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

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 ${Tor}_i^R\left(M{,}^{f^e}R\right)=0$ for $t\le i\le t+dimR$ and infinitely many $e$. This extends results of Herzog, who proved it when $M$ is finitely generated. It is also proved that when $R$ is a Cohen–Macaulay local ring, it suffices that the $Tor$ vanishing holds for one $e\ge {log}_{p}e\left(R\right)$, where $e\left(R\right)$ is the multiplicity of $R$.