  We discuss the question of whether the global dimension is a monoidal
invariant for Hopf algebras, in the sense that if two Hopf algebras have
equivalent monoidal categories of comodules, then their global dimensions
should be equal. We provide several positive new answers to this question,
under various assumptions of smoothness, cosemisimplicity or finite dimension.
We also discuss the comparison between the global dimension and the
Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining
equality in the case the latter is finite. One of our main tools is the new
concept of twisted separable functor.
