Infinite Comatrix Corings

TL;DR

This paper characterizes a class of corings with specific comodules using non-commutative descent theory, generalizes comatrix and Galois corings without finiteness constraints, and identifies conditions for infinite comatrix corings.

Contribution

It introduces a generalized framework for comatrix and Galois corings applicable to infinite cases, expanding understanding of coring structures beyond finiteness assumptions.

Findings

Coalgebras over fields are isomorphic to infinite comatrix corings.

Coring associated to group-graded rings are also infinite comatrix corings.

Conditions for the Galois property of modules over infinite sets of group-like elements.

Abstract

We characterize the corings whose category of comodules has a generating set of small projective comodules in terms of the (non commutative) descent theory. In order to extricate the structure of these corings, we give a generalization of the notions of comatrix coring and Galois comodule which avoid finiteness conditions. A sufficient condition for a coring to be isomorphic to an infinite comatrix coring is found. We deduce in particular that any coalgebra over a field and the coring associated to a group-graded ring are isomorphic to adequate infinite comatrix corings. We also characterize when the free module canonically associated to a (not necessarily finite) set of group like elements is Galois.