Mathematics, Department of


First Advisor

Srikanth Iyengar

Second Advisor

Roger Wiegand

Date of this Version



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 Professors Srikanth Iyengar and Roger Wiegand. Lincoln, Nebraska: May, 2011
Copyright 2011 Micah Josiah Leamer


This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti numbers of a finitely generated module over an equidimensional local ring are eventually non-decreasing, then the dimensions of sufficiently high syzygies are constant.