Hochschild (co)homology of the second kind I

By Alexander Polishchuk, L. Positselski. 2010

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e41fe5dde7b12ada01fcbb1762ce48b4631fc587

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Global matrix factorizations (2011, methodology): “phism by Lemma 2.7 of [Yek], which states that that when X is smooth over C, the functor Homcont OX2 (Bb q(X),−) : Mod discO X2→ Mod O is exact. Our argument here parallels the argument on page 25 of [PolPos]. GLOBAL MATRIX FACTORIZATIONS 11 Deﬁne Cb(X) := Bb(X) ⊗O X2 OX, with induced diﬀerential al”
Coherent analogues of matrix factorizations and relative singularity categories (2011, methodology): “The rest of the proof is similar to that in [Polishchuk and Positselski 2012].”