Morita theorem for hereditary Calabi-Yau categories

Abstract

We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum $T\oplus \cdots \oplus T[-(d-2)]$ has hereditary endomorphism algebra $H$ is triangle equivalent to the orbit category ${\mathcal{D}}^b(\mathrm{mod}H)/{\tau }^{-1/(d-1)}[1]$ of the derived category of $H$ for a naturally defined $(d-1)$-st root ${\tau }^{1/(d-1)}$ of the AR translation, provided $H$ is of non-Dynkin type. We also show that hereditaryness of $H$ follows from that of $T$ is when $d=3$, that of $T\oplus T[-1]$ when $d=4$, and similarly from a smaller endomorphism algebra for higher dimensions under vanishing of some negative self-extensions of $T$. Our result therefore generalizes the established theorems by Keller--Reiten and Keller--Murfet--Van den Bergh. Furthermore, we show that enhancements of such triangulated categories are unique. Finally we apply our results to Calabi-Yau reductions of a higher cluster category of a finite dimensional algebra and of the singularity category of an invariant subring.

37 pages. Section 4.3 and Section 6 added