THE SUPPORT OF REGULAR ELEMENTS IN
THE COHOMOLOGY RING OF A GROUP
Jon F. Carlson1and Hans-Werner Henn2
The last few years have witnessed a great deal of research on the algebraic s*
*tructure of
the cohomology rings of groups. The reason is that many of the ring-theoretic c*
*onsider-
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 suchrelatio*
*nships has
been building ever since Quillen first proved [Q] that the minimal primes of th*
*e mod-p
cohomology were the radicals of thekernels of restrictions to maximal elementar*
*y abelian
p-subgroups. More generally, any associated prime of the cohomologyring is inva*
*riant
under the Steenrod reduced power operations and hence must be the radical of th*
*e kernel
of restriction to some elementary abelian p-subgroup of the group.
Recently, particularly through Lannes' work [L], it has become evident that t*
*he 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 pro*
*ve 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-sub*
*groups
determined by the regular element. The mainpurp ose of this note is to strengt*
*hen that
theorem (Theorem 1,below) by proving that it has a strong converse, that the de*
*tectability
condition requires the existence of certain homogeneous regular elements and he*
*nce that
the set of homogeneous regular elements is characterized by the detectability c*
*ondition.
Let p be a fixed prime and and let G be either a compact Lie group (which may*
* be
finite) or a discrete group of finite virtual cohomological dimension such that*
* the mod p
cohomology H BG = H (BG;Z=p) of its classifying space BG is noetherian. The set*
* of
elementary abelian p subgroups of G willb e denoted by A(G). For a subgroup H o*
*f G
let CG(H) be its centralizer in G. The starting point of this paper is the resu*
*lt mentioned
above [HLS, Corollary I.5.7].
Theorem 1. Let G be as above.Assume x 2 H BG is not a zero divisor. Denote by Cx
the set of elementary abelian p subgroups such that the restriction of x to H B*
*E is not
nilpotent. Then the map
! Y Y
! resG;CG(E): H BG ! H BCG(E)
!! E2Cx E2Cx
!
whose!components are the restriction homomorphisms is a monomorphism.
!
! 1
Partially supported by a grant from NSF.
2Supported by a Heisenberg grant from DFG.
Typeset by AM S-T*
*EX
2 JON F. CARLSON AND HANS-WERNER HENN
Jon F. Carlson Hans-Werner Henn
Department of Mathematics Mathematisches Institut
University of Georgia der Universit{t
Athens, GA 30602 Im Neuenheimer Feld 288
USA D-69120 Heidelberg