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 com mutative Hopf algebra H over a field, coacting on a commutative H-comodule algebra A. If A(coH) denotes the subalgebra of coinvariant elements of A an d beta : A circle timesA(coll) A --> A circle times H the canonical map, he proved that the following are equivalent: (a) A(coH) subset of A is a faithfully flat Hopf Galois extension; (b) the functor (-)(coH) : M-A(H) --> A(coH) -Mod is an equivalence; (c) A is coflat as a right H-comodule and beta is surjective. Schneider generalized this result in [14, Theorem 1] to the non-commutative situation imposing as a condition the bijectivity of the antipode of the u nderlying Hopf algebra. Interpreting the functor of coinvariants as a Hom-f unctor, Menini and Zuccoli gave in [10] a module-theoretic presentation of parts of the theory. Refining the techniques involved we are able to genera lize Schneiders result to H-comodule-algebras A for a Hopf algebra H (with bijective antipode) over a commutative ring R under fairly weak assumptions .