Localization of equivariant cohomology rings

Abstract

The main result of this paper is the "calculation" of the Borel equivariant cohomology ring H ∗ ( E G × G X , Z / p Z ) {H^{\ast } }(EG \times _G\,X,{\mathbf {Z}}/p{\mathbf {Z}}) localized at one of its minimal prime ideals. In case X X is a point, the work of Quillen shows that the minimal primes P A {\mathfrak {P}_A} are parameterized by the maximal elementary abelian p p -subgroups A A of G G and the result is \[ H ∗ ( B G , Z / p Z ) P A ≅ H ∗ ( B C G ( A ) , Z / p Z ) P A W G ( A ) {H^{\ast } }{(BG,{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}} \cong {H^{\ast } }(B{C_G}(A),{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}^{{W_G}(A)} \] . Here, C G ( A ) {C_G}(A) is the centralizer of A A in G G , and W G ( A ) = N G ( A ) / C G ( A ) {W_G}(A) = {N_G}(A)/{C_G}(A) , where N G ( A ) {N_G}(A) is the normalizer of A A in G G . An example is included.