Blas Torrecillas

Covers in finitely accessible categories

We show that in a finitely accessible additive category every class of objects closed under direct limits and pure epimorphic images is covering. In particular, the classes of flat objects in a locally finitely presented additive category and of absolutely pure objects in a locally coherent category are covering.

Flat Covers and Flat Representations of Quivers

Abstract This paper is devoted to the study of flat module representations. We characterize such representations for quivers not containing the path ·• → • (the so called rooted quivers). Then, we prove that for every such quiver any representation has a flat cover and a cotorsion envelope. We finally observe that if Q is one of those quivers and if ℱ denotes the class of all flat representations of Q, then the pair (ℱ, ℱ⊥) is a cotorsion theory.

Torsion theories for coalgebras

Abstract For any coalgebra C, we can consider the category of all right C-comodules MC, which is an abelian category. We give a complete characterization of hereditary (pre)torsion theories in MC. In the first part of the paper we study monomorphism and epimorphism in the category of coalgebras, since they are related to the torsion theories determined by certain morphism of coalgebras with some additional properties (e.g. coflat morphisms).

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.

Preserving and reflecting covers by functors. Applications to graded modules

Abstract We study C -covers in the context of Grothendieck categories. Namely, we analyse when a functor between two Grothendieck categories preserves or reflects C -covers. We apply our general study to the category of graded modules over a graded ring, by showing that relative injective covers with respect to a torsion theory are preserved and reflected, in some cases, among the categories R-gr, R1-Mod and R-Mod.

Strongly rational comodules and semiperfect Hopf algebras over QF rings

Abstract Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or — equivalently — the (left) integrals form a free R-module of rank 1.

From Hopf Algebras to Tensor Categories

This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the category of representations of a suitable Hopf algebra.

Relative graded Clifford theory

We give a relative version of the ‘Graded Clifford Theorem’. The relative graded Clifford theorem is a powerful tool in the study of C-cocritical objects of the category R-gr where C is a rigid localizing subcategory of R-gr. We apply the result to the study of Gabriel (Krull) dimension of a graded module.

THE BRAIDED MONOIDAL STRUCTURES ON THE CATEGORY OF VECTOR SPACES GRADED BY THE KLEIN GROUP

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Twisted Smash Products and L-R Smash Products for Biquasimodule Hopf Quasigroups

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