Depth and transfer maps in the cohomology of groups
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 \(d\)?
scientific article; zbMATH DE number 778957
scientific article