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Slices of motivic Landweber spectra. (English) Zbl 1249.14008
In this interesting paper, the author discusses the relationship between conjectures on the slices of motivic spectra. The mother of this type of conjecture is Voevodsky’s conjecture on the slices of the motivic Thom spectrum representing algebraic cobordism in [V. Voevodsky, “Open problems in the motivic stable homotopy theory. I”, in: Bogomolov, Fedor (ed.) et al., Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. Int. Press Lect. Ser. 3, No. I, 3–34 (2002; Zbl 1047.14012)]. The slices of a spectrum emerge via the so-called slice filtration in the stable motivic homotopy category. As Voevodsky explained in loc. cit., to understand the shape of these slices is of fundamental importance in motivic homotopy theory. In the paper under review, the author shows that, under a certain assumption, Voevodsky’s conjecture allows the computation of the slices of motivic Landweber spectra. Motivic Landweber spectra are the motivic analogs of Landweber spectra in topology and have been studied by the author in joint work with N. Naumann and M. Østvær [“Motivic Landweber exactness”, Doc. Math., J. DMV 14, 551–593 (2009; Zbl 1230.55005)].

MSC:
14F42 Motivic cohomology; motivic homotopy theory
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
18E30 Derived categories, triangulated categories (MSC2010)
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