Mixed motives and the slice filtration

Pablo Pelaez

doi:10.1016/j.crma.2009.02.028

Algebraic Geometry/Topology

Mixed motives and the slice filtration
[Motifs mixtes et la filtration par les tranches]

Présenté par : Christophe Soulé

Pablo Pelaez ¹

¹ Département de mathématiques, Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 541-544.

Résumé
Abstract

Nous construisons plusieurs structures des modèles de Quillen dans la catégorie de Jardine Spt des T-spectres symétriques motiviques [J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445–553], tel que leur catégories d'homotopie associées sont naturellement isomorphiques à la filtration par les tranches de Voevodsky [V. Voevodsky, Open problems in the motivic stable homotopy theory. I, in: Motives, Polylogarithms and Hodge Theory, Part I, Int. Press Lect. Ser., Irvine, CA, 1998]. Nous prouvons une conjecture de Voevodsky [V. Voevodsky, Open problems in the motivic stable homotopy theory. I, in: Motives, Polylogarithms and Hodge Theory, Part I, Int. Press Lect. Ser., Irvine, CA, 1998], laquelle affirme que sur un corps parfait tous les tranches $s_{q}$ sont canoniquement modules dans Spt sur le spectre motivique d'Eilenberg–MacLane $H Z$ . Si le corps est de charactéristique zéro, nous obtenons que les tranches $s_{q}$ sont motifs grands au sens de Voevodsky. Nous montrons aussi que le produit « smash » dans Spt induit des structures multiplicatives sur la suite spectrale motivique de Atiyah–Hirzebruch.

We construct several Quillen model structures in Jardine's category Spt of motivic symmetric T-spectra [J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445–553], such that their associated homotopy categories are naturally isomorphic to Voevodsky's slice filtration [V. Voevodsky, Open problems in the motivic stable homotopy theory. I, in: Motives, Polylogarithms and Hodge Theory, Part I, Int. Press Lect. Ser., Irvine, CA, 1998]. We prove a conjecture of Voevodsky [V. Voevodsky, Open problems in the motivic stable homotopy theory. I, in: Motives, Polylogarithms and Hodge Theory, Part I, Int. Press Lect. Ser., Irvine, CA, 1998], which says that over a perfect field all the slices $s_{q}$ have a canonical structure of modules in Spt over the motivic Eilenberg–MacLane spectrum $H Z$ . Restricting the field even further to the case of characteristic zero, we get that the slices $s_{q}$ may be interpreted as big motives in the sense of Voevodsky. We also show that the smash product in Spt induces pairings in the motivic Atiyah–Hirzebruch spectral sequence.

Reçu le : 2008-12-14
Accepté le : 2009-02-17
Publié le : 2009-03-19

DOI : 10.1016/j.crma.2009.02.028

Affiliations des auteurs :

Pablo Pelaez ¹

¹ Département de mathématiques, Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

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     author = {Pablo Pelaez},
     title = {Mixed motives and the slice filtration},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {541--544},
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     year = {2009},
     doi = {10.1016/j.crma.2009.02.028},
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Pablo Pelaez. Mixed motives and the slice filtration. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 541-544. doi : 10.1016/j.crma.2009.02.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.028/

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