# Frobenius and Homological Dimensions of Complexes

## TL;DR

This paper establishes new criteria for finite flat dimension of modules over Noetherian local rings using Frobenius endomorphism-based Tor vanishing conditions, generalizing and improving previous results.

## Contribution

It proves that Tor vanishing conditions involving Frobenius powers imply finite flat dimension, extending known theorems to broader classes of modules and rings.

## Key findings

- Finite flat dimension follows from Tor vanishing over prime characteristic rings.
- Single Tor vanishing at specific degrees suffices for complete intersection rings.
- Results generalize previous theorems to arbitrary modules and improve existing conditions.

## Abstract

It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $e\geq \log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as generalizing a theorem due to C. Miller from finitely generated modules to arbitrary modules. We also show that if $R$ is a complete intersection ring then the vanishing of Tor$_i^R({}^e R, M)$ for single positive values of $i$ and $e$ is sufficient to imply $M$ has finite flat dimension. This extends a result of L. Avramov and C. Miller.