Tensor product of maximal Cohen-Macaulay modules

Question

Let $(R,\frak{m})$ be a commutative Cohen-Macaulay local ring and $M$ and $N$ two maximal Cohen-Macaulay modules of Krull dimension $d$. Is $M\otimes_RN$ Cohen-Macaulay?

Answer 1

No. Tensor products resist being Cohen-Macaulay outside of exceedingly strong hypotheses. For a class of examples, let $R$ be a local domain of dimension $1$ that is not a DVR and let $I$ be a nonprincipal ideal. Then $I$ is maximal Cohen-Macaulay as an $R$-module since it embeds into $R$ and we are in dimension $1$. But $I \otimes_R I$ cannot be Cohen-Macaulay. Indeed, if $x,y$ is part of a minimal generating set of $I$, one may observe that $x \otimes y-y \otimes x$ is a nonzero torsion element of $I \otimes_R I$.