Rigidity of the frobenius, matlis reflexivity, and minimal flat resolutions

Abstract

Let R be a commutative, Noetherian ring of characteristic p > 0. Denote by f R → R the Frobenius endomorphism, and let R^(e) denote the ring R viewed as an R-module via f^e. Following on classical results of Peskine, Szpiro, and Herzog, Marley and Webb use flat, cotorsion module theory to show that if R has finite Krull dimension, then an R-module M has finite flat dimension if and only if Tor_i^R(R ^(e),M) = 0 for all i > 0 and infinitely many e > 0. Using methods involving the derived category, we show that one only needs vanishing for dim R +1 consecutive values of i and infinitely many values of e to conclude that M has finite flat dimension. We also study a general notion of Matlis duality and give a change of rings result for Matlis reflexive modules.