[Sur quelques propriétés de finitude des groupes algébriques sur des corps de type fini]
Nous présentons plusieurs résultats de finitude pour les groupes algébriques absolument presque simples définis sur des corps de type fini plus généraux que les corps globaux. Nous discutons aussi des liens entre les propriétés de finitude diverses qui entrent dans le cadre de notre analyse, telles que la propreté de l'application globale–locale dans la cohomologie galoisienne d'un K-groupe G par rapport à un ensemble convenable V de valuations discrètes de K, et la finitude du nombre de K-formes de G ayant, d'une part, bonne réduction en V et possédant, d'autre part, les même classes d'isomorphisme de K-tores maximaux que G.
We present several finiteness results for absolutely almost simple algebraic groups over finitely generated fields that are more general than global fields. We also discuss the relations between the various finiteness properties involved in these results, such as the properness of the global-to-local map in the Galois cohomology of a given K-group G relative to a certain natural set V of discrete valuations of K, and the finiteness of the number of isomorphism classes of K-forms of G having, on the one hand, smooth reduction with respect to all places in V and, on the other hand, the same isomorphism classes of maximal K-tori as G.
Accepté le :
Publié le :
Vladimir I. Chernousov 1 ; Andrei S. Rapinchuk 2 ; Igor A. Rapinchuk 3
@article{CRMATH_2016__354_9_869_0,
author = {Vladimir I. Chernousov and Andrei S. Rapinchuk and Igor A. Rapinchuk},
title = {On some finiteness properties of algebraic groups over finitely generated fields},
journal = {Comptes Rendus. Math\'ematique},
pages = {869--873},
publisher = {Elsevier},
volume = {354},
number = {9},
year = {2016},
doi = {10.1016/j.crma.2016.07.012},
language = {en},
}
TY - JOUR AU - Vladimir I. Chernousov AU - Andrei S. Rapinchuk AU - Igor A. Rapinchuk TI - On some finiteness properties of algebraic groups over finitely generated fields JO - Comptes Rendus. Mathématique PY - 2016 SP - 869 EP - 873 VL - 354 IS - 9 PB - Elsevier DO - 10.1016/j.crma.2016.07.012 LA - en ID - CRMATH_2016__354_9_869_0 ER -
%0 Journal Article %A Vladimir I. Chernousov %A Andrei S. Rapinchuk %A Igor A. Rapinchuk %T On some finiteness properties of algebraic groups over finitely generated fields %J Comptes Rendus. Mathématique %D 2016 %P 869-873 %V 354 %N 9 %I Elsevier %R 10.1016/j.crma.2016.07.012 %G en %F CRMATH_2016__354_9_869_0
Vladimir I. Chernousov; Andrei S. Rapinchuk; Igor A. Rapinchuk. On some finiteness properties of algebraic groups over finitely generated fields. Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 869-873. doi : 10.1016/j.crma.2016.07.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.07.012/
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