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 ${\mathsf{D}}^b(\mathsf{qgr}eAe)$, the bounded derived category of graded tails, and of levelled mutations on a tilting object of ${\mathsf{D}}^b(\mathsf{qgr}eAe)$.

18 pages. Comments welcome