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A Hopf algebra having a separable Galois extension is finite dimensional


Author: Juan Cuadra
Journal: Proc. Amer. Math. Soc. 136 (2008), 3405-3408
MSC (2000): Primary 16W30
DOI: https://doi.org/10.1090/S0002-9939-08-09557-9
Published electronically: May 29, 2008
MathSciNet review: 2415022
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Abstract: It is shown that a Hopf algebra $ H$ over a field admitting a Galois extension $ A$ separable over its subalgebra of coinvariants $ B$ is of finite dimension. This answers in the affirmative a question posed by Beattie et al. in [Proc. Amer. Math. Soc. 128, No. 11 (2000), 3201-3203]. It is also proven that this result holds true if $ H$ has bijective antipode and the extension $ A/B$ is Frobenius.


References [Enhancements On Off] (What's this?)

  • 1. M. Beattie, S. Dăscălescu, and Ş. Raianu, A Co-Frobenius Hopf Algebra with a Separable Galois Extension Is Finite. Proc. Amer. Math. Soc. 128, No. 11 (2000), 3201-3203. MR 1690974 (2001b:16040)
  • 2. S. Caenepeel, G. Militaru, and S. Zhu, Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations. Lecture Notes in Mathematics 1787, Springer-Verlag, Berlin, 2002. MR 1926102 (2003h:16061)
  • 3. S.U. Chase and M.E. Sweedler, Hopf Algebras and Galois Theory. Lecture Notes in Mathematics 97, Springer-Verlag, Berlin, 1969. MR 0260724 (41:5348)
  • 4. M. Cohen and D. Fischman, Semisimple Extensions and Elements of Trace 1. J. Algebra 149 (1992), 419-437. MR 1172438 (93c:16038)
  • 5. S. Dăscălescu, C. Năstăsescu, and Ş. Raianu, Hopf Algebras. An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 235, Marcel-Dekker, New York, 2001. MR 1786197 (2001j:16056)
  • 6. D. Fischman, S. Montgomery, and H.-J. Schneider, Frobenius Extensions of Subalgebras of Hopf Algebras. Trans. Amer. Math. Soc. 349, No. 12 (1997), 4857-4895. MR 1401518 (98c:16049)
  • 7. H.F. Kreimer and M. Takeuchi, Hopf Algebras and Galois Extensions of an Algebra. Indiana Univ. Math. J. 30 (1981), 675-692. MR 625597 (83h:16015)
  • 8. S. Montgomery, Hopf Algebras and Their Actions on Rings. CBMS Regional Conf. Ser. in Math. 82, AMS, 1993. MR 1243637 (94i:16019)
  • 9. C. Năstăsescu, M. Van den Bergh, and F. Van Oystaeyen, Separable Functors Applied to Graded Rings. J. Algebra 123 (1989), 397-413. MR 1000494 (90j:16001)
  • 10. H.-J. Schneider, Principal Homogeneous Spaces for Arbitrary Hopf Algebras. Israel J. Math. 72, Nos. 1-2 (1980), 167-195. MR 1098988 (92a:16047)
  • 11. M.E. Sweedler, Integrals for Hopf Algebras. Ann. of Math. (2) 89 (1969), 323-335. MR 0242840 (39:4167)

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Additional Information

Juan Cuadra
Affiliation: Universidad de Almería, Depto. Álgebra y Análisis Matemático, E-04120 Almería, Spain
Email: jcdiaz@ual.es

DOI: https://doi.org/10.1090/S0002-9939-08-09557-9
Received by editor(s): January 17, 2007
Received by editor(s) in revised form: March 1, 2007, and March 13, 2007
Published electronically: May 29, 2008
Additional Notes: This research was supported by projects MTM2005-03227 from MCYT and FEDER and P06-FQM-1889 from Junta de Andalucía
Dedicated: To José Luis Gómez Pardo on the occasion of his 60th birthday
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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