preserved under taking the direct product). Thus the $T_1\times T_2$-action on $M_1 \times M_2$ is Cohen-Macaulay, and so is the $G_1\times G_2$-action.

We will need an equivalent characterization of the Cohen-Macaulay condition above, this time exclusively in terms of the ring structure of $H^*_G(M)$. The latter is in general not a commutative ring, but it is nevertheless graded commutative, in the sense that $x\cdot y =(-1)^{({\rm deg} x)({\rm deg} y)}y\cdot x$, for all homogeneous $x, y \in H^*_G(M)$. Although a self-contained treatment of graded commutative rings is not immediately available in the literature, there is no essential difference relative to the (usual) commutative case. A systematic and thorough study of graded commutative rings has been undertaken by M. Poulsen in Appendix A of his Master Thesis [@Po]. For instance, by using the remark following Prop. A.5 in [@Po] and also taking into account that $H^*_G(M)$ is finitely generated as an algebra over its degree zero component $H^0_G(M)\simeq \bQ$ (see for instance [@Quillen Cor. 2.2]), we deduce that $H^*_G(M)$ is a Noetherian ring. For graded commutative Noetherian rings of the type $R=\oplus_{i\ge 0}R^i$ whose degree zero component $R^0$ is a field, one can see in [@Po] that the concepts of Krull 