Hochschild (co)homology of the second kind I

Polishchuk, Alexander; Positselski, Leonid

doi:10.1090/S0002-9947-2012-05667-4

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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