## Hochschild (co)homology of the second kind I

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- by Alexander Polishchuk and Leonid Positselski PDF
- Trans. Amer. Math. Soc.
**364**(2012), 5311-5368 Request permission

## 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. In particular, we show that the two kinds of Hochschild (co)homology are isomorphic for the DG-category of matrix factorizations of a regular function on a smooth affine variety over a perfect field provided that the function has no other critical values but zero.

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## Additional Information

**Alexander Polishchuk**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 339630
- Email: apolish@uoregon.edu
**Leonid Positselski**- Affiliation: Department of Mathematics, National Research University “Higher School of Economics”, Moscow 117312, Russia – and – Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow 127994, Russia
- Email: posic@mccme.ru
- Received by editor(s): October 15, 2010
- Published electronically: May 30, 2012
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 5311-5368 - MSC (2010): Primary 16E40; Secondary 18G10, 18G15, 18E30, 13D99
- DOI: https://doi.org/10.1090/S0002-9947-2012-05667-4
- MathSciNet review: 2931331