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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Stangle, Josh John"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Graham J. Leuschke"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-06-30T07:00:00Z","date_published":"2017-06-30T07:00:00Z","updated_at":"2026-07-24T04:55:19Z","subjects":["Commutative Algebra","Representation Theory","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/678","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Graham J. Leuschke"]},{"key":"dc:creator","label":"Author","values":["Stangle, Josh John"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Commutative Algebra","Representation Theory","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/678"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Representation Theory of Orders over Cohen-Macaulay Rings"]}]}],"canonical_facts":{"dc:contributor":["Graham J. Leuschke"],"dc:creator":["Stangle, Josh John"],"dc:description.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.
"],"dc:identifier":["https://surface.syr.edu/etd/678"],"dc:subject":["Commutative Algebra","Representation Theory","Physical Sciences and Mathematics"],"dc:title":["Representation Theory of Orders over Cohen-Macaulay Rings"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:55:19Z"}