Quillen stratification for modules

Let G be a finite group and k a fixed algebraically closed field of characteristic p>O. If p is odd, let H, be the subring of H*(G, k) consisting of elements of even degree; following [20-221 we take H, = H*(G, k) if p=2, though one could just as well use the subring of elements of even degree for all p. H, is a finitely generated commutative k-algebra [13], and we let V, denote its associated affine variety Max H,. If M is any finitely generated kG-module, then the cohomology variety V,(M) of M may be defined as the support in V, of the H,-module H*(G, M) if G is a p-group, and in general as the largest support of H*(G, L@ M), where L is any kG-module [4, 91. A module L with each irreducible kc-module as a direct summand will serve. D. Quillen [20-221 proved a number of beautiful results relating k;; to the varieties V, associated with the various elementary abelian p-subgroups E of G, culminating in his stratification theorem [20, 221. This theorem gives a piecewise description of V, almost explicitly in terms of the subgroups E and their normalizers in G. A well-known corollary is that dim V, =max dim V,, where E