Godement resolutions and sheaf homotopy theory

arXiv:1302.2442v4 [math.AG] 13 Sep 2014 GODEMENT RESOLUTIONS AND SHEAF HOMOTOPY THEORY BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG Abstract. The Godement cosimplicial resolution is available for a wide range of categories of sheaves. In this paper we investigate under which conditions of the Grothendieck site and the category of coefficients it can be used to obtain fibrant models and hence to do sheaf homotopy theory. For instance, for which Grothendieck sites and coefficients we can define sheaf cohomology and derived functors through it. Contents 1. Introduction 2 2. Homotopical preliminaries 4 2.1. Descent categories 4 2.2. Cartan-Eilenberg categories 7 3. Categories of sheaves 8 3.1. Sheaves of sets 8 3.2. Sheaves with general coefficients. 10 3.3. The cosimplicial Godement resolution 11 4. Cartan-Eilenberg categories of sheaves 12 4.1. Cartan-Eilenberg fibrant sheaves 12 4.2. The hypercohomology sheaf 13 4.3. Characterization 17 4.4. Derived functors for sheaves 20 5. Examples 21 5.1. Bounded complexes of sheaves 22 Date: September 16, 2014. First named author partially supported by ERC Starting Grant project TGASS and by contracts SGR-119 and FQM-218. Second named author partially supported by projects MTM2009-09557, 2009 SGR 119 and MTM201238122-C03-01/FEDER. To appear in Collectanea Mathematica. The final publication is available at Springer via http://dx.doi.org/10.1007/s13348-014-0123-x. 1 2 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG 5.2. Unbounded complexes of sheaves 22 5.3. Sheaves of fibrant simplicial sets 25 5.4. Sheaves of fibrant spectra 27 5.5. Sheaves of filtered complexes 27 6. Varying Xand D28 6.1. Varying X28 6.2. Varying D31 References 32 1. Introduction 1.0.1. Godement resolutions have been an essential tool in sheaf homotopy theory and its applications almost from the start [Go] and keep cropping up in different contexts: see for instance [SGA4] for abelian sheaves on a Grothendieck site, [Th] for sheaves of spectra on a Grothendieck site, [N] for sheaves of (filtered) dg commutative algebras over topological spaces, [MV] for simplicial sheaves on a Grothendieck site, [SdS] for sheaves of OX-modules over schemes, or [GL] and [Ba] for sheaves of DG-categories over schemes..., to name but a few. In particular, the great flexibility of the cosimplicial Godement resolution, together with its excellent functorial properties, appear to account for its omnipresence: in fact, in order to define it for a sheaf F:Xop −→ D on a Grothendieck site with enough points Xand values in some category of coefficients D, we only need Dto have filtered colimits and arbitrary products. In this situation, we obtain a functor G•:Sh(X,D)−→ ∆Sh(X,D) from sheaves on Xwith values in Dto cosimplicial ones. The question we address in this paper is the following: under which conditions for the Grothendieck site Xand the category of coefficients Dcan the cosimplicial Godement construction be used to transfer homotopical structure from Dto the category of sheaves Sh(X,D)? 1.0.2. Let us elaborate a little further. Making use of a (realization of the) homotopy limit s: ∆D −→ D, which we call a simple functor, we can “reassemble” all the cosimplicial pieces of GpFobtaining a single sheaf which might be entitled to be a “model” for F. To get anchorage for her ideas, the reader may think of Das being the category of cochain complexes of abelian groups C∗(Ab) and sthe total complex of a double complex. In this way we obtain a sheaf together a universal map ρF:F −→ HX(F) = sG•(F),(1.0.1) called here the hypercohomology sheaf of Ffollowing Thomason and Mitchell ([Th], [Mit]). GODEMENT RESOLUTIONS 3 So a particular instance of our initial question is the following: assume that Xhas a final object X, when would it make sense to define sheaf cohomology of Xwith coefficients in Fas Γ(X, HX(F))? More precisely, we are asking when this formula would define a right derived functor in the sense of Quillen [Q]; that is, a left Kan extension. 1.0.3. In order to talk about derived functors and homotopy categories, we need to specify the class of morphisms with respect to which we localize. In all the examples we are aware of, this is the class that keeps track of the topology of X, the one of local equivalences: we have a distinguished class of morphisms E, or “equivalences”, in the category of coefficients D; and, for a morphism of sheaves ϕ:F −→ G to be called a local equivalence, we require every morphism induced on stalks ϕx:Fx−→ Gxto be in E. Let us note this class of local equivalences as W. For instance, for D=C∗(Ab) we could take E to be the class of quasi-isomorphisms,quis, morphisms which induce isomorphisms in cohomology. 1.0.4. A first approach could be to study our question in the context of Quillen model categories. That is, to assume that our coefficient category (D,E) supports a Quillen model structure and that it induces one on (Sh(X,D),W) in such a way that the Godement resolution becomes a fibrant model for every sheaf. As proved in [Be1] and [Be2], this is indeed possible under certain (non-trivial) hypotheses on the model category (D,E). Instead, we opt here to keep to the minimum the amount of structure on (Sh(X,D),W) necessary to have the sheaves HX(F) as fibrant models. This allows us to 1) cover a more general class of coefficient categories (e.g. filtered complexes over any (AB4)∗and (AB5) abelian category) and 2) have more flexibility in the transference of the resulting technique to the multiplicative setting. This last point is the subject of a forthcoming sequel to this paper, where we transfer the results obtained here for Sh(X,D) to sheaf of operads and algebras (over any operad) on D, and their corresponding filtered versions. One such minimal amount of structure is attained with Cartan-Eilenberg categories, or CEcategories, for short: an approach to homotopical algebra started in [GNPR1] and further developed in [P], [C1] and [C2]. A (right) CE-category consists of a category Cendowed with two classes of distinguished morphisms, strong and weak equivalences, S ⊂ W, and a CE-fibrant model for each object (see 2.2.2 for the precise definition). The name of these structures comes from the classic book [CE], where, in modern parlance, the homotopy theory of the category of cochain complexes C∗(Ab) is developed around two classes of distinguished morphisms: homotopy equivalences (S) and quis (W). But CE-structures allow more freedom of choice for classes Sand Wthan classical “homotopy equivalences” and “weak equivalences”. This is particularly interesting for our categories of sheaves, for which the natural choices are: •global equivalences, as S: those morphisms of sheaves such that ϕ(U) : F(U)−→ G(U) belongs to the class of equivalences E in Dfor every object (open set) U∈ X , and •local equivalences, as W: already mentioned above. In order to provide our categories of sheaves Sh(X,D) with a CE-structure, we need very few elements in our category of coefficients D: essentially, our needs reduce to a class of equivalences 4 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG E and a simple functor s: ∆D −→ D which is a realization of the homotopy limit. This is summarized in the notion of descent category (see [Rod1], [Rod2] and the second section in this paper; cf. also [GN]). 