# Singularity categories of Gorenstein monomial algebras

Research paper by Ming Lu, Bin Zhu

Indexed on: 01 Aug '17Published on: 01 Aug '17Published in: arXiv - Mathematics - Representation Theory

#### Abstract

In this paper, we study the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ of a Gorenstein monomial algebra $A$. Firstly, for a general positively graded $1$-Gorenstein algebra, we prove that its ${\mathbb Z}$-graded singularity category admits a silting object. Secondly, for $A=kQ/I$ being a $1$-Gorenstein monomial algebra, which is viewed as a ${\mathbb Z}$-graded algebra by setting each arrow to be degree one, we prove that $D_{sg}(\mod^{\mathbb Z} A)$ has a tilting object. In particular, $D_{sg}(\mod^{\mathbb Z}A)$ is triangle equivalent to the derived category of a hereditary algebra $H$ which is of finite representation type. Finally, we give a characterization of $1$-Gorenstein monomial algebra $A$, and describe its singularity category by using the triangulated orbit categories of type ${\mathbb A}$.