The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures

Abstract

Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras (=dg) with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\ZZ$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. In this article, we prove a variant of our general correspondence for bimodule right Calabi--Yau dg algebras. A novel ingredient is a new cohomology theory which contains obstructions to the existence and uniqueness of minimal $A_\infty$-bimodule structures on a graded bimodule. As an application of our results, we obtain, to our knowledge, the first example of an algebraic Calabi--Yau triangulated category whose graded Serre duality pairing is not induced by a bimodule right Calabi--Yau structure on any of its dg enhancements.