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Hochschild (co)homology of the second kind I

Authors: Alexander Polishchuk and Leonid Positselski
Journal: Trans. Amer. Math. Soc. 364 (2012), 5311-5368
MSC (2010): Primary 16E40; Secondary 18G10, 18G15, 18E30, 13D99
Published electronically: May 30, 2012
MathSciNet review: 2931331
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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

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

Received by editor(s): October 15, 2010
Published electronically: May 30, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.