Varieties and cohomology of infinitely generated modules.
type: Item
instance of: scholarly article
title: Varieties and cohomology of infinitely generated modules.
zbMATH Open document ID: 1151.20040
published in: Archiv der Mathematik
publication date: 2008-09-12T00:00:00Z
review text: Let \(G\) be a finite group and let \(H^*(G,k)\) be the cohomology ring of \(G\) with values in a field \(k\). Suppose that the \(p\)-rank of \(G\) is at least \(3\) and that \(k\) is of characteristic \(p\). For any \(kG\)-module \(M\) the ring \(H^*(G,k)\) acts on the Ext-algebra \(\text{Ext}^*_{kG}(M,M)\). Suppose that there is a regular sequence of length \(2\), given by \((x,y)\), say. Using these elements the authors construct an infinite \(kG\)-module \(F(x,y)\) with the following property. The set of primes in \(H^*(G,k)\) containing the annihilator of \(\text{Ext}^*_{kG}(F(x,y),F(x,y))\) in \(H^*(G,k)\) is exactly the set of primes not containing the ideal generated by \(x\) and \(y\). The module \(F(x,y)\) is one of Rickard's idempotent modules. The small support of a module \(N\) is the set of primes occurring in a minimal injective resolution of \(N\). The authors show, heavily using arguments due to Rickard, that the small support of \(F(x,y)\) in \(H^*(G,k)\) consists of all primes of \(H^*(G,k)\). For finitely generated modules the small support coincides with the variety defined as annihilator of the Ext-algebra.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5343883
zbMATH Keywords: cohomology varieties
zbMATH Keywords: support varieties
zbMATH Keywords: idempotent modules
zbMATH Keywords: cohomology rings
author: David Benson
author: Jon F. Carlson
MaRDI profile type: Publication
full work available at URL: s00013-008-2731-8
OpenAlex ID: W2078638163
DOI: 10.1007/S00013-008-2731-8
description: scientific article; zbMATH DE number 5343883