Description
The Hochschild (co)homology theory for associative algebras projective over a commutative ring k have a rich structure. Namely, the cup and the cap products, the Gerstenhaber bracket and Connes differential, this is summarized as differential calculus or Tamarkin-Tsygan calculus. There are interesting connections of this theory with derived categories: for instance, D. Happel proved the derived invariance of the cup product. Later on, B. Keller showed the derived invariance of the Gerstenhaber bracket.
First I will extend the derived invariance of the cup product to allow coefficients in an arbitrary bimodule. Then I will present two proofs of the derived invariance of the cap product, one of them is a joint work with B. Keller. They will be part of my thesis, prepared under the direction of C. Cibils (IMAG UM Montpellier) and J.A. de la Pena (CIMAT México).
As a main tool, I will provide interpretations of these operations merely in terms of derived categories, through a canonical morphism between the Hochschild homology and the k-dual of Hochschild cohomology with coefficients in the k-dual of the algebra. Finally I will present related results concerning the Batalin-Vilkovisky structure on Hochschild cohomology.
