Morita theory for coring extensions and cleft bicomodules

Abstract

Listen

Translate article

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 M C → M D . 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 Galois C -comodule and isomorphic to End C ( Σ ) ⊗ L D , in the category of left modules for the ring End C ( 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.