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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • 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:
  • 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:
  • MathSciNet review: 2931331