Morita theory for coring extensions and cleft bicomodules

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)⊗LD, 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.