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 triangulated category with a triangulated Calabi--Yau structure that cannot be lifted to a bimodule right Calabi--Yau structure on any of its dg enhancements.

103 pages. v2: Corrected several small typos. Added new Section 7.3 on a non-enhanceable triangulated Calabi--Yau structure. v3: Editorial improvements. v4: Enhanced exposition, several small corrections. Subsection 7.3 is now Section 8