Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism

TL;DR

This paper establishes criteria for modules over Noetherian rings of positive characteristic to have finite flat dimension, based on vanishing of certain Tor groups involving iterates of the Frobenius endomorphism, extending previous results.

Contribution

It generalizes Herzog's finite generation case and improves conditions needed for detecting finite flat dimension via Frobenius iterates.

Findings

Finite flat dimension characterized by Tor vanishing for infinitely many Frobenius powers.

Extension of Herzog's result to arbitrary modules, not just finitely generated.

In Cohen-Macaulay rings, a single Frobenius iterate suffices for detection.

Abstract

It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and infinitely many $e$. This extends a result of Herzog, who proved it when $M$ is finitely generated, and strengthens a result of the third author and Webb in the case $M$ is arbitrary. It is also proved that when $R$ is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one $e\ge \log_{p}e(R)$, where $e(R)$ is the multiplicity of $R$.