Derived Invariance of Higher Structures on the Hochschild Complex

We show that derived equivalences preserve the homotopy type of the (cohomological) Hochschild complex as a B∞-algebra. More generally, we prove that, as an object of the homotopy category of B∞-algebras, the Hochschild complex is contravariant with respect to fully faithful derived tensor functors. We also show that the Hochschild complexes of a Koszul algebra and its dual are homotopy equivalent as B∞-algebras. In particular, their Hochschild cohomologies are isomorphic as algebras, which is a recent result by R.-O. Buchweitz [4], and as Lie algebras. Our methods also yield a derived invariant definition of the Hochschild complex of an exact category.