Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties

Abstract

Abstract. El Kaoutit and Gómez-Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.

Morita theory for coring extensions and cleft bicomodules

“…Generalizing constructions in [21], [23], [1] and [20], Caenepeel, De Groot and Vercruysse in [19,Section 4] associated Morita contexts to comodules of an A-coring C. For any right C-comodule Σ, they constructed a Morita context, connecting the algebras End C (Σ) and the right dual C * of C (cf. M ′ (Σ) below).…”

“…In the next generalization of [19,Proposition 4.7] the relationship between the Morita contexts N(Σ) in (2.7) and M(Σ) in (2.4) is investigated. (1) If the connecting map in (2.5) is surjective then C is a finitely generated projective left A-module.…”

“…Recall that the existence of a grouplike element in an A-coring C is equivalent to the existence of a (left or right) C comodule structure in A. In the paper [19] the particular C-comodule A in [20] was replaced by any C-comodule Σ, which is finitely generated and projective as an A-module. In all of the listed (more and more general) situations, strictness of the Morita context was related to properties of the extension (or the coring) behind.…”

“…The first aim of the present paper is to generalize the construction of a Morita context further. As a first step, in Section 2 we remove the finitely generated projectivity condition in [19] and construct a Morita context for an arbitrary comodule Σ of an A-coring C. It connects the algebra of C-comodule endomorphisms of Σ and the A-dual algebra of C.…”

Morita Theory for Group Corings

“…As the generalization of both contexts, Caenepeel, Vercruysse and Wang associate different types of Morita contexts to a coring with a fixed grouplike element, which was generalized by Caenepeel, Janssen and Wang to group coring with a grouplike family [5]. Without the assumption of a coring with a fixed grouplike element, Caenepeel, De Groot and Vercruysse associated a Morita context to a comodule over a coring in [8]. Morita theory for group corings with fixed grouplike family is a remarkable tool to discuss Hopf-Galois extensions.…”