Singularity Categories of B\"ackstr\"om Orders

TL;DR

This paper characterizes the singularity categories of Bäckström orders, linking their algebraic properties to von Neumann regular algebras and classifying various Gorenstein conditions.

Contribution

It provides explicit descriptions of singularity categories and establishes singular equivalences for Bäckström orders, advancing understanding of their homological properties.

Findings

Explicit descriptions of singularity categories via von Neumann regular algebras

Singular equivalences with finite dimensional radical square zero algebras

Classification of Gorenstein and related properties of Bäckström orders

Abstract

B\"ackstr\"om 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 B\"ackstr\"om orders via certain von Neumann regular algebras and associated bimodules. We further provide singular equivalences between B\"ackstr\"om orders and specific finite dimensional radical square zero algebras. We also classify weakly regular, Gorenstein, Iwanaga-Gorenstein and sg-Hom-finite B\"ackstr\"om orders.