Singularity categories of locally bounded categories with radical square zero

TL;DR

This paper characterizes various singularity categories of locally bounded categories with radical square zero, establishing triangle equivalences with orbit categories and providing methods to compute generators from quivers.

Contribution

It provides a complete description of singularity categories for such categories, extending prior work and offering practical computation techniques.

Findings

Established triangle equivalences with orbit categories.

Provided explicit descriptions of singularity categories.

Offered examples for computing generators from quivers.

Abstract

This paper studies several singularity categories of a locally bounded $k-$linear category $\mathscr{C}$ with radical square zero. Following the work of Bautista and Liu [6], we give a complete description of $D^{b}_{sg}(\mathscr{C})$, $D^{b}_{sg}(\mathscr{C}^{op})$, $D^{-}_{sg}(proj$-$\mathscr{C})$, and $D^{+}_{sg}(inj$-$\mathscr{C})$ by proving a triangle equivalences between these categories and certain orbit categories of the bounded derived categories of certain semisimple abelian categories of representations. In the end, we will give some examples to show how one can easily compute the generators of $D_{sg}(\mathscr{C})$ from the quiver of $\mathscr{C}$.