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}.     

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.     

Faithfully Flat Hopf Bi-Galois Extensions

This article is devoted to faithfully flat Hopf bi-Galois extensions defined by Fischman, Montgomery, and Schneider. Let H be a Hopf algebra with bijective antipode. Given a faithfully flat right H-Galois extension A/R and a right H-comodule subalgebra C ? A such that A is faithfully flat over C, we provide necessary and sufficient conditions for the existence of a Hopf algebra W so that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra. As a consequence, we prove that if R = k, there is a Hopf algebra W such that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra if and only if C is an H-submodule of A with respect to the Miyashita–Ulbrich action.     

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(■).     

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.     

A Morita Context and Galois Extensions for Quasi-Hopf Algebras

If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we show that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A ^H . We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case.     

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

Galois theory for Hopf algebroids

An extensionB⊂A of algebras over a commutative ringk is anH-extension for anL-bialgebroidH ifA is anH-comodule algebra andB is the subalgebra of its coinvariants. It isH-Galois if the canonical mapA ⊗_B A →A ⊗_L H is an isomorphism or, equivalently, if the canonical coringA ⊗_L H:A is a Galois coring. In the case of Hopf algebroid anyH _R-extension is shown to be also anH _L-extension. If the antipode is bijective then also the notions ofH _R-Galois extensions and ofH _L-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroidH with bijective antipode. It states that anyH-Galois extensionB⊂A is projective, and ifA isk-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

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Sunto Un'estensioneB(A di algebre su un anello commutativok è unaH-estensione per unL-bialgebroideH seA è unaH-comodulo algebra eB è la sottoalgebra dei suoi coinvarianti. Essa èH-Galois se l'applicazione canonicaA ⊗_A B→A ⊗_L H è un isomorfismo o, equivalentemente, se il coanello canonicoA ⊗_L H:A è un coanello di Galois. Nel caso di un algebroide di Hopf si dimostra che ogniH _R-estensione è unaH _L-estensione. Se l'antipode è biiettivo allora si dimostra che anche le nozioni di estensioniH _R-Galois eH _L-Galois coincidono. I risultati per le strutture biiettive entwining sono estesi alle strutture entwining su algebre non commutative, al fine di dimostrare un teorema simile al Teorema dii Kreimer-Takeuchi per un Hopf algebroideH proiettivo finitamento generato con antipode biiettivo. Il teorema afferma che ogni estensioneH-GaloisB ⊂A è proiettiva e seA èk-piatto allora la suriettività dell'applicazione canonica è sufficiente a garantire la proprietà di Galois. La teoria di Morita, sviluppata per i coanelli da Caenepeel, Vercruysse e Wang, viene applicata per ottenere criteri equivalenti per la proprietà di Galois per estensioni di algebroidi di Hopf. Questo conduce a risultati analoghi, per algebroidi di Hopf, a quelli ottenuti da Doi per estensioni di algebre di Hopf e da Cohen Fishman e Montgomery nel caso degli algebroidi di Hopf Frobenius.

     

Galois-type correspondences for Hopf Galois extensions

In order to obtain a Galois correspondence between intermediate subalgebras of a Hopf-Galois extension and corresponding Hopf subalgebras, we have to reconstruct such Hopf algebras from the given Hopf-Galois extension. The construction of a suitable Hopf algebra associated to a commutativeH-Galois extension is provided; the results are applicable to field extensions.This author is supported by a grant of the European Liaisons Committee.     

On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra,and A/AHa right H*-Galois extension. The authors prove that, if AHis a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants A~B= {a ∈ A |b · a = ε(b)a, b∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AHas in the classical case. The results are applied to the case H =(kG)*for a finite group G to get a Galois 1-1 correspondence.     

Hilbert's Theorem 90 for Hopf Galois Extensions

For a faithfully flat extension A/B and a right A-module M, we give a new characterization of the set of descent data on M. Assuming that B is a simple Artinian ring and A/B is H-Galois, for a certain finite dimensional Hopf algebra H, we prove that Sweedler's noncommutative cohomology H ¹(H?, A) is trivial as a pointed set.     

