G-algebras, Jacobson radical and almost split sequences
type: Item
instance of: scholarly article
title: G-algebras, Jacobson radical and almost split sequences
zbMATH Open document ID: 0668.20015
DOI: 10.1007/BF01393690
author: Jacques Thévenaz
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143592
review text: Let k be an algebraically closed field of characteristic \(p\neq 0\) and G a finite group; let H be a subgroup of G. Let M be an indecomposable kG- module and X the almost split exact sequence terminating in M. The behaviour of X under restriction to H is looked at; in particular, as to when the restriction is an SAS-sequence (a direct sum of split and almost split exact sequences). Induction up to G of an almost split exact sequence of kH-modules (terminating in an indecomposable kH-module) is also looked at. For restriction, J. A. Green obtained a criterion for the restriction to be an SAS-sequence in terms of various relative Jacobson radicals ideals lying in \(End_ kM\). One generalization is to the corresponding condition for G-algebra A (instead of \(End_ kM).\) However here a very local criterion for restriction to behave is obtained. This is in terms of the multiplicity module \(V_{p_{\gamma}}\) corresponding to a vertex P and source of M; \(V_{p_{\gamma}}\) is an indecomposable projective module over a twisted group algebra on the group \(\overline{N_ G(P_{\gamma})}=N_ G(P_{\gamma})/P\) and the condition is that Res\(\begin{matrix} \overline{N_ G(P_{\gamma})}\\ \overline{N_ H(P)}\end{matrix} (Soc(V_{P_{\gamma}}))\) be semisimple. A similar analysis is given for the behaviour of almost split exact sequences under induction. Indeed the whole behaviour and result of induction and restriction on almost split exact sequences are given. The analysis and terminology follow that introduced by \textit{L. Puig} [Math. Z. 176, 265-292 (1981; Zbl 0464.20007)].
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 4093736
zbMATH Keywords: finite group
zbMATH Keywords: indecomposable kG-module
zbMATH Keywords: SAS-sequence
zbMATH Keywords: almost split exact sequences
zbMATH Keywords: restriction
zbMATH Keywords: relative Jacobson radicals
zbMATH Keywords: local criterion
zbMATH Keywords: multiplicity module
zbMATH Keywords: vertex
zbMATH Keywords: source
zbMATH Keywords: indecomposable projective module
zbMATH Keywords: twisted group algebra
zbMATH Keywords: induction
MaRDI profile type: Publication
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OpenAlex ID: W161489496
description: scientific article; zbMATH DE number 4093736