Cohen-Macaulay property is preserved under orbit equivalence. This turns out to be true: see Thm. \[orbit\], which represents the main result of Sect. \[oea\]. The proof of Thm. \[thm:realmain\] is also given in full detail in that section.

[**Acknowledgement.**]{} We would like to thank the referee for numerous valuable comments on a previous version of this paper.

General considerations {#sec:general}
======================

The action of $G$ on $M$ {#cmsec}
------------------------

Throughout this subsection $G$ will always be a compact and connected Lie group which acts smoothly on a closed manifold $M$, although many of the results presented below hold in a larger generality. To any such action one attaches the equivariant cohomology $H^*_G(M)$. This can be defined as the usual cohomology of the Borel construction $EG\times_G M$, where $EG\to BG$ is the classifying principal bundle of $G$. This construction originally belongs to A. Borel [@Bo] and it became gradually an important tool in the theory of transformation groups by work of M. F. Atiyah and R. Bott [@At-Bo], D. Quillen [@Quillen], and many others. The book [@AlldayPuppe] by C. Allday and V. Puppe is a useful reference for this topic. We confine ourselves to mentioning the basic fact that $H^*_G(M)$ has 