\author{David Benson}
\address{Department of Mathematics, University of Georgia}
\email{/b$\backslash$e/n$\backslash$s/o$\backslash$n/d%
$\backslash$j/ (without the slashes) at math dot uga dot edu}
\thanks{This research was partly supported by NSF grant DMS-0242909}
\title[Regularity conjecture for cohomology]
{On the regularity conjecture for the cohomology of finite groups}
\begin{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 zero.
We prove that if the regularity conjecture holds for a finite
group $H$ then it holds for the wreath product $H \wr \Z/p$. As a corollary,
we prove the regularity conjecture for the symmetric groups
$\Sigma_n$. 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 two,
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 form a PORC function in the sense of Higman.
\end{abstract}