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Journal of Noncommutative Geometry

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Volume 13, Issue 1, 2019, pp. 59–85
DOI: 10.4171/JNCG/323

Published online: 2019-03-11

Hochschild (co)homology of Koszul dual pairs

Estanislao Herscovich[1]

(1) Université Grenoble Alpes, Gières, France and Universidad de Buenos Aires, Argentina

In this article we prove that the Tamarkin–Tsygan calculus of an Adams connected augmented dg algebra and of its Koszul dual are dual to each other. This uses the fact that the Hochschild cohomology and homology may be regarded as a twisted convolution dg algebra and as a twisted tensor product, respectively. As an immediate application of this latter point of view we also show that the cup product on Hochschild cohomology and the cap product on Hochschild homology of a Koszul algebra is directly computed from the coalgebra structure of $\operatorname{Tor}_{\bullet}^{A}(k,k)$ (the first of these results is proved differently in [2]).

Keywords: Homological algebra, Hochschild (co)homology, dg algebras, Koszul algebras, Yoneda algebras

Herscovich Estanislao: Hochschild (co)homology of Koszul dual pairs. J. Noncommut. Geom. 13 (2019), 59-85. doi: 10.4171/JNCG/323