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Cohomological detection and regular elements in group cohomology

Authors: Jon F. Carlson and Hans-Werner Henn
Journal: Proc. Amer. Math. Soc. 124 (1996), 665-670
MSC (1991): Primary 20J05, 20J06, 55R40
MathSciNet review: 1327000
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Abstract: 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 $\mathcal{O}$ 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 $\mathcal{O}$. Assume also that $\mathcal{O}$ is closed under conjugation and that $E$ is in $\mathcal{O}$ whenever some subgroup of $E$ is in $\mathcal{O}$. Then there exists a regular element $\zeta $ in the cohomology ring $H^*(BG,k)$ such that the restriction of $\zeta $ to an elementary abelian $p$-subgroup $E$ is not nilpotent if and only if $E$ is in $\mathcal{O}$. 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.

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Additional Information

Jon F. Carlson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: jfc@sloth.math.uga.ed

Hans-Werner Henn
Affiliation: Mathematisches Institut der Universität, Im Neuenheimer Feld 288, D–69120 Heidelberg, Federal Republic of Germany

Received by editor(s): December 22, 1993
Additional Notes: The first author was partially supported by a grant from NSF.
The second author was supported by a Heisenberg grant from DFG.
Communicated by: Eric Friedlander
Article copyright: © Copyright 1996 American Mathematical Society