Comatrix corings: Galois corings, descent theory, and a structure theorem for cosemisimple corings

Constructing Infinite Comatrix Corings from Colimits

“…This was observed by Brzeziński in his paper [4], see also [8] for a detailed discussion. A more general theory was proposed by El Kaoutit and Gómez Torrecillas [10]. We start from two rings A and B, connected by a (B, A)-module P .…”

“…The condition that P is finitely generated and projective as a right A-module is crucial in the theory. Nevertheless, El Kaoutit and Gómez Torrecillas [11] proposed an infinite version of comatrix corings, starting from an infinite collection of finitely generated projective right A-modules {P i | i ∈ I}. They consider the direct sum P of the P i , and the direct sum P † of the P * i .…”

“…The bimodules in a comatrix coring context connecting firm rings are not necessarily finitely generated projective. The comatrix coring contexts from [11] are of this type: only one of the two rings involved has units, the other one has only local units (a complete set of orthogonal idempotents). In this paper, we propose some classes of infinite comatrix corings.…”

“…In this situation, we can define a canonical coring map from the associated comatrix coring to D, and M is called a system of Galois C-comodules if this map is an isomorphism. The comatrix corings introduced in [11] are special cases.…”

The Brauer Group of Azumaya Corings and the Second Cohomology Group

“…If I = S, then we obtain Sweedler's canonical coring, introduced in [23]; in general, Can R (I; S) is an example of a comatrix coring, as introduced in [11]. We also compute that S Hom(C, S) = S Hom(I * ⊗ R I, S) ∼ = R Hom(I, I) = R End(I).…”