Relative projectivity and ideals in cohomology rings
type: Item
instance of: scholarly article
title: Relative projectivity and ideals in cohomology rings
zbMATH Open document ID: 0870.20032
author: Chuang Peng
author: Jon F. Carlson
published in: Journal of Algebra
publication date: 1997-09-08T00:00:00Z
review text: Let \(G\) be a finite group and \(\kappa\) an algebraically closed field of characteristic \(p>2\). All \(\kappa G\)-modules are assumed to be finitely generated. It is well known that if \(M\), \(N\) are \(\kappa G\)-modules then \(\text{Ext}^*_{\kappa G}(M,N)\) is a finitely generated module over the group cohomology ring \(\text{Ext}_{\kappa G}^*(\kappa,\kappa)=H^*(G,\kappa)\). It follows from \textit{J. Carlson} and \textit{W. Wheeler} [Homomorphisms in higher complexity quotient categories (preprint)] that if \(I\) is an ideal of \(H^*(G,\kappa)\) generated by a set \(S\) of homogeneous elements and \(C_I\) the full subcategory of all modules whose cohomology is annihilated by the ideal \(I\), then the ideal itself can be represented by a single element of \(\text{Ext}_{\kappa G}^1(U,\kappa)\) for a suitably chosen \(U=U(S)\). That is, a module \(M\) is in \(C_I\) if and only if it annihilates the given element of \(\text{Ext}^1_{\kappa G}(U,\kappa)\). In the above mentioned joined work of J. Carlson and W. Wheeler it is also shown that the infinitely generated idempotent modules discovered by \textit{J. Rickard} [in J. Lond. Math. Soc. (to appear)] are colimits of the modules \(U(S)\). In this paper a relative cohomology theory is defined which turns out to be the natural setting for these results. In this context the sequences representing the ideal \(I\) are steps in the relative projective resolution of the trivial module \(\kappa\). This has some interesting consequences one of which is that the module \(U(S)\) has an essential direct summand \(U_I\) which is independent of the choice of generators \(S\) and depends only on the ideal \(I\). The notion of relative projectivity was originally developed by \textit{T. Okuyama} [in ``A generalization of projective covers of modules over finite group algebras'' (unpublished manuscript)]. It is based on the choice of a single module \(V\) which generates the projectivity by tensor products. A module \(M\) is rela…
reviewed by: Olympia Talelli
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 923710
zbMATH Keywords: finitely generated modules
zbMATH Keywords: finite groups
zbMATH Keywords: cohomology rings
zbMATH Keywords: homogeneous elements
zbMATH Keywords: idempotent modules
zbMATH Keywords: relative projective resolutions
zbMATH Keywords: essential direct summands
zbMATH Keywords: relative projectivity
zbMATH Keywords: tensor products
MaRDI profile type: Publication
full work available at URL: jabr.1996.0245
OpenAlex ID: W2014710621
DOI: 10.1006/JABR.1996.0245
description: scientific article; zbMATH DE number 923710