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Morita theory for coring extensions and cleft bicomodules | | Gabriella Böhm
; Joost Vercruysse
; | | Date: |
19 Jan 2006 | | Subject: | Rings and Algebras | | Abstract: | A {em Morita context} is constructed for any comodule of a coring and, more generally, for an $L$-${mathcal C}$ bicomodule $Sigma$ for a coring extension $({mathcal D}:L)$ of $({mathcal C}:A)$. It is related to a 2-object subcategory of the category of $k$-linear functors ${mathcal M}^{mathcal C} o{mathcal M}^{mathcal D}$. Strictness of the Morita context is shown to imply the Galois property of $Sigma$ as a ${mathcal C}$-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold.
{em Cleft} property of an $L$-${mathcal C}$ bicomodule $Sigma$ -- implying strictness of the associated Morita context -- is introduced. It is shown to be equivalent to being a {em Galois} ${mathcal C}$-comodule and isomorphic to ${
m End}^{mathcal C}(Sigma)otimes_{L} {mathcal D}$, in the category of left modules for the ring ${
m End}^{mathcal C}(Sigma)$ and right comodules for the coring ${mathcal D}$, i.e. satisfying the {em normal basis} property.
Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a 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. | | Source: | arXiv, math/0601464 | | Services: | Forum | Review | PDF | Favorites | |
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