Karoubi’s relative Chern character and Beilinson’s regulator

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1. [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. Zbl0588.14013MR760999
2. [2] A. J. Berrick, An approach to algebraic $K$-theory, Research Notes in Math. 56, Pitman (Advanced Publishing Program), 1982. Zbl0479.18006MR649409
3. [3] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup.7 (1974), 235–272. Zbl0316.57026MR387496
4. [4] A. Borel, Cohomologie de ${\mathrm{SL}}_n$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci.4 (1977), 613–636. Zbl0382.57027MR506168
5. [5] A. K. Bousfield & D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, 1972. Zbl0259.55004MR365573
6. [6] J. I. Burgos Gil, The regulators of Beilinson and Borel, CRM Monograph Series 15, Amer. Math. Soc., 2002. Zbl0994.19003MR1869655
7. [7] B. C. Carlson, Special functions of applied mathematics, Academic Press, 1977. Zbl0394.33001MR590943
8. [8] P. Deligne, Théorie de Hodge. II, Publ. Math. I.H.É.S. 40 (1971), 5–57. Zbl0219.14007MR498551
9. [9] P. Deligne, Théorie de Hodge. III, Publ. Math. I.H.É.S. 44 (1974), 5–77. Zbl0237.14003MR498552
10. [10] J. L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology15 (1976), 233–245. Zbl0331.55012MR413122
11. [11] J. L. Dupont, Curvature and characteristic classes, Lecture Notes in Math. 640, Springer, 1978. Zbl0373.57009MR500997
12. [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. Zbl0976.14005MR1736876
13. [13] H. Esnault & E. Viehweg, Deligne-Beĭlinson cohomology, in Beĭlinson’s conjectures on special values of $L$-functions, Perspect. Math. 4, Academic Press, 1988, 43–91. Zbl0656.14012MR944991
14. [14] E. M. Friedlander, Étale homotopy of simplicial schemes, Annals of Math. Studies 104, Princeton Univ. Press, 1982. Zbl0538.55001MR676809
15. [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. Zbl0285.18010MR382398
16. [16] H. Gillet, Riemann-Roch theorems for higher algebraic $K$-theory, Adv. in Math.40 (1981), 203–289. Zbl0478.14010MR624666
17. [17] H. Gillet, Universal cycle classes, Compositio Math.49 (1983), 3–49. Zbl0538.14009MR699858
18. [18] H. Gillet, On the $K$-theory of surfaces with multiple curves and a conjecture of Bloch, Duke Math. J.51 (1984), 195–233. Zbl0557.14003MR744295
19. [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. MR1175629
20. [20] P. Griffiths & J. Harris, Principles of algebraic geometry, Wiley-Interscience, 1978. Zbl0836.14001MR507725
21. [21] A. Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France86 (1958), 137–154. Zbl0091.33201MR116023
22. [22] N. Hamida, Description explicite du régulateur de Borel, C. R. Acad. Sci. Paris Sér. I Math.330 (2000), 169–172. Zbl0947.19003MR1748302
23. [23] A. Hatcher, Algebraic topology, Cambridge Univ. Press, 2002. Zbl1044.55001MR1867354
24. [24] G. Hochschild & G. D. Mostow, Cohomology of Lie groups, Illinois J. Math.6 (1962), 367–401. Zbl0111.03302MR147577
25. [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. Zbl0553.18006MR686108
26. [26] M. Karoubi, Homologie cyclique et régulateurs en $K$-théorie algébrique, C. R. Acad. Sci. Paris Sér. I Math.297 (1983), 557–560. Zbl0532.18009MR735692
27. [27] M. Karoubi, Homologie cyclique et $K$-théorie, Astérisque149 (1987), 1–147. Zbl0648.18008MR913964
28. [28] M. Karoubi, Théorie générale des classes caractéristiques secondaires, $K$-Theory 4 (1990), 55–87. Zbl0716.57018MR1076525
29. [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. Zbl0889.19001MR1478702
30. [30] J. W. Milnor & J. D. Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press, 1974. Zbl0298.57008MR440554
31. [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. Zbl0292.18004MR338129
32. [32] M. Rapoport, Comparison of the regulators of Beĭlinson and of Borel, in Beĭlinson’s conjectures on special values of $L$-functions, Perspect. Math. 4, Academic Press, 1988, 169–192. Zbl0667.14005MR944994
33. [33] P. Schneider, Introduction to the Beĭlinson conjectures, in Beĭlinson’s conjectures on special values of $L$-functions, Perspect. Math. 4, Academic Press, 1988, 1–35. Zbl0673.14007MR944989
34. [34] C. Soulé, Régulateurs, Séminaire Bourbaki, vol. 1984/85, exp. no 644, Astérisque 133-134 (1986), 237–253. Zbl0617.14008
35. [35] C. Soulé, Connexions et classes caractéristiques de Beilinson, in Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., 1989, 349–376. Zbl0695.14003MR991985
36. [36] G. Tamme, The relative Chern character and regulators, Thèse, Universität Regensburg, 2010. 
37. [37] G. Tamme, Comparison of Karoubi’s regulator and the $p$-adic Borel regulator, J. K-Theory9 (2012), 579–600. Zbl1272.19003MR2955976
38. [38] G. Tamme, Karoubi’s relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map, preprint arXiv:1111.4109. 
39. [39] U. Tillmann, Relation of the van Est spectral sequence to $K$-theory and cyclic homology, Illinois J. Math.37 (1993), 589–608. Zbl0790.19002MR1226784
40. [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. Zbl0669.18007MR991991