Abstract. Let R be a commutative Noetherian local ring of prime characteristic p and f : R−→R the Frobenius ring homomorphism. For e ≥ 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 Tor R i (R (e) , M ) = 0 for all i > 0 and all (equivalently, infinitely many) e ≥ 1. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.
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