On the Constructive Inverse Problem in Differential Galois Theory#

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G ⁰ = G ₁ · ··· · G _r , where each G _i is a simple group of type A_?, C_?, D_?, E₆, or E₇, we construct a differential equation over C(x) having Galois group G.     

Central invariants of H-module algebras

Let H be a Hopf algebra over a field k:, and A an H-module algebra, with subalgebra of H-invariants denoted by A^H . When (H, R) is quasitriangular and A is quantum commutative with respect to (H,R), (e.g. quantum planes, graded commutative superalgebras), then A^H ? center of A = Z(A). In this paper we are mainly concerned with actions of H for which A^H ? Z(A). We show that under this hypothesis there exists strong relations between the ideal structures of A^H A and A#H.

We demonstrate the theorems by constructing an example of a quantum commutative A, so that A/A^H is H ^?-Galois. This is done by giving (C G)^? G = Z_n × Z_n , a nontrivial quasitriangular structure and defining an action of it on a localization of the quantum plane.     

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X ₁₁, . . . , X _nn , T) be the characteristic polynomial of the generic matrix (X _ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.     

Depth Two,Normality, and a Trace Ideal Condition for Frobenius Extensions

A ring extension A ‖ B is depth two if its tensor-square satisfies a projectivity condition w.r.t. the bimodules _A A _B and _B A _A . In this case the structures (A ? _B A) ^B and End  _B A _B are bialgebroids over the centralizer C _A (B) and there is a certain Galois theory associated to the extension and its endomorphism ring. We specialize the notion of depth two to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse, we observe that normal Hopf subalgebras over a field are depth two extensions. A generalized Miyashita–Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.     

Biring Theory and Its Galois Theory of Rings

Biring theory is about birings (A, P), that is, algops (A, P) of an associative algebra A and (A, A)-biring P acting on A via a morphism γ: P → Pres _F A from P to the terminal (A, A)-biring Pres _F A of preservations of A. (The word biring is used in a theory for a structure with unit, product, counit, coproduct subject to conditions of the theory.) Biring theory has its central simple theory and its Galois theory of rings. Its Galois birings are the reduced simple birings.     

Complex algebras of subalgebras

Let V be a variety of algebras. We specify a condition (the so-called generalized entropic property), which is equivalent to the fact that for every algebra A ∈ V, the set of all subalgebras of A is a subuniverse of the complex algebra of the subalgebras of A. The relationship between the generalized entropic property and the entropic law is investigated. Also, for varieties with the generalized entropic property, we consider identities that are satisfied by complex algebras of subalgebras. Dedicated to George Gr?tzer on the occasion of his 70th birthday Supported by INTAS grant No. 03-51-4110. Supported by MŠMTČR (project MSM 0021620839) and by the Grant Agency of the Czech Republic (grant No. 201/05/0002). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 655–686, November–December, 2008.     

Codepth Two and Related Topics

A depth two extension A | B is shown to be weak depth two over its double centralizer V _A (V _A (B)) if this is separable over B. We consider various examples and non-examples of depth one and two properties. Depth two and its relationship to direct and tensor product of algebras as well as cup product of relative Hochschild cochains is examined. Section 6 introduces a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End ^D C ^D forms a right bialgebroid over the centralizer subalgebra g ^* : D ^* → C ^* of the dual algebra C ^*. Dedicated to Daniel Kastler on his eightieth birthday.     

On Galois Coverings of the Enveloping Algebras of Self-Injective Nakayama Algebras

Abstract

There is constructed a Galois covering F of the enveloping K-algebra A ^e of a self-injective Nakayama K-algebra A such that the right A ^e -module A is of the first kind with respect to F. Then, with the help of the constructed Galois covering, the Auslander-Reiten translation period of A is computed.