  Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has
positive dimension and prime characteristic. Li proved that the tensor product
of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism
${}^{\varphi^n}\!R$ is not maximal Cohen-Macaulay provided that $M$ has rank
and $n\gg 0$. We replace the rank hypothesis with the weaker assumption that
$M$ is locally free on the minimal prime ideals of $R$. As a consequence, we
obtain, if $R$ is a one-dimensional non-regular complete reduced local ring
that has a perfect residue field and prime characteristic, then
${}^{\varphi^n}\!R \otimes_{R}{}^{\varphi^n}\!R$ has torsion for all $n\gg0$.
This property of the Frobenius endomorphism came as a surprise to us since,
over such rings $R$, there exist non-free modules $M$ such that $M\otimes_{R}M$
is torsion-free.
