aa r X i v : . [ m a t h . A T ] O c t Representation of relative sheaf cohomology
Tatsuo Suwa ∗ Abstract
We study the cohomology theory of sheaf complexes for open embeddings of topolog-ical spaces and related subjects. The theory is situated in the intersection of the generalˇCech theory and the theory of derived categories. That is to say, on the one hand thecohomology is described as the relative cohomology of the sections of the sheaf complex,which appears naturally in the theory of ˇCech cohomology of sheaf complexes. On theother hand it is interpreted as the cohomology of a complex dual to the mapping coneof a certain morphism of complexes in the theory of derived categories. We prove a “rel-ative de Rham type theorem” from the above two viewpoints. It says that, in the casethe complex is a soft or fine resolution of a certain sheaf, the cohomology is canonicallyisomorphic with the relative cohomology of the sheaf. Thus the former provides a handyway of representing the latter. Along the way we develop various theories and establishescanonical isomorphisms among the cohomologies that appear therein. The second view-point leads to a generalization of the theory to the case of cohomology of sheaf morphisms.Some special cases together with applications are also indicated.
Keywords : relative sheaf cohomology; flabby, soft and fine sheaves; cohomology for openembeddings; relative de Rham type theorem; ˇCech cohomology; relative cohomology forthe sections of a sheaf complex; co-mapping cone; cohomology of sheaf morphisms.
Mathematics Subject Classification (2010) : 14B15, 14F05, 18G40, 32C35, 32C36, 35A27,46M20, 55N05, 55N30, 58A12, 58J15.
The relative cohomology of a sheaf is usually defined by taking its flabby resolution. Thetheme of this paper is how to represent this cohomology. To be a little more precise, let S be a sheaf of Abelian groups on a topological space X . For an open set X ′ in X , therelative cohomology H q ( X, X ′ ; S ) is defined, letting 0 → S → F • be a flabby resolutionof S , as the cohomology of the complex F • ( X, X ′ ) of sections of F • on X that vanishon X ′ . Theoretically it works well as the flabbiness implies the exactness of the sequence0 −→ F • ( X, X ′ ) −→ F • ( X ) i − −→ F • ( X ′ ) −→ , (1.1)where F • ( X ) and F • ( X ′ ) denote the complexes of sections of F • on X and X ′ , respec-tively, and i − the restriction of sections. In practice we would like to have some concreteways of representing the cohomology. One possibility is to adopt the ˇCech method. In ∗ Supported by JSPS Grant 16K05116. X ′ = ∅ , this is commonly used in such areas as algebraic ge-ometry, complex analytic geometry and analytic functions of several complex variables.The relative version is used, for instance, in algebraic analysis. Another way is to usesoft or fine resolutions. Again in the absolute case, this has been done successfully asculminated in such theorems as de Rham’s and Dolbeault’s. They make it possible torepresent a cohomology class by a C ∞ differential form and the former provides a bridgebetween topology and geometry and the latter between geometry and analysis. In therelative case, this method is not directly applicable, as the morphism corresponding to i − in (1.1) fails to be surjective. However it is possible to remedy the situation by in-corporating the ˇCech philosophy. In this paper we pursue this direction and present asystematical way of representing the cohomology via soft or fine resolutions. Along theway we also establish various canonical isomorphisms.In general let K • be a complex of fine sheaves on a paracompact space X . For anopen set X ′ of X , we let V = X ′ and V a neighborhood of the closed set S = X r X ′ and consider the coverings V = { V , V } and V ′ = { V } of X and X ′ . In the sequencecorresponding to (1.1) for K • , we replace K • ( X ) by the complex K • ( V ) of triples ξ = ( ξ , ξ , ξ ) with ξ , ξ and ξ sections of K • on V , V and V = V ∩ V , respectively,the differential being defined in an appropriate manner (cf. Section 4 below for details).Then the morphism i − corresponds to the assignment ξ ξ and K • ( X, X ′ ) is replacedby the subcomplex K • ( V , V ′ ) of triples ξ with ξ = 0 so that a cochain is a pair ( ξ , ξ ).Then we have the exact sequence0 −→ K • ( V , V ′ ) −→ K • ( V ) i − −→ K • ( X ′ ) −→ . The cohomology of K • ( V , V ′ ) a priori depends on the choice of V . However it is shownthat the cohomology of K • ( V ) is canonically isomorphic with that of K • ( X ) so thatthe cohomology of K • ( V , V ′ ) is determined uniquely modulo canonical isomorphisms,independently of the choice of V . Thus the cohomology is denoted by H qD K ( X, X ′ ) andis called the relative cohomology of the sections of K • . In the case K • gives a resolutionof a sheaf S , H qD K ( X, X ′ ) is canonically isomorphic with H q ( X, X ′ ; S ), more preciselywe have : Theorem (Relative de Rham type theorem)
Suppose X and X ′ are paracompact.Then, for any fine resolution → S → K • of a sheaf S on X such that each K q | X ′ isfine, there is a canonical isomorphism : H qD K ( X, X ′ ) ≃ H q ( X, X ′ ; S ) . In fact we give two proofs for the above theorem. Namely we first introduce thecohomology H qd K ( i ) of an arbitrary complex K • of sheaves for the open embedding i : X ′ ֒ → X (cf. Subsection 2.3). On the one hand it is nothing but H qD K ( X, X ′ ) with V = X , if K • is a complex of fine sheaves. On the other hand it is interpreted as thecohomology of a “co-mapping cone”, a notion dual to the mapping cone in the theory ofderived categories (cf. Section 5). The latter viewpoint fits nicely with soft resolutionsand we prove the above theorem in this context (cf. Theorems 2.23 and 5.16). Whilethis first proof is a little abstract, the second proof, which is for fine resolutions, employscoverings and is more direct (cf. Theorem 4.14). In any case the cohomology H qD K ( X, X ′ )2oes well with derived functors. Furthermore the above theorem is generalized to the caseof cohomology for sheaf morphisms (cf. Theorem 6.16).Historically this combination of soft or fine resolutions with the ˇCech method startedwith the introduction of the ˇCech-de Rham cohomology theory (cf. [25], [2]). In partic-ular, the relative version together with its integration theory has been effectively used invarious problems related to localization of characteristic classes (cf. [3], [17], [19], [20] andreferences therein). Likewise we may develop the ˇCech-Dolbeault cohomology theory andon the way we naturally come up with the relative Dolbeault cohomology. This cohomol-ogy again has a number of applications (cf. [1], [21] and [22]). The above theorem appliedto this case shows that it is canonically isomorphic with the local (relative) cohomologyof A. Grothendieck and M. Sato with coefficients in the sheaf of holomorphic forms (cf.[9] and [18]). In particular, if we apply this to the Sato hyperfunction theory, we havesimple explicit expressions of hyperfunctions, some fundamental operations on them andrelated local duality theorems. This approach also gives a new insight into the theory andleads to a number of results that can hardly be achieved by the conventional way (cf. [11]and [24]).The paper is organized as follows. In Section 2, we first recall the cohomology theoryfor sheaf complexes. Although the materials are rather well-known, we outline them inorder to fix notation and conventions and also to describe the isomorphisms explicitly.We then introduce the cohomology H qd K ( i ) of a sheaf complex K • for an open embedding i : X ′ ֒ → X . This is the basic object we study in this paper and later it is interpreted intwo ways, as mentioned above. One is as the relative cohomology H qD K ( X, X ′ ) of sectionsof a sheaf complex and is done from the ˇCech theoretical viewpoint in Section 4. Theother is as the co-mapping cone of a certain morphism of complexes, which is done in Sec-tion 5. We prove the aforementioned relative de Rham type theorem for soft resolutions(Theorem 2.23). Although it is a special case of a more general result (Theorem 6.16),which is a direct consequence of a theorem proved in [16], we state it and give a proof forits independent interest.We develop, in Section 3, a general theory of ˇCech cohomology of sheaf complexesand discuss canonical isomorphisms among various cohomologies which come up in theconstruction. This is more or less a straightforward generalization of the ˇCech-de Rhamcohomology theory. We present the theory so that the isomorphisms are canonical andthe correspondences in them are trackable. We then specialize the theory to the case ofcomplexes of fine sheaves and state the isomorphisms above in this case (Theorem 3.29).In Section 4, we introduce the relative cohomology H qD K ( X, X ′ ) for the sections of a sheafcomplex K • . As mentioned above it gives an interpretation of the cohomology H qd K ( i ).We also give an alternative proof of the relative de Rham type theorem for fine resolutions(Theorem 4.14).In Section 5, we introduce the aforementioned notion of co-mapping cone. We thensee that the complex K • ( i ) introduced in Section 2 is given as the co-mapping cone of acertain morphism of complexes. This leads to a statement of the relative de Rham typetheorem in terms of derived functors (Theorem 5.16). In Section 6 we introduce, following[16], the cohomology for sheaf morphisms, which generalizes the relative sheaf cohomology.Then we give a representation theorem (Theorem 6.16) generalizing Theorem 2.23. Finallywe discuss, in Section 7, some special cases and indicate applications in each case.3he author would like to thank Naofumi Honda for stimulating discussions and valu-able comments during the preparation of the paper. In the sequel, by a sheaf we mean a sheaf with at least the structure of Abelian groups.For a sheaf S on a topological space X and a subset A of X , we denote by S ( A ) thegroup of sections of S on A . Also, for a subset A ′ of A , we denote by S ( A, A ′ ) thesubgroup of S ( A ) consisting of sections that vanish on A ′ .A complex K of sheaves is a collection ( K q , d q K ) q ∈ Z , where K q is a sheaf on X and d q K : K q → K q +1 is a morphism, called differential , with d q +1 K ◦ d q K = 0. We omit thesubscript or superscript on d if there is no fear of confusion. The complex is also denotedby ( K • , d ) or K • . We only consider the case K q = 0 for q <
0. We say that K is aresolution of S if there is a morphism ι : S → K such that the following sequence isexact : 0 −→ S ι −→ K d −→ · · · d −→ K q d −→ · · · . We abbreviate this by saying that 0 → S → K • is a resolution. We come back togeneralities on complexes in Subsection 5.1 below. As reference cohomology theory, we adopt the one via flabby resolutions (cf. [4], [6], [13],[15]). Recall that a sheaf F is flabby if the restriction F ( X ) → F ( V ) is surjective forevery open set V in X .Let S be a sheaf on X . We may use any flabby resolution of S to define thecohomology of S , however we take the canonical resolution (Godement resolution), tofix the idea : 0 −→ S −→ C ( S ) d −→ · · · d −→ C q ( S ) d −→ · · · . The q -th cohomology H q ( X ; S ) of X with coefficients in S is the q -th cohomology ofthe complex ( C • ( S )( X ) , d ). For a subset A of X , H q ( A ; S ) denotes H q ( A ; S | A ), where S | A is the restriction of S to A .More generally, for an open set X ′ in X , we denote by H q ( X, X ′ ; S ) the q -th coho-mology of ( C • ( S )( X, X ′ ) , d ). Note that H q ( X, ∅ ; S ) = H q ( X ; S ). Setting S = X r X ′ ,it will also be denoted by H qS ( X ; S ). This cohomology in the first expression is referredto as the relative cohomology of S on ( X, X ′ ) (cf. [18]) and in the second expression the local cohomology of S on X with support in S (cf. [9]).We recall some of the basic facts : Proposition 2.1
The above cohomology has the following properties :(1) H ( X, X ′ ; S ) = S ( X, X ′ ) . (2) For a flabby sheaf F , H q ( X, X ′ ; F ) = 0 for q ≥ . For a triple ( X, X ′ , X ′′ ) with X ′′ an open set in X ′ , there is an exact sequence · · · → H q − ( X ′ , X ′′ ; S ) δ → H q ( X, X ′ ; S ) j − → H q ( X, X ′′ ; S ) i − → H q ( X ′ , X ′′ ; S ) → · · · . (4) (Excision) For any open set V in X containing S , there is a canonical isomorphism : H q ( X, X r S ; S ) ≃ H q ( V, V r S ; S ) . (5) For an exact sequence −→ S ′ −→ S −→ S ′′ −→ of sheaves, there is an exact sequence · · · → H q − ( X, X ′ ; S ′′ ) → H q ( X, X ′ ; S ′ ) → H q ( X, X ′ ; S ) → H q ( X, X ′ ; S ′′ ) → · · · . Note that the exact sequence in (3) above arises from the exact sequence0 −→ C • ( S )( X, X ′ ) j − −→ C • ( S )( X, X ′′ ) i − −→ C • ( S )( X ′ , X ′′ ) −→ , (2.2)where i − and j − denote the morphisms induced by the inclusions i : ( X ′ , X ′′ ) ֒ → ( X, X ′′ )and j : ( X, X ′′ ) ֒ → ( X, X ′ ) (cf. Proposition 5.1 below). Also, (5) follows from the factsthat C • ( ) is an exact functor and that the following sequence is exact :0 −→ C • ( S ′ )( X, X ′ ) −→ C • ( S )( X, X ′ ) −→ C • ( S ′′ )( X, X ′ ) −→ . Remark 2.3
The cohomology H q ( X, X ′ ; S ) is determined uniquely modulo canonicalisomorphisms, independently of the flabby resolution. Although this fact is well-known,we indicate a proof below in order to make the correspondence explicit (cf. Corollary 2.10). Let K = ( K q , d K ) be a complex of sheaves on a topological space X . For each q , we takethe canonical resolution 0 → K q → C • ( K q ) whose differential is denoted by δ G . Thedifferential d K : K q → K q +1 induces a morphism of complexes C • ( K q ) → C • ( K q +1 ),which is denoted also by d K . Thus we have a double complex ( C • ( K • ) , δ G , ( − • d K ).We consider the associated single complex ( C ( K ) • , D G K ), where C ( K ) q = M q + q = q C q ( K q ) , D G K = δ G + ( − q d K . Then there is an exact sequence of complexes :0 −→ K • κ −→ C ( K ) • , (2.4)which is given by K q ֒ → C ( K q ) ⊂ C ( K ) q for each q . Definition 2.5
Let X ′ be an open set in X . The cohomology H q ( X, X ′ ; K • ) of K • on( X, X ′ ) is the cohomology of ( C ( K ) • ( X, X ′ ) , D G K ).5f X ′ = ∅ , we denote H q ( X, X ′ ; K • ) by H q ( X ; K • ). We have H ( X, X ′ ; K • ) = S ( X, X ′ ), where S is the kernel of d : K → K .In the above situation, we have a complex ( K • ( X, X ′ ) , d K ), whose cohomology isdenoted by H qd K ( X, X ′ ). From (2.4), we have an exact sequence of complexes :0 −→ K • ( X, X ′ ) κ −→ C ( K ) • ( X, X ′ ) , (2.6)which induces a morphism ϕ ( X,X ′ ) : H qd K ( X, X ′ ) −→ H q ( X, X ′ ; K • ) . Proposition 2.7
Suppose H q ( X, X ′ ; K q ) = 0 for q ≥ and q ≥ . Then ϕ ( X,X ′ ) isan isomorphism for all q , i.e., κ in (2.6) is a quasi-isomorphism (cf. Subsection 5.3) , inthis case. Proof:
We consider one of the spectral sequences associated with the double complex C • ( K • )( X, X ′ ) : ′ E q ,q = H q d H q δ ( C • ( K • )( X, X ′ )) = ⇒ H q ( X, X ′ ; K • ) , where we denote d K and δ G simply by d and δ . By assumption, H q δ ( C • ( K q )( X, X ′ )) = H q ( X, X ′ ; K q ) = 0 for q ≥ q ≥
1, while H δ ( C • ( K q )( X, X ′ )) = K q ( X, X ′ ). ✷ Let S denote the kernel of d : K → K . Then there is an exact sequence ofcomplexes 0 −→ C • ( S ) −→ C ( K ) • , which is given by C q ( S ) ֒ → C q ( K ) ⊂ C ( K ) q . It induces0 −→ C • ( S )( X, X ′ ) −→ C ( K ) • ( X, X ′ )and ψ ( X,X ′ ) : H q ( X, X ′ ; S ) −→ H q ( X, X ′ ; K • ) . Proposition 2.8 If → S → K • is a resolution, then ψ ( X,X ′ ) is an isomorphism. Proof:
We consider the other spectral sequence associated with C • ( K • )( X, X ′ ) : ′′ E q ,q = H q δ H q d ( C • ( K • )( X, X ′ )) = ⇒ H q ( X, X ′ ; K • ) . From the assumption, 0 → C q ( S ) → C q ( K • ) is an exact sequence of flabby sheaves andthus 0 → C q ( S )( X, X ′ ) → C q ( K • )( X, X ′ ) is exact. Hence H q d ( C q ( K • )( X, X ′ )) = 0for q ≥
1, while H d ( C q ( K • )( X, X ′ )) = C q ( S )( X, X ′ ). ✷ From Propositions 2.7 and 2.8 we have :
Theorem 2.9 1.
