Calabi-Yau structures on derived and singularity categories of symmetric orders

Abstract

We construct left and right Calabi-Yau structures on derived respectively singularity categories of symmetric orders $\Lambda$ 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 $\Lambda$ 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 $\Lambda$. 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 $\Lambda$ is triangle equivalent to a generalized cluster category in the sense of Amiot.