Depth of modular invariant rings
type: Item
instance of: scholarly article
title: Depth of modular invariant rings
zbMATH Open document ID: 0961.13003
DOI: 10.1007/BF01237176
author: H. E. A. Campbell
author: I. P. Hughes
author: Gregor Kemper
author: R. J. Shank
author: David Wehlau
published in: Transformation Groups
publication date: 2000-06-06T00:00:00Z
review text: This paper investigates the depth of modular rings of invariants for certain finite groups. Although the ring of invariants of a non-modular representation of a finite group is Cohen-Macaulay, and hence has depth equal to the dimension of the representation, the ring of invariants of a modular representation may fail to be Cohen-Macaulay. In this case computing the depth of the ring of invariants can be a difficult problem. In this paper the authors introduce the concept of a shallow representation and prove that the depth of the ring of invariants of a shallow representation, \(V\), is given by \(\min(\dim (V^P)+2, \dim(V))\) where \(P\) is any \(p\)-Sylow subgroup. For an abelian group with a cyclic \(p\)-Sylow subgroup, every representation is shallow. Thus the paper provides a relatively elementary proof of the well known result of \textit{G. Ellingsrud} and \textit{T. Skjelbred} [Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)]. Additional examples of shallow representations are given in the paper and the observation is made that the direct sum of \(m\) copies of a shallow representation is still shallow. The final section of the paper gives an alternate proof of the fact, originally due to \textit{H. Nakajima} [Tsukuba J. Math. 3, 109-122 (1979; Zbl 0418.20041)], that the ring of invariants of a \(p\)-group is Cohen-Macaulay if and only if it is Buchsbaum. The authors then conjecture that this should hold for any finite linear group acting on a polynomial algebra. The conjecture has since been proven by one of the authors [see \textit{G. Kemper}, ``Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaum property'' (preprint, Univ. Heidelberg 2000)].
reviewed by: Peter Fleischmann
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 1435994
zbMATH Keywords: \(p\)-Sylow subgroups
zbMATH Keywords: Cohen-Macaulay ring
zbMATH Keywords: Buchsbaum ring
zbMATH Keywords: modular rings of invariants
zbMATH Keywords: shallow representation
describes a project that uses: Magma
MaRDI profile type: Publication
cites work: Q4273960
cites work: The Magma algebra system. I: The user language
cites work: Depth and the Steenrod algebra (with an appendix by J. Lannes)
cites work: Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
cites work: Q3879338
cites work: Q4270317
cites work: Q3659962
cites work: Calculating invariant rings of finite groups over arbitrary fields
cites work: On the Cohen-Macaulay property of modular invariant rings
cites work: Q4270322
cites work: Invariants of finite groups generated by pseudo-reflections in positive characteristic
cites work: Q5580133
cites work: Q4838154
cites work: Homological codimension of modular rings of invariants and the Koszul complex
cites work: Invariants of finite groups and their applications to combinatorics
cites work: Q3744270
cites work: Q4004234
cites work: Q5732721
full work available at URL: https://doi.org/10.1007/bf01237176
OpenAlex ID: W2002007546
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description: scientific article; zbMATH DE number 1435994