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

Ginot, Grégory

doi:10.5802/aif.2081

Formules explicites pour le caractère de Chern en

K

-théorie algébrique

Ginot, Grégory ¹

¹ Université Paris 13, ENS Cachan, CMLA-LAGA, 61 avenue du Président Wilson, 94235 Cachan Cedex (France)

Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2327-2355.

Résumé
Abstract

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.

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.

MR Zbl

DOI : 10.5802/aif.2081

Classification : 19D55, 16E40, 18H10, 19D45, 19C20
Mot clés : homologie cyclique, $K$-théorie algébrique, caractère de Chern, symboles de Steinberg, symboles de Loday
Keywords: Cyclic homology, algebraic $K$-theory, Chern character, Steinberg symbols, Loday Symbols

Affiliations des auteurs :

Ginot, Grégory ¹

¹ Université Paris 13, ENS Cachan, CMLA-LAGA, 61 avenue du Président Wilson, 94235 Cachan Cedex (France)

@article{AIF_2004__54_7_2327_0,
     author = {Ginot, Gr\'egory},
     title = {Formules explicites pour le caract\`ere de {Chern} en $K$-th\'eorie alg\'ebrique},
     journal = {Annales de l'Institut Fourier},
     pages = {2327--2355},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {7},
     year = {2004},
     doi = {10.5802/aif.2081},
     mrnumber = {2139695},
     zbl = {1068.19005},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/aif.2081/}
}

TY  - JOUR
AU  - Ginot, Grégory
TI  - Formules explicites pour le caractère de Chern en $K$-théorie algébrique
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 2327
EP  - 2355
VL  - 54
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2081/
DO  - 10.5802/aif.2081
LA  - fr
ID  - AIF_2004__54_7_2327_0
ER  -

%0 Journal Article
%A Ginot, Grégory
%T Formules explicites pour le caractère de Chern en $K$-théorie algébrique
%J Annales de l'Institut Fourier
%D 2004
%P 2327-2355
%V 54
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2081/
%R 10.5802/aif.2081
%G fr
%F AIF_2004__54_7_2327_0

Ginot, Grégory. Formules explicites pour le caractère de Chern en $K$-théorie algébrique. Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2327-2355. doi : 10.5802/aif.2081. http://archive.numdam.org/articles/10.5802/aif.2081/

Bibliographie
Cité par

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

[2] J.-L. Cathelineau $λ$ -structures in algebraic $K$ -theory and cyclic homology, $K$ -Theory, Volume 4 (1990-1991) no. 6, pp. 591-606 | DOI | MR | Zbl

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

[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 $K_{2}$ of discrete valuation rings, Advances in Math, Volume 18 (1975) no. 2, pp. 182-238 | DOI | MR | Zbl

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

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

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

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

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

[12] J. D. S. Jones Cyclic homology and equivariant homology, Invent. Math, Volume 87 (1987), pp. 403-424 | DOI | EuDML | MR | Zbl

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

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

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

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

[17] Ch. Kratzer $λ$ -structure en $K$ -théorie algébrique, Comment. Math. Helv, Volume 55 (1980) no. 2, pp. 233-254 | DOI | EuDML | MR | Zbl

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

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

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

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

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

[23] H. Maazen; J. Stienstra A presentation for $K_{2}$ of split radical pairs, J. Pure Appl. Algebra, Volume 10 (1977/78) no. 3, pp. 271-294 | DOI | MR | Zbl

[24] R. McCarthy The cyclic homology of an exact category, J. Pure Appl. Algebra, Volume 93 (1994) no. 3, pp. 251-296 | DOI | MR | Zbl

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

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

[27] C. Soulé Éléments cyclotomiques en $K$ -théorie (Astérisque), Volume 147-148 (1987) | Numdam | Zbl

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

[28] B.L. Tsygan Homology of matrix algebras over rings and Hochschild homology, Russ. Math. Surveys, Volume 38 (1983), pp. 198-199 | DOI | MR | Zbl

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

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

Cité par Sources :