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