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Godeaux–Serre varieties and the étale index

Published online by Cambridge University Press:  04 April 2013

Benjamin Antieau  and
Ben Williams
Benjamin Antieau
Affiliation:
UCLA, Department of Mathematics, 520 Portola Plaza, Los Angeles CA 90095-1555, [email protected]
Ben Williams
Affiliation:
USC, Department of Mathematics, 3620 South Vermont Avenue, Los Angeles CA 90089-2532, [email protected]
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Abstract

We use Godeaux–Serre varieties of finite groups, projective representation theory, the twisted Atiyah–Segal completion theorem, and our previous work on the topological period-index problem to compute the étale index of Brauer classes α ∈ Brét(X) in some specific examples. In particular, these computations show that the étale index of α differs from the period of α in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Godeaux–Serre varieties in terms of projective representation theory.

Keywords

Projective representation theoryBrauer groupstwisted K-theoryperiod-index problemsfinite groupsPrimary: 14F2220C25Secondary: 19L50
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 2 , April 2013 , pp. 283 - 295
Copyright
Copyright © ISOPP 2013 

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References

1.Antieau, B., Čech approximation to the Brown-Gersten spectral sequence, Homology Homotopy Appl., 13(1) (2011), 319–348.Google Scholar
2.Antieau, B., Cohomological obstruction theory f or Brauer classes and the period-index problem, Journal of K-Theory 8(3) (2011), 419–435.Google Scholar
3.Antieau, B., Williams, B., The period-index problem for twisted topological K-theory ArXiv e-prints, http://arxiv.org/abs/1104.4654, 2011.Google Scholar
4.Antieau, B., Williams, B., The period-index problem over 6-complexes, ArXiv e-prints, http://arxiv.org/abs/1208.4430, 2012.Google Scholar
5.Atiyah, F., Hirzebruch, F., Analytic cycles on complex manifolds, Topology 1 (1962), 25–45.CrossRefGoogle Scholar
6.Atiyah, F., Segal, G., Twisted K-theory, Ukr. Mat. Visn. 1 (3) (2004), 287–330, Ukr. Math. Bull., 1 (3) (2004), 291–334.Google Scholar
7.Atiyah, F., Segal, G., Twisted K-theory and cohomology, Nankai Tracts Math. 11, World Sci. Publ., Hackensack, NJ, 2006, 5–43.Google Scholar
8.Boardman, M., Conditionally convergent spectral s equences, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math. 239 (1999), 49–84.Google Scholar
9.Freed, D. S., Hopkins, M. J., Teleman, C.. Loop groups and twisted K-theory I, J. Topol. 4(4) (2011), 737–798.Google Scholar
10.Grothendieck, A., Le groupe de Brauer. I. Algèbres d'Azumaya et interprétations diverses, Séminaire Bourbaki 9, Soc. Math. France, Paris, 1995, Exp. No. 290, 199–219.Google Scholar
11.Higgs, R. J., On the degrees of projective representations, Glasgow Math. J. 30 (2) (1988), 133–135.Google Scholar
12.Higgs, R. J., Projective representations of abelian groups, J. Algebra, 242 (2) (2001), 769–781.CrossRefGoogle Scholar
13.Jouanolou, J. P., Une suite exacte de Mayer-Vietoris en K-théorie algébrique, Lecture Notes in Math. 341 (1973), 293–316, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin.Google Scholar
14.Karpilovsky, G., Projective representations of finite groups, Monographs and Textbooks in Pure and Applied Mathematics 94, Marcel Dekker Inc., New York, 1985, xiii+644.Google Scholar
15.Lahtinen, A., The Atiyah–Segal completion t heorem in twisted K-theory, Alg. Geo. Topology 12 (4) (2012), 1925–1940.Google Scholar
16.Milnor, J., Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430–436.Google Scholar
17.Milnor, J., On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341.Google Scholar
18.Schröer, S., Topological methods for complex-analytic Brauer groups, Topology 44 (5) (2005), 875–894.CrossRefGoogle Scholar
19.Serre, J.-P., Sur la topologie des variétés algébriques en caractéristique p, Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, 24–53.Google Scholar
20.Totaro, B., The Chow ring of a classifying s pace, Algebraic K-theory, Seattle, WA, 1997, Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI 1999, 249–281.Google Scholar
21.Totaro, B., Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (2) (1997), 467–493.Google Scholar