The singularity category of an algebra with radical square zero

TL;DR

This paper characterizes the singularity category of artin algebras with radical square zero using associated regular algebras and bimodules, providing criteria for Hom-finiteness based on the algebra's quiver.

Contribution

It introduces a new description of the singularity category for these algebras via triangulated categories and establishes a criterion for Hom-finiteness based on the valued quiver.

Findings

Complete description of the singularity category as a triangulated category

Criterion for Hom-finiteness in terms of the algebra's quiver

Connection between algebraic data and categorical properties

Abstract

To an artin algebra with radical square zero, a regular algebra in the sense of von Neumann and a family of invertible bimodules over the regular algebra are associated. These data describe completely, as a triangulated category, the singularity category of the artin algebra. A criterion on the Hom-finiteness of the singularity category is given in terms of the valued quiver of the artin algebra.