On Generalized Hopf Galois Extensions

is an isomorphism, which can be interpreted as the correct algebraic formulation of the condition that the G-action of X should be free, and transitive on the fibers of the map X → Y . In many applications surjectivity of the Galois map β, which, in the commutative case, means freeness of the action of G, is obvious, or at least easy to prove (it is sufficient to find 1 ⊗ h in the image for each h in a generating set for the algebra H). It is usually much harder to decide whether β is injective. The present paper has two main topics: When does surjectivity of β already imply bijectivity? What can we conclude about the module structure of A over B, or the comodule structure of A, or general Hopf modules, over H? Both questions will be studied for more general extensions. The Kreimer-Takeuchi Theorem [16, Thm. 1.7] says that if β is onto and H is finite, then β is bijective and A is a projective B-module. This generalizes a Theorem of Grothendieck [8, III, §2, 6.1] on the actions of finite group schemes. Theorem 3.5 in [32] implies that if β is surjective and A is a relative injective H-comodule, then β is bijective and A is a faithfully flat B-module. This generalizes results of Oberst [26], and Cline, Parshall, and Scott [6] for the case where H represents a closed subgroup of an affine group scheme represented by A; in this situation the canonical map is trivially surjective, while injectivity of the H-comodule A means that the induction functor from the subgroup in question is exact.