Singularity Categories of Bäckström Orders
Hongrui Wei
TL;DR
The paper develops a concrete bridge between singularity theory of Bäckström orders and radical square zero algebras by constructing von Neumann regular models that capture $D_{sg}(Λ)$ as a category of projective modules over a regular algebra $V(Λ)$. It then proves a singular equivalence between Bäckström orders and their associated radical square zero algebras via the trivial extension $A(Λ)=D⊕M$, and provides precise quiver-based criteria to classify sg-Hom-finite, weakly regular, Gorenstein, and Iwanaga–Gorenstein Bäckström orders. The results unify and extend known statements for Artin algebras to the one-dimensional, order-theoretic setting, with explicit functors linking CM Λ to module categories over a finite-dimensional hereditary algebra $H$. The framework leverages stabilization, GP theory, and endomorphism-algebra techniques to yield both structural descriptions and actionable classification criteria, with several illustrative examples. Overall, the work deepens the interplay between CM representation theory, singularity categories, and finite-dimensional quiver mutations in the context of Bäckström orders.
Abstract
Bäckström orders are a class of algebras over complete discrete valuation rings. Their Cohen-Macaulay representations are in correspondence with the representations of certain quivers/species by Ringel and Roggenkamp. In this paper, we give explicit descriptions of the singularity categories of Bäckström orders via certain von Neumann regular algebras and associated bimodules. We further provide singular equivalences between Bäckström orders and specific finite dimensional radical square zero algebras. We also classify weakly regular, Gorenstein, Iwanaga-Gorenstein and sg-Hom-finite Bäckström orders.