Hochschild (co)homology of the second kind I
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.
Keywords
Cite
@article{arxiv.1010.0982,
title = {Hochschild (co)homology of the second kind I},
author = {Alexander Polishchuk and Leonid Positselski},
journal= {arXiv preprint arXiv:1010.0982},
year = {2012}
}
Comments
LaTeX 2e, 67 pages. v.2: The case of matrix factorizations discussed in detail in the new subsections 4.8 and 4.10