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 ${\mathbb A}$.

中文翻译：

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Gorenstein 单项式代数的奇点范畴

在本文中，我们研究了奇点范畴 $D_{sg}(\mod A)$ 和 $\mathbb{Z}$-graded 奇点范畴 $D_{sg}(\mod^{\mathbb Z} A)$ Gorenstein 单项式代数 $A$。首先，对于一般正分级的 $1$-Gorenstein 代数，我们证明其 ${\mathbb Z}$-graded 奇点范畴允许一个淤泥物体。其次，对于 $A=kQ/I$ 是 $1$-Gorenstein 单项式代数，通过将每个箭头设置为度 1 将其视为 ${\mathbb Z}$-graded 代数，我们证明 $D_{sg }(\mod^{\mathbb Z} A)$ 有一个倾斜的物体。特别地，$D_{sg}(\mod^{\mathbb Z}A)$ 是等价于有限表示类型遗传代数$H$ 的派生范畴的三角形。最后，我们给出 $1$-Gorenstein 单项式代数 $A$ 的表征，