For any resolution → S → K • , there is a canonical morphism χ ( X,X ′ ) : H qd K ( X, X ′ ) −→ H q ( X, X ′ ; S ) , where χ ( X,X ′ ) = ( ψ ( X,X ′ ) ) − ◦ ϕ ( X,X ′ ) . Moreover, if H q ( X, X ′ ; K q ) = 0 for q ≥ and q ≥ , then χ ( X,X ′ ) is an isomor-phism. orollary 2.10 For any flabby resolution → S → F • , there is a canonical isomor-phism : χ ( X,X ′ ) : H qd F ( X, X ′ ) ∼ −→ H q ( X, X ′ ; S ) . Remark 2.11
Suppose H q ( X ; K q ) = 0 and H q ( X ′ ; K q ) = 0 for q ≥ q ≥ H q ( X, X ′ ; K q ) = 0 for q ≥ q ≥
2. Thus in this case, if H ( X, X ′ ; K q ) = 0for q ≥
0, the hypothesis of Theorem 2.9. 2 is fulfilled (cf. [16, Theorem 3.3]).Let K • be a complex of sheaves on X . We come back to the double complex C • ( K • )and the associated single complex C ( K ) • . Proposition 2.12 1.
The morphism κ in (2.4) induces an isomorphism H q ( K • ) ∼ −→ H q ( C ( K ) • ) , i.e., it is a quasi-isomorphism. Suppose → S ι → K • is a resolution. Then the composition of ι : S → K and κ : K → C ( K ) leads to a flabby resolution → S → C ( K ) • of S so that thefollowing diagram is commutative :0 / / S / / K • κ (cid:15) (cid:15) / / S / / C ( K ) • . Proof:
1. Consider one of the spectral sequences associated with the double complex C • ( K • ) : ′ E q ,q = H q d H q δ ( C • ( K • )) = ⇒ H q ( C ( K ) • ) . We have H q δ ( C • ( K q )) = 0 for q ≥ q ≥
1, while H δ ( C • ( K q )) = K q .2. For each q , the sheaf C ( K ) q is flabby, being a direct sum of flabby sheaves. The restfollows from 1. Note that the other spectral sequence leads to the same conclusion. ✷ Remark 2.13 1.
The cohomology H q ( X, X ′ ; K • ) in Definition 2.5 is sometimes referredto as the hypercohomology of K • . We may explicitly describe the correspondence in each of the above isomorphisms,as explained more in detail in the case of ˇCech cohomology below. For example, inTheorem 2.9 we think of a cocycle s in K q ( X, X ′ ) and a cocycle γ in C q ( S )( X, X ′ )as being cocycles in C ( K ) q ( X, X ′ ). The classes [ s ] and [ γ ] correspond in the aboveisomorphism, if and only if s and γ define the same class in H q ( X, X ′ ; K • ), i.e., thereexists a ( q − χ in C ( K ) q − ( X, X ′ ) such that s − γ = D G K χ, see the remark after Theorem 3.19 and Remark 3.25 below.7 .3 Cohomology for open embeddings Let X be a topological space and X ′ an open set in X with inclusion i : X ′ ֒ → X . For acomplex of sheaves K • on X , we construct a complex K • ( i ) as follows. We set K q ( i ) = K q ( X ) ⊕ K q − ( X ′ )and define the differential d : K q ( i ) = K q ( X ) ⊕ K q − ( X ′ ) −→ K q +1 ( i ) = K q +1 ( X ) ⊕ K q ( X ′ )by d ( s, t ) = ( ds, i − s − dt ) , where i − : K q ( X ) → K q ( X ′ ) denotes the pull-back of sections by i , the restriction to X ′ in this case. Obviously we have d ◦ d = 0. Definition 2.14
The cohomology H qd K ( i ) of K • for i : X ′ ֒ → X is the cohomology of( K • ( i ) , d ). Remark 2.15 1.
This kind of cohomology is considered in [2] for the de Rham complexeson C ∞ manifolds (cf. Remark 6.10. 2 below). The complex K • ( i ) is nothing but the co-mapping cone M ∗ ( i − ) of the morphism i − : K • ( X ) → K • ( X ′ ) (cf. Subsection 5.4 below). It is also identical with the complex K • ( V ⋆ , V ′ ) considered in Section 4 and the cohomology H qd K ( i ) is equal to H qD K ( X, X ′ ),the relative cohomology for the sections of K • on ( X, X ′ ) (cf. (4.5) and (5.12)). The above cohomology is generalized to that of sheaf complex morphisms in Section 6.Denoting by K [ − • the complex with K [ − q = K q − and d q K [ − = − d q − K , wedefine morphisms α ∗ : K • ( i ) → K • ( X ) and β ∗ : K [ − • ( X ′ ) → K • ( i ) by α ∗ : K q ( i ) = K q ( X ) ⊕ K q ( X ′ ) q − −→ K q ( X ) , ( s, t ) s, and β ∗ : K [ − q ( X ′ ) = K q − ( X ′ ) −→ K q ( i ) = K q ( X ) ⊕ K q − ( X ′ ) , t (0 , t ) . Then we have the exact sequence of complexes0 −→ K [ − • ( X ′ ) β ∗ −→ K • ( i ) α ∗ −→ K • ( X ) −→ , (2.16)which gives rise to the exact sequence · · · −→ H q − d K ( X ′ ) β ∗ −→ H qd K ( i ) α ∗ −→ H qd K ( X ) i − −→ H qd K ( X ′ ) −→ · · · . (2.17)If we define ρ : K • ( X, X ′ ) → K • ( i ) by s ( s, Proposition 2.18
Let F • be a complex of flabby sheaves on X . Then the above mor-phism induces an isomorphism ρ : H qd F ( X, X ′ ) ∼ −→ H qd F ( i ) . roof: We have the commutative diagram with exact rows :0 / / F • ( X, X ′ ) ρ (cid:15) (cid:15) j − / / F • ( X ) i − / / F • ( X ′ ) / / / / F [ − • ( X ′ ) β ∗ / / F • ( i ) α ∗ / / F • ( X ) / / , which gives rise to the diagram · · · / / H q − d F ( X ′ ) δ / / H qd F ( X, X ′ ) ρ (cid:15) (cid:15) j − / / H qd F ( X ) i − / / H qd F ( X ′ ) / / · · ·· · · / / H q − d F ( X ′ ) β ∗ / / H qd F ( i ) α ∗ / / H qd F ( X ) i − / / H qd F ( X ′ ) / / · · · , where δ assigns to the class of t the class of d ˜ t with ˜ t an extension of t to X . The rowsare exact and the diagram is commutative, except for the rectangle at the left, where ρ ◦ δ = − β ∗ . Thus we may apply the five lemma to prove the proposition. ✷ Note that the above is a special case of Proposition 5.9 below.
Corollary 2.19 If → S → F • is a flabby resolution, there is a canonical isomorphism H qd F ( i ) ∼ −→ H q ( X, X ′ ; S ) , which is given by χ ( X,X ′ ) ◦ ρ − with χ ( X,X ′ ) the isomorphism of Corollary 2.10. Theorem 2.20
Suppose → S → K • is a resolution of S such that H q ( X ; K q ) = 0 and H q ( X ′ ; K q ) = 0 for q ≥ and q ≥ . Then there is a canonical isomorphism : H qd K ( i ) ≃ H q ( X, X ′ ; S ) . Proof:
By Proposition 2.12, there exist a flabby resolution 0 → S → F • and amorphism κ : K • → F • such that the following diagram is commutative :0 / / S / / K • κ (cid:15) (cid:15) / / S / / F • . The morphism κ induces morphisms κ : K • ( X ) → F • ( X ) and κ ′ : K • ( X ′ ) → F • ( X ′ ).They in turn induce a morphism κ ( i ) : K • ( i ) → F • ( i ), given by κ ( i ) q = κ q ⊕ ( κ ′ ) q − .We have the following diagram : · · · / / H q − d K ( X ′ ) / / χ ∼ % % ❑❑❑❑❑❑ κ ′ (cid:15) (cid:15) H qd K ( i ) / / κ ( i ) (cid:15) (cid:15) H qd K ( X ) κ (cid:15) (cid:15) χ ∼ ●●●●● / / H qd K ( X ′ ) κ ′ (cid:15) (cid:15) / / χ ∼ $ $ ❍❍❍❍❍ · · ·· · · / / H q − ( X ′ ; S ) / / H q ( X, X ′ ; S ) / / H q ( X ; S ) / / H q ( X ′ ; S ) / / · · ·· · · / / H q − d F ( X ′ ) / / χ ∼ ssssss H qd F ( i ) χ ◦ ρ − ∼ : : ✉✉✉✉✉✉ / / H qd F ( X ) / / χ ∼ ; ; ✇✇✇✇✇✇ H qd F ( X ′ ) / / χ ∼ ; ; ✈✈✈✈✈✈ · · · , where the top and bottom sequences are the ones in (2.17) for K • and F • , the middlesequence is the one in Proposition 2.1 (3) with X ′′ = ∅ , the vertical morphisms are9he ones induced by κ and the χ ’s are the ones in Theorem 2.9. The triangles and therectangles are commutative. The parallelograms are commutative except for the one atthe left bottom, which is anti-commutative. By assumption, all the χ ’s are isomorphismsso that κ and κ ′ are isomorphisms. Hence by the five lemma, κ ( i ) is an isomorphism. ✷ Later the theorem above is reproved as Theorem 5.15 and is generalized as Theo-rem 6.14.
