A Schneider type theorem for Hopf algebroids

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B⊆A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B⊆A and also to the equivalence of the category of relative Hopf modules to the category of B-modules. This extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf algebra. Our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor, a notion introduced and analysed hereby.