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Comparison of Karoubi's regulator and the p-adic Borel regulator

Published online by Cambridge University Press:  04 November 2011

Georg Tamme
Georg Tamme
Affiliation:
California Institute of Technology, Department of Mathematics, MC 253-37, 1200 East California Boulevard, Pasadena CA 91125, [email protected]
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Abstract

In this paper we prove the p-adic analogue of a result of Hamida [11], namely that the p-adic Borel regulator introduced by Huber and Kings for the K-theory of a p-adic number field equals Karoubi's p-adic regulator up to an explicit rational factor.

Keywords

p-adic regulatorrelative Chern characterKaroubi's regulatorlocal fieldLazard isomorphism
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 3 , June 2012 , pp. 579 - 600
Copyright
Copyright © ISOPP 2011

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References

1.Beĭlinson, A. A., Higher regulators and values of L-functions, Current problems in mathematics 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238. MR 760999 (86h:11103)Google Scholar
2.Besser, Amnon, Syntomic regulators and p-adic integration. I. Rigid syntomic regulators, Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), vol. 120, 2000, pp. 291–334. MR 1809626 (2002c:14035)Google Scholar
3.Borel, Armand, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 0387496 (52 #8338)CrossRefGoogle Scholar
4.Borel, Armand, Cohomologie de SL n et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 4 (1977), no. 4, 613–636. MR 0506168 (58 #22016)Google Scholar
5.Gil, Jose I. Burgos, The regulators of Beilinson and Borel, CRM Monograph Series 15, American Mathematical Society, Providence, RI, 2002. MR 1869655 (2002m:19002)Google Scholar
6.Calvo, Adina, K-théorie des anneaux ultramétriques, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 14, 459–462.Google Scholar
7.Choo, Zacky and Snaith, Victor, p-adic cocycles and their regulator maps, J. K-theory 8 (2011), 241–249.CrossRefGoogle Scholar
8.Connes, Alain and Karoubi, Max, Caractère multiplicatif d'un module de Fredholm, K-Theory 2 (1988), no. 3, 431–463.CrossRefGoogle Scholar
9.Dupont, Johan L., Curvature and characteristic classes, Lecture Notes in Mathematics 640, Springer-Verlag, Berlin, 1978. MR MR0500997 (58 #18477)Google Scholar
10.Guichardet, A. and Wigner, D., Sur la cohomologie réelle des groupes de Lie simples réels, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 277–292. MR 510552 (80g:22023a)CrossRefGoogle Scholar
11.Hamida, Nadia, Description explicite du régulateur de Borel, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 3, 169–172. MR MR1748302 (2001a:20073)CrossRefGoogle Scholar
12.Hamida, Nadia, Les régulateurs en K-théorie algébrique, Ph.D. thesis, Université Paris VII - Denis Diderot, 2002.Google Scholar
13.Hamida, Nadia, Le réegulateur p-adique, C. R. Math. Acad. Sci. Paris 342 (2006), no. 11, 807–812.Google Scholar
14.Huber, Annette and Kings, Guido, A p-adic analogue of the Borel regulator and the Bloch-Kato exponential map, J. Inst. Math. Jussieu 10 (2011), no. 1, 149–190. MR 2749574CrossRefGoogle Scholar
15.Huber, Annette, Kings, Guido, and Naumann, Niko, Some complements to the Lazard isomorphism, Compos. Math. 147 (2011), no. 1, 235–262. MR 2771131CrossRefGoogle Scholar
16.Karoubi, Max, Homologie cyclique et régulateurs en K-théorie algébrique, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 10, 557–560.Google Scholar
17.Karoubi, Max, Homologie cyclique et K-théorie, Astérisque 149 (1987), 1–147.Google Scholar
18.Karoubi, Max, Sur la K-théorie multiplicative, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun. 17, Amer. Math. Soc., Providence, RI, 1997, pp. 59–77.Google Scholar
19.Karoubi, Max and Villamayor, Orlando, K-théorie algébrique et K-théorie topologique. I, Math. Scand. 28 (1971), 265–307 (1972). MR MR0313360 (47 #1915)CrossRefGoogle Scholar
20.Loday, Jean-Louis, Cyclic homology, Grundlehren der Mathematischen Wissenschaften 301, Springer-Verlag, Berlin, 1992.Google Scholar
21.Quillen, Daniel, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586. MR 0315016 (47 #3565)CrossRefGoogle Scholar
22.Schneider, Peter, Nonarchimedean Functional Analysis, Springer Monographs in Mathematics, Springer, 2002.Google Scholar
23.Schneider, Peter, p-Adic Analysis and Lie Groups, 2008, Lecture notes from a course given in Münster 2007/2008, available at http://wwwmath.uni-muenster.de/u/pschnei/publ/lectnotes/p-adic-analysis.pdf.Google Scholar
24.Tamme, Georg, Comparison of the Karoubi regulator and the p-adic Borel regulator, preprint 17/2007 DFG-Forschergruppe Regensburg/Leipzig, available at http://epub.uni-regensburg.de/13817/, 2007.Google Scholar
25.Tamme, Georg, The relative Chern character and regulators, Ph.D. thesis, Universität Regensburg, 2010, available at http://epub.uni-regensburg.de/15595/.Google Scholar
26.Weibel, Charles A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.Google Scholar