F ) is an isomorphism for any F ∈ D ( X ) and any i . (2) There is a basis B of open subsets such that H − iω ( X, Z U ⊗ D ωX ) → H i ( U, Z ) is anisomorphism for any U ∈ B and any i . (3) The natural morphism Z → R Hom • ( D ωX , D ωX ) is an isomorphism. In any of these cases, we shall say that (
X, ω ) is a
P V -space (a Poincar´e-Verdier space).
Proof.
Notice that ω ( F L ⊗ D ωX ) ∨ = R Hom • ( F L ⊗ D ωX , D ωX ) = R Hom • ( F, R Hom • ( D ωX , D ωX ))and then ω ( Z U ⊗ D ωX ) ∨ = R Γ( U, R Hom • ( D ωX , D ωX )) . Then, Φ F : ω ( F L ⊗ D ωX ) → L ( X, F ) is an isomorphism for any F if and only ifΦ ∨ F : R Hom • ( F, Z ) → R Hom • ( F, R Hom • ( D ωX , D ωX ))is an isomorphism for any F , i.e., iff Z → R Hom • ( D ωX , D ωX )) is an isomorphism. This givesthe equivalence of (1) and (3). Analogously, Φ U : ω ( Z U ⊗ D ωX ) → L ( U, Z ) is an isomorphismfor any U ∈ B if and only if Φ ∨ U : R Γ( U, Z ) → R Γ( U, R Hom • ( D ωX , D ωX )) is an isomorphismfor any U ∈ B , i.e., iff Z → R Hom • ( D ωX , D ωX )) is an isomorphism. (cid:3) Remark 2.39. (1) Any finite dimensional topological manifold X (with or without bound-ary) is a P V -space, for ω = R Γ c ( X, ), since it satisfies condition (3) of Theorem 2.38.(2) Let ( X, ω ) be a topological space with a duality theory and U an open subset. If( X, ω ) is a
P V -space, so is (
U, ω | U ). Conversely, if ( U i , ω | U i ) is a P V -space for an opencovering X = ∪ i U i , then ( X, ω ) is a
P V -space.(3) Any topological space X is a P V -space with respect to ω = L ( X, ).If Y is the boundary of a topologial manifold X with boundary, then R Γ Y Z = 0. Thismotivates the following: Corollary 2.40.
Let ( X, ω ) be a topological space with a duality theory. Assume that D ωX is supported on an open subset j : U ֒ → X (i.e., D X = j ! j − D ωX ) and that ( U, ω | U ) is a P V -space. The following conditions are equivalent (let us denote Y = X − U ): (1) ( X, ω ) is a P V -space. (2) R Γ Y Z = 0 . (3) For any F ∈ D ( X ) and any i , H Yi ( X, F ) = 0 .Proof.
Let us denote ω ′ = ω | U . By hypothesis, D ωX = j ! j − D ωX = j ! D ω ′ U and Z = R Hom • ( D ω ′ U , D ω ′ U ). Then R Hom • ( D ωX , D ωX ) = R Hom • ( j ! D ω ′ U , j ! D ω ′ U ) = R j ∗ R Hom • ( D ω ′ U , D ω ′ U ) = R j ∗ Z . Moreover one has the exact triangle R Γ Y Z → Z → R j ∗ Z . Hence, X is a P V -space ⇔ Z → R j ∗ Z is an isomorphism ⇔ R Γ Y Z = 0.The equivalence of (2) and (3)follows from the equality L Y ( X, F ) ∨ = R Hom • ( F, R Γ Y Z ). (cid:3) OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 25
Homological manifolds.
In this section we assume that we are in case (A) or (B).We leave to the reader to give the natural generalizations to a topological space X with aduality theory ω . Definition 2.41. (See [1, Ch.V, Def. 9.1]) Let X be a topological space of type (A) or(B). We say that X is an n -dimensional homological manifold if D X ≃ T X [ n ], where T X is a sheaf on X locally isomorphic to the constant sheaf Z . Remark 2.42.