Soft sheaves :
Let X be a paracompact topological space, i.e., it is Hausdorff andevery open covering of X admits a locally finite refinement. A sheaf G on X is soft if therestriction G ( X ) → G ( S ) is surjective for every closed set S in X . A flabby sheaf is soft.If G is soft, then H q ( X ; G ) = 0 for q ≥ X is paracompact. Foe example, this is the case if X is alocally compact Hausdorff space with a countable basis, in particular, a manifold with acountable basis. In this case, for a soft sheaf S and a locally closed set A in X , the sheaf S | A is soft (cf. [6]).Under the above assumption on X , let X ′ be an open set in X and G a soft sheaf on X . Then from Proposition 2.1 (3) with X ′′ = ∅ , we see that H q ( X, X ′ ; G ) = 0 for q ≥ H ( X, X ′ ; G ) = 0 in general. In fact we have the exact sequence0 −→ G ( X, X ′ ) j − −→ G ( X ) i − −→ G ( X ′ ) δ −→ H ( X, X ′ ; G ) −→ H ( X, X ′ ; G ) is the obstruction to i − being surjective.From Theorem 2.9 with X ′ = ∅ , we have : Theorem 2.22 (de Rham type theorem)
Let X be a paracompact topological spaceand S a sheaf on X . Then, for any soft resolution → S → K • of S , there is acanonical isomorphism : H qd K ( X ) ≃ H q ( X ; S ) . More generally, let X ′ be an open set in X . We say that ( X, X ′ ) is a paracompact pair if X and X ′ are paracompact. From Theorem 2.20, we have : Theorem 2.23 (Relative de Rham type theorem)
Let ( X, X ′ ) be a paracompact pairand S a sheaf on X . Then, for any soft resolution → S → K • such that each K q | X ′ is soft, there is a canonical isomorphism : H qd K ( i ) ≃ H q ( X, X ′ ; S ) . The above theorem is restated as Theorem 5.16 and is generalized as Theorem 6.16below. An alternative proof is given in Theorem 4.14 for fine resolutions.
We briefly recall the usual ˇCech cohomology theory for sheaves.10et X be a topological space, S a sheaf on X and W = { W α } α ∈ I an open coveringof X . We set W α ...α q = W α ∩ · · · ∩ W α q and consider the direct product C q ( W ; S ) = Y ( α ,...,α q ) ∈ I q +1 S ( W α ...α q ) . The differential ˇ δ : C q ( W ; S ) → C q +1 ( W ; S ) is defined by(ˇ δσ ) α ...α q +1 = q +1 X ν =0 ( − ν σ α ... c α ν ...α q +1 . Then we have ˇ δ ◦ ˇ δ = 0 and the q -th ˇCech cohomology H q ( W ; S ) of S on W is the q -thcohomology of the complex ( C • ( W ; S ) , ˇ δ ).Let X ′ be an open set in X . Let W = { W α } α ∈ I be a covering of X such that W ′ = { W α } α ∈ I ′ is a covering of X ′ for some I ′ ⊂ I . In the sequel we refer to such a pair( W , W ′ ) as a pair of coverings of ( X, X ′ ). We set C q ( W , W ′ ; S ) = { σ ∈ C q ( W ; S ) | σ α ...α q = 0 if α , . . . , α q ∈ I ′ } . Then the operator ˇ δ restricts to C q ( W , W ′ ; S ) → C q +1 ( W , W ′ ; S ). The ˇCech cohomol-ogy H q ( W , W ′ ; S ) of S on ( W , W ′ ) is the cohomology of ( C • ( W , W ′ ; S ) , ˇ δ ). We havethe properties (1) - (3) in Proposition 2.1, replacing H q ( X, X ′ ; S ) by H q ( W , W ′ ; S ). Let ( K • , d K ) be a complex of sheaves on a topological space X and W = { W α } α ∈ I anopen covering of X . Also let X ′ be an open set in X and W ′ a subcovering of W asbefore. Then we have a double complex ( C • ( W , W ′ ; K • ) , ˇ δ, ( − • d K ) :... ˇ δ (cid:15) (cid:15) ... ˇ δ (cid:15) (cid:15) · · · ( − q d / / C q ( W , W ′ ; K q ) ( − q d / / ˇ δ (cid:15) (cid:15) C q ( W , W ′ ; K q +1 ) ( − q d / / ˇ δ (cid:15) (cid:15) · · ·· · · ( − q d / / C q +1 ( W , W ′ ; K q ) ( − q d / / ˇ δ (cid:15) (cid:15) C q +1 ( W , W ′ ; K q +1 ) ( − q d / / ˇ δ (cid:15) (cid:15) · · · ... ... . (3.1)We consider the associated single complex ( K • ( W , W ′ ) , D K ). Thus K q ( W , W ′ ) = M q + q = q C q ( W , W ′ ; K q ) , D K = ˇ δ + ( − q d K . Definition 3.2
The ˇCech cohomology H q ( W , W ′ ; K • ) of K • on ( W , W ′ ) is the coho-mology of ( K • ( W , W ′ ) , D K ).In the case X ′ = ∅ , we take ∅ as I ′ and denote H q ( W , W ′ ; K • ) by H q ( W ; K • ).11 emark 3.3 In the case K p = 0 for p > K q ( W , W ′ ) = C q ( W , W ′ ; K ) so that H q ( W , W ′ ; K • ) = H q ( W , W ′ ; K ) . We describe the differential D K a little more in detail. A cochain ξ in K q ( W , W ′ )may be expressed as ξ = ( ξ q ) ≤ q ≤ q with ξ q in C q ( W , W ′ ; K q − q ). In the sequel ξ q α ...α q will also be written as ξ α ...α q . Then D = D K : K q ( W , W ′ ) → K q +1 ( W , W ′ ) is givenby ( Dξ ) q = dξ q = 0ˇ δξ q − + ( − q dξ q ≤ q ≤ q ˇ δξ q q = q + 1 . (3.4)In particular, for q = 0 , Dξ ) α = dξ α , ( Dξ ) α α = ξ α − ξ α − dξ α α . (3.5)Thus the condition for ξ being a cocycle is given by dξ = 0ˇ δξ q − + ( − q dξ q = 0 1 ≤ q ≤ q ˇ δξ q = 0 . (3.6)We have H ( W , W ′ ; K • ) = S ( X, X ′ ), where S is the kernel of d K : K → K .For a triple ( W , W ′ , W ′′ ), we have the exact sequence0 −→ K • ( W , W ′ ) −→ K • ( W , W ′′ ) −→ K • ( W ′ , W ′′ ) −→ · · · −→ H q − ( W ′ , W ′′ ; K • ) δ −→ H q ( W , W ′ ; K • ) j − −→ H q ( W , W ′′ ; K • ) i − −→ H q ( W ′ , W ′′ ; K • ) −→ · · · . (3.7)The inclusion K q ( X, X ′ ) ֒ → C ( W , W ′ ; K q ) ⊂ K q ( W , W ′ ) is compatible with thedifferentials and induces a morphism ϕ ( W , W ′ ) : H qd K ( X, X ′ ) −→ H q ( W , W ′ ; K • ) . (3.8)In the case I ′ = ∅ , we denote the above morphism by ϕ W : H qd K ( X ) → H q ( W ; K • ).Here is a special case where this is an isomorphism : Proposition 3.9
Suppose W α = X for some α ∈ I , then ϕ W is an isomorphism. In factthe map π : K • ( W ) → K • ( X ) given by ξ ξ α is a morphism of complexes and inducesthe inverse of ϕ W . roof: By the first identity in (3.5), π is a morphism of complexes and induces π : H q ( W ; K • ) → H qd K ( X ). By definition we have π ◦ ϕ ( W , W ′ ) = 1, the identity. Thus itsuffices to show that ϕ W ◦ π = 1. For this, take ξ ∈ K q ( W ) with Dξ = 0. We claim thatthere exists a cochain η ∈ K q − ( W ) such that ξ − ξ α = Dη, (3.10)which will prove the proposition. Indeed, let η be defined by η α ...α p = ξ αα ...α p , ≤ p ≤ q − . (3.11)Then ( Dη ) α = dη α = dξ αα = ξ α − ξ α = ( ξ − ξ α ) α . For q with 1 ≤ q ≤ q −
1, by(3.4), ( Dη ) α ...α q = (ˇ δη ) α ...α q + ( − q dη α ...α q . (3.12)The first term in the right is equal to q X ν =0 ( − ν η α ... c α ν ...α q = q X ν =0 ( − ν ξ αα ... c α ν ...α q . By the cocycle condition (3.6), the second term in the right hand side of (3.12) is equalto( − q dξ αα ...α q = − ( − q +1 dξ αα ...α q = (ˇ δξ ) αα ...α q = ξ α ...α q − q X ν =0 ( − ν ξ αα ... c α ν ...α q . Thus ( Dη ) α ...α q = ξ α ...α q = ( ξ − ξ α ) α ...α q . Finally, by (3.4) for η and the last identityin (3.6), ( Dη ) α ...α q =(ˇ δη ) α ...α q = q X ν =0 ( − ν η α ... c α ν ...α q = q X ν =0 ( − ν ξ αα ... c α ν ...α q = ξ α ...α q = ( ξ − ξ α ) α ...α q . Therefore we have (3.10) and the proposition. ✷ Note that, in the situation of the proposition, ϕ ( W , W ′ ) may not be an isomorphism, asthe cochain η defined by (3.11) may not be in K q − ( W , W ′ ).In general, we have : Proposition 3.13
Suppose H q ( W , W ′ ; K q ) = 0 for q ≥ and q ≥ . Then ϕ ( W , W ′ ) is an isomorphism. Proof:
We consider one of the spectral sequences associated with the double complex C • ( W , W ′ ; K • ) : ′ E q ,q = H q d H q δ ( C • ( W , W ′ ; K • )) = ⇒ H q ( W , W ′ ; K • ) , where we denote ˇ δ by δ . We have H q δ ( C • ( W , W ′ ; K q )) = H q ( W , W ′ ; K q ) = 0 for q ≥ q ≥
1, by assumption, while H δ ( C • ( W , W ′ ; K q )) = K q ( X, X ′ ). ✷ K q ( X, X ′ ) may be thought of as a cochain in K q ( W , W ′ ) and this identification induces an isomorphism between the cohomologies.A cocycle s in K q ( X, X ′ ) and a cocycle ξ in K q ( W , W ′ ) determine the same class if andonly if there exists a ( q − η in K q − ( W , W ′ ) such that ξ = s + Dη.
Setting η q = 0, we may rephrase this as (cf. (3.4)) ( ξ = s + dη ,ξ q = ˇ δη q − + ( − q dη q , ≤ q ≤ q. (3.14)Let S denote the kernel of d K : K → K . Then the inclusion C q ( W , W ′ ; S ) ֒ → C q ( W , W ′ ; K ) ⊂ K q ( W , W ′ ) is compatible with the differentials and induces a mor-phism ψ ( W , W ′ ) : H q ( W , W ′ ; S ) −→ H q ( W , W ′ ; K • ) . (3.15) Proposition 3.16
Suppose H q ( K • ( W α ...α q )) = 0 for q ≥ and q ≥ . Then ψ ( W , W ′ ) is an isomorphism. Proof:
We consider the other spectral sequence associated with the double complex C • ( W , W ′ ; K • ) : ′′ E q ,q = H q δ H q d ( C • ( W , W ′ ; K • )) = ⇒ H q ( W , W ′ ; K • ) , where we denote ˇ δ by δ . We claim that the sequence0 −→ C q ( W , W ′ ; S ) ι −→ C q ( W , W ′ ; K ) d −→ C q ( W , W ′ ; K ) d −→ · · · (3.17)is exact for q ≥
0, which would imply the proposition. For this, note that the assumptionimplies that the following sequence is exact :0 −→ S ( W α ...α q ) ι −→ K ( W α ...α q ) d −→ K ( W α ...α q ) d −→ · · · . From this we see that (3.17) is exact up to the term C q ( W , W ′ ; K ). Let q ≥ ξ ∈ C q ( W , W ′ ; K q ) with dξ = 0. Then there exists η ∈ C q ( W ; K q − )such that ξ = dη . If α , . . . , α q are in I ′ , dη α ...α q = ξ α ...α q = 0. Thus there exists χ ∈ C q ( W ′ ; K q − ) such that η = dχ , where we set K − = S and d − = ι . By setting χ α ...α q = 0, if α ν ∈ I r I ′ for some ν ∈ { , . . . , q } , we may think of χ as a cochain in C q ( W ; K q − ). If we set η ′ = η − dχ , it is in C q ( W , W ; K q − ) and ξ = dη ′ . Hence(3.17) is exact. ✷ A cochain in C q ( W , W ′ ; S ) may be thought of as a cochain in K q ( W , W ′ ) and acocycle σ in C q ( W , W ′ ; S ) and a cocycle ξ in K q ( W , W ′ ) determine the same class ifand only if there exists a ( q − ζ in K q − ( W , W ′ ) such that ξ = σ + Dζ . ζ − = 0, we may rephrase this as ( ξ q = ˇ δζ q − + ( − q dζ q , ≤ q ≤ q − ,ξ q = σ + ˇ δζ q − . (3.18)From Propositions 3.13 and 3.16 we have : Theorem 3.19
Let ( K • , d K ) be a complex of sheaves on X and let S be the kernel of d K : K → K . Suppose H q ( W , W ′ ; K q ) = 0 and H q ( K • ( W α ...α q )) = 0 , for q ≥ and q ≥ . Then there is a canonical isomorphism : H qd K ( X, X ′ ) ≃ H q ( W , W ′ ; S ) . In the above, we think of a cocycle s in K q ( X, X ′ ) and a cocycle σ in C q ( W , W ′ ; S )as cocycles in K q ( W , W ′ ). The classes [ s ] and [ σ ] correspond in the above isomorphism, ifand only if s and σ define the same class in H qD K ( W , W ′ ), i.e., there exists a ( q − χ in K q − ( W , W ′ ) such that s − σ = Dχ.
