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The equivariant Brauer group of a group


Authors: S. Caenepeel, F. Van Oystaeyen and Y. H. Zhang
Journal: Proc. Amer. Math. Soc. 134 (2006), 959-972
MSC (2000): Primary 16H05, 16W50
DOI: https://doi.org/10.1090/S0002-9939-05-08041-X
Published electronically: August 16, 2005
MathSciNet review: 2196026
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Abstract: We consider the Brauer group ${\operatorname{BM}'}(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence

\begin{displaymath}1\longrightarrow \operatorname{Br}'(k)\longrightarrow \oper... ...}'(k,G)\longrightarrow\operatorname{Gal}(k,G) \longrightarrow 1\end{displaymath}

is obtained. This generalizes the Fröhlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of $G$ to the infinite case of $G$. Here $\operatorname{Br}'(k)$ is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.

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

  • 1. M. Beattie, A direct sum decomposition for the Brauer group of $H$-module algebras, J. Algebra 43 (1976), 686-693. MR 0441942 (56:333)
  • 2. S. Caenepeel, ``Brauer groups, Hopf algebras and Galois theory", K-Monographs in Mathematics 4, Kluwer Academic Publishers, Dordrecht, 1998. MR 1610222 (99b:16028)
  • 3. R. Blattner, M. Cohen, and S. Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986), 672-711. MR 0860387 (87k:16012)
  • 4. S. Caenepeel and F. Grandjean, A note on Taylor's Brauer group, Pacific J. Math 186 (1998), 13-27. MR 1665054 (2000f:14027)
  • 5. S. Caenepeel, F. Van Oystaeyen and Y.H. Zhang, Quantum Yang-Baxter module algebras, K-Theory, 8(1993), 231-255. MR 1291020 (95e:16031)
  • 6. S. Caenepeel, F. Van Oystaeyen and Y.H. Zhang, The Brauer group of Yetter-Drin'feld module algebras, Trans. Amer. Math. Soc. 349 (1997), 3737-3771. MR 1454120 (98c:16047)
  • 7. L.N. Childs, The Brauer group of graded Azumaya algebras II: graded Galois extensions, Trans. Amer. Math. Soc. 204 (1975), 137-160. MR 0364216 (51:471)
  • 8. A. Fröhlich, Orthogonal and symplectic representations of groups, Proc. London Math. Soc. (3) 24 (1971), 470-506. MR 0308248 (46:7362)
  • 9. A. Fröhlich, C.T.C. Wall, Equivariant Brauer groups in algebraic number theory, Bull. Soc. Math. France 25 (1971), 91-96. MR 1803361 (2001m:18008)
  • 10. A. Fröhlich, C.T.C. Wall, Equivariant Brauer groups, Contemp. Math. 272 (2000), 57-71. MR 1803361 (2001m:18008)
  • 11. O. Gabber, Some theorems on Azumaya algebras, in ``Groupe de Brauer", M. Kervaire and M. Ojanguren (Eds.), Lecture Notes Math. 844, Springer-Verlag, Berlin, 1981. MR 0611868 (83d:13004)
  • 12. M.A. Knus, M. Ojanguren, Cohomologie étale et groupe de Brauer, in ``Groupe de Brauer", M. Kervaire and M. Ojanguren (Eds.), Lecture Notes in Math. 844, Springer-Verlag, Berlin, 1981. MR 0611869 (83k:16006)
  • 13. Y. Miyashita, An exact sequence associated with a generalized crossed product, Nagoya Math. J. 49 (1973), 21-51. MR 0320080 (47:8621)
  • 14. D.J. Picco, M.I. Platzeck, Graded algebras and Galois extensions, Bol. Un. Mat. Argentina 25 (1971), 401-415. MR 0332894 (48:11219)
  • 15. I. Raeburn, J. L. Taylor, The bigger Brauer group and étale cohomology, Pacific J. Math. 119 (1985), 445-463. MR 0803128 (87a:16007)
  • 16. J.L. Taylor, A bigger Brauer group, Pacific J. Math. 103 (1982), 163-203. MR 0687968 (84g:13007)
  • 17. A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), 917-932. MR 1220906 (94h:16075)
  • 18. A. Van Daele, Y.H. Zhang, Galois theory for multiplier Hopf algebras with integrals, Algebras Representation Theory, 2 (1999), 83-106. MR 1688472 (2000f:16050)
  • 19. A. Van Daele, Y.H. Zhang, A survey on multiplier Hopf algebras, in ``Hopf algebras and quantum groups", S. Caenepeel and F. Van Oystaeyen (eds.), Lecture Notes Pure Appl. Math. 209, Marcel Dekker, New York, 2000, 269-309. MR 767620 (2001e:16077)

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Additional Information

S. Caenepeel
Affiliation: Faculty of Applied Sciences, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium
Email: scaenepe@vub.ac.be

F. Van Oystaeyen
Affiliation: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerp, Belgium
Email: fred.vanoystaeyen@ua.ac.be

Y. H. Zhang
Affiliation: School of Mathematics and Computing Science, Victoria University of Wellington, Wellington, New Zealand
Email: yinhuo.zhang@vuw.ac.nz

DOI: https://doi.org/10.1090/S0002-9939-05-08041-X
Keywords: Equivariant Brauer group, Taylor Azumaya algebra
Received by editor(s): December 16, 2003
Received by editor(s) in revised form: August 16, 2004, and November 1, 2004
Published electronically: August 16, 2005
Additional Notes: The third named author was supported by the Marsden Fund
Communicated by: Martin Lorenz
Article copyright: © Copyright 2005 American Mathematical Society

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