A-H-bimodules and equivalences

In [6, Theorem 2.2] Doi gave a Hopf-algebraic proof of a generalization of Oberst’s theorem on affine quotients of affine schemes. He considered a commutative Hopf algebra H over a field, coacting on a commutative H-comodule algebra A. If AcoH denotes the subalgebra of coinvariant elements of A and β : A ⊗AcoH A −→ A ⊗H the canonical map, he proved that the following are equivalent: (a) AcoH ⊂ A is a faithfully flat Hopf Galois extension; (b) the functor (−)coH :MA −→ AcoH -Mod is an equivalence; (c) A is coflat as a right H-comodule and β is surjective. Schneider generalized this result in [14, Theorem 1] to the noncommutative situation imposing as a condition the bijectivity of the antipode of the underlying Hopf algebra. Interpreting the functor of coinvariants as a Hom-functor, Menini and Zuccoli gave in [10] a moduletheoretic presentation of parts of the theory. Refining the techniques involved we are able to generalize Schneiders result to H-comodulealgebras A for a Hopf algebra H (with bijective antipode) over a commutative ring R under fairly weak assumptions.