[@Ki]) or isotropy actions of compact symmetric spaces (see [@Go]). A larger class consists of the so-called Cohen-Macaulay actions, which are defined in Sect. \[sec:general\] below. Their relevance in the context of equivariant topology was noticed by several authors already in the 70s and early 80s: for example, M. F. Atiyah used the Cohen-Macaulay condition to study equivariant $K$-theory for torus actions in [@Atiyah] and his ideas were adapted to equivariant cohomology by G. E. Bredon in [@Bredon]; see also the paper [@Duflot] by J. Duflot for actions of discrete groups. More recently, Lie group actions whose equivariant cohomology satisfies the requirement above have been considered in [@FranzPuppe2003], before being thoroughly investigated in [@Go-Ro] and [@Go-To]. They are natural generalizations of equivariantly formal group actions to the more general situation when all isotropy subgroups may have their ranks strictly smaller than the rank of the acting group. Cohen-Macaulay actions are worth studying due to their interesting features: for example, just like in the equivariantly formal case, a compact Lie group action and the induced action of a maximal torus satisfy simultaneously this requirement; furthermore, Cohen-Macaulay actions are characterized by the exactness of a certain Atiyah-Bredon long sequence; for more details 