Karoubi's relative Chern character and Beilinson's regulator
[Le caractère de Chern relatif de Karoubi et le régulateur de Beilinson]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 4, pp. 601-636.

Nous construisons une variante du caractère de Chern relatif de Karoubi pour les variétés lisses sur 𝐂 et prouvons un résultat de comparaison avec le régulateur de Beilinson à valeurs dans la cohomologie de Deligne-Beilinson. En corollaire, nous obtenons une nouvelle preuve du théorème de Burgos que, pour un corps de nombres, le régulateur de Beilinson est deux fois le régulateur de Borel.

We construct a variant of Karoubi’s relative Chern character for smooth varieties over 𝐂 and prove a comparison result with Beilinson’s regulator with values in Deligne-Beilinson cohomology. As a corollary we obtain a new proof of Burgos’ Theorem that for number fields Borel’s regulator is twice Beilinson’s regulator.

DOI : https://doi.org/10.24033/asens.2174
Classification : 19F27,  19D55,  14F43,  19E20,  19L10,  55R40,  57R20
Mots clés : régulateur, caractère de Chern relatif, classe caractéristique secondaire, régulateur de Borel
@article{ASENS_2012_4_45_4_601_0,
     author = {Tamme, Georg},
     title = {Karoubi's relative Chern character and Beilinson's regulator},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {601--636},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {4},
     year = {2012},
     doi = {10.24033/asens.2174},
     zbl = {1266.19004},
     mrnumber = {3059242},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2174/}
}
Tamme, Georg. Karoubi's relative Chern character and Beilinson's regulator. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 4, pp. 601-636. doi : 10.24033/asens.2174. http://www.numdam.org/articles/10.24033/asens.2174/

[1] A. A. Beĭlinson, Higher regulators and values of L-functions, in Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1984, 181-238. | MR 760999 | Zbl 0588.14013

[2] A. J. Berrick, An approach to algebraic K-theory, Research Notes in Math. 56, Pitman (Advanced Publishing Program), 1982. | MR 649409 | Zbl 0479.18006

[3] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. 7 (1974), 235-272. | EuDML 81938 | Numdam | MR 387496 | Zbl 0316.57026

[4] A. Borel, Cohomologie de SL n et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 613-636. | EuDML 83764 | Numdam | MR 506168 | Zbl 0382.57027

[5] A. K. Bousfield & D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, 1972. | MR 365573 | Zbl 0259.55004

[6] J. I. Burgos Gil, The regulators of Beilinson and Borel, CRM Monograph Series 15, Amer. Math. Soc., 2002. | MR 1869655 | Zbl 0994.19003

[7] B. C. Carlson, Special functions of applied mathematics, Academic Press, 1977. | MR 590943 | Zbl 0394.33001

[8] P. Deligne, Théorie de Hodge. II, Publ. Math. I.H.É.S. 40 (1971), 5-57. | EuDML 103914 | Numdam | MR 498551 | Zbl 0219.14007

[9] P. Deligne, Théorie de Hodge. III, Publ. Math. I.H.É.S. 44 (1974), 5-77. | EuDML 103935 | Numdam | MR 498552 | Zbl 0237.14003

[10] J. L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), 233-245. | MR 413122 | Zbl 0331.55012

[11] J. L. Dupont, Curvature and characteristic classes, Lecture Notes in Math. 640, Springer, 1978. | MR 500997 | Zbl 0373.57009

[12] J. L. Dupont, R. Hain & S. Zucker, Regulators and characteristic classes of flat bundles, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lecture Notes 24, Amer. Math. Soc., 2000, 47-92. | MR 1736876 | Zbl 0976.14005

[13] H. Esnault & E. Viehweg, Deligne-Beĭlinson cohomology 1988, 43-91. | MR 944991 | Zbl 0656.14012

[14] E. M. Friedlander, Étale homotopy of simplicial schemes, Annals of Math. Studies 104, Princeton Univ. Press, 1982. | MR 676809 | Zbl 0538.55001

[15] S. M. Gersten, Higher K-theory of rings, in Algebraic K-theory, I: Higher K-theories (Proc. Conf. Seattle Res. Center, Battelle Memorial Inst., 1972), Lecture Notes in Math. 341, Springer, 1973, 3-42. | MR 382398 | Zbl 0285.18010

