The Avrunin and Scott theorem and a truncated polynomial algebra.
type: Item
instance of: scholarly article
title: The Avrunin and Scott theorem and a truncated polynomial algebra.
zbMATH Open document ID: 1102.16007
author: Karin Erdmann
author: Miles Holloway
published in: Journal of Algebra
publication date: 2006-08-04T00:00:00Z
review text: Let \(G\) be a finite group and \(k\) be an algebraically closed field of characteristic \(p>0\). Over the past 30 years, the cohomology and representation theory of \(kG\)-modules has been greatly influenced by the idea of cohomological support varieties which provide a geometric way to study cohomology. The success of this tool in the modular representation theory of finite groups has led to its use in many other areas. One key to this success has been the ability to provide non-cohomological descriptions of support varieties. The first such description, in the case of finite groups, was conjectured by J. Carlson. More precisely, for an elementary Abelian \(p\)-group, Carlson introduced the notion of a rank variety for a module which is defined representation-theoretically without using cohomology. For the trivial module \(k\), the support variety and rank variety are naturally isomorphic. For any finite-dimensional module, the support (rank) variety is a subvariety of the support (rank) variety for the trivial module. Carlson conjectured that under this isomorphism, for any finite-dimensional module, the support variety and rank varieties should correspond. This conjecture was first proven by G. Avrunin and L. Scott and is the theorem referred to in the title of the paper. The goal of this paper is to investigate the analogous question in the context of modules for a truncated polynomial algebra \(\Lambda_m=k[X_1,X_2,\dots,X_m]/(X_i^2)\) over an algebraically closed field \(k\) of arbitrary characteristic. For \(\Lambda_m\), support varieties are defined using Hochschild cohomology. This is based on work of \textit{N. Snashall} and \textit{Ø. Solberg} [Proc. Lond. Math. Soc. (3) 88, No. 3, 705--732 (2004; Zbl 1067.16010)] and of the authors with \textit{N. Snashall, Ø. Solberg} and \textit{R. Taillefer} [\(K\)-Theory 33, No. 1, 67--87 (2004; Zbl 1116.16007)]. Rank varieties were introduced by the authors [in Math. Z. 247, No. 3, 441--460 (2004; Zbl 1078.16008)]. …
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 5043671
zbMATH Keywords: rank varieties
zbMATH Keywords: support varieties
zbMATH Keywords: group algebras
zbMATH Keywords: truncated polynomial algebras
zbMATH Keywords: Avrunin-Scott theorem
zbMATH Keywords: stable maps
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.12.031
OpenAlex ID: W2000919709
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DOI: 10.1016/J.JALGEBRA.2005.12.031
description: scientific article