Tilting objects in singularity categories and levelled mutations

We show the existence of tilting objects in the singularity category ${\mathsf{D}}_{\mathsf{S}\mathsf{g}}^{\mathsf{g}\mathsf{r}}(eAe)$ associated to certain noetherian AS-regular algebras $A$ and idempotents $e$. This gives a triangle equivalence between ${\mathsf{D}}_{\mathsf{S}\mathsf{g}}^{\mathsf{g}\mathsf{r}}(eAe)$ 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 $$\mathsf{D}${\mathsf{ Sg}}^{$\mathsf{g}\mathsf{r}$}(S^G)$,where$S$isaKoszulAS-regularalgebraand$G$ is a finite group acting on $S$, found by Iyama-Takahashi and Mori-Ueyama. Our method involves the use of Orlov’s embedding of $$\mathsf{D}${\mathsf{ Sg}}^{$\mathsf{g}\mathsf{r}$}(eAe)$into$$\mathsf{D}$^{\operatorname{b}}($\mathsf{q}\mathsf{g}\mathsf{r}$ eAe)$, the bounded derived category of graded tails, and of levelled mutations on a tilting object of ${\mathsf{D}}^{\mathrm{b}}(\mathsf{q}\mathsf{g}\mathsf{r}eAe)$.