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Hochschild (co)homology of the second kind I
Transactions of the American Mathematical Society. 2012. Vol. 364. No. 10. P. 5311–5368.
Polishchuk A., Positselski L.
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.
Keywords: гомологии и когомологии ХохшильдаHochschild homologyHochschild cohomologycurved DG-algebrascurved DG-categoriesderived categories of the second kindresolution of the diagonalmatrix factorizationsискривленные DG-алгебрыискривленные DG-категориипроизводные категории второго родаматричные факторизации
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