(cf., e.g., [@Go-Ma Prop. 2.1]), and on the other hand the previous map factors as $H^*_G(M) \to H^*_H(M) \to H^*(M)$. However, it is possible for the $G$-action to be Cohen-Macaualy and the $H$-action not to be like that, see [@Go-Ro Ex. 2.13].

From the previous proposition one can deduce:

\[dirprod\] If two smooth actions on closed manifolds are Cohen-Macaulay then their direct product is Cohen-Macaulay as well.

Let the two group actions be $G_i \times  M_i \to M_i$ and let $T_i \subset G_i$ be maximal tori, where $i=1,2$. By Prop. \[maxtor\] above, the restricted $T_i$-action on $M_i$ is Cohen-Macaulay. By [@Go-To Rem. 2], the restricted $T_i$-action on $M_i$ is Cohen-Macaulay if and only if there exists a subtorus $S_i \subset T_i$ of rank equal to the minimal dimension of a $T_i$-orbit on $M_i$, which acts locally freely on $M_i$ such that the induced action of $T/S_i$ on the orbit space $M/S_i$ is equivariantly formal. But then $S_1 \times S_2$ acts locally freely on $M_1 \times M_2$, has rank equal to the minimal dimension of a $T_1\times T_2$-orbit on $M_1\times M_2$ and the induced action of $T_1/S_1 \times T_2/S_2$ on $M_1/S_1 \times M_2/S_2$ is equivariantly formal (note that equivariant formality is 