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In our paper we heavily used the result that two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule categories. This statement was based on [T. Brzeziński, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3, Theorem 2.6], whose proof turned out to contain an unjustified step...
Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B⊆A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B⊆A and also to the equivalence of the category of relative Hopf modules...
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