On module varieties and quotient groups
type: Item
instance of: scholarly article
title: On module varieties and quotient groups
zbMATH Open document ID: 0686.20008
DOI: 10.1016/0021-8693(89)90095-1
author: Hiroaki Kawai
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let G be a finite group and K an algebraically closed field of characteristic \(p>0\). Write \(V_ G(K)\) for the spectrum of the even cohomology ring \(E_ G^{ev}(K)\). If M is a (finitely generated) KG- module, \(V_ G(M)\) is the subvariety of \(V_ G(K)\) coming from the annihilator \(J_ G(M)\) of the \(E_ G^{ev}(K)\)-module \(Ext^*_{KG}(M,M)\). If H is a subgroup of G, the restriction map induces \(\psi_{H,G}: V_ H(K)\to V_ G(K)\). It has been shown that \(\Psi^{-1}_{H,G}(V_ G(M))=V_ H(M).\) In this paper the map \(\lambda_{G,G/H}: V_ G(K)\to V_{G/H}(K)\) is considered when H is normal in G. This is induced by the inflation map \(\inf_{G/H,G}: E^{ev}_{G/K}(K)\to E_ G^{ev}(K)\). The principal theorem is that \(\lambda^{-1}_{G,G/H}(V_{G/H}(M))=V_ G(M)\), for a K(G/H)-module M. Let M be an indecomposable KG-module such that p does not divide \(\dim_ KN\) for a source N of M. Conditions are given so that \(V_ G(M)\) is an irreducible variety or again equal to \(V_ G(K)\) itself.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 4123970
zbMATH Keywords: associated affine variety
zbMATH Keywords: finite group
zbMATH Keywords: cohomology ring
zbMATH Keywords: restriction map
zbMATH Keywords: inflation map
zbMATH Keywords: indecomposable KG-module
zbMATH Keywords: source
zbMATH Keywords: irreducible variety
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90095-1
OpenAlex ID: W2065375669
cites work: Quillen stratification for modules
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description: scientific article; zbMATH DE number 4123970