On the test properties of the Frobenius endomorphism

TL;DR

This paper proves new theorems about the Frobenius endomorphism's test properties in prime characteristic rings, extending previous results and providing applications for characterizing regularity.

Contribution

It generalizes existing theorems on Ext and Tor vanishing and Cohen-Macaulay tensor products using modules viewed through Frobenius iterations.

Findings

Generalized vanishing results for Ext and Tor modules.

Extended Cohen-Macaulay tensor product results.

Provided new characterizations of regular local rings.

Abstract

In this paper, we prove two theorems concerning the test properties of the Frobenius endomorphism over commutative Noetherian local rings of prime characteristic $p$. Our first theorem generalizes a result of Funk-Marley on the vanishing of Ext and Tor modules, while our second theorem generalizes one of our previous results on maximal Cohen-Macaulay tensor products. In these earlier results, we replace $^{e}R$ with a more general module $^{e}M$, where $R$ is a Cohen-Macaulay ring, $M$ is a Cohen-Macaulay $R$-module with full support, and $^{e}M$ is the module viewed as an $R$-module via the $e$-th iteration of the Frobenius endomorphism. We also provide examples and present applications of our results, yielding new characterizations of the regularity of local rings.