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 get a finite free resolution of the counit of $\mathcal B(E)$, the Hopf algebra of the bilinear form associated to an invertible matrix $E$, generalizing an ealier 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 Poincar\'e duality. Combining this with results of Vergnioux, it also follows that when $E$ is an antisymetric 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.