Any n -dimensional homological manifold is a P V -space, with respect to ω = R Γ c ( X, ) in case (A) (resp. ω = R Γ( X, ) in case (B)). Theorem 2.43 (See [1, Ch.V, Thm. 15.1]) . Let X be a connected topological of type (A) or (B) . Then X is an n -dimensional homological manifold (for some n ) if and only if D X is perfect.Proof. The direct is immediate. For the converse, it suffices to see that for any closed point p ∈ X there is a neighbourhood U ∋ p such that ( D X ) | U ≃ Z [ n ]. First notice that R Γ p ( X, D X ) = R Hom • ( Z { p } , D X ) = R Hom • ( R Γ( p, Z ) , Z ) = Z , hence H • p ( X, D X ) = Z . Now, by hypothesis, there is a neighbourhood U of x such that( D X ) | U ≃ π − U D , where π U : U → {∗} is the projection to a point and D is a boundedcomplex of free Z -modules of finite rank. Then Z = H • p ( X, D X ) = H • p ( U, ( D X ) | U ) = H • p ( U, Z ) ⊗ D. This implies (it is an exercise on complexes of abelian groups) that, for some integer n , H • p ( U, Z ) ≃ Z [ − n ] and D ≃ Z [ n ]. Then ( D X ) | U ≃ Z [ n ]. (cid:3) Homology of sheaves and homology of groups
In this section we make an additional hypothesis on the topological space: we assumethat X is connected and locally simply connected (by simply connected we mean a topo-logical space such that every covering is trivial). Let φ : e X → X be a universal cover and G = Aut X e X the fundamental group.Let us briefly summarize the content of this section. Let us denote LC( X ) the categoryof locally constant sheaves on X and D (LC( X )) its derived category. Let us consider theinclusion functor i : D (LC( X )) → D ( X ) and the commutative diagram D ( Z ) π − / / π − ❍❍❍❍❍❍❍❍❍ D (LC( X )) i x x rrrrrrrrrr D ( X ) Homology of groups is nothing but the left adjoint of π − : D ( Z ) → D (LC( X )), takinginto account the equivalence D (LC( X )) = D ( G ), where D ( G ) is the derived categoryof the category of G -modules. Homology of sheaves is, by definition, the left adjoint of π − : D ( Z ) → D ( X ). We shall see that i has also a left adjoint LC : D ( X ) → D (LC( X )).Hence, homology of sheaves is the composition of LC with homology of groups. We shallcompute the functor LC in terms of homology of sheaves in the universal covering.3.1. Let us begin with the underived version of the above situation. Let us denote Mod( G )the category of (left) Z [ G ]-modules, i.e., the category of G -modules. For any abelian group A , we denote by A tr the trivial G -module (i.e., it is the Z -module A with the trivial actionof G ). For any G -module M , we denote by M G the quotient module of coinvariants: M G = Z ⊗ Z [ G ] M. Then we have functorsTr : Mod( Z ) → Mod( G ) ,A A tr ( ) G : Mod( G ) → Mod( Z ) M M G and ( ) G is left adjoint of Tr.One has a well known equivalence between the category of locally constant sheaves on X and the category of G -modules; for each locally constant sheaf M on X , we shall denoteby M the corresponding G -module and viceversa. ThusLC( X ) = Mod( G ) M ↔ M (*)that interchanges constant sheaves with trivial G -modules: A ↔ A tr . The assignation M 7→ M may be described in terms of cosections in the following way: M = L( e X, φ − M ),which is a G -module because φ − M is a G -sheaf on e X .Via the equivalence (*), the functor Tr coincides with π − and then, by adjunction,( ) G coincides with L( X, ); that isL( X, M ) = M G . (1)Let us now consider the inclusion functor LC( X ) ֒ → Shv( X ). Proposition 3.1.
The inclusion functor
LC( X ) ֒ → Shv( X ) has a left adjoint LC : Shv( X ) → LC( X ) . Thus, for any sheaf F and any locally constant sheaf M one has Hom( F, M ) = Hom(LC( F ) , M ) . Proof.
Since the inclusion functor is exact, it suffices to see that it commutes with directproducts. Indeed, if {M i } i ∈ I is a collection of locally constant sheaves, we have to see thatthe direct product sheaf Q i M i is also locally constant. Restricting to a simply connectedopen subset, we are reduced to prove that the direct product of constant sheaves is constant.This is given by Proposition 1.1. (cid:3) OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 27
For any sheaf F one has a natural morphism F → LC( F ), and for any locally constantsheaf M a natural isomorphism LC( M ) → M . It is direct from the representability thatLC is right exact and commutes with direct sums (more generally, with direct limits).Via the equivalence LC( X ) = Mod( G ), we can see LC as a functor LC : Shv( X ) → Mod( G ), whose description in terms of the universal cover is given by the following: Proposition 3.2.
