THE SUPPORT OF REGULAR ELEMENTS IN THE

The last few years have witnessed a great deal of research on the algebraic structure of the cohomology rings of groups. The reason is that many of the ring-theoretic consider-ations such as varieties, associated primes and depth reflect properties of the groups and also of the topology of their classifying spaces. The evidence for such relationships has been building ever since Quillen rst proved [Q] that the minimal primes of the mod-p cohomology were the radicals of the kernels of restrictions to maximal elementary abelian p-subgroups. More generally, any associated prime of the cohomology ring is invariant under the Steenrod reduced power operations and hence must be the radical of the kernel of restriction to some elementary abelian p-subgroup of the group. Recently, particularly through Lannes ’ work [L], it has become evident that the ele-mentary abelian p-groups play a central role in the theory of unstable modules over the Steenrod algebra. The new developments in this theory were used in [HLS] to prove that the existence of regular elements (non-divisors of zero) in cohomology implies a detectabil-ity of mod-p cohomology on the centralizers of certain elementary abelian p-subgroups determined by the regular element. The main purpose of this note is to strengthen that theorem (Theorem 1, below) by proving that it has a strong converse, that the detectability condition requires the existence of certain homogeneous regular elements and hence that the set of homogeneous regular elements is characterized by the detectability condition. Let p be a xed prime and and let G be either a compact Lie group (which may be nite) or a discrete group of nite virtual cohomological dimension such that the mod p cohomology HBG = H(BG;Z=p) of its classifying space BG is noetherian. The set of elementary abelian p subgroups of G will be denoted by A(G). For a subgroup H of G let CG(H) be its centralizer in G. The starting point of this paper is the result mentioned above [HLS, Corollary I.5.7]. Theorem 1. Let G be as above. Assume x 2 HBG is not a zero divisor. Denote by Cx the set of elementary abelian p subgroups such that the restriction of x to HBE is not nilpotent. Then the map