  In this paper we are concerned with the following question: if the tensor
product of finitely generated modules $M$ and $N$ over a local complete
intersection domain is maximal Cohen-Macaulay, then must $M$ or $N$ be a
maximal Cohen-Macaulay? Celebrated results of Auslander, Lichtenbaum, and
Huneke and Wiegand, yield affirmative answers to the question when the ring
considered has codimension zero or one, but the question is very much open for
complete intersection domains that have codimension at least two, even open for
those that are one-dimensional, or isolated singularities. Our argument
exploits Tor-rigidity and proves the following, which seems to give a new
perspective to the aforementioned question: if $R$ is a complete intersection
ring which is an isolated singularity such that dim($R$) > codim($R$), and the
tensor product $M\otimes_R N$ is maximal Cohen-Macaulay, then $M$ is maximal
Cohen-Macaulay if and only if $N$ is maximal Cohen-Macaulay.
