Comatrix Corings Research Articles

Galois theory for comatrix group corings

In this paper, we introduce comatrix group corings and define the generalized Galois group corings. Then we give the generalized Galois group coring Structure Theorem.

Corings with Decomposition and Semiperfect Corings

We give a characterization, in terms of Galois infinite comatrix corings, of the corings that decompose as a direct sum of left comodules which are finitely generated as left modules. Then we show that the associated rational functor is exact. This is the case of a right semiperfect coring which is locally projective and whose Galois comodule is a projective left unital module with superfluous radical.

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gómez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 (2007), 271–306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 (2006), 2683–2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.

Comatrix Corings and Galois Comodules over Firm Rings

We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory.

Comonadicity and invertible bimodules

The purpose of this paper is to exhibit the main results of A. Masuoka [A. Masuoka, Corings and invertible bimodules, Tsukuba J. Math. 13 (1989) 353–362] and of L. El Kaoutit and J. Gómez-Torrecillas [L. El Kaoutit, J. Gómez-Torrecillas, Comatrix corings and invertible bimodules, Ann. Univ. Ferrara Ser. VII 51 (2005) 263–280] as special cases of a more general result.

Constructing Infinite Comatrix Corings from Colimits

We propose a class of infinite comatrix corings, and describe them as colimits of systems of usual comatrix corings. The infinite comatrix corings of El Kaoutit and Gómez Torrecillas are special cases of our construction, which in turn can be considered as a special case of the comatrix corings introduced recently by Gómez Torrecillas and the third author.

Galois corings and a Jacobson–Bourbaki type correspondence

The Jacobson–Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in [M.E. Sweedler, The predual theorem to the Jacobson–Bourbaki Theorem, Trans. Amer. Math. Soc. 213 (1975) 391-406]. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler's result to simple artinian rings using a particular class of corings, comatrix corings. A Jacobson–Bourbaki like correspondence for simple artinian rings is then obtained by duality.

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

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.

Comatrix corings and invertible bimodules

We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite comatrix corings introduced in [6].

Infinite comatrix corings

We characterize the corings whose category of comodules has a generating set of small projective comodules in terms of the (noncommutative) 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.

On Comatrix Corings and Bimodules

To any bimodule that is finitely generated and projective on one side one can associ- ate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associated comatrix corings. In particular it is shown that separable bimodules give rise to coseparable comatrix corings, while Frobenius bimodules induce Frobenius comatrix corings.

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

We study the corings whose category of right comodules has a finitely generated projective generator. In order to extricate the structure of these corings, we introduce the notion of comatrix coring. Comatrix corings allow as well to give a complete description of all cosemisimple corings.