Duality in G-algebras
type: Item
instance of: scholarly article
title: Duality in G-algebras
zbMATH Open document ID: 0667.20008
DOI: 10.1007/BF01161746
author: Jacques Thévenaz
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
review text: Let G be a finite group, and let R be a complete discrete valuation ring of characteristic zero with field of fractions K and residue field k of prime characteristic p. The author introduces the notion of a symmetric G-algebra which is a G-algebra A (free of finite rank over R) together with an R-linear map \(\tau\) : \(A\to R\) satisfying \(\tau (ab)=\tau (ba)\) and \(\tau (^ ga)=\tau (a)\) for \(a,b\in A\), \(g\in G\) such that the symmetric bilinear form \(\phi\) : \(A\times A\to R\), (a,b)\(\to \tau (ab)\), is non-degenerate. Then \(\phi\) extends to a non-degenerate symmetric bilinear form (again denoted by \(\phi)\) on \(K\otimes_ RA\), and the restriction \(\phi_ G\) of \(| G|^{-1}\phi\) is a non-degenerate symmetric bilinear form on \(K\otimes_ RA^ G\). Any full R-lattice L in \(K\otimes_ RA^ G\) has a dual \(L^*:=\{a:\) \(\phi_ G(a,L)\subseteq R\}\). If \(L=L^*\) then L is called unimodular. Suppose now that A itself is unimodular. Then one has \((A^ G)^*=A^ G_ 1\), the projective ideal of \(A^ G\). This implies that \(A^ G/A^ G_ 1\) is a symmetric algebra over the artinian ring R/\(| G| R\). The major part of the paper is devoted to properties of the almost projective ideal \(L_ G:=(J_ G)^*\) of \(A^ G\) where \(J_ G=J(A^ G)+A^ G_ 1\). We have \(A^ G_ 1\subseteq L_ G\subseteq A^ G\), and \(L_ G/A^ G_ 1\) is the socle of \(A^ G/A^ G_ 1.\) If M is an RG-lattice then \(A:=End_ R(M)\) is a unimodular symmetric G- algebra, and the almost projective ideal is closely related to almost split sequences. For example, it can be used to give an easy proof of the existence and properties of almost split sequences. Similarly, any block A of RG is a unimodular symmetric G-algebra, and in this case the almost projective ideal can be used to prove properties of defect groups and the central character of A. There are also interesting connections to virtually irreducible lattices (as introduced by R. Knörr) and splitting trace lattices (as defined by M. Auslander and J. F. C…
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 4091728
zbMATH Keywords: finite group
zbMATH Keywords: complete discrete valuation ring
zbMATH Keywords: symmetric G-algebra
zbMATH Keywords: symmetric bilinear form
zbMATH Keywords: projective ideal
zbMATH Keywords: symmetric algebra
zbMATH Keywords: RG-lattice
zbMATH Keywords: unimodular symmetric G-algebra
zbMATH Keywords: almost projective ideal
zbMATH Keywords: almost split sequences
zbMATH Keywords: block
zbMATH Keywords: defect groups
zbMATH Keywords: virtually irreducible lattices
zbMATH Keywords: splitting trace lattices
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01161746
OpenAlex ID: W1555593722
description: scientific article; zbMATH DE number 4091728