Hochschild cohomology and Atiyah classes

Details Hochschild cohomology and Atiyah classes Hochschild cohomology and Atiyah classes

2007-08-20

arxiv.org/abs/0708.2725v4

Advances in Mathematics 224 (2010), no. 5, 1839--1889

Mathematics

K-Theory and Homology

Reference to work of Cattaneo, Felder and Willwacher added

Scientific paper

10.1016/j.aim.2010.01.012

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.

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