[16] H. Gillet, Riemann-Roch theorems for higher algebraic K-theory, Adv. in Math. 40 (1981), 203-289. | MR 624666 | Zbl 0478.14010

[17] H. Gillet, Universal cycle classes, Compositio Math. 49 (1983), 3-49. | Numdam | MR 699858 | Zbl 0538.14009

[18] H. Gillet, On the K-theory of surfaces with multiple curves and a conjecture of Bloch, Duke Math. J. 51 (1984), 195-233. | MR 744295 | Zbl 0557.14003

[19] H. Gillet, Comparing algebraic and topological K-theory, in Higher algebraic K-theory: an overview, Lecture Notes in Math. 1491, Springer, 1992, 55-99. | MR 1175629

[20] P. Griffiths & J. Harris, Principles of algebraic geometry, Wiley-Interscience, 1978. | MR 507725 | Zbl 0836.14001

[21] A. Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137-154. | Numdam | MR 116023 | Zbl 0091.33201

[22] N. Hamida, Description explicite du régulateur de Borel, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 169-172. | MR 1748302 | Zbl 0947.19003

[23] A. Hatcher, Algebraic topology, Cambridge Univ. Press, 2002. | MR 1867354 | Zbl 1044.55001

[24] G. Hochschild & G. D. Mostow, Cohomology of Lie groups, Illinois J. Math. 6 (1962), 367-401. | MR 147577 | Zbl 0111.03302

[25] M. Karoubi, Connexions, courbures et classes caractéristiques en K-théorie algébrique, in Current trends in algebraic topology, Part 1 (London, Ont., 1981), CMS Conf. Proc. 2, Amer. Math. Soc., 1982, 19-27. | MR 686108 | Zbl 0553.18006

[26] M. Karoubi, Homologie cyclique et régulateurs en K-théorie algébrique, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), 557-560. | MR 735692 | Zbl 0532.18009

[27] M. Karoubi, Homologie cyclique et K-théorie, Astérisque 149 (1987), 1-147. | MR 913964 | Zbl 0648.18008

[28] M. Karoubi, Théorie générale des classes caractéristiques secondaires, K-Theory 4 (1990), 55-87. | MR 1076525 | Zbl 0716.57018

[29] M. Karoubi, Sur la K-théorie multiplicative, in Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun. 17, Amer. Math. Soc., 1997, 59-77. | MR 1478702 | Zbl 0889.19001

[30] J. W. Milnor & J. D. Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press, 1974. | MR 440554 | Zbl 0298.57008

[31] D. Quillen, Higher algebraic K-theory. I, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math. 341, Springer, 1973, 85-147. | MR 338129 | Zbl 0292.18004

[32] M. Rapoport, Comparison of the regulators of Beĭlinson and of Borel 1988, 169-192. | MR 944994 | Zbl 0667.14005

[33] P. Schneider, Introduction to the Beĭlinson conjectures 1988, 1-35. | MR 944989 | Zbl 0673.14007

[34] C. Soulé, Régulateurs, Séminaire Bourbaki, vol. 1984/85, exp. no 644, Astérisque 133-134 (1986), 237-253. | Numdam | Zbl 0617.14008

[35] C. Soulé, Connexions et classes caractéristiques de Beilinson 1987), Contemp. Math. 83, Amer. Math. Soc., 1989, 349-376. | MR 991985 | Zbl 0695.14003

[36] G. Tamme, The relative Chern character and regulators, Thèse, Universität Regensburg, 2010.

[37] G. Tamme, Comparison of Karoubi’s regulator and the p-adic Borel regulator, J. K-Theory 9 (2012), 579-600. | MR 2955976 | Zbl 1272.19003

[38] G. Tamme, Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map, preprint arXiv:1111.4109.

[39] U. Tillmann, Relation of the van Est spectral sequence to K-theory and cyclic homology, Illinois J. Math. 37 (1993), 589-608. | MR 1226784 | Zbl 0790.19002

[40] C. A. Weibel, Homotopy algebraic K-theory, in Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., 1989, 461-488. | MR 991991 | Zbl 0669.18007