Classification of upper motives of algebraic groups of inner type $ {A}_{n}$

Charles De Clercq

doi:10.1016/j.crma.2011.02.020

Algebraic Geometry

Classification of upper motives of algebraic groups of inner type

A_{n}

[Classification des motifs supérieurs des groupes algébriques intérieurs de type

A_{n}

]

Présenté par : the Editorial Board

Charles De Clercq ¹

¹ Université Pierre-et-Marie-Curie (Paris 6), équipe topologie et géométrie algébriques, 4, place Jussieu, 75252 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 433-436.

Résumé
Abstract

Soient A, $A^{'}$ deux algèbres centrales simples sur un corps F et $F$ un corps fini de caractéristique p. Nous prouvons que les facteurs directs indécomposables supérieurs des motifs de deux variétés anisotropes de drapeaux dʼidéaux à droite $X (d_{1}, \dots, d_{k}; A)$ et $X (d_{1}^{'}, \dots, d_{k^{'}}^{'}; A^{'})$ à coefficients dans $F$ sont isomorphes si et seulement si les valuations p-adiques de $pgcd (d_{1}, \dots, d_{k})$ et $pgcd (d_{1}^{'}, \dots, d_{k^{'}}^{'})$ sont égales et les classes des composantes p-primaires $A_{p}$ et $A_{p}^{'}$ de A et $A^{'}$ engendrent le même sous-groupe dans le groupe de Brauer de F. Ce résultat mène à une surprenante dichotomie entre les motifs supérieurs des groupes algébriques absolument simples, adjoints et intérieurs de type $A_{n}$ .

Let A, $A^{'}$ be two central simple algebras over a field F and $F$ be a finite field of characteristic p. We prove that the upper indecomposable direct summands of the motives of two anisotropic varieties of flags of right ideals $X (d_{1}, \dots, d_{k}; A)$ and $X (d_{1}^{'}, \dots, d_{k^{'}}^{'}; A^{'})$ with coefficients in $F$ are isomorphic if and only if the p-adic valuations of $\gcd (d_{1}, \dots, d_{k})$ and $\gcd (d_{1}^{'}, \dots, d_{k^{'}}^{'})$ are equal and the classes of the p-primary components $A_{p}$ and $A_{p}^{'}$ of A and $A^{'}$ generate the same group in the Brauer group of F. This result leads to a surprising dichotomy between upper motives of absolutely simple adjoint algebraic groups of inner type $A_{n}$ .

Reçu le : 2011-01-24
Accepté le : 2011-02-28
Publié le : 2011-03-26

DOI : 10.1016/j.crma.2011.02.020

Affiliations des auteurs :

Charles De Clercq ¹

¹ Université Pierre-et-Marie-Curie (Paris 6), équipe topologie et géométrie algébriques, 4, place Jussieu, 75252 Paris cedex 05, France

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     title = {Classification of upper motives of algebraic groups of inner type $ {A}_{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {433--436},
     publisher = {Elsevier},
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     year = {2011},
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     language = {en},
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Charles De Clercq. Classification of upper motives of algebraic groups of inner type $ {A}_{n}$. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 433-436. doi : 10.1016/j.crma.2011.02.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.020/

Bibliographie
Cité par

[1] S.A. Amitsur Generic splitting fields of central simple algebras, Ann. of Math., Volume 62 (1955), pp. 8-43

[2] Y. André Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, vol. 17, Société Mathématique de France, 2004

[3] V. Chernousov; A. Merkurjev Motivic decomposition of projective homogeneous varieties and the Krull–Schmidt theorem, Transform. Groups, Volume 11 (2006), pp. 371-386

[4] C. De Clercq Motivic decompositions of projective homogeneous varieties and change of coefficients, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010) no. 17–18, pp. 955-958

[5] R. Elman; N. Karpenko; A. Merkurjev The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society, Providence, 2008

[6] N. Karpenko, Canonical dimension, in: Proceedings of the ICM 2010, vol. II, pp. 146–161.

[7] N. Karpenko On the first Witt index of quadratic forms, Invent. Math., Volume 153 (2003) no. 2, pp. 455-462

[8] N. Karpenko, Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties, Linear Algebraic Groups and Related Structures (preprint server) 333, 2009.

[9] M.-A. Knus; A. Merkurjev; M. Rost; J.-P. Tignol The Book of Involutions, AMS Colloquium Publications, vol. 44, American Mathematical Society, 1998

[10] A. Merkurjev; A. Panin; A. Wadsworth Index reduction formulas for twisted flag varieties, I, J. K-Theory, Volume 10 (1996), pp. 517-596

[11] V. Petrov, N. Semenov, Higher Tits indices of algebraic groups, preprint, 2007.

[12] V. Petrov; N. Semenov; K. Zainoulline J-invariant of linear algebraic groups, Ann. Sci. École Norm. Sup., Volume 41 (2008), pp. 1023-1053

[13] A. Vishik, Motives of quadrics with applications to the theory of quadratic forms, in: Proceedings of the Summer School “Geometric Methods in the Algebraic Theory of Quadratic Forms, Lens, 2000”, in: Lect. Notes in Math., vol. 1835, 2004, pp. 25–101.

[14] A. Vishik, Excellent connections in the motives of quadrics, Annales Scientifiques de LʼENS, 2010, in press.

[15] A. Vishik Fields of u-invariant $2^{r} + 1$ , Algebra, Arithmetic and Geometry – In Honor of Yu.I. Manin, Birkhäuser, 2010, pp. 661-685

[16] A. Vishik; N. Yagita Algebraic cobordism of a Pfister quadric, J. Lond. Math. Soc. (2), Volume 76 (2007) no. 3, pp. 586-604

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