Dickson invariants, regularity and computation in group cohomology

By D. Benson. 2003

47 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7295195fbd3cf80a0ebc4b7deb8231c1bcc90cbe

How researchers see this paper

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In Algebra and Number Theory

1. Peter Symonds
2. Adam Van Tuyl
3. Jessica Sidman
4. Huy Tài Hà
5. Russ Woodroofe
6. K. Kimura
7. Arindam Banerjee

In Geometry and Topology

1. Peter Symonds
2. Adam Van Tuyl
3. Jessica Sidman
4. Huy Tài Hà
5. Russ Woodroofe
6. K. Kimura

In Mathematical Physics

1. Peter Symonds
2. Adam Van Tuyl
3. Jessica Sidman

Citations in the corpus

• On the Castelnuovo-Mumford regularity of rings of polynomial invariants (2011, background): “Benson [2] §10 defines τNS and proves a version of 2.”
• Multigraded Castelnuovo-Mumford Regularity (2003, unknown): “A C ] 1 5 M ay 2 00”
• Multigraded Regularity: Syzygies and Fat Points (2004, background): “If M is a finitely generated graded module over a Z-graded polynomial ring over k, its CastelnuovoMumford regularity, denoted reg(M), is an invariant that measures the difficulty of computations involving M. Recently, several authors (cf. [1, 18, 19]) have proposed extensions of the notion of…”