### Abstract

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 GaloisC -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.

Original language | English |
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Pages (from-to) | 611-648 |

Number of pages | 38 |

Journal | Advances in Mathematics |

Volume | 209 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2007 |

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### Keywords

- Cleft bicomodule
- Coring extension
- Morita theory
- Weak and strong structure theorems

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*209*(2), 611-648. https://doi.org/10.1016/j.aim.2006.05.010