On Galois cohomology of connected reductive groups and Kottwitz exact sequence
type: Item
instance of: scholarly article
title: On Galois cohomology of connected reductive groups and Kottwitz exact sequence
zbMATH Open document ID: 1473.11086
author: Nguyêñ Quôć Thǎńg
published in: Bulletin des Sciences Mathématiques
publication date: 2019-02-27T00:00:00Z
review text: Let \(k\) be a global field and let \(G\) be a connected reductive algebraic group over \(k\). Write \(G'\) for the derived group of \(G\) and \(\hat{G}\) for the Langlands dual of \(G\). Let \(V\) denote the set of places of \(k\), let \(\mathbf{A}\) denote the \(k\)-adeles and put \(\mathbf{A}_s=\mathbf{A}\otimes_k k_s\), where \(k_s\) is a separable closure of \(k\). Define \(\mathscr{A}(G)= (\pi_0(Z(\hat{G})^{\operatorname{Gal}(k_s/k)}))^D\cong \mathrm{Pic}(G)^D\), where \((?)^D=\operatorname{Hom}_{\mathbb{Z}}(?,\mathbb{Q}/\mathbb{Z})\). The author generalizes several results of Kottwitz [\textit{R. E. Kottwitz}, Duke Math. J. 51, 611--650 (1984; Zbl 0576.22020); Math. Ann. 275, 365--399 (1986; Zbl 0577.10028)] concerning the Galois cohomology set \(\mathrm{H}^1(k,G)\) from the case of number fields to all global fields. The main result of the paper (Theorem~4.7) states that there exists an exact sequence \[ 1\to \operatorname{Sha}^1(G)\to \mathrm{H}^1(k,G )\to \bigoplus_{v\in V}\mathrm{H}^1(k_v,G) \to \mathscr{A}(G)\to \operatorname{Sha}^2(G/G')\to 1 \] which is functorial in \(G\); here, \(\mathrm{Sha}^i\) denotes the \(i\)-th Shafarevich-Tate group, or pointed set. In characteristic \(0\), the existence and exactness of the sequence formed by the left four terms were shown by Kottwitz [loc. cit., Prop. 2.6]. Further related results are proved along the way. Note first that for a \(k\)-torus \(T\), it is known that there is a natural isomorphism \(\mathrm{H}^1(k,T(\mathbf{A}_s)/T(k_s))\to \mathscr{A}(T)\) (Section~1.5). The author shows that this isomorphisms extends uniquely to a natural transformation \(\mathrm{H}^1(k,G(\mathbf{A}_s)/Z_G(k_s))\to \mathscr{A}(G)\), where \(Z_G\) is the center of \(G\). Moreover, the latter isomorphism fits into an exact sequence \( \mathrm{H}^1(k,G^{\mathrm{ad}})\to \mathrm{H}^1(k,G(\mathbf{A}_s)/Z(k_s)) \to \mathscr{A}(G)\), which is functorial in \(G\) (Theorem~3.14). It is further shown that \(\operatorname{Sha}^1(G)\cong …
reviewed by: Uriya A. First
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 7030204
zbMATH Keywords: Galois cohomology
zbMATH Keywords: local and global fields
zbMATH Keywords: reductive groups
zbMATH Keywords: Langlands dual
zbMATH Keywords: weak approximation
zbMATH Keywords: Hasse principles
MaRDI profile type: Publication
full work available at URL: j.bulsci.2019.01.002
OpenAlex ID: W2910496780
cites work: Q3999894
cites work: Abelianization of the second nonabelian Galois cohomology
cites work: Abelian Galois cohomology of reductive groups
cites work: Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields
cites work: Q3972725
cites work: Q3804918
cites work: Résolutions flasques des groupes linéaires connexes
cites work: Arithmetic of linear algebraic groups over 2-dimensional geometric fields
cites work: La $R$-équivalence sur les tores
cites work: Q3877792
cites work: Le défaut d'approximation forte dans les groupes linéaires connexes
cites work: Q5602064
cites work: G-torseurs en théorie de Hodgep-adique
cites work: LA COURBE
cites work: On Picard groups of algebraic fibre spaces
cites work: Q5507554
cites work: Quasi-abelian crossed modules and nonabelian cohomology
cites work: Q5595003
cites work: Über die Galoiskohomologie halbeinfacher Matrizengruppen II
cites work: Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen
cites work: Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III.
cites work: The geometry of algebraic groups
cites work: Lectures on Galois cohomology of classical groups. With an appendix by T. Springer
cites work: Stable trace formula: Cuspidal tempered terms
cites work: Stable trace formula: Elliptic singular terms
cites work: Q4243095
cites work: Q4248227
cites work: Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale
cites work: Q4015219
cites work: Q3413444
cites work: The fundamental lemma for Lie algebras
cites work: Counting of \(G\)-shtukas: the elliptic part
cites work: On some Hasse principles for algebraic groups over global fields
cites work: On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic
cites work: Nombres de Tamagawa et groupes unipotents en caractéristique p
cites work: A.D. Alexandrov spaces with curvature bounded below
cites work: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.
cites work: Galois Cohomology
cites work: Q2544721
cites work: Cohomology of artinian group schemes over local fields
cites work: The cohomological dimension of certain Grothendieck topologies
cites work: Counting points on Igusa varieties
cites work: The Cohomology Groups of Tori in Finite Galois Extensions of Number Fields
cites work: On Galois cohomology of semisimple groups over local and global fields of positive characteristic
cites work: On Galois cohomology and weak approximation of connected reductive groups over fields of positive characteristic
cites work: On Galois cohomology of semisimple groups over local and global fields of positive characteristic. III
cites work: Tate-Shafarevich kernel, weak Brauer and R-equivalence on connected reductive groups over local and global fields
cites work: Weak approximation, Brauer and \(R\)-equivalence in algebraic groups over arithmetical fields
cites work: Weak approximation, Brauer and \(R\)-equivalence in algebraic groups over arithmetical fields. II.
cites work: A norm principle for class groups of reductive group schemes over Dedekind rings of integers of local and global fields
cites work: Q3884236
DOI: 10.1016/J.BULSCI.2019.01.002
description: scientific article; zbMATH DE number 7030204