On Carlson's conjecture for cohomology rings of modules
type: Item
instance of: scholarly article
title: On Carlson's conjecture for cohomology rings of modules
zbMATH Open document ID: 0678.16019
DOI: 10.1016/0022-4049(89)90096-0
author: Takashi Niwasaki
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let K be an algebraically closed field of characteristic \(p>0\) and G be a finite group. For a finitely generated left KG module M denote the cohomology ring \({\mathcal E}_ G(M)=Ext^*_{KG}(M,M)\simeq H^*(G,End_ K(M))\). The purpose of this paper is to give a complete proof to a conjecture that \textit{J. F. Carlson} raised in two papers on the cohomology ring of modules [published in J. Pure Appl. Algebra 36, 105-121 (1985; Zbl 0565.20003) and 44, 85-97 (1987; Zbl 0617.20029)]. Carlson proved in the case when M was induced from a linear KH module for a normal subgroup H of G that every maximal two sided ideal contains the kernel of the restriction to some cyclic shifted subgroup. The principal method the author uses is to regard \({\mathcal E}_ G(M)\) as a finitely generated module over \({\mathcal E}_ G(K)\). As an application it is proved a theorem of Avrunin and Scott which asserts the equality of the module variety and the rank variety.
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 4110880
zbMATH Keywords: Carlson's conjecture
zbMATH Keywords: finite group
zbMATH Keywords: finitely generated left KG module
zbMATH Keywords: cohomology ring
zbMATH Keywords: maximal two sided ideal
zbMATH Keywords: kernel
zbMATH Keywords: restriction
zbMATH Keywords: cyclic shifted subgroup
zbMATH Keywords: module variety
zbMATH Keywords: rank variety
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90096-0
OpenAlex ID: W2081811048
Wikidata QID: Q123198451
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description: scientific article; zbMATH DE number 4110880