Calabi-Yau structures on derived and singularity categories of symmetric orders
Norihiro Hanihara, Junyang Liu
TL;DR
This work constructs and analyzes Calabi–Yau structures on derived and singularity categories of symmetric $R$-orders by lifting Amiot’s quotient Serre construction to the dg level and tracing base-change behavior from a commutative Gorenstein ring $R$ to a field $k$. It proves the existence of a right Calabi–Yau structure on the dg singularity category and a left Calabi–Yau structure on the dg bounded derived category, generalizing Brav–Dyckerhoff and connecting to cluster categories via Keller–Liu-type structure theorems. Under suitable hypotheses, the singularity category is triangle equivalent to a generalized cluster category associated with a deformed dg preprojective algebra. The paper also develops a robust dg framework tying connecting morphisms in dual Hochschild homology to dg-enhancements of Amiot’s construction and establishes comprehensive base-change results for transferring CY-structures across base rings.
Abstract
We construct left and right Calabi-Yau structures on derived respectively singularity categories of symmetric orders $Λ$ over commutative Gorenstein rings $R$. For this, we first construct Calabi-Yau structures over $R$ by lifting Amiot's construction of Calabi-Yau structures on Verdier quotients to the dg level. Then we prove base change properties relating Calabi-Yau structures over $R$ to those over the base field $k$. As a result, we prove the existence of a right Calabi-Yau structure on the dg singularity category associated with $Λ$ which is a cyclic lift of the weak Calabi-Yau structure constructed by the first-named author and Iyama. We also show the existence of a left Calabi-Yau structure on the dg bounded derived category of $Λ$. This is a non-commutative generalization of a result by Brav and Dyckerhoff. By combining the existence of the right Calabi-Yau structure on the dg singularity category with a structure theorem by Keller and the second-named author, we deduce that under suitable hypotheses, the singularity category associated with $Λ$ is triangle equivalent to a generalized cluster category in the sense of Amiot.