Joost Vercruysse

Galois theory for comatrix corings: descent theory, Morita theory, Frobenius and separability properties

El Kaoutit and Gomez Torrecillas introduced comatrix corings, generalizing Sweedlers canonical coring, and proved a new version of the Faith- fully 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 con- text 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 cor- ing is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.

MULTIPLIER BI- AND HOPF ALGEBRAS

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

Local Units Versus Local Projectivity Dualisations: Corings with Local Structure Maps

We unify and generalize different notions of local units and local projectivity. We investigate the connection between these properties by constructing elementary algebras from locally projective modules. Dual versions of these constructions are discussed, leading to corings with local comultiplications, corings with local counits, and rings with local multiplications.

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.

A Torsion Theory in the Category of Cocommutative Hopf Algebras

The purpose of this article is to prove that the category of cocommutative Hopf K-algebras, over a field K of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie K-algebras, respectively.

Morita Theory for Comodules Over Corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End 𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.

A Note on the Categorification of Lie Algebras

In this short note we study Lie algebras in the framework of symmetric monoidal categories. After a brief review of the existing work in this field and a presentation of earlier studied and new examples, we examine which functors preserve the structure of a Lie algebra.

The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived.

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.

Corrigendum to "Morita theory for coring extensions and cleft bicomodules" (Adv. Math. 209 (2) (2007) 611-648)

Abstract The results in our paper heavily rely on the journal version of [T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005) 15–27; a corrected version is available at http://arxiv.org/abs/math/0410020v3 , Theorem 2.6]. Since it turned out recently that in the proof of the quoted theorem there are some assumptions missing, our derived results are not expected to hold at the stated level of generality either. Here we supplement the constructions in our article with the missing assumptions and show that they hold in most of our examples. In order to handle also the non-fitting case of cleft extensions by arbitrary Hopf algebroids, Morita contexts are constructed that do not necessarily correspond to coring extensions. They are used to prove a Strong Structure Theorem for cleft extensions by arbitrary Hopf algebroids. In this way we obtain in particular a corrected form of the journal version of [G. Bohm, Integral theory for Hopf algebroids, Algebr. Represent. Theory 8 (4) (2005) 563–599; Corrigendum, to be published; see also http://arxiv.org/abs/math/0403195v4 , Theorem 4.2], whose original proof contains a similar error.