Hochschild cohomology of the second kind: Koszul duality and Morita invariance
Ai Guan, Julian Holstein, Andrey Lazarev
TL;DR
The paper defines Hochschild cohomology of the second kind for dg and curved algebras as a derived endomorphism in the compactly generated second-kind framework and proves Morita invariance in this setting. It develops bimodule Koszul duality and demonstrates preservation of the second-kind Hochschild cohomology under nonconilpotent Koszul duality, establishing connections with ordinary Hochschild cohomology of dg categories in geometry. The authors provide a robust set of model-categorical tools (coderived and second-kind derived categories) and derive explicit equivalences for cobar/bar constructions, along with curved-case generalizations. They then illustrate the machinery through concrete examples: Dolbeault and de Rham/DG-algebras reflecting coherent sheaves, infinity local systems, and matrix factorizations, showing that $\mathrm{HH}^{\mathrm{II}}_{\mathrm{c}}$ often recovers the standard Hochschild cohomology of associated dg categories and thus encodes geometrically meaningful invariants.
Abstract
We define Hochschild cohomology of the second kind for differential graded (dg) or curved algebras as a derived functor in a compactly generated derived category of the second kind, and show that it is invariant under Morita equivalence of the second kind. A bimodule version of Koszul duality is constructed and used to show that Hochschild cohomology of the second kind is preserved under (nonconilpotent) Koszul duality. We show that Hochschild cohomology of the second kind of an algebra often computes the ordinary Hochschild cohomology of geometrically meaningful dg categories. Examples include the category of infinity local systems on a topological space, the bounded derived category of a complex algebraic manifold and the category of matrix factorizations.