José Gómez-Torrecillas

Separable functors in corings

We develop some basic functorial techniques for the study of the categories of comodules over corings. In particular, we prove that the induction functor stemming from every morphism of corings has a left adjoint, called ad-induction functor. This construction generalizes the known adjunctions for the categories of Doi-Hopf modules and entwined modules. The separability of the induction and ad-induction functors are characterized, extending earlier results for coalgebra and ring homomorphisms, as well as for entwining structures.

IDEMPOTENTS AND MORITA-TAKEUCHI THEORY

ABSTRACT We offer an approach to basic coalgebras with inspiration in the classical theory of idempotents for finite dimensional algebras. Our theory is based upon the fact that the co-hom functors associated to direct summands of the coalgebra can be easily described in terms of idempotents of the convolution algebra. Our approach is shown to be equivalent to that given by W. Chin and S. Montgomery by using co-endomorphism coalgebras of minimal injective cogenerators.

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.

Dual coalgebras of algebras over commutative rings

Abstract In the study of algebraic groups the representative functions related to monoid algebras over fields provide an important tool which also yields the finite dual coalgebra of any algebra over a field. The purpose of this note is to transfer this basic construction to monoid algebras over commutative rings R. As an application we obtain a bialgebra (Hopf algebra) structure on the finite dual of the polynomial ring R[x] over a noetherian ring R. Moreover, we give a sufficient condition for the finite dual of any R-algebra A to become a coalgebra. In particular, this condition is satisfied provided R is noetherian and hereditary.

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.

Quasi-co-Frobenius Coalgebras. II

Abstract We prove new characterizations of Quasi-co-Frobenius (QcF) coalgebras and co-Frobenius coalgebras. Among them, we prove that a coalgebra is QcF if and only if C generates every left and every right C-comodule. We also prove that every QcF coalgebra is Morita-Takeuchi equivalent to a co-Frobenius coalgebra.

Basic Module Theory over Non-commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.

LOCALIZATION IN COALGEBRAS: APPLICATIONS TO FINITENESS CONDITIONS

We introduce the notion of right strictly quasi-finite coalgebras, as coalgebras with the property that the class of quasi-finite right comodules is closed under factor comodules, and study its properties. A major tool in this study is the local techniques, in the sense of abstract localization.

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.

A New Perspective of Cyclicity in Convolutional Codes

In this paper, we propose a new way of providing cyclic structures to convolutional codes. We define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non-commutative polynomial ring. In contrast to the previous approaches to cyclicity for convolutional codes, we use Ore polynomials with coefficients in a field (the rational function field over a finite field), so their arithmetic is very well known and we may proceed similarly to cyclic block codes. In particular, we show how to obtain easily skew cyclic convolutional codes of a given dimension, and we compute an idempotent generator of the code and its dual.