Illinois Journal of Mathematics

Dickson invariants, regularity and computation in group cohomology

Dave Benson

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Abstract

In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler.

As a by-product of this investigation, we give a new proof of the Bourguiba--Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra.

Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

Article information

Source
Illinois J. Math., Volume 48, Number 1 (2004), 171-197.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258136180

Digital Object Identifier
doi:10.1215/ijm/1258136180

Mathematical Reviews number (MathSciNet)
MR2048221

Zentralblatt MATH identifier
1041.20036

Subjects
Primary: 20J06: Cohomology of groups
Secondary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 13D45: Local cohomology [See also 14B15]

Citation

Benson, Dave. Dickson invariants, regularity and computation in group cohomology. Illinois J. Math. 48 (2004), no. 1, 171--197. doi:10.1215/ijm/1258136180. https://projecteuclid.org/euclid.ijm/1258136180


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