Formules explicites pour le caractère de Chern en $K$-théorie algébrique

Ginot, Grégory

Annales de l'Institut Fourier, Tome 54 (2004), p. 2327-2355 / Harvested from Numdam

Résumé

Dans cet article on donne une formule explicite pour le caractère de Chern reliant la $K$- théorie algébrique et l’homologie cyclique négative. On calcule le caractère de Chern des symboles de Steinberg et de Loday et on donne une preuve élémentaire du fait que le caractère de Chern est multiplicatif.

Abstract

In this paper we give an explicit formula for the Chern character from algebraic $K$- theory to negative cyclic homology. We compute formulas for the Chern character of Steinberg, Dennis-Stein and Loday symbols. From the previous results we get a new proof of the compatibility of the Chern character with products.

Détail

Publié le : 2004-01-01

MR 2139695 | Zbl 1068.19005

DOI : https://doi.org/10.5802/aif.2081
Classification:  19D55,  16E40,  18H10,  19D45,  19C20
Mots clés: homologie cyclique, $K$-théorie algébrique, caractère de Chern, symboles de Steinberg, symboles de Loday

Bibliographie

[1] S.K. Brown; S. Kenneth Cohomology of groups, Springer-Verlag, New York-Berlin, Graduate Texts in Mathematics, Tome 87 (1982) | MR 672956 | Zbl 0584.20036

[2] J.-L. Cathelineau $\lambda$-structures in algebraic $K$-theory and cyclic homology, $K$-Theory, Tome 4 (1990-1991) no. 6, pp. 591-606 | Article | MR 1123180 | Zbl 0735.19005

[3] A. Connes Noncommutative differential geometry, I, II, Publ. Math. Inst. Hautes Étud. Sci, Tome 62 (1985), pp. 41-144 | Article | Numdam | MR 823176 | Zbl 0592.46056

[4] R. K. Dennis Differentials in algebraic $K$-theory (1975) (non publié, circa)

[5] R. K. Dennis Algebraic $K$-theory and Hochschild homology (1975-1976) (non publié)

[6] R. K. Dennis; M. R. Stein $K\sb 2$ of discrete valuation rings, Advances in Math, Tome 18 (1975) no. 2, pp. 182-238 | Article | MR 437620 | Zbl 0318.13017

[7] R. H. Fox Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2), Tome 57 (1953), pp. 547-560 | Article | MR 53938 | Zbl 0050.25602

[8] Ph. Gaucher Produit tensoriel de matrices, homologie cyclique, homologie des algèbres de Lie, Ann. Inst. Fourier, Grenoble, Tome 44 (1994) no. 2, pp. 413-431 | Article | Numdam | MR 1296738 | Zbl 0803.19003

[9] S. C. Geller; Ch. A. Weibel Hodge decompositions of Loday symbols in $K$-theory and cyclic homology, $K$-Theory, Tome 8 (1994) no. 6, pp. 587-632 | Article | MR 1326752 | Zbl 0824.19002

[10] Th. G. Goodwillie Relative algebraic $K$-theory and cyclic homology, Ann. of Math. (2), Tome 124 (1986) no. 2, pp. 347-402 | Article | MR 855300 | Zbl 0627.18004

[11] Ch. E. Hood; J. D. S. Jones Some algebraic properties of cyclic homology groups, $K$-Theory, Tome 1 (1987) no. 4, pp. 361-384 | Article | MR 920950 | Zbl 0636.18005

[12] J. D. S. Jones Cyclic homology and equivariant homology, Invent. Math, Tome 87 (1987), pp. 403-424 | Article | MR 870737 | Zbl 0644.55005

[13] M. R. Kantorovitz Adams operations and the Dennis trace map, J. Pure Appl. Algebra, Tome 144 (1999) no. 1, pp. 21-27 | Article | MR 1723189 | Zbl 0937.19006

[14] M. Karoubi Homologie cyclique et $K$-théorie, Soc. Math. France, Paris, Astérisque, Tome 149 (1987) | MR 913964 | Zbl 0648.18008

[15] Ch. Kassel Cyclic homology, comodules, and mixed complexes, J. Algebra, Tome 107 (1987) no. 1, pp. 195-216 | Article | MR 883882 | Zbl 0617.16015

[16] Ch. Kassel Homologie cyclique, caractère de Chern et lemme de perturbation, J. Reine Angew. Math, Tome 408 (1990), pp. 159-180 | Article | MR 1058987 | Zbl 0691.18002

[17] Ch. Kratzer $\lambda $-structure en $K$-théorie algébrique, Comment. Math. Helv, Tome 55 (1980) no. 2, pp. 233-254 | Article | MR 576604 | Zbl 0444.18008

[18] J.-L. Loday $K$-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4), Tome 9 (1976) no. 3, pp. 309-377 | Numdam | MR 447373 | Zbl 0362.18014

[19] J.-L. Loday Symboles en $K$-théorie algébrique supérieure, C. R. Acad. Sci. Paris, Sér. I Math., Tome 292 (1981) no. 18, pp. 863-866 | MR 623517 | Zbl 0493.18006

[20] J.-L. Loday Cyclic homology, Springer-Verlag, Berlin (1998) | MR 1600246 | Zbl 0885.18007

[21] J.-L. Loday; C. Procesi Cyclic homology and lambda operations, Kluwer Acad. Publ., Dordrecht (NATO Adv. Sci. Inst. Sér. C, Math. Phys. Sci.) Tome 279 (1989), pp. 209-224 | Zbl 0719.19002

[22] J.-L. Loday; D. Quillen Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv, Tome 59 (1984) no. 4, pp. 569-591 | MR 780077 | Zbl 0565.17006

[23] H. Maazen; J. Stienstra A presentation for $K\sb 2$ of split radical pairs, J. Pure Appl. Algebra, Tome 10 (1977/78) no. 3, pp. 271-294 | Article | MR 472795 | Zbl 0393.18013

[24] R. Mccarthy The cyclic homology of an exact category, J. Pure Appl. Algebra, Tome 93 (1994) no. 3, pp. 251-296 | Article | MR 1275967 | Zbl 0807.19002

[25] J. Milnor Introduction to algebraic $K$-theory, Princeton University Press and University of Tokyo Press, Princeton, N.J. and Tokyo, Annals of Mathematics Studies (1971) | MR 349811 | Zbl 0237.18005

[26] Th. Mulders Generating the tame and wild kernels by Dennis-Stein symbols, $K$-Theory, Tome 5 (1991/92) no. 5, pp. 449-470 | Article | MR 1166514 | Zbl 0761.11040

[27] C. Soulé Éléments cyclotomiques en $K$-théorie, Soc. Math. France (Astérisque) (1987), p. 147-148 | Zbl 0632.12014

[28] B.L. Tsygan Homology of matrix algebras over rings and Hochschild homology, Uspekhi Mat. Nauk, Tome 38 (1983), p. 217-218 | MR 695483 | Zbl 0518.17002 | Zbl 0526.17006

[28] B.L. Tsygan Homology of matrix algebras over rings and Hochschild homology, Russ. Math. Surveys, Tome 38 (1983), p. 198-199 | Article | MR 695483 | Zbl 0526.17006

[29] Ch. A. Weibel Nil $K$-theory maps to cyclic homology, Trans. Amer. Math. Soc, Tome 303 (1987) no. 2, pp. 541-558 | MR 902784 | Zbl 0627.18005

[30] Ch. A. Weibel An introduction to algebraic $K$-theory (, http://math.rutgers.edu:80/$\sim$weibel/Kbook.html)