Quillen Stratification for Modules

Let G be a finite group and k a fixed algebraically closed field of characteristic p>0 . If p is odd, let H(; be the subring of H*(G,k) consisting of elements of even degree; following [20-22] 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 a is a finitely generated commutative k-algebra [13], and we let Va denote its associated affine variety Max Hc. If M is any finitely generated kG-module, then the cohomology variety Vc(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| where L is any kG-module [4, 9]. A module L with each irreducible kG-module as a direct summand will serve. D. Quillen [20-22] proved a number of beautiful results relating V~ to the varieties l/t: associated with the various elementary abelian p-subgroups E of G, culminating in his stratification theorem [20, 22]. 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 Va = max dim VE, where E