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Morita theory for coring extensions and cleft bicomodules
- Source :
- Advances in mathematics, 209 (2
- Publication Year :
- 2007
- Publisher :
- Elsevier BV, 2007.
-
Abstract
- A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $\Sigma$ for a pure coring extension $(\cD:L)$ of $(\cC:A)$. It is related to a 2-object subcategory of the category of $k$-linear functors $\Mm^\Cc\to\Mm^\Dd$. Strictness of the Morita context is shown to imply the Galois property of $\Sigma$ as a $\cC$-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an $L$-$\cC$ bicomodule $\Sigma$ -- implying strictness of the associated Morita context -- is introduced. It is shown to be equivalent to being a Galois $\cC$-comodule and isomorphic to $\End^\cC(\Sigma)\otimes_{L} \cD$, in the category of left modules for the ring $\End^\cC(\Sigma)$ and right comodules for the coring $\cD$, i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a pure Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules. Cleft extensions by arbitrary Hopf algebroids are described in terms of Morita contexts that do not necessarily correspond to coring extensions.<br />Comment: 34 pages LaTeX. v2:A missing purity assumption is added throughout Sections 3, 4 and 5
- Subjects :
- Mathematics(all)
Pure mathematics
General Mathematics
Coalgebra
Context (language use)
Algèbre - théorie des anneaux - théorie des corps
Coring extension
Comodule
Mathematics::K-Theory and Homology
Mathematics::Category Theory
Mathematics::Quantum Algebra
16D90
FOS: Mathematics
Mathematics
Subcategory
Ring (mathematics)
Functor
Mathematics::Rings and Algebras
Mathematics - Rings and Algebras
Morita theory
Hopf algebra
Normal basis
Rings and Algebras (math.RA)
16W30
Géométrie non commutative
Weak and strong structure theorems
Cleft bicomodule
Subjects
Details
- ISSN :
- 00018708
- Volume :
- 209
- Database :
- OpenAIRE
- Journal :
- Advances in Mathematics
- Accession number :
- edsair.doi.dedup.....9d5d0b494d8dd3c64bd2093c86905e90