Hochschild (co)homology of the second kind I

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.

💡 Research Summary

The paper introduces and develops a “second kind” of Hochschild (co)homology for curved differential graded (CDG) categories, also known as Borel‑Moore Hochschild homology and compactly supported Hochschild cohomology. Classical Hochschild (co)homology is well‑behaved for flat DG categories, but when a curvature element w of degree + 1 is present, the usual chain complexes may diverge or fail to converge. To overcome this, the authors construct a Borel‑Moore type complex by taking the CDG‑bimodule B (the diagonal bimodule) over a CDG‑category B, equipping it with the total differential D = d +