The bicategories of corings

To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projective on one side, together with a map of corings over the same base ring, lead to the notion of a module-morphism, which extends the notion of a morphism of corings (over different base rings). A module-morphism of corings induces functors between the categories of comodules. These functors are termed pull-back and push-out functors, respectively, and thus relate categories of comodules of different corings. We study when the pull-back functor is fully faithful and when it is an equivalence. A generalised descent associated to a morphism of corings is introduced. We define a category of module-morphisms, and show that push-out functors are naturally isomorphic to each other if and only if the corresponding module-morphisms are mutually isomorphic. All these topics are studied within a unifying language of bicategories and the extensive use is made of interpretation of corings as comonads in the bicategory Bim of bimodules and module-morphisms as 1-cells in the associated bicategories of comonads in Bim.     

Co-Frobenius corings and adjoint functors

We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properties of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterizes quasi-co-Frobenius corings in terms of a pair of adjoint functors (F,G) such that (G,F) is locally quasi-adjoint in a sense defined in this note.     

Monoidal categories of corings

We introduce a monoidal category of corings using two different notions of corings morphisms. The first one is the (right) coring extensions recently introduced by T. Brzeziński in [2], and the other is the usual notion of morphisms defined in [6] by J. Gómez-Torrecillas.

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Sunto Introduciamo una categoria monoidale di coanelli usando due diverse nozioni di morfismi di coanelli. La prima è l'estensione (destra) di coanelli recentemente introdotta da Brzeziński in [2], mentre la seconda è la nozione usuale di morfismo definita in [6] da J. Gómez-Torrecillas.

     

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.

     

A note on coinduction functors between categories of comodules for corings

In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case oflocally projective corings as a composition of suitable “Trace” and “Hom” functors and show how to derive it from a moregeneral coinduction functor between categories of type σ[M]. In special cases (e.g. the corings morphism is part of a morphism of measuringa-pairings or the corings have the same base ring), a version of our functor is shown to be isomorphic to the usual coinduction functor obtained by means of the cotensor product. Our results in this note generalize previous results of the author on coinduction functors between categories of comodules for coalgebras over commutative base rings.     

A local finiteness theorem for corings over semihereditary rings

The fundamental theorem on coalgebras asserts that coalgebras are locally finite in the case where the ground ring is a field. We prove the local finiteness theorem of corings under the semihereditarity condition on the base algebra and the projectivity condition on a coring. This result generalizes not only the fundamental theorem on coalgebras but also Hazewinkel’s result on the local finiteness of coalgebras over a principal ideal domain and Bergman’s unpublished result on the local finiteness of corings over a semisimple Artinian ring.     

Homotopical Morita theory for corings

A coring (A,C) consists of an algebra A in a symmetric monoidal category and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf–Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings (A,C) and (B,D) in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules V _A ^C and V _B ^D are Quillen equivalent. As an illustration of the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.     

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].

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Sunto Estendiamo il Teorema di Masuoka [11] riguardante l'isomorfismo tra il gruppo dei bimoduli invertibili su un'estensione di anelli non commutativa e il gruppo di automorfismi del coanello canonico associato di Sweedler, alla classe dei coanelli di comatrici finiti introdotta in [6].

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Research supported by the grant BFM2001-3141 from the Ministerio de Ciencia y Tecnología of Spain and FEDER.

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Supported by the grant SB2003-0064 from the Ministerio de Educación, Cultura y Deporte of Spain.     

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.      

Group Corings

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodule algebra over a Hopf group coalgebra. This research was supported by the research project G.0622.06 “Deformation quantization methods for algebras and categories with applications to quantum mechanics” from Fonds Wetenschappelijk Onderzoek-Vlaanderen. The third author was partially supported by the SRF (20060286006) and the FNS (10571026).     

Quasi-Frobenius Functors. Applications

We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a quasi-Frobenius bimodule and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of quasi-Frobenius homomorphism of corings is introduced. Finally, a version of the endomorphism ring Theorem for quasi-Frobenius extensions in terms of corings is obtained.     

Adjoint and Frobenius Pairs of Functors for Corings

We investigate adjoint and Frobenius pairs between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtain new results in this setting and in graded ring theory.     

Towers of Corings

Abstract

The notion of a Frobenius coring is introduced, and it is shown that any such coring produces a Jones-like tower of Frobenius corings and Frobenius extensions. This establishes a one-to-one correspondence between Frobenius corings and extensions.     

Comatrix Coring Generalized and Equivalences of Categories of Comodules

We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.     

Comonads and Galois Corings

We investigate which aspects of recent developments on Galois corings and comodules admit a formulation in terms of comonads. The general theory is applied to the study of Galois comodules over corings over firm rings. Supported by the research project “Algebraic Methods in Non Commutative Geometry,” with financial support of the grant MTM2004-01406 from the DGICYT and FEDER.     

The Picard Group of Corings

We introduce and study the Picard group of a coring. We give an exact sequence relating the Picard group of a coring and its automorphisms generalizing the known exact sequences associated to an algebra and a coalgebra over a field. We also extend to corings the Aut–Pic property and we give some new corings satisfying this property.     

Fundamental Constructions for Coalgebras, Corings, and Comodules

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.      

Morita contexts and Galois theory for weak Hopf comodulelike algebras

We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for μ to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.