Representation Theory of Orders over Cohen-Macaulay Rings

<p>ABSTRACT</p> <p>Orders are a certain class of noncommutative algebras over commutative rings. Originally</p> <p>defined by Auslander and Bridger, an R-order is an R-algebra which is a maximal CohenMacaulay</p> <p>R-module. In this thesis we consider orders, Λ, over Cohen-Macaulay local rings</p> <p>R possessing a canonical module, ωR. In this case a great deal of structure is imposed on Λ.</p> <p>In Chapter 3 we focus on the use of orders as noncommutative resolutions of commutative</p> <p>local rings. This idea was introduced by Van den Bergh [45] for R Gorenstein and we</p> <p>investigate the generalization to the case where R is Cohen-Macaulay. We show that if</p> <p>an order is totally reflexive over R and has finite global dimension, then R was already</p> <p>Gorenstein. Further, we investigate Gorenstein orders and give a necessary and sufficient</p> <p>condition for the endomorphism ring EndR(R ⊕ ω) to be a Gorenstein order.</p> <p>The rest of the thesis focuses on various aspects of the representation theory of orders.</p> <p>We investigate orders which have finite global dimension on the punctured spectrum, but</p> <p>are not necessarily isolated singularities. In this case we are able to prove a generalization</p> <p>of Auslander’s theorem about finite CM type [3]. We prove that if an order which satisfies</p> <p>projdimΛop ωΛ 6 n possesses only finitely many indecomposable n</p> <p>th syzygies of MCM Λ-</p> <p>modules, then in fact gldim Λp 6 n + dim Rp for all non-maximal primes p. We are then</p> <p>able to translate this to a condition on R by considering path algebras, since these maintain</p> <p>finiteness of global dimension.</p> <p>Finally, we consider orders which are true isolated singularities and Iyama’s higher</p> <p>Auslander-Reiten theory [27]. We consider the action of τn on n-orthogonal subcategories</p> <p>of CM Λ and on n-cluster tilting subcategories. For the former we are able to characterize</p> <p>the projective dimension of duals of modules. For the latter, we provide an obstruction to a</p> <p>module being τn-periodic, a question of great interest for the representation theory of orders</p> <p>of finite global dimension.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2017

Author dc:creator

• Stangle, Josh John

Contributors dc:contributor

• Graham J. Leuschke

• Commutative Algebra
• Representation Theory
• Physical Sciences and Mathematics

Repository record dc:identifier

https://surface.syr.edu/etd/678

OAI identifier oai:identifier

oai:surface.syr.edu:etd-1678