1.0.5. Our main result (Theorem 4.3.2) provides equivalent conditions guaranteeing that our initial question has a positive answer: Theorem 1.0.1. Let Xbe a Grothendieck site and (D,E) a descent category satisfying the hypotheses (4.1.1). Then, the following statements are equivalent: (1) (Sh(X,D),S,W)is a right Cartan-Eilenberg category and for every sheaf F ∈ Sh(X,D), ρF:F −→ HX(F)is a CE-fibrant model. (2) For every sheaf F ∈ Sh(X,D),ρF:F −→ HX(F)is in W. (3) The simple functor commutes weakly with stalks. (4) For every sheaf F ∈ Sh(X,D),HX(F)satisfies Thomason’s descent; that is, ρHX(F): HX(F)−→ H2 X(F)is in S. This theorem shows, first, that the existence of a CE-structure on the category of sheaves Sh(X,D) boils down to the property that for every sheaf Fthe universal arrow ρF:F −→ HX(F) is a local equivalence (condition (2)). Hence, we need nothing else that this CE-structure to answer our problem; i.e., the fact that the Godement construction can be used to transfer homotopical structure from Dto the category of sheaves Sh(X,D) is equivalent to the existence of this CE-structure. The theorem also shows that the fact of the Godement resolution being a CE-fibrant model is equivalent to Thomason’s classic descent (for sheaves of spectra [Th], condition (4); see also Corollary 4.3.5). So being CE-fibrant is quite a natural and central notion for sheaves. Finally, the theorem gives a down-to-earth equivalent condition for all this to happen, which will be the one we will use in practice: condition (3) says that the simple functor smust commute with stalks up to local equivalence. For instance, for bounded cochain complexes this is a consequence of the commutation of the total complex functor Tot with filtered colimits. 1.0.6. Acknowledgements. This paper develops an idea suggested to us by Vicente Navarro. We owe him a debt of gratitude for sharing it with us. The second named author also benefited from many fruitful conversations with Pere Pascual. We are indebted to Francisco Guill´en, Fernando Muro, Luis Narv´aez and Abd´o Roig for their comments. People at sci.math.research and Mathoverflow made useful suggestions kindly answering our questions there. 2. Homotopical preliminaries We introduce here the definitions and results concerning descent and CE-categories necessary for our paper. The interested reader may consult [Rod1], [Rod2] and [GNPR1] for further details. 2.1. Descent categories. GODEMENT RESOLUTIONS 5 2.1.1. Notations. By ∆we mean the simplicial category. We denote by ∆D(resp. ∆opD) the category of cosimplicial (resp. simplicial) objects in a fixed category D. The diagonal functor D : ∆∆D −→ ∆Dis given by D({Zn,m}n,m≥0) = {Zn,n}n≥0. The constant simplicial object defined by A∈ D will be denoted by c(A) or by A×∆. 2.1.2. A (cosimplicial) descent category consists, roughly, of a category Dendowed with a class E of ‘weak equivalences’ and with a ‘simple’ functor s:∆D −→ D subject to the axioms below. These axioms ensure that sis a realization of the homotopy limit for cosimplicial objects, and that the localized category D[E−1] possesses a rich homotopical structure. Definition 2.1.1. [Rod1, 1.1] A (cosimplicial)descent category is the data (D,E,s, µ, λ) where Dis a category closed under finite products and E is a saturated class of morphisms of D, closed under finite products, called weak equivalences. The triple (s, µ, λ) is subject to the following axioms: (S1) The simple functor s:∆D −→ D commutes with finite products up to equivalence. That is, the canonical morphism s(X×Y)−→ s(X)×s(Y) is in E for all X,Yin ∆D. (S2) µ:ss 99K sD is a zigzag of natural weak equivalences. Recall that sDZdenotes the simple of the diagonal of Z, while ssZ=s(n−→ s(m→Zn,m)). (S3) λ: idD99K s(− × ∆) is a zigzag of natural weak equivalences, which is assumed to be compatible with µin the sense of op.cit.. (S4) If f:X−→ Yis a morphism in ∆Dwith fn∈E for all n, then s(f)∈E. (S5) The image under the simple functor of the cosimplicial map Ad0:A∆[1] −→ Ais a weak equivalence for each object Aof D. For the sake of brevity, we will also denote a descent category by (D,E). Remark 2.1.2. The presence of zigzags in the definition of descent category is needed to ensure its homotopy invariance (see [Rod1], Proposition 1.8). However, every example used in this paper has both µand λas actual natural transformations (see Examples 2.1.4 - 2.1.6). Since this significantly simplifies exposition, we will assume they are so for the descent categories considered throughout the paper. We will also assume that simple functors preserve limits. But this is not a major restriction: it is fulfilled by all our examples of descent categories so far. 2.1.3. Among the hereditary results of descent categories, let us point out one we will be using time and again and whose proof we leave as an easy exercise for the interested reader: Lemma 2.1.3 (Transfer Lemma).Let (D′,E′,s′, µ′, λ′)be a descent category. Given a functor ψ:D −→ D′, consider in Dthe weak equivalences E = ψ−1E′. Assume that Dhas finite products and is equipped with a functor s:∆D −→ D, together with compatible natural weak equivalences µ:ss −→ sDand λ: idD−→ s(−×∆). Then, (D,E,s, µ, λ)is a descent category provided the following statements hold: (FD1)ψcommutes with finite products up to equivalence. That is, the natural map ψ(X× Y)−→ ψ(X)×ψ(Y)is in Efor all X, Y in D. 6 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG (FD2)There exists a natural weak equivalence θ:ψs−→ s′ψfilling the square ∆Dψ// s  ∆D′ s′  Dψ// θ ⇒ D′ 2.1.4. To end with, we describe some examples of descent categories. Example 2.1.4. Bounded complexes [Rod1, (3.4)].Let Abe an abelian category. For a fixed integer b∈Z, denote by C≥b(A) the category of uniformly bounded below cochain complexes of A; that is, An= 0 for all n < b and all A∗∈C≥b(A). We will consider the following descent structure on C≥b(A). The weak equivalences E are the quasi-isomorphism (quis): those maps inducing isomorphism in cohomology. The simple functor s:∆C≥b(A)−→ C≥b(A) at a given cosimplicial cochain complex Ais the (product) total complex of the double complex induced by A: s(A)n=Y p+q=n Apq . Which, in this case, since Ahas finite codiagonals, s(A)n=Lp+q=nApq.