Such a χ is given by χ = ζ − η with η and ζ as in (3.14) and (3.18). The above relationis rephrased as, for χ q in C q ( W , W ′ ; K q − q − ), 0 ≤ q ≤ q − s = dχ , δχ q − + ( − q dχ q , ≤ q ≤ q − − σ = ˇ δχ q − . (3.20)The above correspondence may be illustrated in the following diagram. For simplicity,we consider the absolute case ( W ′ = ∅ ), the relative case being similar. We also denote ˇ δ by δ : K ( X ) d / / _(cid:127) (cid:15) (cid:15) K ( X ) d / / _(cid:127) (cid:15) (cid:15) · · · d q − / / s ∈ K q ( X ) d q / / _(cid:127) (cid:15) (cid:15) · · · C ( W ; S ) (cid:31) (cid:127) / / δ (cid:15) (cid:15) C ( W ; K ) d / / δ (cid:15) (cid:15) C ( W ; K ) d / / δ (cid:15) (cid:15) · · · d q − / / C ( W ; K q ) / / d q / / δ (cid:15) (cid:15) · · · C ( W ; S ) (cid:31) (cid:127) / / δ (cid:15) (cid:15) C ( W ; K ) − d / / δ (cid:15) (cid:15) C ( W ; K ) − d / / δ (cid:15) (cid:15) · · · − d q − / / C ( W ; K q ) − d q / / δ (cid:15) (cid:15) · · · ... δ q − (cid:15) (cid:15) ... δ q − (cid:15) (cid:15) ... δ q − (cid:15) (cid:15) ... σ ∈ C q ( W ; S ) (cid:31) (cid:127) / / δ q (cid:15) (cid:15) C q ( W ; K ) ( − q d / / δ q (cid:15) (cid:15) C q ( W ; K ) ( − q d / / δ q (cid:15) (cid:15) · · · ... ... ... . (3.21)15f we let K • = C • ( S ), the canonical resolution of S , in Theorem 3.19, noting that H q ( C • ( S )( W α ...α q )) = H q ( W α ...α q ; S ) we have : Corollary 3.22 (Relative Leray theorem) If H q ( W α ...α q ; S ) = 0 for q ≥ and q ≥ , there is a canonical isomorphism H q ( W , W ′ ; S ) ≃ H q ( X, X ′ ; S ) . The following proposition, which shows the functoriality of various cohomologies ap-peared in the above, is not difficult to see :
Proposition 3.23
Let f : ( K • , d K ) → ( L • , d L ) be a morphism of complexes of sheaveson X and denote by S and T the kernels of d K : K → K and d L : L → L ,respectively. Then the morphism f induces morphisms H qd K ( X, X ′ ) −→ H qd L ( X, X ′ ) , H q ( W , W ′ ; S ) −→ H q ( W , W ′ ; T ) and H q ( W , W ′ ; K • ) −→ H q ( W , W ′ ; L • ) that are compatible with (3.8) and (3.15) . Remark 3.24
We may use only “alternating cochains” in the above construction andthe resulting cohomology is canonically isomorphic with the one defined above, as in theusual ˇCech theory.In the sequel, we denote H q ( W , W ′ ; K • ) also by H qD K ( W , W ′ ) and H q ( W ; K • ) by H qD K ( W ). Some special cases : I.
In the case W = { X } , we have ( K • ( W ) , D K ) = ( K • ( X ) , d K )and H qD K ( W ) = H qd K ( X ) . II.
In the case W consists of two open sets W and W , we may write (cf. Remark 3.24) K q ( W ) = C ( W , K q ) ⊕ C ( W , K q − ) = K q ( W ) ⊕ K q ( W ) ⊕ K q − ( W ) . Thus a cochain ξ ∈ K q ( W ) is expressed as a triple ξ = ( ξ , ξ , ξ ) and the differential D : K q ( W ) → K q +1 ( W ) is given by D ( ξ , ξ , ξ ) = ( dξ , dξ , ξ − ξ − dξ )(cf. (3.5)). If we set W ′ = { W } , K q ( W , W ′ ) = { ξ ∈ K q ( W ) | ξ = 0 } = K q ( W ) ⊕ K q − ( W ) . Thus a cochain ξ ∈ K q ( W , W ′ ) is expressed as a pair ξ = ( ξ , ξ ) and the differential D : K q ( W , W ′ ) → K q +1 ( W , W ′ ) is given by D ( ξ , ξ ) = ( dξ , ξ − dξ ) . The q -th cohomology of ( K • ( W , W ′ ) , D ) is H qD K ( W , W ′ ).16f we set W ′′ = ∅ , then H q − D K ( W ′ , W ′′ ) = H q − D K ( W ′ ) = H q − d K ( W ) and the connectingmorphism δ in (3.7) assigns to the class of a ( q − ξ on W the class of (0 , − ξ )(restricted to W ) in H qD K ( W , W ′ ).We discuss this case more in detail in the subsequent sections. III.
Suppose W consists of three open sets W , W and W and set W ′ = { W , W } and W ′′ = { W } . Then K q ( W ) = L i =0 K q ( W i ) ⊕ L ≤ i It is possible to establish an isomorphism as in Theorem 3.19 with-out introducing the ˇCech cohomology of sheaf complexes, using the so-called Weil lemmainstead (cf. [12, Lemma 5.2.7]). The latter amounts to performing the “ladder diagramchasing” in (3.21) with all the horizontal differentials with positive sign to find a corre-spondence. However this correspondence is different from the one in Theorem 3.19, thedifference being the sign of ( − q ( q +1)2 .Incidentally, if we perform the ladder diagram chasing in (3.21) with the sign of d as itis, we get another correspondence. However, this correspondence is again different formthe one in Theorem 3.19, the difference being this time the sign of ( − q . We could as well consider the complex ( K • ( W , W ′ ) , D ′ ) with K q ( W , W ′ ) = M q + q = q C q ( W , W ′ ; K q ) , D ′ = ( − q ˇ δ + d. The resulting cohomology is canonically isomorphic with H qD K ( W , W ′ ). We also havean isomorphism as in Theorem 3.19 and the correspondence between H qd K ( X, X ′ ) and H q ( W , W ′ ; S ) remains the same. Similar remarks as above apply to the isomorphism of Theorem 2.9, with K • ( W , W ′ )replaced by C ( K ) • ( X, X ′ ). Let X be a topological space and X ′ an open set in X . For a sheaf S on X , we setˇ H q ( X, X ′ ; S ) = lim −→ ( W , W ′ ) H q ( W , W ′ ; S ) , the direct limit in the set of pairs of coverings ( W , W ′ ) of ( X, X ′ ) directed by the relation ofrefinement. Let 0 → S → C • ( S ) be the canonical resolution. Then by Proposition 3.13,there is an isomorphism H q ( X, X ′ ; S ) ∼ → H q ( W , W ′ ; C • ( S )). On the other hand there isa morphism H q ( W , W ′ ; S ) → H q ( W , W ′ ; C • ( S )) (cf. (3.15)). Thus we have canonicalmorphisms H q ( W , W ′ ; S ) −→ H q ( X, X ′ ; S ) and ˇ H q ( X, X ′ ; S ) −→ H q ( X, X ′ ; S ) . roposition 3.26 Suppose X and X ′ are paracompact. Then the second morphism aboveis an isomorphism. Proof: Recall that it is true in the absolute case so that ˇ H q ( X ; S ) ≃ H q ( X ; S ) andˇ H q ( X ′ ; S ) ≃ H q ( X ′ ; S ). On the other hand the cohomology ˇ H q ( X, X ′ ; S ) also has theproperty (3) in Proposition 2.1. Thus by the five lemma, the above is an isomorphism. ✷ In this subsection we let X be a paracompact topological space and consider only locallyfinite coverings. Fine sheaves : A sheaf G on X is fine if the sheaf H om ( G , G ) is soft. A fine sheaf issoft. If R is a soft sheaf of rings with unity, every R -module is fine. Thus R itself is fine.A sheaf G is fine if and only if it is an R -module, where R is a sheaf of rings with unitysuch that, for any covering W = { W α } of X , there exists a partition of unity subordinateto W , i.e., a collection { ρ α } , ρ α ∈ R ( X ), such that supp ρ α ⊂ W α and P α ρ α ≡ 1. Wemay use this to show that for a fine sheaf G and any covering W , H q ( W ; G ) = 0 for q ≥ σ is in C q ( W ; G ) and ˇ δσ = 0, then σ = ˇ δτ , where τ ∈ C q − ( W ; G ) is definedby τ α ...α q − = X α ρ α σ αα ...α q − . (3.27) Canonical isomorphisms : We introduce the following : Definition 3.28 Let K • be a complex of sheaves on X . A covering W = { W α } of X is good for K • if the hypothesis of Proposition 3.16 holds, i.e., H q ( K • ( W α ...α q )) = 0 for q ≥ q ≥ Theorem 3.29 Let K • be a complex of fine sheaves on a paracompact space X and S the kernel of d K : K → K . For any covering W , there is a canonical isomorphism H qd K ( X ) ∼ −→ H q ( W ; K • ) . If W is good for K • , there is a canonical isomorphism H q ( W , W ′ ; K • ) ∼ ←− H q ( W , W ′ ; S ) . Suppose every open set in X is paracompact. If W is good for K • and if → S → K • is a resolution, H q ( W , W ′ ; S ) ≃ H q ( X, X ′ ; S ) . Proof: This follows from Proposition 3.13 with W ′ = ∅ . This is the contentof Proposition 3.16. By Theorem 2.22, H q ( W α ...α q ; S ) ≃ H q ( K • ( W α ...α q )). Thusthe isomorphism follows from Corollary 3.22. ✷ We now come back to the case II in Subsection 3.2.18 ase of coverings with two open sets : In the case W = { W , W } , we have K q ( W ) = K q ( W ) ⊕ K q ( W ) ⊕ K q − ( W )and the inclusion K q ( X ) ֒ → C ( W ; K q ) ⊂ K q ( W ) is given by s ( s | W , s | W , H qd K ( X ) ∼ → H qD K ( W ). Proposition 3.30 The inverse of the above isomorphism is given by assigning to theclass of ξ = ( ξ , ξ , ξ ) the class of s given by ξ + d ( ρ ξ ) on W and by ξ − d ( ρ ξ ) on W . Proof: Given a cocycle ξ = ( ξ , ξ , ξ ) in K q ( W ). We have ˇ δξ (1) = 0 and thus ξ (1) = ˇ δτ , τ α = P β ρ β ξ (1) βα (cf. (3.27)). In particular, τ = − ρ ξ (1)01 and τ = ρ ξ (1)01 . Thenletting s = ω and η (0) = τ in (3.14), we have the proposition. ✷ Remark 3.31 1. The two expressions above coincide on W by the cocycle condition. In the case W = X , we may set ρ ≡ ρ ≡ ξ = ( ξ , ξ , ξ ) the class of ξ , which isconsistent with Proposition 3.9.If we set Z q ( W ) = Ker D q and B q ( W ) = Im D q − , then H qD K ( W ) = Z q ( W ) /B q ( W )by definition. In fact we may somewhat simplify the coboundary group B q ( W ) : Proposition 3.32 We have B q ( W ) = { ξ ∈ K q ( W ) | ξ = ( dη , dη , η − η ) , for some η i ∈ K q − ( W i ) , i = 0 , } . Proof: It suffices to show that the left hand side is in the right hand side. For ξ ∈ B q ( W ), there exists η = ( η , η , η ) such that ξ = Dη . Take a partition of unity { ρ , ρ } subordinate to W and set η ′ = η + d ( ρ η ) , η ′ = η − d ( ρ η ) . Then we see that ξ = ( dη ′ , dη ′ , η ′ − η ′ ). ✷ Let X be a topological space and X ′ an open set in X . Also let K • be a complex ofsheaves on X . Letting V = X ′ and V a neighborhood of the closed set S = X r X ′ ,consider the coverings V = { V , V } and V ′ = { V } of X and X ′ (cf. the case II inSubsection 3.2). We have the cohomology H qD K ( V , V ′ ) as the cohomology of the complex( K • ( V , V ′ ) , D K ), where K q ( V , V ′ ) = K q ( V ) ⊕ K q − ( V ) , V = V ∩ V , (4.1)19nd D : K q ( V , V ′ ) → K q +1 ( V , V ′ ) is given by D ( ξ , ξ ) = ( dξ , ξ − dξ ). Noting that K q ( { V } ) = K q ( X ′ ), we have the exact sequence0 −→ K • ( V , V ′ ) j − −→ K • ( V ) i − −→ K • ( X ′ ) −→ , (4.2)where j − ( ξ , ξ ) = (0 , ξ , ξ ) and i − ( ξ , ξ , ξ ) = ξ . This gives rise to the exactsequence (cf. (3.7)) · · · −→ H q − d K ( X ′ ) δ −→ H qD K ( V , V ′ ) j − −→ H qD K ( V ) i − −→ H qd K ( X ′ ) −→ · · · , (4.3)where δ assigns to the class of θ the class of (0 , − θ ).Now we consider the special case where V = X . Thus, letting V = X ′ and V ⋆ = X ,we consider the coverings V ⋆ = { V , V ⋆ } and V ′ = { V } of X and X ′ . Definition 4.4 We denote H qD K ( V ⋆ , V ′ ) by H qD K ( X, X ′ ) and call it the relative cohomol-ogy for the sections of K • on ( X, X ′ ).In the case