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

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 $t\ge 0$suchthat$\operatorname{Tor}_i^R(M, {}^{f^{e}}!R)=0$for$t\le i\le t+\dim R$andinfinitelymany$e$\mathrm{.}ThisextendsaresultofHerzog,whoproveditwhen$M$ 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 $R$.