in [@BrunsHerzog Sect. 1.5]. Let us just briefly recall that to each such module one can associate its depth, that is, the maximal length of a regular sequence of elements in the maximal graded ideal. One can also associate its Krull dimension, which is the Krull dimension of the ring modulo the annihilator of the module. In general, the latter number is at least equal to the depth. We say that the module is Cohen-Macaulay if it is equal to 0 or its Krull dimension is equal to its depth. For a more detailed account of this topic, suitable for applications to equivariant cohomology, we refer to [@Go-To Sect. 5].

\[def:cm\] A smooth action $G\times M \to M$ is called [Cohen-Macaulay]{} if $H^*_G(M)$, considered as a module over $H^*(BG)$, is Cohen-Macaulay.

A systematic study of this type of actions was undertaken in [@Go-Ro] and [@Go-To] and we will often rely on results obtained there. However, some caution is necessary, since in the two references above the coefficient field for cohomology is $\bR$, whereas in this paper it is $\bQ$. We will next show that the two seemingly different Cohen-Macaulayness conditions are in fact equivalent. This will be done by using 