Augmentation Ideals of Equivariant Cohomology Rings

The purpose of this note is to establish a number of useful results about the augmentation ideal J for the coefficient ring F ∗ G of a Noetherian complex orientable equivariant cohomology theory. The results show that various naturally occurring substitutes for the ideal have the same radical, and can therefore be used instead of the augmentation ideal in all geometric constructions. 1. Statement of results. Let G be a finite group, and let F denote a ring G-spectrum, representing an equivariant cohomology theory F ∗ G(X). When comparing the equivariant theory with the associated non-equivariant theory it is natural to measure the difference by the augmentation ideal J(G) = ker ( F ∗ G = F ∗ G(S ) −→ F ∗ G(G+) = F ∗ ) . When the ambient group is clear from the context it is common to simply write J , and it is one of the purposes of this note to show that it is reasonable to do this. For example if H is a subgroup of G the restriction map resH : F ∗ G = F ∗ G(S ) −→ F ∗ G(G/H+) = F ∗ H is a ring homomorphism which allows us to regard any F ∗ H-module M as a F ∗ G-module. It is fundamental to inductive proofs to know that completions and local cohomology for the ideals J(G) and J(H) coincide. It is convenient to work entirely with ideals of F ∗ H ; the image of J(G) will not generally be an ideal, so we follow the convention that (resHJ(G)) denotes the ideal it generates. We work throughout with homogeneous ideals; for suitably periodic cohomology theories such as considered in [3] it would be sufficient to work entirely in degree 0. We assume that the non-equivariant coefficient ring F ∗ is an integral domain, so that J(G) is a prime ideal; this holds in the applications, but the assumption can be relaxed at the expense of taking radicals. Much more important is that our results all depend on a finiteness condition. Finiteness Hypothesis 1.1. The ring F ∗ G is Noetherian, and for any finite complex X the module F ∗ G(X) is finitely generated over F ∗ G. This is equivalent to asking that F ∗ G is Noetherian, and that for each subgroup H, the restriction map F ∗ G −→ F ∗ H makes F ∗ H into a finitely generated module over F ∗ G. This is a very natural hypothesis, and satisfied in many cases of interest: for example it is satisfied for equivariant K-theory. It is also satisfied by the Borel theories F ∗ G(X) = F (EG+ ∧G X) associated to any non-equivariant multiplicative theory F ∗(·) which is