On the Duflot filtration for equivariant cohomology rings

Abstract

We study the Fp-cohomology rings of the classifying space of a compact Lie group G using methods from equivariant cohomology. Building on ideas of Duflot and Symonds we study a “rank filtration” on the p-toral equivariant cohomology of a smooth manifold. We analyze the structure induced by this filtration and construct a well behaved chain complex that controls the local cohomology of H∗BG. We also refine the Duflot filtration to a filtration by a ranked poset, and from this get a detection result and restrictions on associated primes that generalize some of the work of Carlson and of Okuyama from finite groups to general compact Lie groups. We also use our methods to give new local cohomology computations for the cohomology of p-Sylow subgroups of Spn . In the final chapter we show that the derived category of cochains on the Borel construc- tion of a finite G-CW complex is stratified in the sense of Benson, Iyengar, and Krause by the equivariant cohomology ring.