Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

David J. Bensona1

a1 Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK (bensondj@maths.abdn.ac.uk)

Abstract

Let $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.

For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

(Online publication July 28 2008)

(Received August 29 2005)

Keywords

  • Primary 20J06;
  • Secondary 13D45;
  • cohomology of groups;
  • local cohomology;
  • Castelnuovo–Mumford regularity;
  • wreath product