X ′ = ∅ , it coincides with H qd K ( X ). If we denote by i : X ′ ֒ → X theinclusion, by construction we see that (cf. Subsection 2.3) : K • ( V ⋆ , V ′ ) = K • ( i ) and H qD K ( X, X ′ ) = H qd K ( i ) . (4.5)By Proposition 3.9, there is a canonical isomorphism H qd K ( X ) ∼ → H qD K ( V ⋆ ), whichassigns to the class of s the class of ( s | X ′ , s, 0) . Its inverse assigns to the class of ( ξ , ξ , ξ )the class of ξ . Thus from (4.3) we have the exact sequence · · · −→ H q − d K ( X ′ ) δ −→ H qD K ( X, X ′ ) j − −→ H qd K ( X ) i − −→ H qd K ( X ′ ) −→ · · · , (4.6)where j − assigns to the class of ( ξ , ξ ) the class of ξ and i − assigns to the class of s the class of s | X ′ . It coincides with the sequence (2.17), except δ = − β ∗ . Proposition 4.7 For a triple ( X, X ′ , X ′′ ) , there is an exact sequence · · · −→ H q − D K ( X ′ , X ′′ ) δ −→ H qD K ( X, X ′ ) j − −→ H qD K ( X, X ′′ ) i − −→ H qD K ( X ′ , X ′′ ) −→ · · · . Proof: We show that the above sequence is obtained by setting W = { X ′′ , X ′ , X } , W ′ = { X ′′ , X ′ } and W ′′ = { X ′′ } in (3.7).First we have H q ( W ′ , W ′′ ; K • ) = H qD K ( X ′ , X ′′ ) by definition. Second, applying (3.7)to the triple ( W , W ′ , ∅ ), we have the exact sequence · · · −→ H q − ( W ′ ; K • ) δ −→ H q ( W , W ′ ; K • ) j − −→ H q ( W ; K • ) i − −→ H q ( W ′ ; K • ) −→ · · · . By Proposition 3.9, H q ( W ; K • ) ∼ ← H qd K ( X ) and H q ( W ′ ; K • ) ∼ ← H qd K ( X ′ ). If we set V = { X ′ , X } and V ′ = { X ′ } , the restriction induces a morphism of complexes K • ( V , V ′ ) → K • ( W , W ′ ), which in turn induces a morphism H qD K ( X, X ′ ) → H q ( W , W ′ ; K • ). Com-paring the above sequence with (4.6) and using the five lemma, we see that H qD K ( X, X ′ ) ∼ → H q ( W , W ′ ; K • ). 20hird, applying (3.7) to the triple ( W , W ′′ , ∅ ), we have the exact sequence · · · −→ H q − ( W ′′ ; K • ) δ −→ H q ( W , W ′′ ; K • ) j − −→ H q ( W ; K • ) i − −→ H q ( W ′′ ; K • ) −→ · · · . By Proposition 3.9, we have isomorphisms H q ( W ; K • ) ∼ ← H qd K ( X ) and H q ( W ′′ ; K • ) ∼ ← H qd K ( X ′′ ). If we set V = { X ′′ , X } and V ′′ = { X ′′ } , the restriction induces a morphism ofcomplexes K • ( V , V ′′ ) → K • ( W , W ′′ ), which in turn induces a morphism H qD K ( X, X ′′ ) → H q ( W , W ′′ ; K • ). Comparing the above sequence with (4.6) and using the five lemma, wesee that H qD K ( X, X ′′ ) ∼ → H q ( W , W ′′ ; K • ). ✷ We note that by (4.5), we may rephrase Theorem 2.23 as : Theorem 4.8 Let ( X, X ′ ) be a paracompact pair and S a sheaf on X . Then, for any softresolution → S → K • such that each K q | X ′ is soft, there is a canonical isomorphism : H qD K ( X, X ′ ) ≃ H q ( X, X ′ ; S ) . Complexes of fine sheaves : In the rest of this section, we assume that X is para-compact and that K • is a complex of fine sheaves on X .Let V = { V , V } be as in the beginning of this section, with V an arbitrary openset containing X r X ′ . By Theorem 3.29. 1, there is a canonical isomorphism H qD K ( V ) ≃ H qd K ( X ) and in (4.3), j − assigns to the class of ( ξ , ξ ) the class of (0 , ξ , ξ ) or the classof ξ − d ( ρ ξ ) (or the class of ξ if V = X ) (cf. Proposition 3.30 and Remark 3.31. 2). Proposition 4.9 The restriction K • ( V ⋆ , V ′ ) → K • ( V , V ′ ) induces an isomorphism H qD K ( X, X ′ ) ∼ −→ H qD K ( V , V ′ ) . Proof: Comparing (4.3) and (4.6), we have the proposition by the five lemma . ✷ Corollary 4.10 The cohomology H qD K ( V , V ′ ) is uniquely determined modulo canonicalisomorphisms, independently of the choice of V . Remark 4.11 This freedom of choice of V is one of the advantages of expressing H qd K ( i )as H qD K ( X, X ′ ). Proposition 4.12 (Excision) Let S be a closed set in X . Then, for any open set V in X containing S , there is a canonical isomorphism H qD K ( X, X r S ) ∼ −→ H qD K ( V, V r S ) . Proof: We denote by V the covering of X consisting of V = X r S and V = V and by V the covering of V consisting of V r S and V . Then we may identify K q ( V , { V } ) and K q ( V , { V r S } ). Thus we have H qD K ( V , { V } ) = H qD K ( V , { V r S } ) ≃ H qD K ( V, V r S ). ✷ Now we give an alternative proof of Theorem 2.23 for fine resolutions. Let W = { W α } α ∈ I be a covering of X and W ′ = { W α } α ∈ I ′ a covering of X ′ , I ′ ⊂ I . Letting V ⋆ = X as before, we define a morphism ϕ : K q ( V ⋆ , V ′ ) −→ C ( W , W ′ ; K q ) ⊕ C ( W , W ′ ; K q − ) ⊂ K q ( W , W ′ )21y setting, for ξ = ( ξ , ξ ), ϕ ( ξ ) α = ( α ∈ I ′ ξ | W α α ∈ I r I ′ , ϕ ( ξ ) αβ = ξ | W αβ α ∈ I ′ , β ∈ I r I ′ − ξ | W αβ α ∈ I r I ′ , β ∈ I ′ . Theorem 4.13 Let ( X, X ′ ) be a paracompact pair and K • a complex of fine sheaves on X such that each K q | X ′ is fine. Then the above morphism ϕ induces an isomorphism H qD K ( X, X ′ ) ∼ −→ H q ( W , W ′ ; K • ) . Proof: We define a morphism ψ : K q ( V ⋆ ) −→ C ( W ; K q ) ⊕ C ( W ; K q − ) ⊂ K q ( W )by setting, for ξ = ( ξ , ξ , ξ ), ψ ( ξ ) α = ( ξ | W α α ∈ I ′ ξ | W α α ∈ I r I ′ , and defining ψ ( ξ ) αβ similarly as for ϕ ( ξ ) αβ . We also define χ : K q ( V ) → K q ( W ′ ) bysetting χ ( ξ ) α = ξ | W α for α ∈ I ′ . Then we have the following commutative diagram withexact rows : 0 / / K q ( V ⋆ , V ′ ) / / ϕ (cid:15) (cid:15) K q ( V ⋆ ) ψ (cid:15) (cid:15) / / K q ( V ) / / χ (cid:15) (cid:15) / / K q ( W , W ′ ) / / K q ( W ) / / K q ( W ′ ) / / . It is not difficult to see that each of the vertical morphisms is compatible with the differ-entials so that we have morphisms, which we denote by the same letters ϕ : H qD K ( V ⋆ , V ′ ) −→ H q ( W , W ′ ; K • ) ,ψ : H qD K ( V ⋆ ) −→ H q ( W ; K • ) , χ : H qd K ( V ) −→ H q ( W ′ ; K • ) . By Theorem 3.29. 1, χ is an isomorphism. We also see that ψ is an isomorphism byconsidering the commutative triangle K q ( X ) / / (cid:15) (cid:15) K q ( V ⋆ ) ψ x x ♣♣♣♣♣♣♣♣ K q ( W )and using Theorem 3.29. 1. Then the theorem follows from (3.7) with W ′′ = ∅ , (4.6) andthe five lemma. ✷ Using the above we have an alternative proof of the relative de Rham type theoremfor fine resolutions (cf. Theorems 2.23 and 4.8) : Theorem 4.14 Let ( X, X ′ ) be a paracompact pair and S a sheaf on X . Then, forany fine resolution → S → K • such that each K q | X ′ is fine, there is a canonicalisomorphism : H qD K ( X, X ′ ) ≃ H q ( X, X ′ ; S ) . roof: Let ( W , W ′ ) be a pair of coverings for ( X, X ′ ). There is a canonical morphism H q ( W , W ′ ; S ) → H q ( W , W ′ ; K • ) (cf. (3.15)). By Theorem 4.13 and Proposition 3.26,there are canonical morphisms H q ( W , W ′ ; S ) −→ H qD K ( X, X ′ ) and H q ( X, X ′ ; S ) −→ H qD K ( X, X ′ ) . In the absolute case the second is an isomorphism (Theorem 2.22). Thus by the fivelemma, we have the theorem. ✷ Remark 4.15 In the case K • admits a good covering, which usually happens in thecases we are interested in (cf. Section 7), the theorem follows from Theorems 3.29 and4.13, without referring to Proposition 3.26.The sequence in Proposition 4.7 is compatible with the one in Proposition 2.1 (3) andthe excision of Proposition 4.12 is compatible with that of Proposition 2.1 (4), both viathe isomorphism of Theorem 4.14. Also the isomorphism is functorial in the followingsense : Proposition 4.16 Let ( X, X ′ ) be a paracompact pair. Suppose we have a commutativediagram / / S / / (cid:15) (cid:15) K • (cid:15) (cid:15) / / T / / L • , where each row is a fine resolution as in Theorem 4.14. Then we have the commutativediagram H q ( X, X ′ ; S ) ∼ (cid:15) (cid:15) H qD K ( X, X ′ ) (cid:15) (cid:15) H q ( X, X ′ ; T ) ∼ H qD L ( X, X ′ ) , where the vertical morphism on the right is the last one in Proposition 3.23 for W = V ⋆ . Remark 4.17 The sequence0 −→ K • ( X, X ′ ) −→ K • ( X ) −→ K • ( X ′ ) −→ X ′′ = ∅ and (2.21)). However, replacing K • ( X ) by K • ( V ), we may “flabbify”the situation and obtain an exact sequence as (4.2), which allows us to naturally definethe relative cohomology, as explained in Introduction. For generalities on derived categories and functors we refer to [14].23 .1 Category of complexes We start by a brief review of basics on complexes. Let C be an additive category. Acomplex K in C is a collection ( K q , d qK ) q ∈ Z , where K q is an object in C and d qK : K q → K q +1 a morphism with d q +1 ◦ d q = 0. A morphism ϕ : K → L of complexes is a collection ( ϕ q )of morphisms ϕ q : K q → L q with d qL ◦ ϕ q = ϕ q +1 ◦ d qK . With these the complexes form anadditive category which is denoted by C ( C ). We denote a complex K also by K • and amorphism ϕ by ϕ • .For a complex K and an integer k , we denote by K [ k ] the complex with K [ k ] q = K k + q and d qK [ k ] = ( − k d k + qK . For a morphism ϕ : K → L , ϕ [ k ] : K [ k ] → L [ k ] is defined by ϕ [ k ] q = ϕ k + q . This way we have an additive functor [ k ] : C ( C ) → C ( C ). Considering anobject K in C as a complex given by K = K , K q = 0 for q = 0 and d q = 0, we may thinkof C as a subcategory of C ( C ). Identifying two morphisms in C ( C ) that are “homotopic”,we have an additive category K ( C ).Suppose C is an Abelian category. For a complex K in C , its q -th cohomology isdefined by H q ( K ) = Ker d qK / Im d q − K . Then it gives additive functors H q : C ( C ) → C and H q : K ( C ) → C . Proposition 5.1 Let → J ι → K ϕ → L → be an exact sequence in C ( C ) . Then thereexists an exact sequence · · · −→ H q − ( L ) δ −→ H q ( J ) ι −→ H q ( K ) ϕ −→ H q ( L ) −→ · · · , where ι and ϕ denotes H q ( ι ) and H q ( ϕ ) , respectively, and δ assigns to the class of y ∈ L q − , d ( y ) = 0 , the class of z ∈ J q such that ι ( z ) = d ( x ) for some x ∈ K q − with ϕ ( x ) = y . Let C be an additive category. For a morphism ϕ : K → L of C ( C ), we define a complex M ∗ ( ϕ ) called the co-mapping cone of ϕ . We set M ∗ ( ϕ ) = K ⊕ L [ − d : M ∗ ( ϕ ) q = K q ⊕ L q − → M ∗ ( ϕ ) q +1 = K q +1 ⊕ L q by d ( x, y ) = ( d K x, ϕ q ( x ) − d L y ) . We define morphisms α ∗ = α ∗ ( ϕ ) : M ∗ ( ϕ ) → K and β ∗ = β ∗ ( ϕ ) : L [ − → M ∗ ( ϕ ) by α ∗ : M ∗ ( ϕ ) q = K q ⊕ L q − −→ K q , ( x, y ) x, and β ∗ : L [ − q = L q − −→ M ∗ ( ϕ ) q = K q ⊕ L q − , y (0 , y ) . Then we have a sequence of morphisms L [ − β ∗ −→ M ∗ ( ϕ ) α ∗ −→ K ϕ −→ L. We have α ∗ ◦ β ∗ = 0 in C ( C ). Moreover, we may prove that β ∗ ◦ ϕ [ − 1] and ϕ ◦ α ∗ arehomotopic to 0 so that β ∗ ◦ ϕ [ − ϕ ◦ α ∗ = 0 in