µZis just the Alexander-Whitney map ssZ−→ sDZand λn X:Xn−→ s(X×∆)nis the canonical inclusion. Example 2.1.5. Unbounded complexes. The category C∗(A) of unbounded cochain complexes of Ais also a descent category with weak equivalences, simple functor, µand λdefined as in the bounded case provided axiom (S4) holds. For instance, this is the case when A=Rmodules. Example 2.1.6. Simplicial model categories [Rod1, Theorem 3.2].The subcategory of fibrant objects Mfof a model category Mis a descent category where E is the class of weak equivalences of Mand the simple functor is the Bousfield-Kan homotopy [BK] limit, holim ←− :∆Mf−→ Mf, as defined in [Hir]. If Mis a simplicial model category, the homotopy limit of a cosimplicial object Xis the end of the bifunctor XN(∆↓·):∆op ×∆−→ Mf, (n, m)7→ (Xm)N(∆↓n), that is, holim ←− X=Zn (Xn)N(∆↓n). Morphisms µand λare easily defined using that a functor F:B −→ C induces a natural map holim ←− CX−→ holim ←− BF∗X. Two particular instances of this example are relevant when talking about sheaf cohomology theories. First, the category sSfof pointed Kan complexes, with weak equivalences the weak homotopy equivalences. Secondly, the category Spfof pointed fibrant spectra, as defined in [Th, 5.2]. The weak equivalences for the descent structure are then the stable weak equivalences; that is, morphisms of spectra inducing bijections in all homotopy groups. GODEMENT RESOLUTIONS 7 Example 2.1.7. Filtered complexes. Denote by FC≥b(A) the category of filtered complexes, with objects the pairs (A, F) where Ais in C≥b(A) and F is a decreasing filtration of A. Given r≥0, consider the class Erof weak equivalences given by the Er-quasi-isomorphisms of FC≥b(A), that is, morphisms of filtered complexes such that the induced morphism between the Er+1-terms of the spectral sequences associated with the filtrations is an isomorphism. It holds that (FC≥b(A),Er) is a descent category with simple functor (s, δr) : ∆FC≥b(A)→ FC≥b(A) defined as (s, δr)(A, F) = (s(A), δr(F)) where δr(F)k(s(A)n) = M i+j=n Fk−riAi,j , and with natural transformations λand µgiven at the level of complexes by those of C≥b(A). If r= 0, note that an E0-isomorphism is the same thing as a graded quasi-isomorphism. Also, (s, δ0)(A, F) is just (s(A),s(F)). The fact that this is a simple functor for (FC≥b(A),E0) is an easy consequence of the transfer lemma applied to the graded functor Gr : FC≥b(A)→ C≥b(A)Z. To treat the general case, consider the decalage filtration functor Dec :FC≥b(A)→FC≥b(A), (A, F) 7→ (A, DecF), where (DecF)kAn= ker{d: Fk+nAn→Fk+nAn+1/Fk+n+1An+1}. Since Dec (s, δr+1) = (s, δr)Dec, by applying the transfer lemma inductively, we can conclude that (s, δr) is a simple functor for (FC≥b(A),Er), for each r≥0. 2.2. Cartan-Eilenberg categories. 2.2.1. Cartan-Eilenberg categories are a new approach to homotopical algebra developed in [GNPR1]. They use, we believe, a minimum amount of data in order to derive functors, so its conditions can be fulfilled by a wider class of categories, as we are going to show. Definition 2.2.1. Let (C,S,W) be a category with two classes Sand Wof distinguished morphisms, called respectively strong and weak equivalences, and such that S ⊂ W. An object Mof Cis called Cartan-Eilenberg fibrant,CE-fibrant for short, if for each weak equivalence w:Y−→ X∈ W and every morphism f∈ C[S−1], there is a unique morphism g∈ C[S−1] making the following triangle commutative: Yw// f  X g ~~ M Remark 2.2.2. Here Wdenotes the saturation of W. Classes Sand Wof strong and weak equivalences considered later in the study of sheaves are saturated, i.e. S=Sand W=W. In this case, Whitehead’s theorem holds: a weak equivalence between CE-fibrant objects is a strong one. 2.2.2. A right CE-fibrant model of an object Xof Cis a morphism w:X−→ Mof C[S−1] that becomes an isomorphism in C[W−1], and such that Mis CE-fibrant. If Xadmits a CE-fibrant model, it is unique up to unique isomorphism of C[S−1]. 8 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG Definition 2.2.3. A category with strong and weak equivalences (C,S,W) is called a right Cartan-Eilenberg category, or CE-category for short, if each object Xof Chas a CE-fibrant model. In this case, we will also say that Chas enough CE-fibrant models. Example 2.2.4. If Cis a Quillen model category and S,Ware the classes of its right homotopy equivalences and weak equivalences, respectively, then (Cc,S,W) is a right Cartan-Eilenberg category. Here Ccis the full subcategory of Quillen cofibrant objects. In this case, every Quillen fibrant object is CE-fibrant, but the converse needs not be true: by its very definition, CE-fibrant objects are homotopically invariant, while Quillen fibrant objects are not. Remark 2.2.5. So, CE-categories naturally include Quillen model ones and the inclusion is “strict” in the sense that, for instance, the class Smust not be any class of “homotopy equivalences”. This is particularly important for us because, in the case of sheaves, the global equivalences cannot indeed be the homotopy equivalences of any Quillen model structure, as shown in [GNPR2]. Since these global equivalences are such a natural ingredient for sheaves, this seems to be significant. Global equivalences are needed, for instance, to talk about sheaves satisfying Thomason descent, which are precisely CE-fibrant models, to close the circle. 2.2.3. In CE-categories, the derivability criterion of functors reads as follows (see [GNPR1, 3.2.1]). Proposition 2.2.6. Let (C,S,W)be a Cartan-Eilenberg category and F:C −→ D a functor such that F(s)is an isomorphism for every strong equivalence s∈ S. Then Fhas a right derived functor RF:C[W−1]−→ D whose value on objects may be computed as RF(X) = F(M), where Mis a fibrant model of X. 2.2.4. In the CE-categories considered later on, the CE-fibrant model of an object Xwill be functorial in the sense of [GNPR1, 2.5]: what we call a resolvent functor. One of the advantages of having a resolvent functor is that, if Cfib denotes the full subcategory of Cof CE-fibrant objects, there is an equivalence of categories ([GNPR1, Proposition 2.5.3(2)]) Cfib[S−1]i ∼//C[W−1] R oo 3. Categories of sheaves We recall some general definitions and results about sheaves of sets on a Grothendieck site.. Our main objective is to point out formulas (3.1.3) and (3.1.4) for stalks and skyscraper sheaves, respectively. Then we observe that these formulas still make sense for sheaves with values in any category with filtered colimits and arbitrary products, and that they do indeed form a pair of adjoint functors. The associated triple gives us the cosimplicial Godement resolution. We also show that the category of sheaves with values in a descent category inherits a natural descent structure, which will be used repeatedly in the rest of the paper. 