Hochschild (co)homology of the second kind I

TL;DR

This paper introduces and investigates the Hochschild (co)homology of the second kind for curved DG-categories, establishing isomorphisms with classical Hochschild (co)homology under certain conditions and exploring various examples.

Contribution

It defines the Hochschild (co)homology of the second kind for curved DG-categories and provides conditions for their isomorphism with standard Hochschild (co)homology.

Findings

Established isomorphism between Hochschild (co)homology of the second kind and classical Hochschild (co)homology.

Formulated conditions based on derived categories for when the two types of Hochschild (co)homology coincide.

Discussed several classes of examples illustrating the theory.

Abstract

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG-categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG-category C of right CDG-modules over B, projective and finitely generated as graded B-modules, is constructed. Sufficient conditions for an isomorphism of the two kinds of Hochschild (co)homology of a DG-category are formulated in terms of the two kinds of derived categories of DG-modules over it. In particular, a kind of "resolution of the diagonal" condition for the diagonal CDG-bimodule B over a CDG-category B guarantees an isomorphism of the two kinds of Hochschild (co)homology of the corresponding DG-category C. Several classes of examples are discussed.