On the test properties of the Frobenius endomorphism

Olgur Celikbas, Arash Sadeghi and Yongwei Yao

Vol. 342 (2026), No. 2, 235–257

DOI: 10.2140/pjm.2026.342.235

Abstract

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 and 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.

Keywords

Cohen–Macaulay module, depth and torsion properties of tensor products of modules, Frobenius endomorphism, ring of prime characteristic, Ext and Tor

Mathematical Subject Classification

Primary: 13A35, 13C14, 13D07

Secondary: 13C10, 13C11

Milestones

Received: 13 March 2025

Revised: 31 January 2026

Accepted: 1 February 2026

Published: 23 April 2026