3.1. Sheaves of sets. GODEMENT RESOLUTIONS 9 3.1.1. Let Xbe a category. Let b X=PrSh(X,Set) denote the category of presheaves on X with values in the category of sets Set. By the Yoneda embedding, every object U∈ X can be thought of as the representable presheaf yU=X(−, U)∈b X. 3.1.2. If Xis a Grothendieck site,e X=Sh(X,Set) will denote the full subcategory of PrSh(X,Set) whose objects are sheaves. Sheaves may be characterized by the following property (see [McLM], page 122): a presheaf F ∈ b Xis a sheaf if and only if for every object U∈ X and every cover S={Uα−→ U}of U, the diagram F(U)//QαF(Uα)////Qαβ F(Uαβ) (3.1.1) is an equalizer of sets. Here the second product ranges over all composable pairs Uαβ −→ Uα, Uα−→ Uwith Uα−→ U∈S(hence also its composition Uαβ −→ Ubelongs to S). It follows that a functor of presheaves that commutes with limits will send sheaves to sheaves. 3.1.3. Let f:X −→ Y be a morphism of sites; that is, a functor between the underlying categories going in the opposite direction f−1:Y −→ X which is continuous. This means that the direct image functor f∗:b X −→ b Y,F 7→ F ◦ f−1, restricts to a functor between sheaves f∗:e X −→ e Y. 3.1.4. Recall that a point of a site Xis by definition a pair of adjoint functors x= (x∗, x∗) e Xx∗ //Set x∗ oo,Set(x∗F, D) = e X(F, x∗D) such that x∗commutes with finite limits. The right adjoint x∗:Set −→ e Xgives for every set Dthe so called skyscraper sheaf x∗Dof Dat the point x. The left adjoint x∗:e X −→ Set gives for every sheaf Fthe fibre or stalk x∗F=Fxof Fat x. The following “computational” formulas for x∗and x∗are for us of utmost importance, since they allow us to extend them for our categories of coefficients D. First, we have a canonical and functorial isomorphism x∗F=Fx= colim −→ (U,u)F(U),(3.1.2) where (U, u) runs over the opposite category of neighbourhoods of x([SGA4], expos´e IV, 6.8). This colimit is a filtered one. For a set D∈Set, the sheaf x∗Dalso admits the following description: for U∈ X , (x∗D)(U) = Y u∈x∗(yU) Du,(3.1.3) where Du=Dfor all u∈x∗(yU). 16 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG Hence, applying the simple functor we deduce that s◦(θ′◦ G•(F))s◦(η◦ s•G•(F)) = s◦s•(η◦ G•(F)). Assume it proved that s◦s•(η◦ G•(F))∈ S. In this case, φ=s◦s•(η◦ G•(F))λs•G•(F)=s◦(θ′◦ G•(F))s◦(η◦ s•G•(F))λs•G•(F)=s◦(θ′◦ G•(F))ρHX(F) is an isomorphism of Sh(X,D)[S−1], so σF=φ−1s◦(θ′◦ G•(F)) is a section of ρHX(F). To finish, it remains to be shown that s◦s•(η◦ G•(F))∈ S. This happens if and only if s•s◦(η◦ G•(F))∈ S. For a fixed n≥0, the coaugmentation η◦ Gn(F)=η◦ Tn+1(F):c◦Tn+1(F)−→ G◦Tn+1(F) has an extra degeneracy. Hence we infer that s◦(η◦ Gn(F)) is in Sfor each n≥0. But then it follows from axiom (S4) that s(n→s◦(η◦ Gn(F))) = s•s◦(η◦ G•(F))∈ S as required.  4.2.4. The class Wof local equivalences is by definition equal to (p∗)−1E. Below we prove that W=T−1S=H−1 XSas well. Proposition 4.2.6. Assume that Xand (D,E) satisfy the hypotheses (4.1.1). Then, for a morphism f:F −→ G of sheaves, the following conditions are equivalent: (1) fis a local equivalence. (2) T(f) : T(F)−→ T(G)is a global equivalence. (3) HX(f) : HX(F)−→ HX(G)is a global equivalence. Proof. (1) implies (2) since T(W)⊂ S. Conversely, if T(f) is a global equivalence, it is in particular a local one, so p∗T(f)∈E. On the other hand, it follows from the triangle identities of the adjoint pair (p∗, p∗) that p∗(f) is a retract of p∗T(f) = p∗p∗p∗(f). But E being saturated, it is closed under retracts, and we deduce that p∗(f)∈E as well. But this is the same as saying that f∈ S. Therefore, (1) and (2) are equivalent. Let us see that (2) implies (3). Assume that T(f)∈ S. Since T(S)⊂T(W)⊂ S, then Gn(f) = Tn+1(f)∈ S for all n≥0, and it follows from (S4) that HX(f) = sG•(f)∈ S as required. Finally, if HX(f)∈ S then also THX(f)∈ S. By Lemma 4.2.4 T(f) is a retract of THX(f), so T(f)∈ S and (2) and (3) are equivalent as well.  4.2.5. As announced, we deduce that the hypercohomology sheaf is always CE-fibrant. Proposition 4.2.7. Assume that Xand (D,E) satisfy hypotheses (4.1.1). Then, for any sheaf F,HX(F)is a CE-fibrant sheaf. Proof. Hypothesis 4.1.1 guarantee that T(S)⊂ S, hence HX(S)⊂ S. By Lemma 4.2.4, it is equipped with natural transformations ρ: id −→ HXand σ:H2 X−→ HXsuch that σ ρ = id. As a first consequence, a morphism g:G −→ HX(F) of Sh(X,D)[S−1] is uniquely determined by HX(g). Indeed, from the commutative diagram Gg// ρG  HX(F)1// ρHX(F)  HX(F) HX(G)HX(g)//HX(F)2 σF 88 q q q q q q q q q q GODEMENT RESOLUTIONS 17 we deduce that g=σFHX(g)ρGas claimed. Consider now a lifting problem Gw// f  G′ HX(F) where fis a morphism of Sh(X,D)[S−1] and wis a morphism of Sh(X,D) that is a local equivalence. Since HX(W)⊂ S, given two solutions g, g′:G′−→ HX(F) of this lifting problem, we would have HX(g) = HX(f) (HX(w))−1=HX(g′). Hence g=g′, and we need only see that there is at least one lifting for the above diagram. But g=σFHX(f) (HX(w))−1ρHX(F)is easily seen to satisfy g w =f, so we are done.  4.3. Characterization. In view of the last proposition, we conclude that if for any sheaf ηF:F −→ HX(F) were in W, then (Sh(X,D),S,W) would be a Cartan-Eilenberg category with (HX, ρ) as a resolvent functor. Below we show that this fact is indeed equivalent to two other conditions: one of them is Thomason’s descent property for hypercohomology sheaves, while the other one consists of a weak commutation between the simple functor and stalks. 4.3.1. Let us state precisely what we mean by the later condition. Definition 4.3.1. Let Xbe a Grothendieck site and (D,E) a descent category. We say that the simple functor commutes weakly with stalks if for each sheaf Fthe map θG•F:p∗HX(F) = p∗sG•(F)−→ sp∗G•(F) in (4.2.1) belongs to E. Equivalently, scommutes weakly with stalks if for each point x∈Xthe canonical map θG•F(x) : (sG•F)x−→ s(G•F)xis a weak equivalence. 4.3.2. We can now state and prove our first main result. Theorem 4.3.2. Let Xbe a Grothendieck site and (D,E) a descent category satisfying the hypotheses (4.1.1). Then, the following statements are equivalent: (1) (Sh(X,D),S,W)is a right Cartan-Eilenberg category and for every sheaf F ∈ Sh(X,D), ρF:F −→ HX(F)is a CE-fibrant model. (2) For every sheaf F ∈ Sh(X,D),ρF:F −→ HX(F)is in W. (3) The simple functor commutes weakly with stalks. (4) For every sheaf F ∈ Sh(X,D),HX(F)satisfies Thomason’s descent; that is, ρHX(F): HX(F)−→ H2 X(F)is in S. Definition 4.3.3. We say that a descent category (D,E) is compatible with the site Xif the equivalent conditions of this theorem are satisfied. Remark 4.3.4. As we will see in the examples, this is not necessarily the case for general Xand (D,E). Furthermore, it may happen that (Sh(X,D),S,W) is indeed a Cartan-Eilenberg category, but the CE-fibrant model of a sheaf Fdoes not agree with HX(F) in general. However, this does not pose much of a problem, and these drawbacks only occur when Xis “cohomologically big”: a suitable finite cohomological dimension