The acyclicity of the Frobenius functor for modules of finite flat dimension

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