The relative trace ideal and the depth of modular rings of invariants
type: Item
instance of: scholarly article
title: The relative trace ideal and the depth of modular rings of invariants
zbMATH Open document ID: 1056.13005
author: Peter Fleischmann
author: R. J. Shank
published in: Archiv der Mathematik
publication date: 2003-08-18T00:00:00Z
review text: Let \(G\) be a finite group, \(p\) a prime divisor of the order of \(G\), \(k\) a field of characteristic \(p\), and \(V\) a finitely generated \(kG\)-module. Consider the algebra \(A=k[V]\) and the subalgebra of invariants \(A^G\). The depth of \(A^G\) is the length of a maximal regular sequence in \(A^G\). Since \(V\) is a modular representation of \(G\), the algebra \(A^G\) may fail to be Cohen-Macaulay, i.e., the depth might be less than the Krull dimension. The authors prove that the depth of \(A^G\) is the sum of the dimension of the fixed point space of the \(p\)-Sylow subgroup and the grade of the relative trace ideal. They also determine which of the Dickson invariants lie in the radical of the relative trace ideal and describe how to use the Dickson invariants to compute the grade of the relative trace ideal.
reviewed by: Ivan V. Arzhantsev
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 1966457
zbMATH Keywords: finite groups
zbMATH Keywords: modular representations
zbMATH Keywords: \(p\)-Sylow subgroups
zbMATH Keywords: depth
zbMATH Keywords: Dickson invariants
zbMATH Keywords: invariants
MaRDI profile type: Publication
full work available at URL: s00013-003-0794-0
OpenAlex ID: W84190008
DOI: 10.1007/S00013-003-0794-0
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description: scientific article; zbMATH DE number 1966457