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

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 with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\mathbb{Z}$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. For $d=1$ this correspondence specialises to previous work of the second-named author on algebraic triangulated categories of finite type. 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 $A_{\mathrm{\infty }}$-bimodule structures on a graded bimodule.