Morita theory for coring extensions and cleft bicomodules

Details Morita theory for coring extensions and cleft bicomodules Morita theory for coring extensions and cleft bicomodules

2006-01-19

arxiv.org/abs/math/0601464v2

Adv. Math. 209 (2007) 611-648

Mathematics

Rings and Algebras

34 pages LaTeX. v2:A missing purity assumption is added throughout Sections 3, 4 and 5

Scientific paper

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.

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