Tensoring with the Frobenius endomorphism

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 $n \gg 0$. 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.

Keywords

Frobenius endomorphism, tensor product of modules, rank and torsion

2010 Mathematics Subject Classification

13A35, 13D07, 13H10

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Received 28 June 2017

Received revised 17 October 2017

Published 21 February 2018