Comatrix Corings and Galois Comodules over Firm Rings

We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory. Presented by A. Verschoren.     

A Zariski Topology for Bicomodules and Corings

In this paper we introduce and investigate top (bi)comodules of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative) rings. We restrict our attention in this paper to duo (bi)comodules (satisfying suitable conditions) and study the interplay between the coalgebraic properties of such (bi)comodules and the introduced Zariski topology. In particular, we apply our results to introduce a Zariski topology on the fully coprime spectrum of a given non-zero coring considered canonically as duo object in its category of bicomodules. Supported by King Fahd University of Petroleum & Minerals, Research Project # INT/296.     

Corings over rings with local units

We show that the bicategory whose 0‐cells are corings over rings with local units is bi‐equivalent to the bicategory of comonads in (right) unital modules whose underlying functors are right exact and preserve direct sums. A base ring extension of a coring by an adjunction is introduced as well (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

On Galois Comodules

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P _A is finitely generated and projective and the evaluation map μ𝒞:Hom ^𝒞 (P, 𝒞) ?  _S P → 𝒞 is an isomorphism (of corings) where S = End ^𝒞 (P). It has been observed that for such comodules the functors ? ?  _A 𝒞 and Hom _A (P, ?) ?  _S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define Galois comodules without requiring P _A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.     

Fully Coprime Comodules and Fully Coprime Corings

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely fully coprime corings (fully coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relations with the fully coprimeness and the simplicity of these comodules. These notions are applied then to study primeness and coprimeness properties of a given coring, considered as an object in its category of right (left) comodules. Supported by King Fahd University of Petroleum and Minerals, Research Project # INT/296.     

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.     

Descent Cohomology and Corings

A coring approach to non-Abelian descent cohomology of Nuss and Wambst (2007 Nuss , P. , Wambst , M. ( 2007 ). Non-Abelian Hopf cohomology . J. Algebra 312 : 733 – 754 . [Google Scholar]) is described and a definition of a Galois cohomology for partial group actions is proposed.     

Galois averages

In this paper, we introduce a notion of “Galois average” which allows us to give a suitable answer to the question: how can one extend a finite Galois extension E/F by a prime degree extension N/E to get a Galois extension N/F? Here, N/E is not necessarily a Kummer extension.     

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).     

Azumaya Extensions and Galois Correspondence

For H a finite-dimensional Hopf algebra over a field k, we study H*-Galois Azumaya extensions A, i.e., A is an H-module algebra which is H*-Galois with A/A^H separable and A^H Azumaya. We prove that there is a Galois correspondence between a set of separable subalgebras of A and a set of separable subalgebras of C_A(A^H), thus generalizing the work of Alfaro and Szeto for H a group algebra. We also study Galois bases and Hirata systems.1991 Mathematics Subject Classification: 16W30, 16H05     

The Generalized Center Galois Extensions of Rings

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and I_i = {c - g_i(c) | c C} for each g_i G. Then, B is called a center Galois extension with Galois group G if BI_i = B for each g_i 1 in G, and a weak center Galois extension with group G if BI_i = Be_i for some nonzero idempotent e_i in C for each g_i 1 in G. When e_i is a minimal element in the Boolean algebra generated by {e_i | g_i G} Be_i is a center Galois extension with Galois group H_i for some subgroup H_i of G. Moreover, the central Galois algebra B(1 – e_i) is characterized when B is a Galois algebra with Galois group G.AMS Subject Classification (1991): 16S35 16W20Supported by a Caterpillar Fellowship, Bradley University, Peoria, Illinois, USA. We would like to thank Caterpillar Inc. for their support.     

The Invariant Subrings of an Azumaya Galois Extension

Let B be an Azumaya Galois extension or a DeMeyer-Kanzaki Galois extension with Galois group G. Equivalent conditions are given for a separable subextension of a Galois extension in the skew group ring B * G being an invariant subring of a subgroup of the Galois group G.AMS Subject Classification (2000): 16S35, 16W20.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Lifting Galois representations over arbitrary number fields

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero p-adic representation, if local lifting problems at places above p are unobstructed.     

On the set of grouplikes of a coring

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

Hereditary triangular matrix comonads

We recognize Harada’s generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less than or equal to 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.     

Fuzzy Galois Connections

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided.