Noetherian centralizing Hopf algebra extensions and finite morphisms of quantum groups

We study finite centralizing extensions A subset of H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H -->> spec A to coincide with the X-orbits in spec H, where X denotes the finite group of characters of H that restrict to the counit of A. In particular, all of the fibres are X-orbits if and only if the fibre over the augmentation ideal of A is an X-orbit. An application to the representation theory of quantum function algebras,. at roots of unity, is presented.