hypothesis on Xensures that the hypercohomology sheaf HX(F) is always a (CE-fibrant) model for F. 18 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG Proof of Theorem 4.3.2. By Proposition 4.2.7 we know that HX(F) is CE-fibrant for any sheaf F. Hence, the equivalence between (1) and (2) is clear. Let us see that (2) and (3) are equivalent. On the one hand, by definition, (2) holds if and only if p∗(ρF) is in EXfor any sheaf F. On the other hand, the cosimplicial Godement resolution is such that the coaugmentation p∗ηF:cp∗(F)−→ p∗G•(F) has an extra degeneracy. It then follows from Proposition 3.2.3 that sp∗(ηF) belongs to E. Since λG:G −→ sc(G) is also in E for any sheaf G, we have the following commutative diagram in which the arrows decorated with ∼are in E: p∗(F)p∗(λF) ∼// λp∗(F) ∼ %% ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ p∗sc(F)p∗s(ηF)// θc(F)  p∗sG•(F) = p∗HX(F) θG•(F)  sp∗c(F)∼ sp∗(ηF)//sp∗G•(F) Note that the composition of the morphisms in the top row is precisely p∗(ρF) : p∗(F)−→ p∗HX(F). By the 2-out-of-3 property, we conclude that p∗(ρF) is in E if and only if θG•(F)is in E. In other words, (2) and (3) are equivalent. To finish with, we now show that (4) and (2) are equivalent. Because of Proposition 4.2.6, W=H−1 XS. Hence, ρF:F −→ HX(F) is in Wif and only if HX(ρF) is in S. It is then enough to check that ρHX(F)is in Sif and only if HX(ρF) is. As in the proof of Lemma 4.2.4, the iteration of θ′gives a canonical morphism of cosimplicial objects θ′◦ F•:G◦s•(F•)−→ s•G◦(F•) that makes the following diagrams commute s◦s•G◦c•(F)∼ s◦s•G◦(η• F)//s◦s•G◦G•(F)s◦s•c◦G•(F) ∼ s◦s•(η◦ G•(F)) oo s◦(η◦ s•G•(F)) ww♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ s◦G◦(F) HX(ρF) 88 ∼ s◦G◦(λF)// ∼ s◦(λG◦(F)) 88 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ s◦G◦s•c•(F)s◦G◦s•(η• F)// ∼ s◦(θ′◦ c•(F)) OO s◦G◦s•G•(F) s◦(θ′◦ G•(F)) OO s•G•(F) ∼ λs•G•(F) OO ρHX(F) oo Note that all the arrows decorated with ∼are global equivalences: for those arrows involving λ this is clear (in particular this is so for s◦(θ′◦ c•(F))). We already proved that s◦s•(η◦ G•(F))∈ S, and again using an extra degeneracy argument it readily follows that s◦s•G◦(η• F)∈ S. Consequently, ρHX(F)∈ S if and only if HX(ρF)∈ S. 4.3.3. As a toy example, let’s check what our main theorem says for the case of a topological space with just one point. Let X={x}be a topological space with just one point and with its unique possible topology; namely, its open sets are Open(X) = {∅,{x}}. So, every sheaf F ∈ Sh(X, D) is determined by its value on x:F(x)∈ D. The correspondence φ:Sh(X, D)−→ D,F 7−→ F(x) defines an isomorphism of categories whose inverse is ψ:D −→ Sh(X, D), D7−→ D, where Dis the sheaf defined by D(x) = D. GODEMENT RESOLUTIONS 19 Next, in a sober space such as X, the points of the site Open(X) are in a bijective correspondence with the points of Xas a plain topological space. So, we have exactly one Grothendieck point; that is, a couple of adjoint functors x∗:Sh(X, D)⇄D:x∗, defined by x∗(F) = Fx=F(x) and (x∗D)(x) = D. In other words, x∗=φand x∗=ψ. Hence, if we identify Sh(X, D) with Dusing φand ψ,x∗and x∗become the identity functor of C. Hence, the Godement construction G•:D −→ ∆Dis simply the constant cosimplicial functor. Applying the simple functor, we get HX(D) = sG•(D) = scD ≃D, because of axiom (S3) of a descent category. This entails that every object Dshould be fibrant with the CE-structure given on Dby our main theorem. The reader can easily check that it is so: under the identifications φand ψ, classes of local and global equivalences are just E: W=S= E and, with these local and global equivalences, every descent category is a CE-category in which every object is fibrant. So, condition (1) of our main theorem is indeed fulfilled. The reader can check, for instance, that condition (3), the commutation between stalks and simple functor, is also trivially fulfilled too. 4.3.4. The first consequence of our main theorem is the following characterization of Thomason’s descent property for sheaves of spectra. Corollary 4.3.5. If (D,E) is compatible with the site X, then a sheaf F ∈ Sh(X,D)satisfies Thomason’s descent if and only if it is a CE-fibrant sheaf. 4.3.5. The existence of an associated sheaf functor, or sheafification, (−)a:PrSh(X,D)−→ Sh(X,D) guarantees that the homotopy theory of presheaves is the same as the homotopy theory of sheaves, because the adjoint pair (−)a:PrSh(X,D)⇄Sh(X,D) : i, where iis the inclusion functor, induces an equivalence of categories PrSh(X,D)[W−1]≃Sh(X,D)[W−1]. Although an associated sheaf functor may not exist for D, when (D,E) is compatible with the site Xthe hypercohomology sheaf may be thought of as a ‘homotopical’ sheafification functor. More precisely, the adjoint pair (p∗, p∗) is also an adjoint pair PrSh(X,D) p∗ //DX p∗ oo and the induced triple on PrSh(X,D) allows an analogous definition HX(F) = sG•(F) for a presheaf F, which enjoys the same properties as in the sheaf case. In addition, T(F) = p∗p∗(F) is a sheaf, and so is HX(F). Corollary 4.3.6. Let (D,E) be a descent category compatible with the site X. Then PrSh(X,D)[W−1] HX//Sh(X,D)[W−1] i oo are inverse equivalences of categories. Proof. By hypothesis, ρF:F −→ HXi(F) is in W, so it is an isomorphism of Sh(X,D)[W−1] for any sheaf F. It remains to be shown that if Fis now a presheaf then ρF:F −→ iHX(F) is in W. Since HX(F) is a sheaf, ρHX(F)∈ W. But ρHX(F)is a morphism between CE-fibrant sheaves 20 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG and hence belongs to S. By the same proof as in Theorem 4.3.2, we infer that HX(ρF)∈ S as well. Again, this means that ρFis a local equivalence as required.  4.3.6. We have seen that a descent structure on (D,E) always induces one on (Sh(X,D),S) defined objectwise. We have another descent structure, though. Proposition 4.3.7. Assume that a descent category (D,E) is compatible with the site Xand that filtered colimits commute with finite products in D. Then, (Sh(X,D),W)is a descent category with simple functor s′=sHX:∆Sh(X,D)−→ Sh(X,D). Proof. The commutation of finite products with filtered colimits guarantees that WQW ⊂ W. The fact that s′is a simple functor for (Sh(X,D),W) may be proved using that HX(W)⊂ S and that sis a simple functor for (Sh(X,D),S).  It follows from the results in [Rod1] that path and loop functors may be constructed for (Sh(X,D),W) in a natural way. They give rise to well behaved fiber sequences, satisfying the usual properties in Sh(X,D)[W−1]. In particular, Sh(X,D)[W−1] is a triangulated category provided that the loop functor is an equivalence of categories. 4.4. Derived functors for sheaves. 