Rigidity of the Frobenius, Matlis Reflexivity, and Minimal Flat Resolutions

Authors

Douglas J. Dailey, University of Nebraska-LincolnFollow

Date of this Version

Spring 4-14-2016

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Thomas Marley. Lincoln, Nebraska: May, 2016

Copyright (c) 2016 Douglas J. Dailey

Abstract

Let R be a commutative, Noetherian ring of characteristic p >0. Denote by f 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.

Adviser: Thomas Marley