Morita theorem for hereditary Calabi-Yau categories

TL;DR

This paper establishes a structure theorem for certain Calabi-Yau triangulated categories with hereditary cluster tilting objects, linking them to orbit categories of derived categories of hereditary algebras, and generalizes previous results.

Contribution

It provides a new characterization of algebraic Calabi-Yau categories with hereditary cluster tilting objects, extending known theorems and exploring their enhancements and applications.

Findings

Categories are equivalent to orbit categories of derived categories of hereditary algebras.

Hereditaryness of endomorphism algebra follows from that of the cluster tilting object in specific cases.

Enhancements of these categories are shown to be unique.

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 $\mathscr{D}^b(\mathrm{\mathop{mod}}\, H)/\tau^{-1/(d-1)}[1]$ 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.