a canonical structure of an $H^*(BG)$-algebra.

In case $H^*_G(M)$ is free as an $H^*(BG)$-module, we say that the $G$-action is [*equivariantly formal*]{}. Here is a well-known class of examples of such actions (cf. [@GS Thm. 6.5.3]):

\[exodd\] [If $H^{\rm odd}(M)=0$ then any $G$-action on $M$ is equivariantly formal.]{}

The following criterion will be useful later, see [@Hs p. 46].

\[criteqf\] Let $T$ be a torus acting on a closed manifold $M$.

\(a) The total Betti number of the fixed point set $M^T$ is at most equal to the total Betti number of $M$.

\(b) The two numbers mentioned above are equal if and only if the $T$-action is equivariantly formal.

In what follows we will be looking in more detail at the algebraic structure of $H^*_G(M)$. Relative to its structure of an $H^*(BG)$-module, it is a positively graded module over a positively graded ring. The latter, $H^*(BG)$, is a polynomial algebra on generators of even degrees, and thus a commutative, graded, Noetherian ring. It is also a \*local ring, in the sense that it has a unique graded ideal which is maximal among all graded ideals (namely $H^{>0}(BG)$). The theory of graded modules over \*local Noetherian rings is nicely treated 