4.4.1. The second consequence of our characterization of CE-fibrant sheaves, the existence of the right derived direct image functor, follows immediately (cf. [Br, th.6 ]). Corollary 4.4.1. Let f:X −→ Y be a continuous functor of Grothendieck sites and (D,E) a descent category compatible with the site X. Then, f∗:Sh(X,D)−→ Sh(Y,D)admits a right derived functor Rf∗:Sh(X,D)[W−1]−→ Sh(Y,D)[W−1]given by Rf∗(F) = f∗HX(F). Proof. In view of Theorem 4.3.2 and Proposition 2.2.6, we only need to show that f∗sends global equivalences to local equivalences. But this is obvious: if ϕ:F −→ G ∈ S, then, for every object V∈ Y, we have f∗(ϕ)(V) = ϕ(f−1(V)) : F(f−1(V)) −→ G(f−1(V)) ∈E. So f∗(ϕ) is also a global equivalence and hence, a fortiori, a local one.  If Uis an object of X, the same proof works for the U-sections functor Γ(U, −) : Sh(X,D)−→ D because, by definition, Γ(U, F) = F(U) sends global equivalences in Sh(X,D) to equivalences in D. Hence, Corollary 4.4.2. Let (D,E) be a descent category compatible with the site X. Then Γ(U, −) : Sh(X,D)−→ D admits a right derived functor RΓ(U, −) : Sh(X,D)[W−1]−→ D[E−1]given by RΓ(U, F) = Γ(U, HX(F)) . GODEMENT RESOLUTIONS 21 4.4.2. When Xhas a terminal object X, e.g. in case Xis the site associated with a topological space X,sheaf cohomology is by definition the right derived functor of the global sections functor Γ(X, −) : Sh(X,D)−→ D. So, under the above assumptions, sheaf cohomology is well defined and agrees with Γ(X, HX(F)). Following [SGA4, 4.3.6.1], if the coefficient category Dhas limits, the notion of global sections functor Γ(X,−) : Sh(X,D)−→ D generalizes to a general site X, possibly without a terminal object, as: Γ(X,F) = lim ←− U∈X F(U). Note that in this case Γ(X,−) does not necessarily send a global equivalence to a weak equivalence of D. But, being (D,E) a descent category in which arbitrary products are E-exact, the right derived functor of lim ←− X:DX−→ D exists, and is given by the composition of the simple functor with the cosimplicial replacement DX−→ ∆D(see [Rod2]). The resulting functor holim ←− X:Sh(X,D)−→ D sends global equivalences to weak ones; hence, it admits a right derived functor Sh(X,D)[W−1]−→ D[E−1] that may be seen to agree with the right derived functor of Γ(X,−). That is, RΓ(X,−) : Sh(X,D)[W−1]−→ D[E−1] exists and is given by RΓ(X,F) = holim ←− U∈X HX(F)(U). 4.4.3. Recall that when there is a sheafification functor then Sh(X,D) is complete (resp. cocomplete) when Dis. A homotopical version of this fact is that when HXis a ‘homotopical’ sheafification functor (that is, when (D,E) is compatible with the site X) then (Sh(X,D),W) is homotopically complete, and homotopically cocomplete provided (D,E) is. The key points to seeing this are that the resolvent functor (HX, ρ) is also a resolvent functor for presheaves, and that it may be lifted to diagram categories: for each small category I, (HX, ρ) induces objectwise a resolvent functor on (PrSh(X,D)I=PrSh(X,DI),S,W). This in turn implies that there is an adjunction natural in I PrSh(X,D)I[S−1]id //PrSh(X,D)I[W−1]≃Sh(X,D)I[W−1] HX oo where the right adjoint HXis fully faithful. This natural adjunction then transfers homotopy limits and colimits existing for (PrSh(X,D),S) = (DX,EX) to Sh(X,D)[W−1]. In particular (Sh(X,D),W) is homotopically complete and holim ←− (Sh(X,D),W) I= holim ←− (Sh(X,D),S) IHX. 5. Examples In this section we show how the above results apply to classic and not so classic examples of categories of sheaves. More concretely, we will prove that a finite cohomological dimension assumption on the site Xguarantees its compatibility with the natural descent structures seen on categories of coefficients Dsuch as complexes, simplicial sets and spectra. Consequently, from the results of the previous section we conclude that for such Xand Dwe have: 22 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG •For every sheaf F, the natural arrow ρF:F −→ HX(F) is a fibrant model of F. Or, what amounts to the same, (Sh(X,D),S,W) is a CE-category with resolvent functor (HX, ρ). •The localized category Sh(X,D)[W−1] is naturally equivalent to Sh(X,D)fib[S−1]. •The CE-fibrant objects of Sh(X,D) are precisely those sheaves satisfying Thomason’s descent. •Derived sections RΓ(U, −) and derived direct image functor Rf∗may be computed by precomposing with HX. •The hypercohomology sheaf HXis a ‘homotopical’ sheafification functor that gives an equivalence Sh(X,D)[W−1]≃PrSh(X,D)[W−1]. 5.1. Bounded complexes of sheaves. 5.1.1. Consider the descent category structure on the category of uniformly bounded cochain complexes C≥b(A) described in example 2.1.4. In this case the simple functor is s=TotQ=Tot⊕:∆C≥b(A)−→ C≥b(A) by the boundedness assumption. The category of sheaves of uniformly bounded cochain complexes Sh(X,C≥b(A)) is a descent category where the weak equivalences are the global equivalences and the simple functor is the total-sum functor applied objectwise: (TotF)(U) = Tot(F(U)). It follows that scommutes in this case with all colimits, since it is defined degree-wise through a finite direct sum. Hence, scommutes trivially with stalks. Therefore, we deduce from Theorem 4.3.2 Theorem 5.1.1. Assume that Ais an abelian category satisfying (AB4)∗and (AB5)(that is, arbitrary products and filtered colimits exist and are exact). Then, the descent category C≥b(A) is compatible with any site X. In particular, properties 5hold for (Sh(X,C≥b(A)),S,W). In this case a local equivalence f∈ W is just a quasi-isomorphism of Sh(X,C≥b(A)) = C≥b(Sh(X,A)). On the other hand, a global equivalence f∈ S is a morphism f:F −→ G of complexes of sheaves such that f(U) is a quasi-isomorphism of C≥b(A) for each object U∈ X. Consequently, a functor F : Sh(X,C≥b(A)) −→ C sending global equivalences to isomorphisms admits a right derived functor RF : D≥b(Sh(X,A)) = Sh(X,C≥b(A))[W−1]−→ C given by RF(F) = F(HX(F)). Note that this derivability criterion does not assume the existence of enough injectives in A. Particularly, for the case A=R−modules, we recover the classic construction of abelian sheaf hypercohomology and derived direct image of sheaves constructed through canonical Godement resolutions by flasque sheaves. 5.2. Unbounded complexes of sheaves. 