Singularity categories of Gorenstein monomial algebras

In this paper, we consider the singularity category $D_{sg}(\mathrm{m}\mathrm{o}\mathrm{d}A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mathrm{m}\mathrm{o}\mathrm{d}{}^{\mathbb{Z}}A)$ for a Gorenstein monomial algebra $A$. Firstly, for a positively graded $1$-Gorenstein algebra, we prove that its $\mathbb{Z}$-graded singularity category admits silting objects. Secondly, for $A=KQ\mathrm{/}I$ being a Gorenstein monomial algebra, we prove that $D_{sg}(\mathrm{m}\mathrm{o}\mathrm{d}{}^{\mathbb{Z}}A)$ has tilting objects. As a consequence, $D_{sg}(\mathrm{m}\mathrm{o}\mathrm{d}{}^{\mathbb{Z}}A)$ is triangulated 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 algebras, and describe their singularity categories clearly by using the triangulated orbit categories of type $\mathbb{A}$.