A Morita context is constructed for any comodule of a coring and, more generally, for an L - C bicomodule Σ for a coring extension (D : L) of (C : A). It is related to a 2-object subcategory of the category of k-linear functors MC → MD. Strictness of the Morita context is shown to imply the Galois property of Σ as a C-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an L - C bicomodule Σ-implying strictness of the associated Morita context-is introduced. It is shown to be equivalent to being a GaloisC -comodule and isomorphic to EndC (Σ) ⊗L D, in the category of left modules for the ring EndC (Σ) and right comodules for the coring D, i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules. © 2006 Elsevier Inc. All rights reserved.
Böhm, G., & Vercruysse, J. (2007). Morita theory for coring extensions and cleft bicomodules. Advances in Mathematics, 209(2), 611–648. https://doi.org/10.1016/j.aim.2006.05.010