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⊕⋯⊕T[−(d−2)] has hereditary endomorphism algebra H is triangle equivalent to the orbit category Db(modH)/τ−1/(d−1)[1] of the derived category of H for a naturally defined (d−1)-st root τ1/(d−1) of the AR translation, provided H is of non-Dynkin type. We also show that heredity of H follows from that of T when d=3, that of T⊕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.