arXiv subjects
Publications and source records attributed to Mohssin Zarouali-Darkaoui.
In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the Zermelo-Fraenkel system, the axiom of choice is equivalent to each of the following statements: H. Kneser's Lemma, Zorn's Lemma, Zermelo's Lemma
We study adjoint and Frobenius pairs of functors, equivalences, and the Picard group for corings.
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.
We introduce the Picard group of corings. We extend the well-known exact sequence from algebras and coalgebras over fields to corings. We extend the Aut-Pic property to corings and we give some new examples of corings having this property. Finally, we give the corresponding exact sequences for the category of entwined modules over an entwining structure, the category of Doi-Koppinen-Hopf modules over a Doi-Koppinen structure, and the category of graded modules by a $G$-set, where $G$ is a group.