Comptes Rendus
Number theory/Algebra
On some finiteness properties of algebraic groups over finitely generated fields
[Sur quelques propriétés de finitude des groupes algébriques sur des corps de type fini]
Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 869-873.

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.

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DOI : 10.1016/j.crma.2016.07.012

Vladimir I. Chernousov 1 ; Andrei S. Rapinchuk 2 ; Igor A. Rapinchuk 3

1 Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
2 Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
3 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
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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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