$J$-invariant of linear algebraic groups

Petrov, Viktor; Semenov, Nikita; Zainoulline, Kirill

doi:10.24033/asens.2088

J

-invariant of linear algebraic groups
[

J

-invariant des groupes algébriques linéaires]

Petrov, Viktor ; Semenov, Nikita ; Zainoulline, Kirill

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 1023-1053.

Résumé
Abstract

Soit $G$ un groupe algébrique linéaire semi-simple de type intérieur sur un corps $F$ et soit $X$ un $G$ -espace homogène projectif tel que le groupe $G$ soit déployé sur le point générique de $X$ . Nous introduisons le $J$ -invariant de $G$ qui caractérise le comportement motivique de $X$ et généralise le $J$ -invariant défini par A. Vishik dans le cadre des formes quadratiques. Nous utilisons cet invariant pour obtenir les décompositions motiviques de tous les $G$ -espaces homogènes projectifs qui sont génériquement déployés, par exemple les variétés de Severi-Brauer, les quadriques de Pfister, la grassmannienne des sous-espaces totalement isotropes maximaux d’une forme quadratique, la variété des sous-groupes de Borel de $G$ . Nous discutons également les relations avec les indices de torsion, la dimension canonique et les invariants cohomologiques du groupe $G$ .

Let $G$ be a semisimple linear algebraic group of inner type over a field $F$ , and let $X$ be a projective homogeneous $G$ -variety such that $G$ splits over the function field of $X$ . We introduce the $J$ -invariant of $G$ which characterizes the motivic behavior of $X$ , and generalizes the $J$ -invariant defined by A. Vishik in the context of quadratic forms. We use this $J$ -invariant to provide motivic decompositions of all generically split projective homogeneous $G$ -varieties, e.g. Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, varieties of Borel subgroups of $G$ . We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group $G$ .

MR Zbl

DOI : 10.24033/asens.2088

Classification : 14C25, 20G15
Keywords: motive, algebraic group, homogeneous variety
Mot clés : motif, groupe algébrique, espace homogène

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     author = {Petrov, Viktor and Semenov, Nikita and Zainoulline, Kirill},
     title = {$J$-invariant of linear algebraic groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1023--1053},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {6},
     year = {2008},
     doi = {10.24033/asens.2088},
     mrnumber = {2504112},
     zbl = {1206.14017},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2088/}
}

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Petrov, Viktor; Semenov, Nikita; Zainoulline, Kirill. $J$-invariant of linear algebraic groups. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 1023-1053. doi : 10.24033/asens.2088. http://www.numdam.org/articles/10.24033/asens.2088/

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