Morita theory for coring extensions and cleft bicomodules.
type: Item
instance of: scholarly article
title: Morita theory for coring extensions and cleft bicomodules.
zbMATH Open document ID: 1123.16028
author: Joost Vercruysse
author: Gabriella Böhm
published in: Advances in Mathematics
publication date: 2007-02-26T00:00:00Z
review text: The paper is devoted to construction and studies of Morita contexts associated to extensions of corings (over different \(k\)-algebras) and bicomodules. These contexts are shown to arise from a 2-object subcategory of the category of \(k\)-linear functors between categories of comodules. The strictness of such a context implies the Galois property of the comodule involved and the weak structure theorem (i.e.\ faithfulness and fullness of the induced functor). Sufficient conditions for the strong structure theorem (i.e.\ for the induced functor to be an equivalence of categories of comodules) are found. The authors introduce the notion of a cleft bicomodule. A bicomodule is cleft if and only if it is a Galois comodule that in addition enjoys a generalised version of the normal basis property. It is proven that such a bicomodule enforces the associated Morita context to be strict. Examples of cleft bicomodules are provided by cleft extensions of Hopf algebras, coalgebras or Hopf algebroids and by cleft (weak) entwining structures over non-commutative rings.
reviewed by: Tomasz Brzeziński
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5129763
zbMATH Keywords: Morita theory
zbMATH Keywords: coring extensions
zbMATH Keywords: weak structure theorems
zbMATH Keywords: strong structure theorems
zbMATH Keywords: cleft bicomodules
zbMATH Keywords: Morita contexts
zbMATH Keywords: categories of functors
zbMATH Keywords: categories of comodules
zbMATH Keywords: equivalences of categories
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Hopf algebroids
zbMATH Keywords: entwining structures
MaRDI profile type: Publication
full work available at URL: j.aim.2006.05.010
OpenAlex ID: W2166661337
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DOI: 10.1016/J.AIM.2006.05.010
description: scientific article; zbMATH DE number 5129763