Morita theory for coring extensions and cleft bicomodules.
The paper is devoted to construction and studies of Morita contexts associated to extensions of corings (over different \(k\)-algebras) and bicomodules. These contexts are shown to arise from a 2-object subcategory of the category of \(k\)-linear functors between categories of comodules. The strictness of such a context implies the Galois property of the comodule involved and the weak structure theorem (i.e.\ faithfulness and fullness of the induced functor). Sufficient conditions for the strong structure theorem (i.e.\ for the induced functor to be an equivalence of categories of comodules) are found. The authors introduce the notion of a cleft bicomodule. A bicomodule is cleft if and only if it is a Galois comodule that in addition enjoys a generalised version of the normal basis property. It is proven that such a bicomodule enforces the associated Morita context to be strict. Examples of cleft bicomodules are provided by cleft extensions of Hopf algebras, coalgebras or Hopf algebroids and by cleft (weak) entwining structures over non-commutative rings.
scientific article; zbMATH DE number 5129763
scientific article