Let a compact Lie group $G$ act on a manifold $X$. Let $\mathbb Q$ be the field of rational numbers and assume that cohomology $H^{\ast} (X)$ with $\mathbb Q$-coeffiecients is finite-dimensional. It seems to be a folklore fact that the Krull dimension of the even-dimensional $G$-equivariant cohomology $H^{even} _G (X, \mathbb Q)$ coincides with the maximal rank of the stabilizers of $G$.

The corresponding result in cohomology with $\mathbb F_p$ coefficients is proven as Theorem 7.1 in D. Quillen's paper "The spectrum of an equivariant cohomology ring. I", Ann. of Math. (2) 94 (1971), 549-579.

With rational coefficients, I have only found this result in the case where $G$ is a torus. It is quite easy to reduce the case of a general compact Lie group $G$ to the commutative case. But I assume that the result for general compact Lie group $G$ has appeared somewhere.

Does somebody know a reference for this fact?

  • $\begingroup$ With rational coefficients the cohomology should be invariants of the cohomology of the torus by the Weyl group, which shouldn't change the Krull dimension. $\endgroup$ – Dylan Wilson Jun 13 '13 at 13:51
  • $\begingroup$ (Sorry, the above comment is for the case $X = pt$!) $\endgroup$ – Dylan Wilson Jun 13 '13 at 14:48

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