The relationship between cohomology with coefficients in a ℤG-module and RO(G)-graded (genuine) equivariant cohomology is nuanced, and they are not generally special cases of one another. Here's a structured breakdown:

1. Definitions and Grading

• ℤG-module coefficient cohomology (Borel-equivariant): This is a ℤ-graded theory, denoted ( H^\ast_G(X; M) ), defined using the equivariant ch...

Short answer: No. Cohomology with coefficients in a ZG–module (i.e. Borel-equivariant cohomology with local coefficients pulled back from BG) is not a special case of “genuine,” RO(G)-graded equivariant cohomology. It is a different flavor of equivariant theory. It is representable, but not by an RO(G)-graded Eilenberg–MacLane G-spectrum of the sort used in May’s book; rather it is represented eit...

No – the two theories live in different “equivariant worlds’’ and only coincide after one forces the action to be free (or, more generally, after smashing with EG₊). Nevertheless the groups H⁎_{G}(X;M) that Brown defines are representable, and the relation with RO(G)-graded ordinary (Bredon) cohomology can be described very explicitly.

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There are in fact two different “equivariant‐cohomology” machines hiding behind your two bullets, and one only becomes the other in very special cases. Let me try to disentangle them and then explain exactly how and when local‐coefficient cohomology sits inside an RO(G)–theory.

1. Borel cohomology with –coefficients.
   By definition [ H_G^(X;M);=;H^!\bigl(\Hom_{\Z G}(C_*(X)...

Short answer

For an arbitrary left ℤG–module M the “Borel–type” groups
HG(X;M)=H(EG×G X;𝔐)
(where 𝔐 is the local coefficient system determined by M) form only a naive equivariant cohomology theory: they are indexed by the ordinary integers and are representable by a naive G–spectrum EG M:=F(EG+,HM) .
(Here HM is the nonequivariant Eilenberg–Mac Lane spectrum of M and the G–...

Excellent question. It gets to the heart of the distinction between two fundamental approaches to equivariant cohomology, often called the "Borel" and "genuine" (or "homotopical") theories.

The short answer to your main question is:

Yes, ordinary equivariant cohomology with coefficients in a -module is a special case of representation-graded ("genuine") equivariant cohomology....

First, I need ...

Is Equivariant Cohomology with Coefficients in a Module a Special Case of Representation-Graded Equivariant Cohomology?

This question asks about the relationship between two notions of equivariant cohomology: one with coefficients in a -module and the other graded by representations.

The Relationship Between the Two Notions

The short answer is: **in general, cohomology with co...

This is a deep question about the relationship between two different approaches to equivariant cohomology. Let me clarify the relationship and address your specific question.

The Two Approaches

1. Bredon cohomology (your first bullet): This is cohomology with coefficients in a -module, defined as .

2. RO(G)-graded cohomology (y...