Cohen–Macaulay Dimension for Coherent Rings

Authors

• Rebecca Egg, University of Nebraska-LincolnFollow

First Advisor

Thomas Marley

Date of this Version

5-2016

Document Type

Dissertation

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 Professor Tom Marley. Lincoln, Nebraska: May, 2016

Copyright (c) 2016 Rebecca Egg

Abstract

This dissertation presents a homological dimension notion of Cohen-Macaulay for non-Noetherian rings which reduces to the standard definition in the case that the ring is Noetherian, and is inspired by the homological notion of Cohen-Macaulay for local rings developed by Gerko. Under this notion, both coherent regular rings (as defined by Bertin) and coherent Gorenstein rings (as defined by Hummel and Marley) are Cohen-Macaulay.

This work is motivated by Glaz's question regarding whether a notion of Cohen-Macaulay exists for coherent rings which satisfies certain properties and agrees with the usual notion when the ring is Noetherian. Hamilton and Marley gave one answer; we develop an alternative approach using homological dimensions which seems to have more satisfactory properties. We explore properties of coherent Cohen-Macaulay rings, as well as their relationship to non-Noetherian Cohen-Macaulay rings as defined by Hamilton and Marley.

Adviser: Tom Marley