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Erschienen in:
2016 | OriginalPaper | Buchkapitel
10. Cohomology of Complexes of Sheaves
verfasst von : Torsten Wedhorn
Erschienen in: Manifolds, Sheaves, and Cohomology
Verlag: Springer Fachmedien Wiesbaden
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Abstract
In this chapter we generalize for sheaves of abelian groups (or more generally, for \({\mathscr{O}}_{X}\)-modules, where \((X,{\mathscr{O}}_{X})\) is a ringed space) cohomology groups in degree 1 to higher degrees in such a way that we can extend the exact sequence of cohomology. The idea (motivated in Sect. 10.1) is to replace an \({\mathscr{O}}_{X}\)-module by a quasi-isomorphic complex of such \({\mathscr{O}}_{X}\)-modules whose cohomology we expect to be trivial (namely injective \({\mathscr{O}}_{X}\)-modules), then to apply the functor of global sections to this complex to obtain a complex of \(\Gamma(X,{\mathscr{O}}_{X})\)-modules, and finally take the cohomology of this complex. Of course, once one follows the strategy of first replacing an \({\mathscr{O}}_{X}\)-module by a complex, it is only natural to start with an arbitrary complex of \({\mathscr{O}}_{X}\)-modules. This will yield the (hyper-)cohomology \(H^{p}(X,{\mathscr{F}}^{\bullet})\) of a complex \({\mathscr{F}}^{\bullet}\) defined in Sect. 10.2. Much more generally, one may view these constructions as a special case of the right derivation of a left exact functor between abelian categories, which is here applied to the functor of taking global sections of an \({\mathscr{O}}_{X}\)-module.