On the Gorenstein property for modular invariants
type: Item
instance of: scholarly article
title: On the Gorenstein property for modular invariants
zbMATH Open document ID: 1243.13003
author: A. Braun
published in: Journal of Algebra
publication date: 2012-04-10T00:00:00Z
reviewed by: Jebrel M. Habeb
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 6022749
zbMATH Keywords: Gorenstein rings
zbMATH Keywords: modular invariants
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.07.030
OpenAlex ID: W2062327241
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DOI: 10.1016/J.JALGEBRA.2011.07.030
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review text: From the author's abstract: Let \(G\subset \mathrm{GL}(V)\) be a finite group, where \(V\) is a finite dimensional vector space over a field \(F\) of arbitrary characteristic. Let \(S(V)\) be the symmetric algebra of \(V\) and \(S(V)^G\) the ring of \(G\)-invariants. The author proves the followingNEWLINENEWLINETheorem: Suppose \(G\) contains no pseudo-reflection (of any kind). (1) If \(S(V)^G\) is Gorenstein, then \( G\subset GL(V)\). (2) If \(G\subset \mathrm{GL}(V)\), then the Cohen-Macaulay locus of \(S(V)^G\) coincides with its Gorenstein locus. In particular if \(S(V)^G\) is Cohen-Macaulay, then it is also Gorenstein.NEWLINENEWLINEThis extends well-known results of K. Watanabe in case \(\mathrm{gcd}(\mathrm{char}F, |G|) = 1\). It also confirms a special case of a conjecture due to G. Kemper, E. Kording, G. Malle, B.H. Matzat, D. Vogel and G. Wiese. A similar extension is given to D. Benson's theorem about the Gorenstein property of \((S(V)\otimes G(V))\), the polynomial tensor exterior algebra invariants. The author's proof uses non-commutative algebra methods in an essential way.
description: scientific article; zbMATH DE number 6022749