Galois Extensions for Co-Frobenius Hopf Algebras

Beattie, Margaret; Dăscălescu, S.; Raianu, Serban

doi:10.1006/jabr.1997.7146

Cited by 33 publications

(96 citation statements)

References 27 publications

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“…We recall some basic results and propositions that we will need later from Alves and Batista [3] and Beattie et al [4].…”

Section: Preliminariesmentioning

confidence: 99%

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On generalized partial twisted smash products

Guo¹

2014

Czech Math J

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“…We recall some basic results and propositions that we will need later from Alves and Batista [3] and Beattie et al [4].…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section we construct a Morita context between A bicoH and A ⋆ H * rat , where A is a partial bicomodule algebra, generalizing M. Beattie et al's work [4].…”

Section: Morita Contextmentioning

confidence: 99%

On generalized partial twisted smash products

Guo¹

2014

Czech Math J

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“…M ρ). We use Sweedler's notation for coproducts ∆ C (c) = c (1) ⊗c (2) , for right coactions ρ M (m) = m [0] ⊗m [1] , and for left coactions M ρ(m) = m [−1] ⊗m [0] . The cotensor product is denoted by − C −.…”

Section: Notationmentioning

confidence: 99%

“…We need to show that N ρ is associative and unital. Throughout the proof we write σ (c) = c [1] ⊗c [2] ∈ N D M (summation assumed). The right action ρ M of A on M is denoted by ✁ between the elements.…”

Section: A -Coendomorphism Coalgebra and Galois Modulesmentioning

confidence: 99%

The Galois Theory of Matrix C-rings

Brzeziński

Turner

2006

Appl Categor Struct

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ABSTRACT. A theory of monoids in the category of bicomodules of a coalgebra C or Crings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix C-ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasifinite injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider's Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167-195], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed.

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“…An A-coring is a coalgebra (comonoid) in the monoidal category A M A , i.e. a triple (C, ∆, ε), where C is an A-bimodule and the coproduct ∆ : C → C ⊗ A C, ∆(c) =: c (1) ⊗ A c (2) (Sweedler notation, with implicit summation understood) and the counit ε : C → A are A-bimodule maps that satisfy (∆ ⊗ A C) • ∆(c) = (C ⊗ A ∆) • ∆(c) =: c (1) ⊗ A c (2) ⊗ A c (3) and c (1) ε(c (2) ) = c = ε(c (1) )c (2) , for all c ∈ C. A right C-comodule consists of a right A-module M together with a right A-linear…”

Section: Introductionmentioning

confidence: 99%

Morita Theory for Comodules Over Corings

Böhm

Vercruysse

2009

Communications in Algebra

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Abstract. 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 Amodule. 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.

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Galois Extensions for Co-Frobenius Hopf Algebras

Cited by 33 publications

References 27 publications

On generalized partial twisted smash products

On generalized partial twisted smash products

The Galois Theory of Matrix C-rings

Morita Theory for Comodules Over Corings

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