On $ {\mathbb{A}}^{1}$-fundamental groups of isotropic reductive groups

Konrad Voelkel; Matthias Wendt

doi:10.1016/j.crma.2016.01.026

Homological algebra/Algebraic geometry

On

A^{1}

-fundamental groups of isotropic reductive groups

Presented by: the Editorial Board

Konrad Voelkel ¹; Matthias Wendt ²

¹ Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104, Freiburg im Breisgau, Germany
² Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 453-458.

Abstract
Résumé

For an isotropic reductive group G satisfying a suitable rank condition over an infinite field k, we show that the sections of the $A^{1}$ -fundamental group sheaf of G over an extension field $L / k$ can be identified with the second group homology of $G (L)$ . For a split group G, we provide explicit loops representing all elements in the $A^{1}$ -fundamental group. Using $A^{1}$ -homotopy theory, we deduce a Steinberg relation for these explicit loops.

Pour un groupe réductif isotrope G défini sur un corps infini k, satisfaisant une condition de rang approprié, nous montrons que l'ensemble des sections du $A^{1}$ -faisceau de groupe fondamental de G sur une extension des corps $L / k$ s'identifient avec la deuxième homologie des groupes de $G (L)$ . Pour un groupe déployé G, nous définissons des lacets explicites représentant tous les elements du groupe $A^{1}$ -fondamental. En utilisant la théorie de la $A^{1}$ -homotopie, on déduit une rélation de Steinberg pour ces lacets explicites.

Received: 2015-10-23
Accepted: 2016-01-19
Published online: 2016-03-24

DOI: 10.1016/j.crma.2016.01.026

Author's affiliations:

Konrad Voelkel ¹; Matthias Wendt ²

¹ Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104, Freiburg im Breisgau, Germany
² Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

@article{CRMATH_2016__354_5_453_0,
     author = {Konrad Voelkel and Matthias Wendt},
     title = {On $ {\mathbb{A}}^{1}$-fundamental groups of isotropic reductive groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {453--458},
     publisher = {Elsevier},
     volume = {354},
     number = {5},
     year = {2016},
     doi = {10.1016/j.crma.2016.01.026},
     language = {en},
}

TY  - JOUR
AU  - Konrad Voelkel
AU  - Matthias Wendt
TI  - On $ {\mathbb{A}}^{1}$-fundamental groups of isotropic reductive groups
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 453
EP  - 458
VL  - 354
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2016.01.026
LA  - en
ID  - CRMATH_2016__354_5_453_0
ER  -

%0 Journal Article
%A Konrad Voelkel
%A Matthias Wendt
%T On $ {\mathbb{A}}^{1}$-fundamental groups of isotropic reductive groups
%J Comptes Rendus. Mathématique
%D 2016
%P 453-458
%V 354
%N 5
%I Elsevier
%R 10.1016/j.crma.2016.01.026
%G en
%F CRMATH_2016__354_5_453_0

Konrad Voelkel; Matthias Wendt. On $ {\mathbb{A}}^{1}$-fundamental groups of isotropic reductive groups. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 453-458. doi : 10.1016/j.crma.2016.01.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.026/

References
Cited by

[1] A. Asok; M. Hoyois; M. Wendt Affine representability results in $A^{1}$ -homotopy theory I: vector bundles, 2015 (ArXiV preprint) | arXiv

[2] A. Asok; M. Hoyois; M. Wendt Affine representability results in $A^{1}$ -homotopy theory II: principal bundles and homogeneous spaces, 2015 (ArXiV preprint) | arXiv

[3] P.G. Goerss; J.F. Jardine Simplicial Homotopy Theory, Prog. Math., vol. 174, Birkhäuser, 1999

[4] P. Hu; I. Kriz The Steinberg relation in $A^{1}$ -stable homotopy, Int. Math. Res. Not. (2001) no. 17, pp. 907-912

[5] J.F. Jardine On the homotopy groups of algebraic groups, J. Algebra, Volume 81 (1983) no. 1, pp. 180-201

[6] A.Yu. Luzgarev; A.A. Stavrova The elementary subgroup of an isotropic reductive group is perfect, Algebra Anal., Volume 23 (2011) no. 5, pp. 140-154 (translation in St. Petersburg Math. J., 23, 5, 2012, pp. 881-890)

[7] H. Matsumoto Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. Éc. Norm. Super. (4), Volume 2 (1969), pp. 1-62

[8] F. Morel $A^{1}$ -Algebraic Topology over a Field, Lecture Notes in Mathematics, vol. 2052, Springer, 2012

[9] V.A. Petrov; A. Stavrova Elementary subgroups in isotropic reductive groups, Algebra Anal., Volume 20 (2008) no. 4, pp. 160-188 (translation in St. Petersburg Math. J., 20, 4, 2009, pp. 625-644)

[10] U. Rehmann Zentrale Erweiterungen der speziellen linearen Gruppe eines Schiefkörpers, J. Reine Angew. Math., Volume 301 (1978), pp. 77-104

[11] A. Stavrova Homotopy invariance of non-stable $K_{1}$ -functors, J. K-Theory, Volume 13 (2014) no. 2, pp. 199-248

[12] K. Voelkel, Matsumotos Satz und $A^{1}$ -Homotopietheorie, Diplomarbeit, Albert-Ludwigs-Universität Freiburg, 2011.

[13] M. Wendt $A^{1}$ -homotopy of Chevalley groups, J. K-Theory, Volume 5 (2010) no. 2, pp. 245-287

[14] M. Wendt On homology of linear groups over $k [t]$ , Math. Res. Lett., Volume 21 (2014) no. 6, pp. 1483-1500

Cited by Sources:

Comments - Policy