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Rigidity of the frobenius, matlis reflexivity, and minimal flat resolutions
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 fe. 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 ToriR(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.
Dailey, Douglas J, "Rigidity of the frobenius, matlis reflexivity, and minimal flat resolutions" (2016). ETD collection for University of Nebraska - Lincoln. AAI10099960.