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Hochschild (co)homology and Koszul duality

In this article we discuss two different but related results on Hochschild (co)homology and the theory of Koszul duality. On the one hand, we prove essentially that the Tamarkin-Tsygan calculus of an Adams connected augmented dg algebra and of its Koszul dual are dual. This uses the fact that Hochschild cohomology and homology may be regarded as a twisted construction of some natural (augmented) dg algebras and dg modules over the former. In particular, from these constructions it follows that the computation of the cup product on Hochschild cohomology and cap product on Hochschild homology of a Koszul algebra is directly computed from the coalgebra structure of the Tor(k,k) group (the first of these results is proved differently by R.-O. Buchweitz, E. Green, N. Snashall and O. Solberg). We even generalize this situation by studying twisting theory of A_infinity-algebras to compute the algebra structure of Hochschild cohomology of more general algebras.