The cohomology ring of a module
type: Item
instance of: scholarly article
title: The cohomology ring of a module
zbMATH Open document ID: 0565.20003
DOI: 10.1016/0022-4049(85)90064-7
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let G be a finite group, K an algebraically closed field of characteristic \(p>0\) and M a finitely generated KG-module. It was shown by the author [in Lect. Notes Math. 903, 62-67 (1981; Zbl 0475.20040)] that an element of \(Ext^{\lambda}_{KG}(M,M)\) is nilpotent iff its restriction to every elementary abelian p-subgroup of G is nilpotent. Now let \(G=<x_ 1,x_ 2,...,x_ n>\) be an elementary abelian p-group, \(\alpha =(K_ 1^{\alpha},K_ 2^{\alpha},...,K_ n^{\alpha})\in K^ n\) with \(\alpha\) \(\neq 0\) and \(u_{\alpha}=1+\sum_{i}K_ i^{\alpha}(x_ i-1)\). Then \(u_{\alpha}\) is a unit of KG of order p. If \(\{\alpha_ 1,\alpha_ 2,...,\alpha_ m\}\) is a set of linearly independent elements of \(K^ n\), then the group \(H=<u_{\alpha_ 1},...,u_{\alpha_ m}>\) is called a shifted subgroup of KG. The embedding KH\(\to KG\) induces a ''restriction'' map on cohomology. Here the author shows that an element of \(Ext^{\alpha}_{KG}(M,M)\) is nilpotent iff its restriction to certain \(<u_{\alpha}>'s\) is nilpotent. He then characterizes the radical of \(Ext^*_{KG}(M,M)\) in terms of restrictions to shifted subgroups of KG. These results are then used to give a new proof of a theorem of \textit{G. S. Avrunin} and \textit{L. L. Scott} [Invent. Math. 66, 277-286 (1982; Zbl 0489.20042)] which states that the cohomology variety W(M) of M is equal to the rank variety V(M) of M. (W(M) is the zero set of an ideal in a polynomial subring of \(Ext^*_{KG}(K,K)\) and \(V(M)=\{0\}\cup \{\alpha \in K^ n|\) \(M_{<u_{\alpha}>}\) is not a free \(K<u_{\alpha}>\)-module\(\}\).) The main theorem was proved by the author for \(p=2\) [in J. Algebra 85, 104- 143 (1983; Zbl 0526.20040)]. Here the proof is different and uses a nontrivial spectral sequence argument. The author adopts the Avrunin- Scott point of view in regarding KG as the restricted enveloping algebra of a commutative Lie algebra. The corresponding coalgebra structure then gives a cup product action of \(Ext^*_{KG}(K,K)\) on \(Ext^*_{KG}(M,…
reviewed by: Olympia Talelli
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 3901008
zbMATH Keywords: finitely generated KG-module
zbMATH Keywords: elementary abelian p-group
zbMATH Keywords: unit
zbMATH Keywords: shifted subgroup
zbMATH Keywords: nilpotent
zbMATH Keywords: radical
zbMATH Keywords: restrictions
zbMATH Keywords: cohomology variety
zbMATH Keywords: rank variety
zbMATH Keywords: spectral sequence
zbMATH Keywords: restricted enveloping algebra of a commutative Lie algebra
zbMATH Keywords: coalgebra
zbMATH Keywords: cup product action
zbMATH Keywords: Yoneda product
author: Jon F. Carlson
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90064-7
OpenAlex ID: W2044148727
cites work: Representations, resolutions and Quillen's dimension theorem
cites work: Quillen stratification for modules
cites work: The varieties and the cohomology ring of a module
cites work: Q3931618
cites work: The Cohomology of Irreducible Modules Over SL(2,pn )
cites work: The Cohomology Ring of a Finite Group
cites work: Indecomposable representations
cites work: Q3236717
cites work: Q5343374
cites work: Cohomology of finite groups and elementary abelian subgroups
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description: scientific article; zbMATH DE number 3901008