Cohomological detection and regular elements in group cohomology - Carlson, Jon F.; Henn, Hans-Werner (Proceedings of the American Mathematical Society, 1996)
Suppose that G is a compact Lie group or a discrete group of ﬁnite virtual cohomological dimension and that k is a ﬁeld of characteristic p > 0. Suppose that O is a set of elementary abelian p-subgroups G such that the cohomology H∗(BG, k) is detected on the centralizers of the elements of O. Assume also that O is closed under conjugation and that E is in O whenever some subgroup of E is in O. Then there exists a regular element ζ in the cohomology ring H∗(BG, k) such that the restriction of ζ to an elementary abelian p-subgroup E is not nilpotent if and only if E is in O. The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.