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.     

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.     

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.     

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.     

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.

     

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.     

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).     

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.     

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.     

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.     

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.     

模范畴与余模范畴之间的等价

本文研究了环上模范畴与余环上余模范畴.运用可裂叉与余可分余环的性质,得到了以上两个范畴等价的一些充分条件,从而推广了文献[6]中的一些结果.

Abstract：

In this article,we consider the categories of modules over rings and categories of comodules over corings.By properties of split forks and coseparable corings,we get some sufficient conditions for the equivalence between above two categories.As a consequence,we generalize some results in[6].     

On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F_p. Let q be a power of p and let G(q) be the finite group of F_q-rational points of G. Let F be the Frobenius morphism such that G(q) = G^F. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).     

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.      

When ∏ is Isomorphic to ⊕

For a category , we investigate the problem of when the coproduct ⊕ and the product functor ∏ from  ^I to  are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category  this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in  that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of  ^I which is isomorphic to  ^I but is not a full subcategory. If  is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

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.