Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions
The author introduces two definitions of cup product in \(Ext\) groups arising from a tensor product pairing \(A \otimes B \rightarrow C\) in an abelian category \(\mathcal{A}\) which is cocomplete, has a generator, has exact filtered colimits and is equipped with a symmetric monoidal tensor product. The two definitions coincide on flat objects. Insofar as categories of abelian sheaves are examples of such abelian categories \(\mathcal{A}\), two cup product constructions for tensor product pairings of sheaves result, and they coincide on a satisfying class of objects. The first of these constructions involves the introduction of pure injective objects, and effectively generalizes classical constructions for sheaves, while the second amounts to the development of a theory of hypercovers within the chain complex context. The latter is a replacement for the theory of projective resolutions, and is strongly related to the theory of locally trivial fibrations of simplicial sheaves.
scientific article; zbMATH DE number 1398842