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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Projectivity and freeness over comodule algebras


Author: Serge Skryabin
Journal: Trans. Amer. Math. Soc. 359 (2007), 2597-2623
MSC (2000): Primary 16W30
Published electronically: January 25, 2007
MathSciNet review: 2286047
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Abstract: Let $ H$ be a Hopf algebra and $ A$ an $ H$-simple right $ H$-comodule algebra. It is shown that under certain hypotheses every $ (H,A)$-Hopf module is either projective or free as an $ A$-module and $ A$ is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for $ H$-simple $ H$-module algebras.


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Additional Information

Serge Skryabin
Affiliation: Chebotarev Research Institute, Universitetskaya St. 17, 420008 Kazan, Russia
Email: Serge.Skryabin@ksu.ru

DOI: http://dx.doi.org/10.1090/S0002-9947-07-03979-7
PII: S 0002-9947(07)03979-7
Received by editor(s): February 27, 2004
Received by editor(s) in revised form: February 11, 2005
Published electronically: January 25, 2007
Additional Notes: This research was supported by the project “Construction and applications of non-commutative geometry" from FWO Vlaanderen. I would like to thank the Free University of Brussels VUB for their hospitality during the time the work was conducted.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.