  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 $R$.