K ( C ).24 co-triangle in K ( C ) is a sequence of morphisms L [ − −→ J −→ K −→ L. The co-triangle is distinguished if it is isomorphic to L ′ [ − β ∗ −→ M ∗ ( ϕ ) α ∗ −→ K ′ ϕ −→ L ′ for some ϕ in C ( C ).Let C be an Abelian category and ϕ : K → L as above. Then the sequence0 −→ L [ − β ∗ −→ M ∗ ( ϕ ) α ∗ −→ K −→ C ( C ). From Proposition 5.1, we have the exact sequence · · · −→ H q − ( L ) β ∗ −→ H q ( M ∗ ( ϕ )) α ∗ −→ H q ( K ) ϕ −→ H q ( L ) −→ · · · . (5.3)Note that δ is given by ϕ . Proposition 5.4 For any distinguished cotriangle L [ − → J → K → L in K ( C ) , thereis an exact sequence · · · −→ H q − ( L ) −→ H q ( J ) −→ H q ( K ) −→ H q ( L ) −→ · · · . Proof: It suffices to consider the case L [ − β ∗ → M ∗ ( ϕ ) α ∗ → K ϕ → L . For this, the resultfollows from (5.3). ✷ Remark 5.5 In [13], a similar complex as M ∗ ( ϕ ) is considered, except the differential d : K q ⊕ L q − → K q +1 ⊕ L q is defined by d ( x, y ) = ( dx, dy + ( − q ϕ ( x )). Its cohomologyis denoted by H q ( K ϕ → L ) and is called the generalized relative cohomology.We finish this subsection by examining the relation between the co-mapping conedefined above and the mapping cone as defined in [14]. We will see that the former isdual to the latter in the sense that, while the mapping cone is a notion extracted fromthe complex of singular chains of the mapping cone of a continuous map of topologicalspaces, the co-mapping cone is the one corresponding to the complex of singular cochainsof the topological mapping cone. Thus, while the mapping cone is of homological nature,the co-mapping cone is cohomological. In this context, we may also think of a cotriangleas a notion dual to a triangle.Let C be an additive category and ϕ : K → L a morphism in C ( C ). Recall that themapping cone M ( ϕ ) of ϕ is the complex such that M ( ϕ ) q = K q +1 ⊕ L q (5.6)with the differential d : M ( ϕ ) q = K q +1 ⊕ L q → M ( ϕ ) q +1 = K q +2 ⊕ L q +1 defined by d ( x, y ) = ( − dx, ϕ ( x ) + dy ) . 25e define morphisms α : L → M ( ϕ ) and β : M ( ϕ ) → K [1] in C ( C ) by α : L q −→ M ( ϕ ) q = K q +1 ⊕ L q , y (0 , y ) , and β : M ( ϕ ) q = K q +1 ⊕ L q −→ K [1] q = K q +1 , ( x, y ) x. To illustrate the idea, let A be the category of Abelian groups. For an object A in C ( A ), we set A q = A − q and denote d − q : A q → A q − by ∂ q . Then A [ k ] q = A [ k ] − q = A − q + k = A q − k . Also we denote by A ∗ the complex given by ( A ∗ ) q = Hom Z ( A q , Z ) = ( A q ) ∗ and d q : ( A ∗ ) q → ( A ∗ ) q +1 the transpose of ∂ q : h dϕ, a i = h ϕ, ∂a i for ϕ ∈ ( A q ) ∗ and a ∈ A q +1 , where h , i denotes the Kronecker product.Let ϕ : B → A be a morphism in C ( A ). The mapping cone M ( ϕ ) is given, setting K = B and L = A and reversing the order in the direct sum in (5.6), by M ( ϕ ) q = A q ⊕ B q − , with ∂ q : M ( ϕ ) q = A q ⊕ B q − → M ( ϕ ) q − = A q − ⊕ B q − given by ∂ ( a, b ) = ( ∂a + ϕ ( b ) , − ∂b ) . Let ϕ ∗ : A ∗ → B ∗ be the transpose of ϕ . Then the co-mapping cone M ∗ ( ϕ ∗ ) is given by M ∗ ( ϕ ∗ ) q = ( A q ) ∗ ⊕ ( B q − ) ∗ . with d : M ∗ ( ϕ ∗ ) q = ( A q ) ∗ ⊕ ( B q − ) ∗ → M ∗ ( ϕ ∗ ) q +1 = ( A q +1 ) ∗ ⊕ ( B q ) ∗ given by d ( f, g ) = ( df, ϕ ∗ ( f ) − dg ) . Proposition 5.7 The differential d : M ∗ ( ϕ ∗ ) q → M ∗ ( ϕ ∗ ) q +1 is the transpose of the dif-ferential ∂ : M ( ϕ ) q → M ( ϕ ) q − . Proof: Take ( a, b ) ∈ M ( ϕ ) q +1 = A q +1 ⊕ B q . We have on the one hand h d ( f, g ) , ( a, b ) i = h ( df, ϕ ∗ ( f ) − dg ) , ( a, b ) i = f ( ∂a + ϕ ( b )) − g ( ∂b ) . On the other hand h ( f, g ) , ∂ ( a, b ) i = h ( f, g ) , ( ∂a + ϕ ( b ) , − ∂b ) i = f ( ∂a + ϕ ( b )) − g ( ∂b ) . ✷ The morphisms α : A → M ( ϕ ) and β : M ( ϕ ) → B [1] are given by α : A q −→ M ( ϕ ) q = A q ⊕ B q − , a ( a, β : M ( ϕ ) q = A q ⊕ B q − −→ B [1] q = B q − , ( a, b ) b. While the morphisms α ∗ : M ∗ ( ϕ ∗ ) → A ∗ and β ∗ : B ∗ [ − → M ∗ ( ϕ ∗ ) are given by α ∗ : M ∗ ( ϕ ∗ ) q = ( A ∗ ) q ⊕ ( B ∗ ) q − −→ ( A ∗ ) q , ( f, g ) fβ ∗ : B ∗ [ − q = ( B ∗ ) q − −→ M ∗ ( ϕ ∗ ) q = ( A ∗ ) q ⊕ ( B ∗ ) q − , g (0 , g ) . By direct computations as in the proof of Proposition 5.7, we have : Proposition 5.8 The morphisms α ∗ and β ∗ are the transposes of α and β , respectively. .3 Derived categories and derived functors Let C be an Abelian category. A morphism ϕ : K → L in K ( C ) is a quasi-isomorphism , qis for short, if the induced morphisms H q ( K ) → H q ( L ) are isomorphisms for all q .The derived category D ( C ) is the category obtained from K ( C ) by regarding a qis as anisomorphism. We have the functors[ k ] : D ( C ) −→ D ( C ) and H q : D ( C ) −→ C . The following is a dual version of [14, Proposition 1.7.5] and is proved as Proposi-tion 2.18 : Proposition 5.9 Let −→ J ι −→ K ϕ −→ L −→ be an exact sequence in C ( C ) . Let M ∗ ( ϕ ) be the co-mapping cone of ϕ and let ρ q : J q −→ M ∗ ( ϕ ) q = K q ⊕ L q − be defined by z ( ι ( z ) , . Then the following diagram is commutative and ρ is a qis :0 / / J ι / / ρ ≀ (cid:15) (cid:15) K ϕ / / L / / L [ − β ∗ ( ϕ ) / / M ∗ ( ϕ ) . α ∗ ( ϕ ) ; ; ✇✇✇✇✇✇✇✇✇ In the above situation, the distinguished cotriangle L [ − h −→ J −→ K −→ L is called the distinguished cotriangle associated with (5.10), where h = β ∗ ( ϕ ) ◦ ρ − . Theabove distinguished cotriangle gives rise to a long exact sequence (cf. Proposition 5.4) · · · −→ H q − ( L ) h −→ H q ( J ) −→ H q ( K ) −→ H q ( L ) −→ · · · . Note that h = − δ , where δ is the connecting morphism in Proposition 5.1. This signdifference occurs also in the case of mapping cones (cf. [14, p.46]). Proposition 5.11 Suppose we have a commutative diagram of complexes in C ( C ) : K ϕ / / κ (cid:15) (cid:15) L λ (cid:15) (cid:15) K ′ ϕ ′ / / L ′ . Let M ∗ ( ϕ ) and M ∗ ( ϕ ′ ) be co-mapping cones of ϕ and ϕ ′ , respectively. Then the collection µ = ( µ q ) : M ∗ ( ϕ ) → M ∗ ( ϕ ′ ) of morphisms µ q : M ∗ ( ϕ ) q → M ∗ ( ϕ ′ ) q given by ( x, y ) ( κ q ( x ) , λ q − ( y )) is a morphism of complexes. Moreover, if κ and λ are qis’s, so is µ . Proof: The first part is straightforward. For the second part, compare the exactsequences (5.3) for ϕ and ϕ ′ and apply the five lemma. ✷ erived functors : For an Abelian category C , we donote by D + ( C ) the full subcategoryof D ( C ) consisting of complexes bounded below.Let F : C → C ′ be a left exact functor of Abelian categories. If there exists an“ F -injective” subcategory I , we may define the right derived functor R F : D + ( C ) −→ D + ( C ′ ) by R F ( K ) = F ( I ) , K ∼ −→ qis I. We define a functor R q F : C → C ′ as the composition C −→ D + ( C ) R F −→ D + ( C ′ ) H q −→ C ′ , i.e., R q F ( K ) = H q ( R F ( K )) = H q ( F ( I )) . Cohomology of sheaves : For a topological space X , we denote by S h ( X ) the categoryof sheaves of Abelian groups on X . We also denote by A the category of Abelian groups.For an open set X ′ in X , we have the functor Γ ( X, X ′ ; ) : S h ( X ) −→ A defined by Γ ( X, X ′ ; S ) = S ( X, X ′ ). The subcategory of flabby sheaves is injective forthis functor. For S in S h ( X ), R Γ ( X, X ′ ; S ) = Γ ( X, X ′ ; F • ) and R q Γ ( X, X ′ ; S ) = H q ( Γ ( X, X ′ ; F • )) ≃ H q ( X, X ′ ; S ) , where S ∼ −→ qis F • is a flabby resolution. Let K = ( K • , d K ) be a complex of sheaves on a topological space X . Also let X ′ bean open set in X with inclusion i : X ′ ֒ → X . Denoting by i − : K • ( X ) → K • ( X ′ )the pull-back (restriction in this case) of sections, we have the co-mapping cone M ∗ ( i − ).Thus M ∗ ( i − ) q = K q ( X ) ⊕ K q − ( X ′ )and the differential d : M ∗ ( i − ) q → M ∗ ( i − ) q +1 is given by d ( s, t ) = ( ds, i − s − dt ) . Hence the complex M ∗ ( i − ) is identical with K • ( i ) in Subsection 2.3. Moreover, setting V ⋆ = { V , V ⋆ } , V ′ = { V } , V = X ′ and V ⋆ = X , we have the following (cf. Definition 4.4and (4.5)) : Two interpretations of the cohomology H qd K ( i ) : M ∗ ( i − ) = K • ( i ) = K • ( V ⋆ , V ′ ) and H q ( M ∗ ( i − )) = H qd K ( i ) = H qD K ( X, X ′ ) . (5.12)28e have the exact sequence (cf. the proof of Proposition 5.4)0 −→ K • ( X ′ )[ − β ∗ −→ M ∗ ( i − ) α ∗ −→ K • ( X ) −→ , where β ∗ ( t ) = (0 , t ) and α ∗ ( s, t ) = s . From this we have the exact sequence · · · −→ H q − d K ( X ′ ) β ∗ −→ H q ( M ∗ ( i − )) α ∗ −→ H qd K ( X ) i − −→ H qd K ( X ′ ) −→ · · · , (5.13)which is identical with (2.17). Note that the sequences (5.13) and (4.6) are essentiallythe same, except β ∗ = − δ .For a sheaf S on X and an open set X ′ in X , we have the exact sequence0 −→ R Γ ( X, X ′ ; S ) −→ R Γ ( X ; S ) −→ R Γ ( X ′ ; S ) −→ . (5.14)The following are expressions of Theorem 2.20, its proof and Theorem 2.23 in thecontext of this section : Theorem 5.15 Suppose → S → K • is a resolution of S such that H q ( X ; K q ) = 0 and H q ( X ′ ; K q ) = 0 for q ≥ and q ≥ . Then M ∗ ( i − ) ≃ qis R Γ ( X, X ′ ; S ) and H q ( M ∗ ( i − )) ≃ H q ( X, X ′ ; S ) . Proof: By Proposition 2.12, there exist a flabby resolution 0 → S → F • and amorphism κ : K • → F • such that the following diagram is commutative :0 / / S / / K • κ (cid:15) (cid:15) / / S / / F • . Then we have a commutative diagram of complexes : K • ( X ) i − K / / κ (cid:15) (cid:15) K • ( X ′ ) κ ′ (cid:15) (cid:15) F • ( X ) i − F / / F • ( X ′ ) . By Theorem 2.9 with X ′ = ∅ , κ is a qis. Likewise κ ′ is also a qis. Thus by Proposition 5.11, M ∗ ( i − K ) is quasi-isomorphic with M ∗ ( i − F ).On the other hand, the sequence (5.14) is represented by0 −→ F • ( X, X ′ ) −→ F • ( X ) i − F −→ F • ( X ′ ) −→ M ∗ ( i − F ) is quasi-isomorphic with F • ( X, X ′ ). ✷ From Theorem 5.15, we have : Theorem 5.16 Let ( X, X ′ ) be a paracompact pair and S a sheaf on X . If → S → K • is a soft resolution such that each K q | X ′ is soft, then M ∗ ( i − ) ≃ qis R Γ ( X, X ′ ; S ) and H q ( M ∗ ( i − )) ≃ H q ( X, X ′ ; S ) . The cohomology H q ( M ∗ ( i − )) is generalized to the cohomology of sheaf morphisms inthe following section. 