Coactions, Smash Products, and Hopf Modules

Cai, Chuanren; Chen, H.X.

doi:10.1006/jabr.1994.1176

Cited by 12 publications

(3 citation statements)

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“…A Morita theory for rings with local units was developed in [3]. Several Morita contexts were constructed in connection to Galois theory for Hopf algebra actions and coactions (see [4,8,12]), where Hopf-Galois extensions are characterized by the surjectivity of one of the Morita maps. As an application, the finite dimensional version of the duality theorem of Blattner-Montgomery [6] was deduced and explained in a nice way by using Morita contexts and Hopf-Galois theory in [30].…”

Section: Introductionmentioning

confidence: 99%

Wide Morita contexts, relative injectivity and equivalence results

2005

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We extend Morita theory to abelian categories by using wide Morita contexts. Several equivalence results are given for wide Morita contexts between abelian categories, widely extending equivalence theorems for categories of modules and comodules due to Kato, Müller, and Berbec. In the case of Grothendieck categories we derive equivalence results by using quotient categories. We apply the general equivalence results to rings with identity, rings with local units, graded rings, Doi-Hopf modules and coalgebras.

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Section: Introductionmentioning

confidence: 99%

Wide Morita contexts, relative injectivity and equivalence results

2005

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“…[1], [28], [30]) or to the recent monograph [12] for the theory of coalgebras over arbitrary base rings. For an R-coalgebra C, we call a right (respectively a left) C-comodule (M, ̺ M ) counital if its structure map ̺ M is injective, compare [13,Lemma 1.1.]. For an R-coalgebra C we denote with M C (respectively C M) the category of counital right (respectively left) C-comodules.…”

Section: Introductionmentioning

confidence: 99%

Hopf Pairings and (Co)induction Functors Over Commutative Rings

Abuhlail

2005

J. Algebra Appl.

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Abstract(Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings) satisfying the so called α-condition. For those pairings the coinduction functor is studied and nice descriptions of it are obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (Noetherian) ground rings.

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“…Since the pioneering work of Sweedler [22], integrals on Hopf algebras were studied by several authors, e.g., in [1,4,5,[19][20][21]. Among others, Sweedler proved the existence and uniqueness up to a constant of a (non-zero) integral on any finite-dimensional Hopf algebra.…”

Section: Introductionmentioning

confidence: 99%