  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.
