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 | References | Similar Articles | Additional Information

Abstract: It is shown that a Hopf algebra over a field admitting a Galois extension separable over its subalgebra of coinvariants 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 has bijective antipode and the extension is Frobenius.

**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.