On the test properties of the Frobenius endomorphism
Olgur Celikbas, Arash Sadeghi, Yongwei Yao
TL;DR
This work investigates test properties of the Frobenius endomorphism on primes characteristic rings by developing FM-type vanishing criteria with the Frobenius twist $^{e}\!M$ in place of $^{e}\!R$ for Cohen–Macaulay modules $M$ with full support. The authors prove three main results: (i) if $\mathrm{Tor}_i^R(^{e}\!M,N)=0$ for $i=t,\dots,t+d-1$ then $\mathrm{fd}_R(N)\le d$; (ii) if $R$ is $F$-finite and $\mathrm{Ext}^i_R(^{e}\!M,N)=0$ in the same range then $\mathrm{id}_R(N)\le d$; and (iii) if $N$ is finitely generated and $\mathrm{Ext}^i_R(N,^{e}\!M)=0$ for that range, then $\mathrm{pd}_R(N)\le t-1$ (or $\min\{t-1,\mathrm{depth}(R)\}$). These results generalize Funk–Marley and yield new characterizations of regularity via Frobenius test properties, including a CM-generalization of CSY and various global/excellent settings. The methods hinge on careful use of Hom–Tor–Ext dualities, injective resolutions, and Frobenius-threshold invariants to transfer vanishing information into finite homological dimensions, advancing the understanding of how Frobenius detects depth and projective behavior in CM modules.
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.