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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

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  • $\begingroup$ Obviously the answer depends on the ring and the module, so what do you expect? You hope this is true for all Cohen-Macaulay? Do you have any nontrivial example? $\endgroup$
    – user42690
    Nov 24, 2015 at 15:49

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