Group algebra extensions of depth one.
type: Item
instance of: scholarly article
title: Group algebra extensions of depth one.
zbMATH Open document ID: 1236.20001
author: Robert Boltje
author: Burkhard Külshammer
published in: Algebra \& Number Theory
publication date: 2011-11-17T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ant/1513729630
full work available at URL: p03.xhtml
review text: Let \(A\) and \(B\) be rings and let \(f\colon A\to B\) be a ring homomorphism. Following Kadison one says that \(f\) is of depth one if the \(A\)-\(A\)-bimodule \(B\) is a direct factor of \(A^n\) for some integer \(n\). The authors discuss in the paper under review the case of the inclusion \(kH\to kG\) for a commutative ring \(k\), a group \(G\) and a subgroup \(H\) of finite index. From an earlier paper of the authors it follows that if the inclusion is of depth one, then \(H\) is normal in \(G\). If \(k\) is a field of characteristic \(0\), then the authors show that the inclusion is of depth one if and only if \(G=H\cdot C_G(X)\) for every cyclic subgroup \(X\) of \(H\). If \(k\) is a field of characteristic \(p>0\) of a complete discrete valuation ring of characteristic \(0\) with residue field of characteristic \(p>0\), then the authors show that the inclusion is of depth one if and only if \(G=H\cdot C_G(X)\) for each of those subgroups \(X\) of \(H\), so that \(X\) has a normal Sylow \(p\)-subgroup \(S_X\) so that \(X/S_X\) is cyclic. If \(B\) is a block of \(kG\) with defect pointed group \(P_\gamma\) and source algebra \(B_\gamma\), then there is a natural homomorphism \(kP_\gamma\to B_\gamma\). The authors show that this homomorphism is of depth one if and only if \(B\) is a nilpotent block so that the only simple \(B\)-module has trivial source.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5973574
zbMATH Keywords: depths of ring extensions
zbMATH Keywords: nilpotent blocks
zbMATH Keywords: trivial source algebras
MaRDI profile type: Publication
OpenAlex ID: W2104063837
DOI: 10.2140/ANT.2011.5.63
description: scientific article; zbMATH DE number 5973574