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

TL;DR

This paper advances the theory of comatrix corings by extending descent theorems, exploring Galois structures, and analyzing Morita contexts, especially in Frobenius and coseparable cases.

Contribution

It generalizes key descent and Galois theorems for comatrix corings and links Morita theory to Galois properties in new ways.

Findings

New version of Joyal-Tierney Descent Theorem

Generalized Galois Coring Structure Theorem

Established Morita contexts for Frobenius and coseparable corings

Abstract

El Kaoutit and G\'omez 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.