The G -module corresponding to LC( F ) is L( e X, φ − F ) (which is a G -module because φ − F is a G -sheaf on e X ). Thus, we shall put LC( F ) = L( e X, φ − F ) , via the equivalence (*) . In particular, for any simply connected open subset U of X onehas LC( Z U ) = Z [ G ] as G -modules.Proof. This is essentially a consequence of the Galois correspondence between sheaves on X and G -sheaves on e X , that we recall now. If F is a sheaf on X , then φ − F is a G -sheafon e X , since its ´etale space ] φ − F = e X × X e F is endowed with the action of G given by theaction on e X and the trivial action on e F . One has then a Galois equivalenceShv( X ) → G -Shv( e X ) F φ − F whose inverse sends a G -sheaf Q on e X to the sheaf of sections of e F := e Q/G → e X/G = X .If M is a G -module, then π − e X M is a (constant) G -sheaf on e X and, for any G -sheaf Q on e X , L( e X, Q ) is a G -module. The functor M π − e X M is right adjoint of the functor Q 7→ L( e X, Q ).Now, let us prove the proposition. Let us denote M F = L( e X, φ − F ). For any locallyconstant sheaf M on X (with corresponding G -module M ) one hasHom LC( X ) (LC( F ) , M ) = Hom Shv( X ) ( F, M ) = Hom G -Shv( e X ) ( φ − F, φ − M ) . Now, φ − M is a constant G -sheaf on e X , hence φ − M = π − e X M . ThenHom G -Shv( e X ) ( φ − F, φ − M ) = Hom G -mod ( M F , M )and we conclude that M F is the G -module corresponding to LC( F ).If U is a simply connected open subset, then φ − ( U ) = U × G , and thenL( e X, φ − Z U ) = L( e X, Z φ − ( U ) ) = L( U × G, Z ) = Z [ G ] . (cid:3) Corollary 3.3.
For any sheaf F on X one has: L( X, F ) = L( X, LC( F )) = L( e X, φ − F ) G . That is, the cosections of a sheaf F are the coinvariants of the cosections of the lifting of F to the universal covering.Proof. The equality L(
X, F ) = L( X, LC( F )) is obtained by adjunction from the commu-tative diagram Mod( Z ) π − / / π − % % ❑❑❑❑❑❑❑❑❑❑ Shv( X )LC( X ) i sssssssss The second equality follows from Proposition 3.2 and formula (1). (cid:3) X ) = Mod( G ) gives an equivalence D (LC( X )) = D ( G ) . Thus, for any
M ∈ D (LC( X )), we shall denote by M its corresponding object in D ( G )and viceversa.Let Z L ⊗ Z [ G ] : D ( G ) → D ( Z )be the left derived functor of ( ) G : Mod( G ) → Mod( Z ). We shall use the standardnotation H i ( G, M ) := H i [ Z L ⊗ Z [ G ] M ]for any M ∈ D ( G ). The functor Z L ⊗ Z [ G ] is left adjoint of the functorTr : D ( Z ) → D ( G )induced by the exact functor Tr : Mod( Z ) → Mod( G ).In other words, the functor L( X, ) : LC( X ) → Mod( Z ), M 7→ L( X, M ) can be derivedby the left (because LC( X ) has enough projectives), giving a functor L lc ( X, ) : D (LC( X )) → D ( Z )which is left adjoint of the functor π − : D ( Z ) → D (LC( X )). We shall denote H lc i ( X, M ) := H i [ L lc ( X, M )] . OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 29
Via the equivalence D (LC( X )) = D ( G ), one has L lc ( X, M ) = Z L ⊗ Z [ G ] MH lc i ( X, M ) = H i ( G, M ) . Now let us consider the inclusion functor i : D (LC( X )) → D ( X ). Notice that thoughLC( X ) → Shv( X ) is fully faithful, i is not (in fact, it is proved in [12] that i is fully faithfulif and only if X is aspherical). We cannot derive by the left the functor LC : Shv( X ) → LC( X ), since Shv( X ) has not enough projectives. However, we can proceed as we did toconstruct homology: Proposition 3.4.
The functor i : D (LC( X )) → D ( X ) has a left adjoint LC : D ( X ) → D (LC( X )) . Thus, for any F ∈ D ( X ) , M ∈ D (LC( X )) one has Hom D ( X ) ( F, i ( M )) = Hom D (LC( X )) ( LC ( F ) , M ) . Proof.
