On the depth of subgroups and group algebra extensions.
type: Item
instance of: scholarly article
title: On the depth of subgroups and group algebra extensions.
zbMATH Open document ID: 1250.20001
author: Robert Boltje
author: Susanne Danz
author: Burkhard Külshammer
published in: Journal of Algebra
publication date: 2012-01-04T00:00:00Z
review text: If \(B\) is a subring of the ring \(A\), then we may form the \(i\)-fold tensor product \(T_i(B,A)\) of \(A\) over \(B\). The extension of \(B\) to \(A\) is said to have depth \(2i+1\) if \(T_{i+1}(B,A)\) is a direct factor of a direct sum of copies of \(T_i(B,A)\) as \(B\)-\(B\)-bimodules. If the same holds as \(A\)-\(B\)-bimodule and as \(B\)-\(A\)-bimodule, then the extension is said to have depth \(2i\). For depth \(1\), \(2\) and \(3\) and extensions of \(RG\) of \(RH\) for some subgroup \(H\leq G\) nice group theoretical characterisations were given in previous papers of the authors. In the present paper the authors give a definition of combinatorial depth, via bisets. This definition only depends on a group inclusion \(H\leq G\). The authors show that this combinatorial definition gives a concept of depth which is at least as big as the original definition for \(R\) being a field of characteristic \(p\) or a cdvr of characteristic \(0\) with residue field of characteristic \(p>0\).
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 5994288
zbMATH Keywords: depths
zbMATH Keywords: bisets
zbMATH Keywords: group algebras
zbMATH Keywords: ring extensions
zbMATH Keywords: tensor products
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.03.019
OpenAlex ID: W1977806073
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DOI: 10.1016/J.JALGEBRA.2011.03.019
description: scientific article; zbMATH DE number 5994288