Mixed motives and the slice filtration

Pelaez, Pablo

Comptes Rendus. Mathématique, Tome 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.

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