no. 4
Taylor towers for Γ-modules
[Approximation de Taylor pour les Γ-modules]
Richter, Birgit1
1 Mathematisches Institut der Universität Bonn, Beringsstrasse 1, 53115 Bonn (Germany)
Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 995-1023.

Soient Γ la catégorie des ensembles finis pointés et F un Γ-module, donc un foncteur de la catégorie Γ vers une catégorie des modules sur un anneau commutatif. Nous développons une approximation de Taylor pour ces foncteurs. On démontre dans cet article qu’il y a une description explicite de l’homologie d’approximation de Taylor pour les Γ-modules. Nous construisons une suite spectrale pour l’homologie des fibres homotopiques dans cette tour de Taylor et nous faisons des calculs en caractéristique zéro, qui donnent une application pour l’homologie de Hochschild d’ordre supérieur.

We consider Taylor approximation for functors from the small category of finite pointed sets Γ to modules and give an explicit description for the homology of the layers of the Taylor tower. These layers are shown to be fibrant objects in a suitable closed model category structure. Explicit calculations are presented in characteristic zero including an application to higher order Hochschild homology. A spectral sequence for the homology of the homotopy fibres of this approximation is provided.

DOI : 10.5802/aif.1842
Classification : 18A25, 18G10, 55P65
Keywords: Taylor tower, cubical construction, dual of the Steenrod algebra
Mot clés : approximation de Taylor, construction cubique, dual de l'algèbre de Steenrod
Richter, Birgit 1

1 Mathematisches Institut der Universität Bonn, Beringsstrasse 1, 53115 Bonn (Germany) 
@article{AIF_2001__51_4_995_0,
     author = {Richter, Birgit},
     title = {Taylor towers for $\Gamma $-modules},
     journal = {Annales de l'Institut Fourier},
     pages = {995--1023},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     doi = {10.5802/aif.1842},
     mrnumber = {1849212},
     zbl = {0997.18008},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1842/}
}
TY  - JOUR
AU  - Richter, Birgit
TI  - Taylor towers for $\Gamma $-modules
JO  - Annales de l'Institut Fourier
PY  - 2001
SP  - 995
EP  - 1023
VL  - 51
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1842/
DO  - 10.5802/aif.1842
LA  - en
ID  - AIF_2001__51_4_995_0
ER  - 
%0 Journal Article
%A Richter, Birgit
%T Taylor towers for $\Gamma $-modules
%J Annales de l'Institut Fourier
%D 2001
%P 995-1023
%V 51
%N 4
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1842/
%R 10.5802/aif.1842
%G en
%F AIF_2001__51_4_995_0
Richter, Birgit. Taylor towers for $\Gamma $-modules. Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 995-1023. doi : 10.5802/aif.1842. http://archive.numdam.org/articles/10.5802/aif.1842/

[BF] A.K. Bousfield; E.M. Friedlander Homotopy theory of Γ-spaces, spectra, and bisimplicial sets (Lecture Notes in Mathematics), Volume 658 (1978), pp. 80-150 | Zbl

[DP] A. Dold; D. Puppe Homologie nicht-additiver Funktoren. Anwendungen., Annales de l'Institut Fourier (Grenoble), Volume 11 (1961), pp. 201-312 | DOI | Numdam | MR | Zbl

[EM1] S. Eilenberg; S. MacLane Homology theory for multiplicative systems, Transactions of the AMS, Volume 71 (1951), pp. 294-330 | DOI | MR | Zbl

[EM2] S. Eilenberg; S. MacLane On the groups H(π,n), II, Annals of Mathematics, Volume 60 (1954), pp. 49-139 | DOI | MR | Zbl

[G] T. Goodwillie Calculus. I: The first derivative of pseudoisotopy theory (To appear in K-theory 4) | MR | Zbl

[G] T. Goodwillie Calculus. II: Analytic functors (To appear in K-theory 5) | MR | Zbl

[G] T. Goodwillie Calculus. III: The Taylor series of a homotopy functor (To appear) | MR | Zbl

[GoJ] P.G. Goerss; J.F. Jardine Simplicial homotopy theory, Progress in Mathematics, Birhäuser, 1999 | MR | Zbl

[I] L. Illusie Complexe cotangent et déformations II, Lecture Notes in Mathematics, 283, Springer, 1972 | MR | Zbl

[JMcC1] B. Johnson; R. McCarthy Taylor towers for functors of additive categories, Journal of Pure and Applied Algebra, Volume 137 (1999), pp. 253-284 | DOI | MR | Zbl

[JMcC2] B. Johnson; R. McCarthy Deriving calculus with cotripels (1999) (Preprint, http://www.math.uiuc.edu/~randy/) | MR | Zbl

[JP] M. Jibladze; T. Pirashvili Cohomology of algebraic theories, Journal of Algebra, Volume 137 (1991), pp. 253-296 | DOI | MR | Zbl

[L] W. Lück Transformation groups and algebraic K-theory, Lecture Notes in Mathematics, 1408, Springer, 1989 | MR | Zbl

[P1] T. Pirashvili Kan extension and stable homology of Eilenberg and Mac Lane spaces, Topology, Volume 35 (1996), pp. 883-886 | DOI | MR | Zbl

[P2] T. Pirashvili Dold-Kan type theorem for Γ-groups, Mathematische Annalen, Volume 318 (2000), pp. 277-298 | DOI | MR | Zbl

[P3] T. Pirashvili Hodge decomposition for higher order Hochschild homology, Annales Scientifiques de l'École Normale Supérieure, Volume 33 (2000), pp. 151-179 | Numdam | MR | Zbl

[R] B. Richter Dissertation, Bonner Mathematische Schriften, 332 (2000) | Zbl

[Re] O. Renaudin Localisation homotopique et foncteurs entre espaces vectoriels (janvier 2000) (Thèse de doctorat, Université de Nantes)

Cité par Sources :