5.2.1. When the boundedness assumption on complexes of sheaves is dropped, Theorem 5.1.1 is not longer true for a general site X, even in the case A=R−modules. Consider the category C∗(R) of unbounded cochain complexes of R-modules with the descent structure of example 2.1.5. In this case, the simple functor s=TotQ:∆C∗(R)−→ C∗(R) GODEMENT RESOLUTIONS 23 is an infinite product degree-wise, and consequently it does not commute (even weakly) with filtered colimits. This in turn means that the hypercohomology sheaf HX(F) associated with an unbounded complex Fof sheaves of R-modules does not necessarily produce a CE-fibrant model for Fin (Sh(X,C∗(R)),S,W), for a general site X. Example 5.2.1. To illustrate this fact, consider a family {F−k}kof abelian sheaves for which (Qk>0Hk(−,F−k))x6= 0 (for instance those described in [We, A.5] or [MV, 1.30]). Then construct the complex of sheaves Fwith zero differential that is 0 in positive degrees and equal to F−kin negative degrees. It is not hard to verify that ρF:F−→ HX(F) is not a quasi-isomorphism in this case, so it does not provide a CE-fibrant model for F. We remark however that (Sh(X,C∗(R)),S,W) is still a Cartan-Eilenberg category for any site X: K-injective complexes of sheaves are easily seen to be CE-fibrant, and by [Sp] each complex of sheaves is locally equivalent to some K-injective one (see also [We], appendix). Hence the CE-fibrant model of an unbounded complex Fof sheaves does not agree in general with its hypercohomology sheaf HX(F), unless some extra assumption is imposed on site X. 5.2.2. We are going to show that finite cohomological dimension is a sufficient condition for the site Xin order that the hypercohomology sheaf HXproduces a resolvent functor for the Cartan-Eilenberg category (Sh(X,C∗(R)),S,W). Recall that a system of neighbourhoods for a point x∈ X is, by definition, a full cofinal subcategory of the category of neighbourhoods of xin X([SGA4] 6.8.2). Definition 5.2.2 ([GS]).A site Xis said to have finite cohomological dimension if for any point x∈ X there exists d≥0 and a system Λ of neighbourhoods of xsuch that for any sheaf of abelian groups F ∈ Sh(X,Ab) and any neighbourhood U∈Λ it holds that Hn(U;F) = 0 whenever n > d. For instance, the following sites have finite cohomological dimension: (1) The small Zariski site of a noetherian topological space of finite Krull dimension; e.g., the Zariski site of a noetherian scheme of finite Krull dimension. This follows from Grothendieck’s vanishing Theorem ([Har] III, Theorem 2.7). (2) The big Zariski site of a noetherian scheme Xof finite Krull dimension consisting of all schemes of finite type over X, or all noetherian schemes of bounded Krull dimension ([GS], page 6). (3) The small site of a topological manifold of finite dimension. This follows from the vanishing Theorem of [KS]. Theorem 5.2.3. The descent category C∗(R)is compatible with any finite cohomological dimension site X. In this case, properties 5hold for (Sh(X,C∗(R)),S,W). The proof is based on a spectral sequence argument, the Colimit Lemma, for which we need some preliminaries. The same spectral sequence argument will also be used in the examples of simplicial sets and spectra. 24 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG 5.2.3. Let Cbe a category with filtered colimits and Ia filtered indexing set. For us “spectral sequence” means a functorial right half-plane cohomological spectral sequence E∗of abelian groups, commuting with filtered colimits: E∗(colim −→ iXi) = colim −→ iE∗(Xi). For an object X∈ C, we say that the spectral sequence E∗(X) is bounded on the right if there exists dsuch that Ep∗ 2(X) = 0 for p > d. Note that, for conditionally convergent spectral sequences, this implies strong convergence ([Boa], Theorem 7.4). Given a filtered system {Xi}i∈I of objects of C, we say that the family of spectral sequences {E∗(X)}i∈Iis uniformly bounded on the right if there is a fixed dthat works for all i∈I. Proposition 5.2.4 (Colimit Lemma).Assume as given the following data: (1) An object X∈ C and a filtered system X•={Xi}i∈Iof objects Xi∈ C. (2) A cone {fi:Xi−→ X}i∈Ifrom the base X•to the vertex Xand, hence, an induced map f: colim −→ iXi−→ X. Moreover, assume also that: (1) The spectral sequences {E∗Xi}i∈Iand E∗Xconverge conditionally to {Hi}i∈Iand H, respectively. (2) The spectral sequences {E∗Xi}i∈Iare uniformly bounded on the right. (3) The map Er(f) : colim −→ iEr(Xi)−→ Er(X)is an isomorphism for some r≥0. Then the map H(f) : colim −→ iHi−→ His an isomorphism too. Proof. See [Mit], Proposition 3.3.  Proof of Theorem 5.2.3. The first filtration of a double complex K∈C∗∗(R), Fp(TotQK)n=Qs≥pKs,n−sgives us a conditionally convergent spectral sequence Epq 2(TotQK) = Hp hHq v(K) =⇒Hp+q(TotQK), p ≥0. By Theorem 4.3.2, to prove that for any sheaf F ∈ Sh(X,C∗(R)) it holds that ρF:F −→ HX(F) is a CE-fibrant model we may equivalently show that the canonical morphism θF(x) : colim −→ (U,u)∈Nbh(x)TotQ(G∗F)(U)−→ TotQcolim −→ (U,u)∈Nbh(x)(G∗F) is a quis of C∗(R) for any sheaf Fand any point xin the set of enough points X. These colimits may be computed using the neighbourhoods (U, u) in the system of neighbourhoods Λ that exists by assumption. Therefore, we have an object TotQx∗(G∗F)∈C∗(R), a filtered system nTotQ(G∗F)(U)o(U,u), where (U, u) runs over all neighbourhoods of xin Λ and the induced map θF(x). Let us verify the hypotheses of the Colimit Lemma: the spectral sequences Epq 2(U) = Hp vHq h((G∗F)(U)) =⇒Hp+q(TotQ(G∗F)(U)) , p ≥0 GODEMENT RESOLUTIONS 25 and Epq 2(x) = Hp vHq h((G∗F)x) =⇒Hp+q(TotQ((G∗F)x)) , p ≥0 converge conditionally. To compute Epq 2(U) we use that T:Sh(X,C∗(R)) −→ Sh(X,C∗(R)) commutes with cohomology in Sh(X,C∗(R)). At the presheaf level, clearly H∗(T(F)) = T(H∗(F)) for any presheaf F, because cohomology in C∗(R) commutes with products and filtered colimits. Since the stalks of a presheaf Gare isomorphic to the ones of its associated sheaf Ga, then T(G) = T(Ga). Hence, if Fis a sheaf T(H∗F) = T((H∗F)a) = T(H∗F) = H∗(TF). In particular H∗(TF) = T(H∗F) is a sheaf, so it agrees with its associated sheaf. Therefore H∗(TF) = H∗(TF) = T(H∗F), and H∗(G•F) = G•(H∗F). We then have, for all p > d, Epq 2(U) = Hp vHq h(G∗F)(U) = Hp(Γ(U, HqG∗F)) = Hp(Γ(U, G∗HqF)) = Hp(U, HqF) = 0 because of the finite cohomological dimension assumption. Finally, already for r= 0, we have an isomorphism colim −→ Epq 0(U) = colim −→ GpFq(U) = (GpFq)x=Epq 0(x). Hence the Colimit Lemma tells us that Hn(TotQ(G∗F))x−→ HnTotQ((G∗F)x) is an isomorphism for all n. 5.3. Sheaves of fibrant simplicial sets. 5.3.1. Let D=sSfwith the descent structure of 2.1.6. As in the case of unbounded complexes, the simple functor may not commute weakly with stalks. Again, for this to hold we must either restrict to simplicial sets with vanishing higher homotopy groups, or impose some finiteness assumption on the site X. Here we study the second alternative, showing that ρF:F −→ HX(F) is a CE-fibrant model for each Fin (Sh(X, sSf),W,S) if and only if Xis a site of finite type in the sense of [MV]. 5.3.2. By a theorem of Joyal, the category Sh(X, sS) possesses a simplicial model category structure in which all objects are cofibrant and the weak equivalences are the local equivalences [Ja]. The fibrant objects in this model structure are then defined through a lifting property, and they are objectwise fibrant simplicial sets. Therefore, there is a fibrant replacement functor Ex that takes a simplicial sheaf to a fibrant one, in particular Ex(F)∈Sh(X, sSf). Given a simplicial sheaf F ∈ Sh(X, sS) and n≥0, let e P(n)Fbe the simplicial sheaf associated to the presheaf U7→ P(n)F(U) = Im{F(U)−→ cosknF(U)}. It is equipped with natural maps F −→ e P(n)Fand e P(n+1)F −→ e P(n)F. If the stalks of Fare fibrant simplicial sets, the tower {x∗e P(n)F=P(n)x∗F} is precisely the Moore-Postnikov tower of x∗F. In this case the natural map x∗F ≃ lim ←− n≥0x∗e P(n)F −→ holim ←− n≥0x∗e P(n)Fis a weak equivalence. 