Frobenius & homological dimensions

Abstract

Let R be a commutative Noetherian ring of prime characteristic p and f : R → R the Frobenius endomorphism. For e ≥ 1 let R^(e) denote the ring R viewed as an R-module via f^e. Results of Peskine, Szpiro, and Herzog show that a finitely generated R-module 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 that when R has finite Krull dimension, this statement holds for arbitrary modules. The proof makes use of the theory of at covers and minimal at resolutions developed by E. Enochs and J. Xu, and we prove several results concerning minimal at resolutions. We end by using the vanishing of Tor ^R_i (R^f ,M) for modules, M, of finite at dimension to study the action of the Frobenius functor on Artinian modules.