  Let $R$ be a commutative Noetherian local ring of prime characteristic $p$
and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$
denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine,
Szpiro, and Herzog state that for finitely generated modules $M$, $M$ has
finite projective dimension if and only if
$\operatorname{Tor}_i^R(R^{(e)},M)=0$ for all $i>0$ and all (equivalently,
infinitely many) $e\ge 1$. We prove this statement holds for arbitrary modules
using the theory of flat covers and minimal flat resolutions.

Commutative Algebra