  We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines
its Hochschild cohomology, and in particular its Gerstenhaber-Schack
cohomological dimension bounds its Hochschild cohomological dimension, with
equality of the dimensions when the Hopf algebra is cosemisimple of Kac type.
Together with some general considerations on free Yetter-Drinfeld modules over
adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack
cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild
cohomological dimensions of the coordinate algebra of the quantum permutation
group are 3.