By Brown representability for the dual, we have to prove that i commutes withdirect products. Restricting to a simply connected open subset U , and taking into accountthat π − U : D ( Z ) → D (LC( U )) is an equivalence, we conclude by Theorem 2.2. (cid:3) We shall denote: LC i ( F ) = H i [ LC ( F )] . By adjunction, LC is exact and commutes withdirect sums. For any F ∈ D ( X ), M ∈ D (LC( X )) one has an isomorphism R Hom • ( F, i ( M )) = R Hom • ( LC ( F ) , M ) . Then, if F is a sheaf, LC i ( F ) = 0 for i < LC ( F ) = LC( F ). For any F ∈ D ( X ) onehas a morphism F → i ( LC ( F )) and for any M ∈ D (LC( X )) a morphism LC ( i ( M )) → M (which is not an isomorphism in general, in contrast with the underived case). The diagram D (LC( X )) i / / L lc ( X, ) % % ▲▲▲▲▲▲▲▲▲▲ D ( X ) L ( X, ) { { ✈✈✈✈✈✈✈✈✈ D ( Z )is not commutative, but there is a natural morphism L ( X, ) ◦ i → L lc ( X, ). Hence, forany M ∈ D (LC( X )), one has a natural morphism H i ( X, M ) → H lc i ( X, M ) = H i ( G, M )which is not an isomorphism in general (one can prove that it is an isomorphism if X isaspherical, see [12] for the cohomological analog).On the other hand, one has: Theorem 3.5.
For any F ∈ D ( X ) : L ( X, F ) = L lc ( X, LC ( F )) , and then H i ( X, F ) = H lc i ( X, LC ( F )) . Equivalently (viewing LC as a functor in D ( G ) ) L ( X, F ) = Z L ⊗ Z [ G ] LC ( F ) , and then H i ( X, F ) = H i ( G, LC ( F )) . Proof.
The result follows from adjunction from the commutativity of the diagram D ( Z ) π − / / π − ❍❍❍❍❍❍❍❍❍ D (LC( X )) i x x rrrrrrrrrr D ( X ) . (cid:3) Let us see how to compute LC ( F ) in terms of homology of sheaves in the universalcover, i.e., let us give the derived analog of Proposition 3.2. First, let us see that LC canbe computed with (a slight variation of) the standard resolution.For any complex of sheaves F on X , we denote LC( F ) the complex of locally constantsheaves · · · → LC( F n ) → LC( F n +1 ) → · · · and one has a morphism of complexes F → LC( F ) and then a morphism in the derivedcategory LC ( F ) → LC( F ). We say that F is LC -acyclic if this morphism is an isomor-phism. If F is a sheaf, this is equivalent to say that LC i ( F ) = 0 for i >
0. For any simplyconnected open subset U , the sheaf Z U is LC-acyclic, because R Hom • ( LC ( Z U ) , M ) = R Hom • ( Z U , M ) = R Γ( U, M ) = Γ( U, M )for any locally constant sheaf M . Thus, for any sheaf F , let us define C ′ F := ⊕ U Z ⊕ F ( U ) U where U runs over the set of simply connected open subsets of X . One has an epimorphism C ′ F → F and C ′ F is LC-acyclic. Repeating the same construction than for the standardresolution (see 2.7.1), for any complex F one constructs a resolution C ′ • F → F , with C ′ • F LC-acyclic, and then LC ( F ) ≃ LC( C ′ • F ) . By Proposition 3.2, the complex of G -modules corresponding to LC ( F ) by the equivalence D (LC( X )) = D ( G ) is L( e X, φ − C ′ • F ). Now, φ − C • F is a resolution of φ − F and it isL-acyclic; indeed, from the construction of C ′ • F , one is reduced to prove that φ − Z U is L-acyclic for any simply connected open subset U . But φ − Z U = Z φ − ( U ) and φ − ( U ) = U × G ,so Z φ − ( U ) = ⊕ g ∈ G Z U , which is a direct sum of L-acyclic sheaves, hence it is L-acyclic.Finally, notice that for any simply connected open subset we have proved LC ( Z U ) =LC( Z U ) 3 . Z [ G ], which is a projective G -module, hence ( ) G -acyclic. OMOLOGY OF SHEAVES VIA BROWN REPRESENTABILITY 31
We have proved then:
Proposition 3.6.
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Departamento de Matem´aticas and Instituto Universitario de F´ısica Fundamental yMatem´aticas (IUFFyM)Universidad de SalamancaPlaza de la Merced 1-4, 37008 SalamancaSpain
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