Galois Correspondences for Hopf Bigalois Extensions

LetHbe a Hopf algebra with bijective antipode over a commutative ringk. A rightH-Galois extension ofkis a rightH-comodule algebraAsuch thatk = Aco Hand a certain canonical mapA A → A His a bijection. We investigate Galois connections for Hopf–Galois extensions that can be formulated with the help of an additional Hopf algebraLover whichAis also a leftL-Galois extension ofk, and anL-H-bicomodule—such an additional Hopf algebra always exists and is unique up to isomorphism. The Galois connection between quotient coalgebras and left modules ofLandH-subcomodule algebras ofAinduces a bijection between those quotients over whichLis faithfully coflat, and those subalgebras over whichAis faithfully flat. As a consequence, the lattices of Hopf subalgebras ofLandHover which these are faithfully flat are isomorphic.