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

Christophe Soulé

Maksim Zhykhovich ¹

¹Université 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.

Article information
Cite this article

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

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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     title = {Motivic decomposability of generalized {Severi{\textendash}Brauer} varieties},
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     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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