  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.
