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Transactions of the American Mathematical Society

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Localization of equivariant cohomology rings

Author: J. Duflot
Journal: Trans. Amer. Math. Soc. 284 (1984), 91-105
MSC: Primary 57S15; Secondary 20J06, 55N91
Erratum: Trans. Amer. Math. Soc. 290 (1985), 857-858.
MathSciNet review: 742413
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Abstract: The main result of this paper is the "calculation" of the Borel equivariant cohomology ring $ {H^{\ast} }(EG \times_G\,X,{\mathbf{Z}}/p{\mathbf{Z}})$ localized at one of its minimal prime ideals. In case $ X$ is a point, the work of Quillen shows that the minimal primes $ {\mathfrak{P}_A}$ are parameterized by the maximal elementary abelian $ p$-subgroups $ A$ of $ G$ and the result is

$\displaystyle {H^{\ast} }{(BG,{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}} \... ...{H^{\ast} }(B{C_G}(A),{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}^{{W_G}(A)}$

. Here, $ {C_G}(A)$ is the centralizer of $ A$ in $ G$, and $ {W_G}(A) = {N_G}(A)/{C_G}(A)$, where $ {N_G}(A)$ is the normalizer of $ A$ in $ G$. An example is included.

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Article copyright: © Copyright 1984 American Mathematical Society