Formules explicites pour le caractère de Chern en <span class="mathjax-formula">$K$</span>-théorie algébrique

Ginot, Grégory

doi:10.5802/aif.2081

Formules explicites pour le caractère de Chern en

K

-théorie algébrique

Ginot, Grégory

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 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

BibTeX
RIS

@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},
     zbl = {1068.19005},
     mrnumber = {2139695},
     language = {fr},
     url = {http://www.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
DA  - 2004///
SP  - 2327
EP  - 2355
VL  - 54
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2081/
UR  - https://zbmath.org/?q=an%3A1068.19005
UR  - https://www.ams.org/mathscinet-getitem?mr=2139695
UR  - https://doi.org/10.5802/aif.2081
DO  - 10.5802/aif.2081
LA  - fr
ID  - AIF_2004__54_7_2327_0
ER  -

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://www.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 672956 | Zbl 0584.20036

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

[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 | Article | MR 437620 | Zbl 0318.13017

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

[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 | Article | MR 1326752 | Zbl 0824.19002

[10] Th. G. Goodwillie Relative algebraic $K$ -theory and cyclic homology, Ann. of Math. (2), Volume 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, Volume 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, Volume 87 (1987), pp. 403-424 | Article | EuDML 143427 | MR 870737 | Zbl 0644.55005

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

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

[15] Ch. Kassel Cyclic homology, comodules, and mixed complexes, J. Algebra, Volume 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, Volume 408 (1990), pp. 159-180 | Article | EuDML 153243 | MR 1058987 | Zbl 0691.18002

[17] Ch. Kratzer $λ$ -structure en $K$ -théorie algébrique, Comment. Math. Helv, Volume 55 (1980) no. 2, pp. 233-254 | Article | EuDML 139823 | MR 576604 | Zbl 0444.18008

[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 81981 | Numdam | MR 447373 | Zbl 0362.18014

[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 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 (NATO Adv. Sci. Inst. Sér. C, Math. Phys. Sci.), Volume 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, Volume 59 (1984) no. 4, pp. 569-591 | EuDML 139991 | MR 780077 | Zbl 0565.17006

[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 | Article | MR 472795 | Zbl 0393.18013

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

[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 349811 | Zbl 0237.18005

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

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

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

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

[29] Ch. A. Weibel Nil $K$ -theory maps to cyclic homology, Trans. Amer. Math. Soc, Volume 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)

Cité par Sources :