Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit

We show that if A and H are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter-Drinfeld resolution of the counit of A to the same kind of resolution for the counit of H, exhibiting in this way strong links between the Hochschild homologies of A and H. This enables us to obtain a finite free resolution of the counit of B (E), the Hopf algebra of the bilinear form associated with an invertible matrix E, generalizing an earlier construction of Collins, Hartel and Thom in the orthogonal case E = I-n. It follows that B (E) is smooth of dimension 3 and satisfies Poincare duality. Combining this with results of Vergnioux, it also follows that when E is an antisymmetric matrix, the L-2-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of B(E) in the cosemisimple case.