dimension, depth (relative to the ideal $\oplus_{i>0}R^i$), and Cohen-Macaulayness can be defined in the same way as for commutative rings. This enables us to prove the following result. It is obtained by adapting [@Se Prop. 12, Sect. IV.B] to our current set-up.

[(J.-P. Serre)]{} \[serre\] Let $R=\oplus_{i\ge 0}R^i$ and $S=\oplus_{i\ge 0}S^i$ be two graded commutative Noetherian rings with $R^0=S^0=\bQ$ and $\varphi: R\to S$ a homomorphism of graded rings which makes $S$ into an $R$-module which is finitely generated. Let also $A$ be a finitely generated $S$-module. Then $A$ is Cohen-Macaulay as an $S$-module if and only if so is $A$ as an $R$-module.

\[cmcrit\] The $G$-action on $M$ is Cohen-Macaulay if and only if the ring $H^*_G(M)$ is Cohen-Macaulay.

The action of $K$ on $G/H$
--------------------------

Let $G$ be again a compact and connected Lie group and let $K, H\subset G$ be closed subgroups. In this section we list some results concerning the three group actions mentioned in the following proposition.

\[crit\] If $K$ and $H$ are connected, the following assertions are equivalent:

-   the action of $K$ on $G/H$ by left translations is Cohen-Macaulay;

-   the action of $H$ on $G/K$ by left translations is Cohen-Macaulay;

-   the action 