# Tensoring with the Frobenius endomorphism

## TL;DR

This paper investigates the behavior of tensor products involving the Frobenius endomorphism over certain Cohen-Macaulay rings, revealing new torsion properties and relaxing previous assumptions about modules.

## Contribution

It replaces the rank condition with local freeness on minimal primes and shows torsion in tensor products over specific rings, expanding understanding of Frobenius actions.

## Key findings

- Tensor product with Frobenius endomorphism has torsion in certain rings.
- Relaxed conditions allow for broader applicability of torsion results.
- Existence of non-free modules with torsion-free self-tensor products over these rings.

## 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 ${}^{\varphi^n}\!R$ 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.