The Acyclicity of the Frobenius Functor for Modules of Finite Flat Dimension

TL;DR

This paper proves that the vanishing of certain Tor modules involving the Frobenius functor characterizes modules of finite projective dimension over Noetherian local rings of prime characteristic, extending previous results to all modules.

Contribution

It generalizes the characterization of finite projective dimension via Frobenius Tor vanishing from finitely generated modules to all modules using flat cover theory.

Findings

Characterization of finite projective dimension via Frobenius Tor vanishing for all modules.

Extension of classical results to arbitrary modules.

Application of flat cover and minimal flat resolution techniques.

Abstract

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and Herzog state that for finitely generated modules $M$, $M$ has finite projective dimension if and only if $\operatorname{Tor}_i^R(R^{(e)},M)=0$ for all $i>0$ and all (equivalently, infinitely many) $e\ge 1$. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.