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⊗ B A → 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. © 2005 Elsevier B.V. All rights reserved.
Schauenburg, P., & Schneider, H. J. (2005). On generalized Hopf galois extensions. Journal of Pure and Applied Algebra, 202(1–3), 168–194. https://doi.org/10.1016/j.jpaa.2005.01.005