Singularity categories of Gorenstein monomial algebras

Abstract

In this paper, we consider the singularity category $D_{sg}(\mathrm{mod}A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mathrm{mod}{}^{\mathbb{Z}}A)$ 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 $A=KQ/I$ being a Gorenstein monomial algebra, we prove that $D_{sg}(\mathrm{mod}{}^{\mathbb{Z}}A)$ has tilting objects. As a consequence, $D_{sg}(\mathrm{mod}{}^{\mathbb{Z}}A)$ 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 $\mathbb{A}$.

36 pages, some changes, to appear in JPAA

References in corpus (2)

• On triangulated orbit categories
• Singularity categories of representations of algebras over local rings

Cited by in corpus (6)

• Tilting theory for Gorenstein rings in dimension one
• Finite generation for Hochschild cohomology of Gorenstein monomial algebras
• Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras
• Singularity categories of representations of algebras over local rings
• On Cohen-Macaulay Auslander algebras
• Higher gentle algebras