A-H-BIMODULES AND EQUIVALENCES

Abstract

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 A coH denotes the subalgebra of coinvariant elements of A and β : A ⊗ A coH 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 : M H A → A coH -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 non-commutative 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 module-theoretic presentation of parts of the theory. Refining the techniques involved we are able to generalize Schneiders result to H-comodule-algebras A for a Hopf algebra H (with bijective antipode) over a commutative ring R under fairly weak assumptions.