Quotients of MGL, their slices and their geometric parts

  • Marc Levine

  • Girja Shanker Tripathi


Let be a system of homogeneous polynomial generators for the Lazard ring and let denote Voevodsky's algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme [V. Voevodsky, ibid., 417--442 (1998; Zbl 0907.19002)]. Relying on Hopkins-Morel-Hoyois isomorphism [M. Hoyois, J. Reine Angew. Math. 702, 173--226 (2015; Zbl 1382.14006)] of the 0th slice for Voevodsky's slice tower with (after inverting all residue characteristics of ), M. Spitzweck [Homology Homotopy Appl. 12, No. 2, 335--351 (2010; Zbl 1209.14019)] computes the remaining slices of as (again, after inverting all residue characteristics of ). We apply Spitzweck's method to compute the slices of a quotient spectrum for an arbitrary subset of , as well as the version and localizations with respect to a system of homogeneous elements in . In case , a field of characteristic zero, we apply this to show that for a localization of a quotient of as above, there is a natural isomorphism for the theory with support for a closed subscheme of a smooth quasi-projective -scheme , .