The equivariant Brauer group of a group
HTML articles powered by AMS MathViewer
- by S. Caenepeel, F. Van Oystaeyen and Y. H. Zhang PDF
- Proc. Amer. Math. Soc. 134 (2006), 959-972 Request permission
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 \[ 1\longrightarrow \operatorname {Br}’(k)\longrightarrow \operatorname {BM}’(k,G)\longrightarrow \operatorname {Gal}(k,G) \longrightarrow 1\] 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
- Margaret Beattie, A direct sum decomposition for the Brauer group of $H$-module algebras, J. Algebra 43 (1976), no. 2, 686–693. MR 441942, DOI 10.1016/0021-8693(76)90134-4
- Stefaan Caenepeel, Brauer groups, Hopf algebras and Galois theory, $K$-Monographs in Mathematics, vol. 4, Kluwer Academic Publishers, Dordrecht, 1998. MR 1610222, DOI 10.1007/978-94-015-9038-9
- Robert J. Blattner, Miriam Cohen, and Susan Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986), no. 2, 671–711. MR 860387, DOI 10.1090/S0002-9947-1986-0860387-X
- S. Caenepeel and F. Grandjean, A note on Taylor’s Brauer group, Pacific J. Math. 186 (1998), no. 1, 13–27. MR 1665054, DOI 10.2140/pjm.1998.186.13
- S. Caenepeel, F. Van Oystaeyen, and Y. H. Zhang, Quantum Yang-Baxter module algebras, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 231–255. MR 1291020, DOI 10.1007/BF00960863
- S. Caenepeel, F. Van Oystaeyen, and Y. H. Zhang, The Brauer group of Yetter-Drinfel′d module algebras, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3737–3771. MR 1454120, DOI 10.1090/S0002-9947-97-01839-4
- Lindsay N. Childs, The Brauer group of graded Azumaya algebras. II. Graded Galois extensions, Trans. Amer. Math. Soc. 204 (1975), 137–160. MR 364216, DOI 10.1090/S0002-9947-1975-0364216-5
- A. Fröhlich, Orthogonal and symplectic representations of groups, Proc. London Math. Soc. (3) 24 (1972), 470–506. MR 308248, DOI 10.1112/plms/s3-24.3.470
- A. Fröhlich and C. T. C. Wall, Equivariant Brauer groups, Quadratic forms and their applications (Dublin, 1999) Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 57–71. MR 1803361, DOI 10.1090/conm/272/04397
- A. Fröhlich and C. T. C. Wall, Equivariant Brauer groups, Quadratic forms and their applications (Dublin, 1999) Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 57–71. MR 1803361, DOI 10.1090/conm/272/04397
- Ofer Gabber, Some theorems on Azumaya algebras, The Brauer group (Sem., Les Plans-sur-Bex, 1980) Lecture Notes in Math., vol. 844, Springer, Berlin-New York, 1981, pp. 129–209. MR 611868
- M.-A. Knus and M. Ojanguren, Cohomologie étale et groupe de Brauer, The Brauer group (Sem., Les Plans-sur-Bex, 1980) Lecture Notes in Math., vol. 844, Springer, Berlin-New York, 1981, pp. 210–228 (French). MR 611869
- Yôichi Miyashita, An exact sequence associated with a generalized crossed product, Nagoya Math. J. 49 (1973), 21–51. MR 320080
- D. J. Picco and M. I. Platzeck, Graded algebras and Galois extensions, Rev. Un. Mat. Argentina 25 (1970/71), 401–415. MR 332894
- Iain Raeburn and Joseph L. Taylor, The bigger Brauer group and étale cohomology, Pacific J. Math. 119 (1985), no. 2, 445–463. MR 803128
- Joseph L. Taylor, A bigger Brauer group, Pacific J. Math. 103 (1982), no. 1, 163–203. MR 687968
- A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), no. 2, 917–932. MR 1220906, DOI 10.1090/S0002-9947-1994-1220906-5
- A. Van Daele and Y. H. Zhang, Galois theory for multiplier Hopf algebras with integrals, Algebr. Represent. Theory 2 (1999), no. 1, 83–106. MR 1688472, DOI 10.1023/A:1009938708033
- T. N. Chan and M. A. Malik, On Erdős-Lax theorem, Proc. Indian Acad. Sci. Math. Sci. 92 (1983), no. 3, 191–193. MR 767620, DOI 10.1007/BF02876763
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
- MR Author ID: 176900
- Email: fred.vanoystaeyen@ua.ac.be
- Y. H. Zhang
- Affiliation: School of Mathematics and Computing Science, Victoria University of Wellington, Wellington, New Zealand
- MR Author ID: 310850
- ORCID: 0000-0002-0551-1091
- Email: yinhuo.zhang@vuw.ac.nz
- 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
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2196026