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L. El Kaoutit

Publications and source records attributed to L. El Kaoutit.

16 recordsLinked to original sources
arXiv2011

Categories of comodules and chain complexes of modules

Let $\lL(A)$ denote the coendomorphism left $R$-bialgebroid associated to a left finitely generated and projective extension of rings $R \to A$ with identities. We show that the category of left comodules over an epimorphic image of $\lL(A)$ is equivalent to the category of chain complexes of left $R$-modules. This equivalence is monoidal whenever $R$ is commutative and $A$ is an $R$-algebra. This is a generalization, using entirely new tools, of results by B. Pareigis and D. Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the non commutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.

math.RA↗
arXiv2009

Corings over rings with local units

We show that the category of corings over a fixed base ring with local units is equivalent to the category of comonads in (right) unital modules whose underlying functors preserve inductive limits. Changing base rings, we prove a bi-equivalence of bicategories. A base ring extension of corings by adjunctions is also introduced.

math.RA↗
arXiv2009

Wide Morita contexts in Bicategories

We give a formal concept of (right) wide Morita context between two 0-cells in arbitrary bicategory. We then construct a new bicategory with the same 0-cells as the oldest one, and with 1-cells all these (right) wide Morita contexts. An application to the (right) Eilenberg-Moore bicategory of comonads associated to the bimodules bicategory is also given.

math.RA↗
arXiv2009

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.

math.RA↗
arXiv2009

Coinduction functor and simple comodules

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (\emph{coinduction functor}) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of simple right modules over the endomorphism ring. A detailed application to a group-graded modules is also given.

math.RA↗
arXiv2009

On the set of grouplikes of a coring

We focus our attention to the set $\gl{\coring{C}}$ of grouplike elements of a coring $\coring{C}$ over a ring $A$. We do some observations on the actions of the groups $U(A)$ and $\aut{\coring{C}}$ of units of $A$ and of automorphisms of corings of $\coring{C}$, respectively, on $\gl{\coring{C}}$, and on the subset $\galois{\coring{C}}$ of all Galois grouplike elements. Among them, we give conditions on $\coring{C}$ under which $\galois{\coring{C}}$ is a group, in such a way that there is an exact sequence of groups $\{1\} \to U(A^{g}) \to U(A) \to \galois{\coring{C}} \to \{1\},$ where $A^g$ is the subalgebra of coinvariants for some $g \in \galois{\coring{C}}$.

math.RA↗
arXiv2007

Corings with exact rational functors and injective objects

We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules, generalizing previous results in this setting.

math.RA↗
arXiv2007

Compatibility Condition between ring and coring

We introduce the notion of bi-monoid in general monoidal category generalizing by this the notion of bialgebra. In the case of bimodules over a noncommutative algebra, we obtain a compatibility condition between ring and coring whenever both structures admit the same underlying bimodule.

math.RA↗
arXiv2006

Extended Distributive Law: Co-wreath over co-rings

A basic theory of cowreath or extended distributive laws in the bicategory of unital bimodules, is deciphered. Precisely, we give in terms of tensor product over a scalar base ring, a simplest and equivalent definition for cowreath over coring and for comodule over cowreath. An adjunction connecting the category of comodules over the factor coring and the category of comodules over the coring arising from the cowreath products is also given. The dual notions i.e. wreaths over rings extension and their modules are included.

math.RA↗
arXiv2006

Cohomology for bicomodules. Separable and Maschke functors

We introduce the category of bicomodules for a comonad in a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [11], and the characterization of coseparable coalgebras coextensions stated in [16].

math.RA↗
arXiv2005

Comatrix Corings an 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].

math.RA↗
arXiv2004

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 anther is the usual notion of morphisms defined in [5] by J. Gómez-Torrecillas.

math.RA↗
arXiv2004

Infinite Comatrix Corings

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

math.RA↗
arXiv2002

Comatrix corings: Galois corings, Descent Theory, and a Structure Theorem for Cosemisimple corings

In order to extrincate the structure of corings with a finitely generated and projective generator we give the notion of a comatrix coring. As consequences we give generalizations of the main characterizations of faithfully flat Galois corings and extensions which work for corings without grouplike elements, as well as a generalization of the Descent Theorem. We provide also a complete description of all cosemisimple corings. No restrictions are made over the ground noncommutative ring.

math.RA↗
arXiv2002

Semisimple corings

While semisimple artinian rings and semisimple coalgebras over a field can be described in terms of matrices (either matrix ring over division rings or comatrix coalgebras over the ground field), semisimple corings seem to have a more intrincated structure in general. It turns out that some well-known properties of semisimple rings or coalgebras, which are immediately deduced from the aforementioned structure, are not evident over a (left) semi-simple coring. For instance, it is not evident that the notion of semi-simple coring is left-right symmetric. To be precise, if every left comodule decomposes a a direct sum of simple comodules, do the right comodules have such a decomposition? In other words, is every left semi-simple coring a right semi-simple coring? We develope the basic essentials for a theory of semi-simple corings, giving a positive answer for the last question, as well as some information about the structure of semi-simple corings.

math.RA↗