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arXiv · 2502.15048

On the test properties of the Frobenius endomorphism

Olgur Celikbas·Arash Sadeghi·Yongwei Yao
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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.

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Olgur Celikbas, Arash Sadeghi, Yongwei Yao. 2025-02-20. On the test properties of the Frobenius endomorphism. https://doi.org/10.2140/pjm.2026.342.235

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