Almost-split sequences and group rings
type: Item
instance of: scholarly article
title: Almost-split sequences and group rings
zbMATH Open document ID: 0594.20005
DOI: 10.1016/0021-8693(86)90173-0
author: Maurice Auslander
author: Jon F. Carlson
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Almost split sequences are discussed in the category of RG-lattices, i.e., finitely generated R-free RG-modules, where R is either a field of characteristic \(p>0\) or a complete discrete valuation ring of characteristic zero with residue characteristic p, and G is a finite group. In particular, applications to tensor products of such modules are given. Some information is developed about splitting trace lattices, which are those lattices X such that the RG-lattice \(End_ R(X)\) has trivial direct summand. Using these, some facts are obtained about direct summands of tensor products. In the case where \(R=K\) is a field, let P be the projective cover of the trivial simple module, the almost split sequence for P/soc P has the form \[ 0\to \Omega (K)\to P\oplus E\to P/soc P\to 0. \] The authors present a new proof of the criterion of \textit{P. Webb} [Math. Z. 179, 97-121 (1982; Zbl 0479.20008)] for E to be indecomposable. Some results on indecomposable KG-modules are also given, where \(p=2\) and G has dihedral Sylow 2-subgroups.
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 3955191
zbMATH Keywords: category of RG-lattices
zbMATH Keywords: tensor products
zbMATH Keywords: splitting trace lattices
zbMATH Keywords: direct summands
zbMATH Keywords: projective cover
zbMATH Keywords: almost split sequence
zbMATH Keywords: indecomposable KG-modules
Wikidata QID: Q105541901
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90173-0
OpenAlex ID: W2020485563
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description: scientific article; zbMATH DE number 3955191