Motivic decomposability of generalized Severi–Brauer varieties

Maksim Zhykhovich

doi:10.1016/j.crma.2010.07.022

Algebraic Geometry

Motivic decomposability of generalized Severi–Brauer varieties

Presented by: Christophe Soulé

Maksim Zhykhovich ¹

¹ Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu, 4, place Jussieu, 75005, Paris, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 989-992

Abstract (Original language)
Résumé

Let F be an arbitrary field. Let p be a positive prime number and D a central division F-algebra of degree $p^{n}$ , with $n ⩾ 1$ . We write $SB (p^{m}, D)$ for the generalized Severi–Brauer variety of right ideals in D of reduced dimension $p^{m}$ for $m = 0, 1, \dots, n - 1$ . We note by $M (SB (p^{m}, D))$ the Chow motive with coefficients in $F_{p}$ of the variety $SB (p^{m}, D)$ . It was proven by Nikita Karpenko that this motive is indecomposable for any prime p and $m = 0$ and for $p = 2$ , $m = 1$ . We prove decomposability of $M (SB (p^{m}, D))$ in all the other cases.

Soient F un corps arbitraire, p un nombre premier positif et D une F-algèbre de division de degré $p^{n}$ . On écrit $SB (p^{m}, D)$ pour la variété de Severi–Brauer généralisée des idéaux à droite de dimension réduite $p^{m}$ , $m = 0, 1, \dots, n - 1$ . On note par $M (SB (p^{m}, D))$ le motif de Chow à coefficients dans $F_{p}$ de la variété $SB (p^{m}, D)$ . Il a été demontré par Nikita Karpenko que ce motif est indecomposable pour tout nombre premier p arbitraire et $m = 0$ et pour $p = 2$ , $m = 1$ . Nous montrons la décomposabilité de $M (SB (p^{m}, D))$ dans tous les autres cas.

Received: 2010-06-16
Accepted: 2010-07-26
Published online: 2010-09-01

DOI: 10.1016/j.crma.2010.07.022

Author's affiliations:

Maksim Zhykhovich ¹

¹ Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu, 4, place Jussieu, 75005, Paris, France

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     author = {Maksim Zhykhovich},
     title = {Motivic decomposability of generalized {Severi{\textendash}Brauer} varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {989--992},
     year = {2010},
     publisher = {Elsevier},
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     number = {17-18},
     doi = {10.1016/j.crma.2010.07.022},
     language = {en},
}

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Maksim Zhykhovich. Motivic decomposability of generalized Severi–Brauer varieties. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 989-992. doi: 10.1016/j.crma.2010.07.022

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