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.