When is the tensor product of Cohen-Macaulay modules Cohen-Macaulay?

Let M1M_1 and M2M_2 be Cohen-Macaulay modules over a ring RR. When is M1⊗RM2M_1 \otimes_R M_2 a Cohen-Macaulay module over RR?

That is the finite question I have. Motivation: I know that when MM is a Cohen-Macaulay module then M⊗RR[x1,…,xn]M \otimes_R R[x_1,…,x_n] is Cohen-Macaulay. Power series over Cohen-Macaulay modules are also Cohen-Macaulay, if the proof I did just now is correct. These are both examples of positive answers to my question.

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What kind of answer are you expecting? Clearly, if one of them is free, this is correct, why it works for polynomial rings. In general it is false. There are no easy conditions that I know of, except trivial conditions as above. If you have some specific situation in your mind, it would be helpful.
– Mohan
Oct 21 at 1:28

  

 

Let me try to think of what the counterexamples look like in the morning. I will come back to comment when I am done.
– Hari Rau-Murthy
Oct 21 at 4:22

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