Homological codimension of modular rings of invariants and the Koszul complex
type: Item
instance of: scholarly article
title: Homological codimension of modular rings of invariants and the Koszul complex
zbMATH Open document ID: 0951.13002
author: Larry Smith
published in: Journal of Mathematics of Kyoto University
publication date: 2000-02-06T00:00:00Z
review text: The rings of invariants \(R^G\) of a finite group \(G\) acting on a ring of polynomials \(R = F[V],\) where \(F\) denotes a field, is a Cohen-Macaulay ring provided the characteristic of \(F\) does not divide the order of \(G,\) see e.g. the author's book: \textit{L. Smith}, ``Polynomial invariants of finite groups'' (1997; Zbl 0864.13002). This is not true in the modular case, i.e. when the characteristic is a divisor of the order of \(G,\) as it was first noticed by \textit{M.-J. Bertin} [see C. R. Acad. Sci., Paris, Sér. A 264, 653-656 (1967; Zbl 0147.29503)]. The paper under review is concerned with the following problem: Let \(h_1, \ldots, h_k \in R^G\) be invariant polynomials that form an \(R\)-regular sequence. Do they form a regular sequence in \(R^G\), too? As follows from examples discussed by the author in his book cited above, respectively: \textit{L. Smith}, Bull. Am. Math. Soc., New Ser. 34, No.~3, 211-250 (1997; Zbl 0904.13004) that this is not true in general. Here the author studies Koszul complexes, which determine the property of being a regular sequence by the vanishing of their homology. Then he uses this in order to describe the depth of rings of invariants of permutation representations of the cyclic group of order \(p \not= 0,\) and to answer the above problem in this case. The main technical tool is a spectral sequence which is in some sense the dual to the one used by \textit{G. Ellingsrud} and \textit{T. Skjelbred} [see Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)].
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 1319571
zbMATH Keywords: ring of invariants
zbMATH Keywords: homological dimension
zbMATH Keywords: Koszul complex
zbMATH Keywords: regular sequences
zbMATH Keywords: vanishing of homology
zbMATH Keywords: depth
reviewed by: Peter Schenzel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518006
OpenAlex ID: W1938728778
DOI: 10.1215/KJM/1250518006
description: scientific article; zbMATH DE number 1319571