29 Cohomology for sheaf morphisms Although the presentation is somewhat different, the contents of this section are essentiallyin [16], except for Theorem 6.16 below.Throughout this section, we let f : Y → X be a continuous map of topological spaces. Direct and inverse images : For a sheaf T on Y , the direct image f ∗ T is the sheafon X defined by the presheaf U T ( f − U ). We have ( f ∗ T )( U ) = T ( f − U ). Thus as afuctor, f ∗ is left exact and exact on the flabby sheaves. If G is a flabby sheaf on Y , f ∗ G isa flabby sheaf on X . Thus if 0 → T → G • is a flabby resolution of T , 0 → f ∗ T → f ∗ G • is a flabby resolution of f ∗ T .For a sheaf S on X , the inverse image f − S is the sheaf on Y defined by the presheaf V lim −→ S ( U ), where U runs through the open sets in X containing f ( V ). For a point y in Y , we have ( f − S ) y = S f ( y ) , which shows that f − is an exact functor. There arecanonical morphisms S −→ f ∗ f − S and f − f ∗ T −→ T . (6.1)Thus giving a morphism S → f ∗ T is equivalent to giving a morphism f − S → T .In the case Y is a subset of X with the induced topology and f : Y ֒ → X is theinclusion, we have f − S = S | Y . Mapping cylinders : Following [16], we define the mapping cylinder Z ( f ) of f asfollows. As a set, Z ( f ) = X ∐ Y (disjoint union). For an open set U in X , we set˜ U = U ∐ f − U . We endow Z ( f ) with the topology whose basis of open sets consists of { ˜ U | U ⊂ X open sets } and { V | V ⊂ Y open sets } . We have the closed embedding µ : X ֒ → Z ( f ) and the open embedding ν : Y ֒ → Z ( f ). The projection p : Z ( f ) → X isdefined as the map that is the identity on X and f on Y . Cohomology of sheaf morphisms : Let S and T be sheaves on X and Y , respec-tively, and η : S → f ∗ T a morphism. We introduce a sheaf Z ∗ ( T η ← S ), which will beabbreviated as Z ∗ ( η ). It is the sheaf on Z ( f ) defined by the presheaf ˜ U S ( U ) and V T ( V ). The presheaf is a sheaf, i.e., Z ∗ ( η )( ˜ U ) = S ( U ) and Z ∗ ( η )( V ) = T ( V ).The restriction Z ∗ ( η )( ˜ U ) = S ( U ) → Z ∗ ( η )( f − U ) = T ( f − U ) is given by η . Definition 6.2 The cohomology of f with coefficients in η is defined by H q ( Y f → X ; T η ← S ) = H q ( Z ( f ) , Z ( f ) r X ; Z ∗ ( T η ← S )) . In the sequel we abbreviate the cohomology as H q ( f ; η ), if there is no fear of confusion.In the case Y = ∅ , we have H q ( f ; η ) = H q ( X ; S ). Proposition 6.3 There is an exact sequence : · · · −→ H q − ( Y ; T ) −→ H q ( f ; η ) −→ H q ( X ; S ) −→ H q ( Y ; T ) −→ · · · . roof: Let 0 → Z ∗ ( η ) → C • ( Z ∗ ( η )) be the canonical resolution of Z ∗ ( η ). Setting Z ( f ) ′ = Z ( f ) r X (in fact it is equal to Y ), we have the exact sequence0 −→ C • ( Z ∗ ( η ))( Z ( f ) , Z ( f ) ′ ) −→ C • ( Z ∗ ( η ))( Z ( f )) −→ C • ( Z ∗ ( η ))( Z ( f ) ′ ) −→ , which gives rise to the exact sequence · · · → H q − ( Z ( f ) ′ ; Z ∗ ( η )) δ → H q ( f ; η ) → H q ( Z ( f ); Z ∗ ( η )) → H q ( Z ( f ) ′ ; Z ∗ ( η )) → · · · . We have C • ( Z ∗ ( η ))( Z ( f )) = p ∗ C • ( Z ∗ ( η ))( X ). Since p ∗ C • ( Z ∗ ( η )) is a flabby reso-lution of p ∗ Z ∗ ( η ) = S , there is a canonical isomorphism H q ( Z ( f ); Z ∗ ( η )) ≃ H q ( X ; S ).Since ν : Y ֒ → Z ( f ) is an open embedding, C • ( Z ∗ ( η ))( Z ( f ) ′ ) = C • ( ν − Z ∗ ( η ))( Y ) = C • ( T )( Y ), thus H q ( Z ( f ) ′ ; Z ∗ ( η )) = H q ( Y ; T ). ✷ We denote by H q ( f ; η ) the sheaf on X defined by the presheaf U H q ( f | f − U ; η ).In the case T = f − S , there is a canonical morphism S → f ∗ T = f ∗ f − S and,when we take this as η , we denote H q ( f ; η ) and H q ( f ; η ) by H q ( f ; S ) and H q ( f ; S ),respectively.In the case f : Y ֒ → X is an open embedding, we set S = X r Y and denote by H qS ( S ) the sheaf defined by the presheaf U H qS ∩ U ( U ; S ). Proposition 6.4 In the case f : Y ֒ → X is an open embedding. T = f − S and η thecanonical morphism, there exist canonical isomorphisms H q ( f ; S ) ≃ H q ( X, Y ; S ) = H qS ( X ; S ) and H q ( f ; S ) ≃ H qS ( S ) . Proof: In this case the projection p is a map ( Z ( f ) , Z ( f ) r X ) → ( X, Y ) of pairs ofspaces and Z ∗ ( η ) = p − S . Thus there is a canonical morphism H q ( X, Y ; S ) −→ H q ( Z ( f ) , Z ( f ) r X ); Z ∗ ( η )) = H q ( f ; S ) . By Proposition 6.3 and the five lemma, we have the proposition. ✷ Remark 6.5 1. The sheaf Z ∗ ( η ) above is defined in [16, Definition 4.2] with a differentnotation and is called the mapping cylinder of η . Also the cohomology in Definition 6.2 isthe same as the one in [16, Definition 5.1], where it is denoted by H q ( X f ← Y, S η → T ). In [18], the sheaf H qS ( S ) is denoted by Dist q ( S, S ) and is called the sheaf of q -distributions of S . It is a priori a sheaf on X , however it is supported on S . The cohomology in Definition 6.2 is isomorphic with the one defined in [13] with thesame notation. Also the sheaf H q ( f ; η ) above is isomorphic with the one denoted by D ist qf ( S η → T ) in [13] (cf. Remark 6.17 below). Co-mapping cylinder of a sheaf complex morphism : Let K and L be complexesof sheaves on X and Y , respectively, and ϕ : K → f ∗ L a morphism. We introduce acomplex of sheaves ( Z ∗ ( L ϕ ← K ) , d ), which will be called the co-mapping cylinder of ϕ and abbreviated as ( Z ∗ ( ϕ ) , d ). It is the complex of sheaves on Z ( f ) defined as follows.We set Z ∗ ( ϕ ) = µ ∗ K ⊕ µ ∗ f ∗ L [ − ⊕ ν ∗ L 31o that ( Z ∗ ( ϕ ) q ( ˜ U ) = K q ( U ) ⊕ ( L q − ⊕ L q )( f − U ) Z ∗ ( ϕ ) q ( V ) = L q ( V ) . Note that the restriction Z ∗ ( ϕ )( ˜ U ) → Z ∗ ( ϕ )( f − U ) = L q ( f − U ) is given by ( k, ℓ ′ , ℓ ) ℓ . We define the differential d = d Z ∗ : Z ∗ ( ϕ ) q → Z ∗ ( ϕ ) q +1 by ( d ( k, ℓ ′ , ℓ ) = ( dk, ϕ q k − dℓ ′ − ℓ, dℓ ) , ( k, ℓ ′ , ℓ ) ∈ K q ( U ) ⊕ ( L q − ⊕ L q )( f − U ) dℓ = dℓ, ℓ ∈ L q ( V ) . For the complex ( Z ∗ ( ϕ ) , d ), we have the cohomology H qd Z ∗ ( Z ( f ) , Z ( f ) r X ) of thecomplex Z ∗ ( ϕ )( Z ( f ) , Z ( f ) r X ) of sections of Z ∗ ( ϕ ) that vanish on Z ( f ) r X (cf. Sub-section 2.2). In the case Y = ∅ , it reduces to H qd K ( X ).This is essentially the construction given in [16, Definition 4.4], where it is done for amorphism of resolutions and is called the mapping cylinder of the morphism. We adopta slightly different sign convention. Co-mapping cone of a sheaf complex morphism : Let K , L and ϕ : K → f ∗ L be as above. Definition 6.6 The co-mapping cone of ϕ is the complex of sheaves ( M ∗ ( ϕ ) , d ) on X given by M ∗ ( ϕ ) = K ⊕ f ∗ L [ − d : M ∗ ( ϕ ) q = K q ⊕ f ∗ L [ − q −→ M ∗ ( ϕ ) q +1 = K q +1 ⊕ f ∗ L [ − q +1 ( k, ℓ ′ ) ( d K k, ϕk − d L ℓ ′ ) , k ∈ K q , ℓ ′ ∈ L [ − q = L q − . For an open set U in X , we have M ∗ ( ϕ )( U ) = K ( U ) ⊕ L [ − f − U ). In particular,we have M ∗ ( ϕ )( X ) = K ( X ) ⊕ L [ − Y ) = M ∗ ( ϕ ), the co-mapping cone of the inducedmorphism ϕ : K ( X ) → L ( Y ). Thus there is an exact sequence (cf. (5.3)) · · · −→ H q − d L ( Y ) β ∗ −→ H q ( M ∗ ( ϕ )) α ∗ −→ H qd K ( X ) ϕ −→ H qd L ( Y ) −→ · · · . (6.7)From the construction, we have (cf. Subsection 2.3 and (5.12) : Proposition 6.8 In the case f : Y ֒ → X is an open embedding, L = f − K and ϕ thecanonical morphism, M ∗ ( ϕ ) = K ( f ) and H q ( M ∗ ( ϕ )) = H qd K ( f ) . Proposition 6.9 The complex Z ∗ ( ϕ )( Z ( f ) , Z ( f ) r X ) is identical with M ∗ ( ϕ ) so that H qd Z ∗ ( Z ( f ) , Z ( f ) r X ) = H q ( M ∗ ( ϕ )) . roof: Noting that Z ( f ) r X = Y , there is an exact sequence0 −→ Z ∗ ( ϕ )( Z ( f ) , Z ( f ) r X ) −→ Z ∗ ( ϕ )( Z ( f )) ν − −→ Z ∗ ( ϕ )( Y ) . We have ( Z ∗ ( ϕ ) q ( Z ( f )) = K q ( X ) ⊕ L q − ( Y ) ⊕ L q ( Y ) Z ∗ ( ϕ ) q ( Y ) = L q ( Y ) . Since ν − ( k, ℓ ′ , ℓ ) = ℓ , we have the proposition. ✷ Remark 6.10 1. In [16, Definition 4.7] the complex in Definition 6.6 is defined for amorphism of resolutions and is called the mapping cone of the morphism. We againadopt a different sign convention. The cohomology H q ( M ∗ ( ϕ )) coincides with the one considered in [2] in the case f : Y → X is a C ∞ map of C ∞ manifolds, K and L are the de Rham complexes on X and Y , respectively, and ϕ : K → f − L is the pull-back by f of differential forms. Generalized relative de Rham type theorem : Suppose we have two resolutions0 → S ı → K and 0 → T → L and also morphisms η : S → f ∗ T and ϕ : K → f ∗ L such that the following diagram is commutative :0 / / S ı / / η (cid:15) (cid:15) K ϕ (cid:15) (cid:15) / / f ∗ T f ∗ / / f ∗ L . In this case we say that ( K , L , ϕ ) is a resolution of ( S , T , η ). We define a morphism ζ : Z ∗ ( η ) → Z ∗ ( ϕ ) by ( Z ∗ ( η )( ˜ U ) = S ( U ) −→ Z ∗ ( ϕ ) ( ˜ U ) = K ( U ) ⊕ L ( f − U ) , s ( ıs, ( f ∗ ) ηs ) Z ∗ ( η )( V ) = T ( V ) −→ Z ∗ ( ϕ ) ( V ) = L ( V ) , t t. Then the following is proved as [16, Theorem 4.5] : Theorem 6.11 If ( K , L , ϕ ) is a resolution of ( S , T , η ) , then → Z ∗ ( η ) ζ → Z ∗ ( ϕ ) isa resolution of Z ∗ ( η ) . Using Proposition 6.9, Theorem 2.9 in our case reads : Theorem 6.12 1. For any resolution ( K , L , ϕ ) of ( S , T , η ) , there is a canonical mor-phism ˜ χ : H q ( M ∗ ( ϕ )) = H qd Z ∗ ( Z ( f ) , Z ( f ) r X ) −→ H q ( Z ( f ) , Z ( f ) r X ; Z ∗ ( η )) = H q ( f ; η ) . Moreover, if H q ( Z ( f ) , Z ( f ) r X ; Z ∗ ( ϕ ) q ) = 0 , for q ≥ and q ≥ , then ˜ χ is anisomorphism. In particular, if K and L are flabby resolutions, then Z ∗ ( ϕ ) is a flabby resolution.Thus we have : 33 orollary 6.13 For a resolution ( K , L , ϕ ) of ( S , T , η ) such that K and L are flabbyresolutions, there is a canonical isomorphism :˜ χ : H q ( M ∗ ( ϕ )) ∼ −→ H q ( f ; η ) . More generally we have the following theorem. Although it is proved in [16, Theo-rem 5.5], we give a proof in our context. Theorem 6.14 Suppose ( K , L , ϕ ) is a resolution of ( S , T , η ) such that H q ( X ; K q ) =0 and H q ( Y ; L q ) = 0 for q ≥ and q ≥ . Then there is a canonical isomorphism : H q ( M ∗ ( ϕ )) ≃ H q ( f ; η ) . Proof: We have the diagram · · · / / H q − d L ( Y ) β ∗ / / χ ≀ (cid:15) (cid:15) H q ( M ∗ ( ϕ )) ˜ χ (cid:15) (cid:15) α ∗ / / H qd K ( X ) ϕ / / χ ≀ (cid:15) (cid:15) H qd L ( Y ) / / χ ≀ (cid:15) (cid:15) · · ·· · · / / H q − ( Y ; T ) δ / / H q ( f ; η ) / / H q ( X ; S ) / / H q ( Y ; T ) / / · · · , where the rows are exact (cf. (6.7) and Proposition 6.3). The rectangles are commutativeexcept for the left one, which is anti-commutative. By assumption the χ ’s are isomor-phisms. Thus by the five lemma, ˜ χ is an isomorphism, which together with Proposition6.9 implies the theorem. ✷ We may express the conclusion above as M ∗ ( ϕ ) ≃ qis R Γ ( Z ( f ) , Z ( f ) r X ; Z ∗ ( η )) . (6.15)In the case f : Y → X is an open embedding, T = f − S , L = f − K , η and ϕ are canonical morphisms, by Propositions 6.4 and 6.8, the above theorem reduces toTheorem 2.20 and (6.15) is the one in Theorem 5.15.From Theorem 6.14, we have the following, which generalizes Theorem 2.23 : Theorem 6.16 (Generalized relative de Rham type theorem) Suppose X and Y are paracompact. Then, for any resolution ( K , L , ϕ ) of ( S , T , η ) such that K and L are soft resolutions, there is a canonical isomorphism : H q ( M ∗ ( ϕ )) ≃ H q ( f ; η ) . Remark 6.17 In [13] it is shown that, given a triple ( S , T , η ), there exists a resolution( K , L , ϕ ) such that K and L are flabby. Then this is used to define the cohomology H q ( f ; η ) as H q ( M ∗ ( ϕ )) (cf. Corollary 6.13, also Remarks 5.5 and 6.5. 