Tilting objects in singularity categories and levelled mutations

Abstract

We show the existence of tilting objects in the singularity category $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ associated to certain noetherian AS-regular algebras $A$ and idempotents $e$. This gives a triangle equivalence between $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ and the derived category of a finite-dimensional algebra. In particular, we obtain a tilting object if the Beilinson algebra of $A$ is a levelled Koszul algebra. This generalises the existence of a tilting object in $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(S^G)$, where $S$ is a Koszul AS-regular algebra and $G$ is a finite group acting on $S$, found by Iyama-Takahashi and Mori-Ueyama. Our method involves the use of Orlov's embedding of $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ into $\mathsf{D}^{\operatorname{b}}(\mathsf{qgr} eAe)$, the bounded derived category of graded tails, and of levelled mutations on a tilting object of $\mathsf{D}^{\operatorname{b}}(\mathsf{qgr} eAe)$.