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Let $A, B, C$ be Noetherian Rings such that $A$ is a subring of $B$ and there exists a ring homomorphism $A \rightarrow C$. Let $M$ be a $(B,C)$ bimodule, i.e. $M$ is both a $B$-module and a $C$-module. Then $M$ is also an $A$-module.

Question 1: If $M$ is Cohen-Macaulay as an $A$-module, as a $B$-module and as a $C$-module, then is $M$ Cohen-Macaulay as a $B \otimes_A C$ module?

Question 2: Are there any known conditions for $M$ to be Cohen-Macaulay as a $B \otimes_A C$ module?

PS: Remark this question is motivated by my previous question: showing Cohen-Macaulay property is preserved under a ring extension

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Manos
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  • I'm confused. Isn't it possible that your setup could force $M$ to have two different $A$-module structures (one from restriction of scalars from the map $A \subseteq C$, the other from restriction of scalars of $B$ to its subring $A$)? If so, then your question is ambiguous. – neilme Apr 03 '14 at 00:46
  • @neilme: That's a good point. I suppose that we should ask that the two $A$-module structures then coincide. – Manos Apr 03 '14 at 01:13

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