Singularity categories of locally bounded categories with radical square zero

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 $\mathscr{C}$.