Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions
type: Item
instance of: scholarly article
title: Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions
zbMATH Open document ID: 0942.18004
DOI: 10.1016/S0022-4049(98)00056-5
author: Richard G. Swan
published in: Journal of Pure and Applied Algebra
publication date: 2000-08-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 55N30
Mathematics Subject Classification ID: 18E25
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 55N45
Mathematics Subject Classification ID: 18F10
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 1398842
zbMATH Keywords: cup products
zbMATH Keywords: pure injectives
zbMATH Keywords: hypercovers
zbMATH Keywords: categories of abelian sheaves
zbMATH Keywords: simplicial sheaves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1398842