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On the Functoriality of the Slice Filtration

Published online by Cambridge University Press:  11 February 2013

Pablo Pelaez
Pablo Pelaez*
Affiliation:
Universität Duisburg-Essen, Mathematik, 45117 Essen, Germanypablo.pelaez@uni-due.de
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Abstract

Let k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

Keywords

K-theoryMixed MotivesMotivic Atiyah-Hirzebruch Spectral SequenceSlice FiltrationPrimary 14F42
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 1 , February 2013 , pp. 55 - 71
Copyright
Copyright © ISOPP 2013

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References

1.Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I, Astérisque 314 (2007), x+466 pp. (2008).Google Scholar
2.Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II, Astérisque 315 (2007), vi+364 pp. (2008).Google Scholar
3.Cisinski, D.-C., Descente propre en k-théorie invariante par homotopie, preprint http://arxiv.org/abs/1003.1487.Google Scholar
4.Cisinski, D.-C., Déglise, F., Triangulated categories of mixed motives, preprint.Google Scholar
5.Dundas, B. I., Röndigs, O., Østvær, P. A., Motivic functors, Doc. Math. 8 (2003), 489–525 (electronic).Google Scholar
6.Gutiérrez, J. J., Röndigs, O., Spitzweck, M., Østvær, P. A., Motivic slices and colored operads, preprint http://arxiv.org/abs/1012.3301.Google Scholar
7.Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, 2003.Google Scholar
8.Jardine, J. F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553 (electronic).Google Scholar
9.Kelly, S., Triangulated categories of motives in positive characteristic. Thesis, Paris 13 (2012).Google Scholar
10.Levine, M., The homotopy coniveau tower, J. Topol. 1(1) (2008), 217–267.Google Scholar
11.Morel, F., Voevodsky, V., A1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001).Google Scholar
12.Neeman, A., The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9(1) (1996), 205–236.Google Scholar
13.Neeman, A., Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, Princeton, NJ, 2001.Google Scholar
14.Panin, I., Pimenov, K., Röndigs, O., On Voevodsky's algebraic K-theory spectrum, Algebraic topology, Abel Symp. 4, Springer, Berlin, 2009, 279–330.Google Scholar
15.Pelaez, P., Mixed motives and the slice filtration, C. R. Math. Acad. Sci. Paris 347(9-10) (2009), 541–544.CrossRefGoogle Scholar
16.Pelaez, P., Multiplicative properties of the slice filtration, Astérisque 335 (2011), xvi+289Google Scholar
17.Pelaez, P., On the orientability of the slice filtration, Homology, Homotopy Appl. 13(2) (2011), 293–300.Google Scholar
18.Quillen, D. G., Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin, 1967Google Scholar
19.Röndigs, O., Østvær, P. A., Modules over motivic cohomology, Adv. Math. 219(2) (2008), 689–727.Google Scholar
20.Schwede, S., Shipley, B. E., Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80(2) (2000), 491–511.Google Scholar
21.Voevodsky, V., A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Extra Vol. I, 1998, 579–604 (electronic).Google Scholar
22.Voevodsky, V., Open problems in the motivic stable homotopy theory. I, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser. 3, Int. Press, Somerville, MA, 2002, 3–34.Google Scholar
23.Voevodsky, V., On the zero slice of the sphere spectrum, Tr. Mat. Inst. Steklova 246 (Algebr. Geom. Metody, Svyazi i Prilozh.), (2004), 106–115.Google Scholar
24.Weibel, C. A., Homotopy algebraic K-theory, Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989, 461–488.Google Scholar