On generalized Hopf galois extensions

The key part of the definition of a Hopf-Galois extension B⊂A over the Hopf algebra H is bijectivity of a canonical map β:A⊗BA→A⊗H. We develop criteria under which surjectivity of β (which is usually much easier to verify) is sufficient, and we investigate the consequences for the structure of A as a B-module and H-comodule. In particular, we prove equivariant projectivity of extensions in several important cases. We study these questions for generalizations of H-Galois extensions like Q-Galois extensions for a quotient coalgebra and one-sided module of a Hopf algebra H, and coalgebra Galois extensions.