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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 
$\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix 
$E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case 
$E=I_n$. It follows that 
$\mathcal {B}(E)$ is smooth of dimension 3 and satisfies Poincaré 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 
$\mathcal {B}(E)$in the cosemisimple case.
