Date of Award

June 2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Graham J. Leuschke

Keywords

Commutative Algebra, Representation Theory

Subject Categories

Physical Sciences and Mathematics

Abstract

ABSTRACT

Orders are a certain class of noncommutative algebras over commutative rings. Originally

defined by Auslander and Bridger, an R-order is an R-algebra which is a maximal CohenMacaulay

R-module. In this thesis we consider orders, Λ, over Cohen-Macaulay local rings

R possessing a canonical module, ωR. In this case a great deal of structure is imposed on Λ.

In Chapter 3 we focus on the use of orders as noncommutative resolutions of commutative

local rings. This idea was introduced by Van den Bergh [45] for R Gorenstein and we

investigate the generalization to the case where R is Cohen-Macaulay. We show that if

an order is totally reflexive over R and has finite global dimension, then R was already

Gorenstein. Further, we investigate Gorenstein orders and give a necessary and sufficient

condition for the endomorphism ring EndR(R ⊕ ω) to be a Gorenstein order.

The rest of the thesis focuses on various aspects of the representation theory of orders.

We investigate orders which have finite global dimension on the punctured spectrum, but

are not necessarily isolated singularities. In this case we are able to prove a generalization

of Auslander’s theorem about finite CM type [3]. We prove that if an order which satisfies

projdimΛop ωΛ 6 n possesses only finitely many indecomposable n

th syzygies of MCM Λ-

modules, then in fact gldim Λp 6 n + dim Rp for all non-maximal primes p. We are then

able to translate this to a condition on R by considering path algebras, since these maintain

finiteness of global dimension.

Finally, we consider orders which are true isolated singularities and Iyama’s higher

Auslander-Reiten theory [27]. We consider the action of τn on n-orthogonal subcategories

of CM Λ and on n-cluster tilting subcategories. For the former we are able to characterize

the projective dimension of duals of modules. For the latter, we provide an obstruction to a

module being τn-periodic, a question of great interest for the representation theory of orders

of finite global dimension.

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