Vertex and source determine the block variety of an indecomposable module.
type: Item
instance of: scholarly article
title: Vertex and source determine the block variety of an indecomposable module.
zbMATH Open document ID: 1070.20013
author: Markus Linckelmann
author: David Benson
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-18T00:00:00Z
full work available at URL: var-vertex-source.pdf
review text: Let \(k\) be a field of finite characteristic \(p\) and let \(G\) be a finite group with order divisible by \(p\). John Carlson defined and studied in the 1980's the cohomology variety \(V_G(M)\) of a module \(M\). This variety depends only on the principal block of \(kG\). Recently Linckelmann defined a variety for any block \(b\) of \(kG\) by some limit construction over the so-called Frobenius category from the usual group cohomology of a fixed defect group \(P\) of \(b\). The main result of the paper is the following. Let \(i\) be a source idempotent and let \(M\) be an indecomposable \(kGb\)-module. Then there is a vertex \(Q\) of \(M\) contained in \(P\) and a source \(U\) of \(M\) such that \(U\) is a direct summand of the restriction of \(iM\) to \(Q\), and such that \(M\) is a direct summand of \(kGi\otimes_{kG}U\). Then, the variety of \(M\) with respect to the block \(b\) equals \(r_Q^*(V_Q(U))\), where \(r_Q\) is given by the composition of inclusion of the block cohomology of \(G\) in the group cohomology of \(P\) and the restriction mapping of usual group cohomology \(H^*(P,k)\to H^*(Q,k)\). This way, the variety of a module in the non principal block is reduced to a variety coming from the usual variety of an appropriate source with respect to an appropriate vertex. Using this result the authors apply results of Carlson on usual cohomology varieties to block cohomology varieties. Namely for indecomposable modules the cohomology variety is connected, and any of its closed homogeneous subvarieties is a module variety for a module of this block.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 2157516
zbMATH Keywords: cohomology varieties
zbMATH Keywords: block varieties
zbMATH Keywords: group cohomology
zbMATH Keywords: defect groups
zbMATH Keywords: blocks
zbMATH Keywords: idempotents
zbMATH Keywords: vertices
zbMATH Keywords: sources
MaRDI profile type: Publication
OpenAlex ID: W2069331393
cites work: Local methods in block theory
cites work: Modular representation theory: New trends and methods
cites work: Q3997902
cites work: Brauer coefficients of p-subgroups associated with a p-block of a finite group
cites work: Characters and local structure in G-algebras
cites work: Q3950716
cites work: The varieties and the cohomology ring of a module
cites work: The variety of an indecomposable module is connected
cites work: Q3240881
cites work: Q4001605
cites work: The source algebras of blocks with a Klein four defect group
cites work: Transfer in Hochschild cohomology of blocks of finite groups
cites work: Varieties in block theory
cites work: On splendid derived and stable equivalences between blocks and finite groups
cites work: Quillen stratification for block varieties.
cites work: Local fusions in block source algebras
cites work: Q4846455
DOI: 10.1016/J.JPAA.2004.08.032
description: scientific article; zbMATH DE number 2157516