Homology of formal deformations of proper étale Lie groupoids. E-Preprint math.KT/0412462

Homology of formal deformations of proper étale Lie groupoids. E-Preprint math.KT/0412462

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by N. Neumaier , M. J. Pflaum , H. B. Posthuma , X. Tang

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@MISC{Neumaier_homologyof,
    author = {N. Neumaier and M. J. Pflaum and H. B. Posthuma and X. Tang},
    title = {Homology of formal deformations of proper étale Lie groupoids. E-Preprint math.KT/0412462},
    year = {}
}

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Abstract

Abstract. In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all

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