Frobenius criteria of freeness and Gorensteinness

TL;DR

This paper investigates how the Frobenius functor influences module properties, establishing criteria for freeness and Gorensteinness in local rings, and introduces rigidity results for certain modules.

Contribution

It provides new Frobenius-based criteria for Gorensteinness and freeness, improving previous characterizations and exploring module rigidity over Cohen-Macaulay rings.

Findings

Maximal Cohen-Macaulay modules under Frobenius are free if they have a rank.

New Frobenius criteria for Gorensteinness of local rings.

Finite length modules over Cohen-Macaulay rings exhibit rigidity against Frobenius.

Abstract

Let $F^n(-)$ be the Frobenius functor of Peskine and Szpiro. In this note, we show that the maximal Cohen-Macaulayness of $F^n(M)$ forces $M$ to be free, provided $M$ has a rank. We apply this result to obtain several Frobenius related criteria for the Gorensteinness of a local ring $R$, one of which improves a previous characterization due to Hanes and Huneke. We also establish a special class of finite length modules over Cohen-Macaulay rings, which are rigid against Frobenius.