Date of Award
Doctor of Philosophy (PhD)
Graham J. Leuschke
Commutative Algebra, Representation Theory
Physical Sciences and Mathematics
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  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 . 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 . 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.
Stangle, Josh John, "Representation Theory of Orders over Cohen-Macaulay Rings" (2017). Dissertations - ALL. 678.