3). As noted in[16], one of the advantages of defining H q ( f ; η ) as in Definition 6.2 is that we can bypassthe proof of the fact that the definition does not depend on the choice of the resolution( K , L , ϕ ) such that K and L are flabby.34 Some particular cases The manifolds we consider below are assumed to have a countable basis, thus they areparacompact and have only countably many connected components. The coverings areassumed to be locally finite. I. de Rham complex Let X be a C ∞ manifold of dimension m and E ( q ) X the sheaf of C ∞ q -forms on X .The sheaves E ( q ) X are fine and, by the Poincar´e lemma, they give a fine resolution of theconstant sheaf C X :0 −→ C −→ E (0) d −→ E (1) d −→ · · · d −→ E ( m ) −→ , where we omitted the suffix X .The de Rham cohomology H qd ( X ) of X is the cohomology of ( E ( • ) ( X ) , d ). By Theo-rem 2.22, there is a canonical isomorphism (de Rham theorem) : H qd ( X ) ≃ H q ( X ; C X ) . (7.1)Let X ′ be an open set in X and ( W , W ′ ) a pair of coverings for ( X, X ′ ). The ˇCech-de Rham cohomology H qD ( W , W ′ ) on ( W , W ′ ) is the cohomology of ( E ( • ) ( W , W ′ ) , D ) with D = ˇ δ + ( − • d (cf. Definition 3.2).We say that W is good if every non-empty finite intersection W α ...α q is diffeomorphicwith R m . If W is good, then it is good for E ( • ) (cf. Definition 3.28). From Theorem 3.29,we have the following canonical isomorphisms :(1) For any covering W , H qd ( X ) ∼ → H qD ( W ).(2) For a good covering W , H qD ( W , W ′ ) ∼ ←− H q ( W , W ′ ; C X ) ≃ H q ( X, X ′ ; C X ) . The relative de Rham cohomology H qD ( X, X ′ ) is defined as in Section 4 and, fromTheorem 4.14 (see also Theorems 2.23 and 4.8), we have : Theorem 7.2 (Relative de Rham theorem) There is a canonical isomorphism : H qD ( X, X ′ ) ≃ H q ( X, X ′ ; C X ) . Since X always admits a good covering (in fact the good coverings are cofinal in theset of coverings), we have the above theorem without going to the limit in the ˇCechcohomology (cf. Remark 4.15).Note that H q ( X, X ′ ; C X ) is canonically isomorphic with the relative singular (or sim-plicial) cohomology H q ( X, X ′ ; C ) with C -coefficients on finite chains.For more about ˇCech-de Rham cohomology and its applications, we refer to [2], [17],[19], [20] and references therein. II. Dolbeault complex X be a complex manifold of dimension n and E ( p,q ) X the sheaf of C ∞ ( p, q )-formson X . The sheaves E ( p,q ) X are fine and, by the Dolbeault-Grothendieck lemma, they givea fine resolution of the sheaf O ( p ) X of holomorphic p -forms :0 −→ O ( p ) −→ E ( p, 0) ¯ ∂ −→ E ( p, 1) ¯ ∂ −→ · · · ¯ ∂ −→ E ( p,n ) −→ . The Dolbeault cohomology H p,q ¯ ∂ ( X ) of X is the cohomology of the complex ( E ( p, • ) ( X ) , ¯ ∂ ).By Theorem 2.22, there is a canonical isomorphism (Dolbeault theorem) : H p,q ¯ ∂ ( X ) ≃ H q ( X ; O ( p ) ) . (7.3)Let ( W , W ′ ) be as above. The ˇCech-Dolbeault cohomology H p,q ¯ ϑ ( W , W ′ ) on ( W , W ′ )is the cohomology of ( E ( p, • ) ( W , W ′ ) , ¯ ϑ ) with ¯ ϑ = ˇ δ + ( − • ¯ ∂ (cf. Definition 3.2).We say that W is Stein if every non-empty finite intersection W α ...α q is biholomorphicwith a domain of holomorphy in C n (cf. [7]). If W is Stein, then it is good for E ( p, • ) .From Theorem 3.29, we have the following canonical isomorphisms :(1) For any covering W , H p,q ¯ ∂ ( X ) ∼ → H p,q ¯ ϑ ( W ).(2) For a Stein covering W , H p,q ¯ ϑ ( W , W ′ ) ∼ ←− H q ( W , W ′ ; O ( p ) ) ≃ H q ( X, X ′ ; O ( p ) ) . The relative Dolbeault cohomology H p,q ¯ ϑ ( X, X ′ ) is defined as in Section 4 and, fromTheorem 4.14 (see also Theorems 2.23 and 4.8), we have : Theorem 7.4 (Relative Dolbeault theorem) There is a canonical isomorphism : H p,q ¯ ϑ ( X, X ′ ) ≃ H q ( X, X ′ ; O ( p ) ) . Since X always admits a Stein covering (in fact the Stein coverings are cofinal inthe set of coverings), we have the above theorem without going to the limit in the ˇCechcohomology (cf. Remark 4.15).For more about ˇCech-Dolbeault cohomology we refer to [21] and [22]. Applicationsare given in [1] for localization of Atiyah classes and in [11] for the Sato hyperfunctiontheory. Remark 7.5 The seemingly standard proof in the textbooks, e.g., [8], [10], of the iso-morphism as in Theorem 2.22 (thus (7.1) and (7.3)) gives a correspondence same as theone given by the Weil lemma. Thus there is a sign difference as explained in Remark 3.25. III. Mixed complex Let X be a complex manifold. We set E ( p,q )+1 X = E ( p +1 ,q ) X ⊕ E ( p,q +1) X and consider the complex · · · d −→ E ( p − ,q − ∂ + ∂ −→ E ( p − ,q − 1) ¯ ∂∂ −→ E ( p,q ) d −→ E ( p,q )+1 ¯ ∂ + ∂ −→ E ( p +1 ,q +1) ¯ ∂∂ −→ · · · . (7.6)36rom this we have the Bott-Chern, Aeppli and third cohomologies and their relativeversions. For details and applications to the localization problem of Bott-Chern classes,we refer to [5]. IV. Some others Here is another type of complex as considered in [11]. We may discuss this in moregeneral settings, however we consider the following situation for simplicity.Let X be a C ∞ manifold and Ω an open set in X with inclusion j : Ω ֒ → X . Weconsider the sheaf j ! j − C X on X , where j ! denotes the direct image with proper supports(cf. [14, § j ! j − C X | Ω = j − C X = C Ω and j ! j − C X | X r Ω = 0. The complex j ! j − E ( • ) X gives a resolution of j ! j − C X . For each q , the sheaf j − E ( q ) X may be thought ofas the sheaf E ( q )Ω of q -forms on Ω so that it is soft (in fact fine). Thus j ! j − E ( q ) X is a c -softsheaf on the paracompact manifold X and thus it is soft. In fact in our case it is fine, asany of its sections may be thought of as a q -forms on X with support in (the intersection ofits domain of definition and) Ω and thus the sheaf j ! j − E ( q ) X admits a natural action of thesheaf E X of C ∞ functions. If we set d ′ = j ! j − d (it is in fact the usual exterior derivative d on forms with support in Ω) by Theorem 2.22, there is a canonical isomorphism : H qd ′ ( X ) ≃ H q ( X ; j ! j − C X ) . If X ′ is an open set in X , setting D ′ = ˇ δ + ( − • d ′ , from Theorem 4.14 (see alsoTheorems 2.23 and 4.8), we see that there is a canonical isomorphism : H qD ′ ( X, X ′ ) ≃ H q ( X, X ′ ; j ! j − C X ) . Note that each element in H qD ′ ( X, X ′ ) is represented by a pair ( ξ , ξ ), where ξ is aclosed q -form on X (or on any neighborhood V of X r X ′ ) with support in Ω and ξ a( q − X ′ with support in X ′ ∩ Ω such that dξ = ξ on X ′ (or on V ∩ X ′ , cf.Corollary 4.10). References [1] M. Abate, F. Bracci, T. Suwa and F. Tovena, Localization of Atiyah classes , Rev.Mat. Iberoam. (2013), 547-578.[2] R. Bott and L. Tu, Differential Forms in Algebraic Topology , Graduate Texts inMath. , Springer, 1982.[3] J.-P. Brasselet, J. Seade and T. Suwa, Vector Fields on Singular Varieties , LectureNotes in Math. , Springer, 2009.[4] G.E. Bredon, Sheaf Theory , McGraw-Hill, 1967.[5] M. Corrˆea and T. Suwa, Localization of Bott-Chern classes and Hermitian residues ,arXiv:1705.09420.[6] R. Godement, Topologie Alg´ebrique et Th´eorie des Faisceaux , Hermann, Paris, 1958.[7] H. Grauert and R. Remmert, Theory of Stein Spaces , Grundlehren der math. Wiss. , Springer, 1979. 378] P. Griffiths and J. Harris, Principles of Algebraic Geometry , John Wiley & Sons,1978.[9] R. Hartshorne, Local Cohomology, A seminar given by A. Grothendieck, HarvardUniversity, Fall, 1961 , Lecture Notes in Math. , Springer, 1967.[10] F. Hirzebruch, Topological Methods in Algebraic Geometry , Third ed., Grundlehrender math. Wiss. , Springer, 1966.[11] N. Honda, T. Izawa and T. Suwa, Sato hyperfunctions via relative Dolbeault coho-mology , arXiv:1807.01831v2.[12] A. Kaneko, Introduction to Hyperfunctions , New Edition, Univ. of Tokyo Press 1996(in Japanese). English translation: Kluwer Academic Publishers, 1988.[13] M. Kashiwara, T. Kawai and T. Kimura, Foundations of Algebraic Analysis , Ki-nokuniya Shoten 1980 (in Japanese). English translation: Princeton Math. Series , Princeton Univ. Press, 1986.[14] M. Kashiwara and P. Schapira, Sheaves on Manifolds , Grundlehren der math. Wiss. , Springer, 1990.[15] H. Komatsu, Hyperfunctions of Sato and Linear Partial Differential Equations withConstant Coefficients , Seminar Notes , Univ. Tokyo, 1968 (in Japanese).[16] H. Komatsu, Cohomology of morphisms of sheafed spaces , J. Fac. Sci. Univ. Tokyo,Sect. IA (1971), 287-327. The essentials of the paper are in “ Relative cohomologyand its applications ”, Hyperfunctions and Related Topics, RIMS Kˆokyˆuroku ,1-44, 1969 (in Japanese).[17] D. Lehmann, Syst`emes d’alv´eoles et int´egration sur le complexe de ˇCech-de Rham ,Publications de l’IRMA, 23, N o VI, Universit´e de Lille I, 1991.[18] M. Sato, Theory of hyperfunctions I, II , J. Fac. Sci. Univ. Tokyo, (1959), 139-193,387-436.[19] T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations ,Actualit´es Math´ematiques, Hermann, Paris, 1998.[20] T. Suwa, Residue Theoretical Approach to Intersection Theory , Proceedings of the9-th International Workshop on Real and Complex Singularities, S˜ao Carlos, Brazil2006, Contemp. Math. , Amer. Math. Soc., 207-261, 2008.[21] T. Suwa, ˇCech-Dolbeault cohomology and the ¯ ∂ -Thom class , Singularities – Niigata-Toyama 2007, Adv. Studies in Pure Math. , Math. Soc. Japan, 321-340, 2009.[22] T. Suwa, Relative Dolbeault cohomology , in preparation.[23] T. Suwa, Relative cohomology for the sections of a complex of fine sheaves , Proceed-ings of the Kinosaki Algebraic Geometry Symposium 2017, 113-128, 2018.3824] T. Suwa, Relative Dolbeault cohomology and Sato hyperfunctions , to appear in Mi-crolocal Analysis and Asymptotic Analysis, RIMS Kˆokyˆuroku.[25] A. Weil, Sur les th´eor`emes de de Rham , Comment. Math. Helv.26