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


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
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