Tensoring with the Frobenius endomorphism

Abstract

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 ${}^{{\phi }^n}R$ 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 ${}^{{\phi }^n}R{\otimes }_R{}^{{\phi }^n}R$ has torsion for all $n\gg 0$. 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.