Comptes Rendus
Algebraic Geometry/Topology
Mixed motives and the slice filtration
[Motifs mixtes et la filtration par les tranches]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 541-544.

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 sq sont canoniquement modules dans Spt sur le spectre motivique d'Eilenberg–MacLane HZ. Si le corps est de charactéristique zéro, nous obtenons que les tranches sq 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 sq have a canonical structure of modules in Spt over the motivic Eilenberg–MacLane spectrum HZ. Restricting the field even further to the case of characteristic zero, we get that the slices sq 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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.02.028

Pablo Pelaez 1

1 Département de mathématiques, Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France
@article{CRMATH_2009__347_9-10_541_0,
     author = {Pablo Pelaez},
     title = {Mixed motives and the slice filtration},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {541--544},
     publisher = {Elsevier},
     volume = {347},
     number = {9-10},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.028},
     language = {en},
}
TY  - JOUR
AU  - Pablo Pelaez
TI  - Mixed motives and the slice filtration
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 541
EP  - 544
VL  - 347
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crma.2009.02.028
LA  - en
ID  - CRMATH_2009__347_9-10_541_0
ER  - 
%0 Journal Article
%A Pablo Pelaez
%T Mixed motives and the slice filtration
%J Comptes Rendus. Mathématique
%D 2009
%P 541-544
%V 347
%N 9-10
%I Elsevier
%R 10.1016/j.crma.2009.02.028
%G en
%F CRMATH_2009__347_9-10_541_0
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/

[1] E. Friedlander; A. Suslin The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002), pp. 773-875

[2] P.S. Hirschhorn Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003

[3] M. Hovey Model Categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999

[4] M. Hovey Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, Volume 165 (2001), pp. 63-127

[5] M. Hovey, Monoidal model categories, preprint, 1998

[6] J.F. Jardine Motivic symmetric spectra, Doc. Math., Volume 5 (2000), pp. 445-553

[7] M. Levine The Homotopy Coniveau Tower, J. Topol., Volume 1 (2008), pp. 217-267

[8] W.S. Massey Products in exact couples, Ann. of Math. (2), Volume 59 (1954), pp. 558-569

[9] A. Neeman Triangulated Categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001

[10] P. Pelaez, Multiplicative properties of the slice filtration, preprint, 2008

[11] O. Röndigs; P.A. Østvær Modules over motivic cohomology, Adv. Math., Volume 219 (2008), pp. 689-727

[12] V. Voevodsky On the zero slice of the sphere spectrum, Tr. Mat. Inst. Steklova, Volume 246 (2004), pp. 106-115

[13] 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

Cité par Sources :

Commentaires - Politique