32 BEATRIZ RODR´ IGUEZ GONZ´ ALEZ AND AGUST´ I ROIG Corollary 6.2.2. Under the same hypotheses of the previous proposition, for any object U∈ X and any sheaf F ∈ Sh(X,D), we have RΓ(U, φF) = φRΓ(U, F). Example 6.2.3. As we have seen in the previous section of examples, the descent categories (C≥b(A),E) and (FC≥b(A),Er) of (filtered) complexes of Examples 2.1.4, 2.1.7 are compatible with any site, provided Ais (AB4)∗and (AB5). Then the above result applies to both the forgetful functor U: (FC≥b(A),E0)−→ (C≥b(A),E) and the decalage filtration functor Dec : (FC≥b(A),Er+1)→(FC≥b(A),Er). This in particular recovers the classic result that filtered sheaf hypercohomology and filtered higher direct images agree with the usual abelian ones when we forget the filtrations. Consequently, we can now extend 2.1.3 to categories of sheaves, obtaining a transfer lemma for CE-structures between them. Proposition 6.2.4. Assume that Dis closed under products and filtered colimits, and that (D′,E′)is a descent category compatible with the site X. If ψ:D −→ D′satisfies the hypotheses of the transfer lemma 2.1.3 and (1) and (2) of the previous proposition, then (I) (D,E = ψ−1E′)is a descent category compatible with X. (II) ψ:Sh(X,D)−→ Sh(X,D′)is a morphism of CE-categories. Proof. By the transfer lemma 2.1.3 and the previous proposition, the only statement remaining to be proved is the fact that the resulting descent category (D,E = ψ−1E′) is compatible with X. First note that by definition (D,E) satisfies hypotheses (4.1.1). Then, using Theorem 4.3.2, it suffices to show that for any F ∈ Sh(X,D), ρHXFis in S. By definition of E, this holds if and only if ψ(ρHXF) is in S′. But arguing as in the previous proof, this happens if and only if ρHXψ(F)is in S′, which holds because (D′,E′) is compatible with X. Examples 6.2.5. In fact, the functors in the previous example also satisfy these stronger hypotheses and then may be used to transfer compatibility with the site. To close the paper, let us briefly describe a classical situation also covered by this transfer lemma. It’s a well-known fact that, if (X, OX) is a ringed space, then the derived functor RΓ(X, F) naturally inherits a module structure for any sheaf Fof OX-modules. But, if we forget this module structure through the forgetful functor ψ:Mod −→ Ab, this derived functor agrees with the usual cohomology as an abelian sheaf: RΓ(X, ψF) = ψRΓ(X, F). In a forthcoming article, we will show that an analogous result holds for sheaves of operad algebras. References [Ba] A. Banerjee, Tensor structures on smooth motives, Journal of K-theory, 9(2012), 57–101. [Boa] J.M. Boardman, Conditionally convergent spectral sequences, Cont.Math. 239 (1999), 49–84. [Bous] A.K. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geometry & Topology, 7(2003), 1001–1053. [Be1] T. Beke, Sheafifiable homotoy model categories Math. Proc. Cambridge Philos. Soc. 129, (2000), 447–475. [Be2] T. Beke, Sheafifiable homotopy model categories, II J. Pure and Appl. Algebra 164, (2001), 307–324. [BK] A.K. Bousfield, D.M. Kan Homotopy limits, completions and localizations Lecture Notes in Math. 304, (1972). GODEMENT RESOLUTIONS 33 [Br] K.S. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology, Trans. Amer. Math. Soc., 186 (1973), 419–458. [C1] J. Cirici, Cofibrant models of diagrams: mixed Hodge structures in rational homotopy. To appear in Trans. Amer. Math. Soc., Available at arXiv:1307.4968 [C2] J. Cirici, Homotopy Theory of Mixed Hodge Complexes. Available at arXiv:1304.6236 [CE] H. Cartan, S. Eilenberg, Homological Algebra, Princenton University Press, (1956). [GL] T. Geisser, M. Levine The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, Journal fr die reine und angewandte Mathematik (Crelles Journal), 530 (2001), 55-103. [GS] H. Gillet, C. Soul´e, Filtrations on Higher Algebraic K-theory, in Algebraic K-Theory, Proc. of Sym. in Pure Math., 67, AMS (1999), 41–88. [Go] R. Godement, Th´eorie des faisceaux, Actualit´es sci. et ind. 1252, Hermann (1973). [GN] F. Guill´en, V. Navarro Aznar, Un crit`ere d’extension des foncteurs d´efinis sur les sch´emas lisses, Publ. Math. IHES, 95 (2002), 1–91. [GNPR1] F. Guill´en Santos, V. Navarro, P. Pascual, Agust´ı Roig, A Cartan-Eilenberg approach to homotopical algebra, J. Pure and Appl. Algebra 214 (2010), 140–164. [GNPR2] F. Guill´en, V. Navarro, P. Pascual, Agust´ı Roig, The differentiable chain functor is not homotopy equivalent to the continuous chain functor, Topol. App. 156 (2009), 65–680. [Har] R. Hartshorne, Algebraic geometry, Springer GTM 52 (1977). [Hir] P.S. Hirschhorn, Model Categories and Their Localizations, Math. Surveys and Monographs, 99, Amer. Math. Soc., Providence (2002). [Ja] J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math. 55 I (1986), 193–239. [KS] M. Kashiwara, P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften 292 Springer (1990). [McL] S. Mac Lane, Categories for the working mathematician (second edition), Springer GTM 5, (1998). [McLM] S. Mac Lane, I. Moerdijk Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer (1992) [Mit] S.A. Mitchell, Hypercohomology spectra and Thomason’s descent theorem, in Algebraic K-Theory, Fields Institute Communications, (1997), 221-278. [MV] F. Morel and V. Voevodsky, A1-homotopy theory of schemes, Pub. Math. I.H.E.S. 90 (1999), 45–143. [N] V. Navarro Aznar Sur la th´eorie de Hodge-Deligne, Inv. Math. 90 (1987), 11–76. [P] P. Pascual, Some remarks on Cartan-Eilenberg categories, Collect.Math. 63 (2012), 203-216. [Q] D. Quillen, Homotopical algebra, Springer LNM 43, (1967). [Rod1] B. Rodr´ıguez Gonz´alez, Simplicial descent categories, J. Pure and Appl. Algebra 216 no. 4 (2012), 775–788. [Rod2] B. Rodr´ıguez Gonz´alez, Realizable homotopy colimits. Available at arXiv:1104.0646. [SdS] F. Sancho de Salas, P. Sancho de Salas, A direct proof of the theorem on formal functions, Proc. Amer. Math. Soc. 137 (2009), 4083–4088. [SGA4] S´eminaire de G´eometrie Alg´ebrique SGA4 Th´eorie des topos et cohomologie ´etale des sch´emas, Springer LNM 269 (1972) [Sp] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65, (1988), p. 121–154. [Th] R. W. Thomason Algebraic K-theory and etale cohomology, Ann. Sci. ENS 18 (1985), 437–552. [We] C. A. Weibel, Cyclic Homology of Schemes, Proc. AMS 124 (1996), 1655–1662. ICMAT, CSIC-Complutense-UAM-CarlosIII, Campus Cantoblanco, UAM. 28049 Madrid, Spain. Dept. Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya, UPC, Diagonal 647, 08028 Barcelona, Spain.