Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-4btjb Total loading time: 0.259 Render date: 2022-05-28T19:33:55.369Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Godeaux–Serre varieties and the étale index

Published online by Cambridge University Press:  04 April 2013

Benjamin Antieau
Affiliation:
UCLA, Department of Mathematics, 520 Portola Plaza, Los Angeles CA 90095-1555, USAantieau@math.ucla.edu
Ben Williams
Affiliation:
USC, Department of Mathematics, 3620 South Vermont Avenue, Los Angeles CA 90089-2532, USAtbwillia@usc.edu
Get access

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.

Type
Research Article
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Antieau, B., Čech approximation to the Brown-Gersten spectral sequence, Homology Homotopy Appl., 13(1) (2011), 319348.CrossRefGoogle Scholar
2.Antieau, B., Cohomological obstruction theory f or Brauer classes and the period-index problem, Journal of K-Theory 8(3) (2011), 419435.CrossRefGoogle 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), 2545.CrossRefGoogle Scholar
6.Atiyah, F., Segal, G., Twisted K-theory, Ukr. Mat. Visn. 1 (3) (2004), 287330, 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, 543.Google Scholar
8.Boardman, M., Conditionally convergent spectral s equences, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math. 239 (1999), 4984.CrossRefGoogle Scholar
9.Freed, D. S., Hopkins, M. J., Teleman, C.. Loop groups and twisted K-theory I, J. Topol. 4(4) (2011), 737798.CrossRefGoogle 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, 199219.Google Scholar
11.Higgs, R. J., On the degrees of projective representations, Glasgow Math. J. 30 (2) (1988), 133135.CrossRefGoogle Scholar
12.Higgs, R. J., Projective representations of abelian groups, J. Algebra, 242 (2) (2001), 769781.CrossRefGoogle Scholar
13.Jouanolou, J. P., Une suite exacte de Mayer-Vietoris en K-théorie algébrique, Lecture Notes in Math. 341 (1973), 293316, 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), 19251940.CrossRefGoogle Scholar
16.Milnor, J., Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430436.CrossRefGoogle Scholar
17.Milnor, J., On axiomatic homology theory, Pacific J. Math. 12 (1962), 337341.CrossRefGoogle Scholar
18.Schröer, S., Topological methods for complex-analytic Brauer groups, Topology 44 (5) (2005), 875894.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, 2453.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, 249281.Google Scholar
21.Totaro, B., Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (2) (1997), 467493.CrossRefGoogle Scholar
1
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Godeaux–Serre varieties and the étale index
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Godeaux–Serre varieties and the étale index
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Godeaux–Serre varieties and the étale index
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *