Calabi-Yau property in derived Koszul calculus

TL;DR

This paper establishes a new duality theorem for strong Kc-Calabi-Yau algebras by proving the existence of derived functors through dg category isomorphisms, and demonstrates that polynomial algebras are examples of such algebras.

Contribution

It removes the need for existence assumptions in the duality theorem by linking dg categories to dg modules, providing a new framework for strong Kc-Calabi-Yau algebras.

Findings

Proves the existence of derived functors via dg category isomorphism.

Defines strong Kc-Calabi-Yau algebras without existence assumptions.

Shows polynomial algebra is strong Kc-Calabi-Yau.

Abstract

A Poincar\'e Van den Bergh duality theorem for strong Kc-Calabi-Yau algebras was obtained by R. Taillefer and the first author under the assumption that the derived functors of functors involved in the statement exist. We prove the existence of these derived functors by showing that the dg category defining the derived Koszul calculus is isomorphic to a dg category of dg modules over a dg algebra. Therefore we get a definition of strong Kc-Calabi-Yau algebras and a corresponding duality theorem without any existence assumption. We prove that a polynomial algebra is strong Kc-Calabi-Yau.