# zbMATH — the first resource for mathematics

Resolutions of unbounded complexes. (English) Zbl 0636.18006
In the most general situation $$R\Gamma\Phi$$ (U;-) for a sheaf and a family of supports $$\Phi$$ on a topological space X (U open in X), RHom., R$${\mathcal H}$$om. and $$\otimes^ L$$ are defined. Standard formulae such as RHom.($${\mathcal A}.\otimes^ L{\mathcal B}.,{\mathcal C}.)\cong$$RHom.($${\mathcal A}.,$$R$${\mathcal H}$$om$$.({\mathcal B}.,{\mathcal C}.))$$ are proven. For a morphism f of ringed spaces $$Rf_*$$ and $$Lf^*$$ are defined and an adjunction formula shown. A notion of softness for sheaves (c-softness) on a paracompact topological space is introduced. One assumes that, for an acyclic complex of c-soft sheaves on the fibers of morphisms $$f: X\to Y$$ of paracompact ringed spaces, for each degree the kernel of the differential is c-soft. Then adjoint formulas involving $$Rf_ !$$ and $$f^ !$$ are demonstrated.
Let $${\mathcal K}({\mathcal A})$$ denote the category of $${\mathbb{Z}}$$-graded complexes with differentials of degree $$+1$$ but chain maps taken up to homotopy. Then, for an associative ring R with 1 and a category $${\mathcal A}$$ of left R-modules, $${\mathcal K}({\mathcal A})$$ is shown to have K-projective and K-injective resolutions (defined in terms of the acyclicity of Hom.(A.,X.) and Hom.(X.,A.) in X., respectively).
Reviewer: P.Cherenack

##### MSC:
 18G35 Chain complexes (category-theoretic aspects), dg categories 13D25 Complexes (MSC2000) 14F20 Étale and other Grothendieck topologies and (co)homologies 55U15 Chain complexes in algebraic topology
Full Text:
##### References:
 [1] A. Borel et al.: Seminar on Intersection Cohomology , Progress in Math. 50, Birkhäuser (1984). · Zbl 0553.14002 [2] P. Deligne : Cohomologie à supports propres, SGA4 , Lecture Notes in Math. 305, Springer-Verlag (1973). · Zbl 0255.14011 [3] A. Dold : Zur Homotopietheorie der Kettenkomplexe , Math. Ann. 140 (1960) 278-298. · Zbl 0093.36903 [4] R. Godement : Topologie algébrique et théorie des faisceaux , Act. Sci. Ind. 1252, Hermann, Paris (1958). · Zbl 0080.16201 [5] R. Hartshorne : Residues and duality , Lecture Notes in Math. 20, Springer-Verlag (1966). · Zbl 0212.26101 [6] J.L. Verdier : Catégories dérivées, état 0, SGA 41/2 , Lecture Notes in Math. 569, Springer-Verlag (1977). · Zbl 0407.18008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.