Depth and transfer maps in the cohomology of groups
type: Item
instance of: scholarly article
title: Depth and transfer maps in the cohomology of groups
zbMATH Open document ID: 0837.20062
DOI: 10.1007/BF02571916
published in: Mathematische Zeitschrift
publication date: 1996-04-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174741
review text: Let \(G\) be a finite group and let \(k\) be a field of characteristic \(p > 0\) dividing the group order. There are a number of results relating the structure of the cohomology algebra \(H^* (G,k)\) to the structure of the group \(G\). For example it is known that the depth of \(H^* (G,k)\) (length of the longest regular sequence) is at least equal to the \(p\)- rank of the center of \(G\). Here, using \textit{D. J. Benson}'s transfer theorem [Arch. Math. 61, 7-11 (1993; Zbl 0815.20040)], the following extension of this result is proved: if there exists an element in the cohomology \(H^*(G, k)\) whose restriction to the centralizers of elementary abelian subgroups of rank \(t\) vanish then the depth of \(H^* (G, k)\) is less than \(t\), and there is a prime ideal of dimension less than \(t\). In an earlier paper of Benson and Carlson is was proved that for groups of \(p\)-rank 2 the algebra \(H^* (G,k)\) is Cohen-Macaulay (has depth 2) if and only if all associated primes have dimension 2. Led by this result the author states as an open question: Let \(H^*(G,k)\) have depth \(d\). Is there an associated prime in \(H^* (G,k)\) of dimension \(d\)?
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 778957
zbMATH Keywords: finite groups
zbMATH Keywords: cohomology algebra
zbMATH Keywords: depth
zbMATH Keywords: \(p\)-rank
zbMATH Keywords: elementary Abelian subgroups
author: Jon F. Carlson
reviewed by: Urs Stammbach
MaRDI profile type: Publication
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OpenAlex ID: W1985714047
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description: scientific article; zbMATH DE number 778957