Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-module. One operation we can perform is to construct $M^{[x]}$, which consists of elements of $M$ which are killed by high power of $x$. Another operation is to construct $M / IM$, where $I$ is a maximal ideal in $A$.
My question is, what conditions on $M$ would guarantee that the operations commute; i.e. $M^{[x]} / I M^{[x]} \to (M / IM)^{[x]}$ is an isomorphism. Is it related somehow to $M$ being Cohen-Macaulay over $A$?
Thank you, Sasha