Quotients of MGL, their slices and their geometric parts

  • Marc Levine

  • Girja Shanker Tripathi


Let x1,x2,x_1, x_2,\dots be a system of homogeneous polynomial generators for the Lazard ring L=MU2\mathbb L^\ast=MU^{2\ast} and let MGLSMGL_S denote Voevodsky's algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme SS [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 s0MGLSs_0MGL_S for Voevodsky's slice tower with MGLS/(x1,x2,)MGL_S/(x_1, x_2,\dots) (after inverting all residue characteristics of SS), M. Spitzweck [Homology Homotopy Appl. 12, No. 2, 335--351 (2010; Zbl 1209.14019)] computes the remaining slices of MGLSMGL_S as snMGLS=TnHZLns_nMGL_S=\sum^n_TH\mathbb Z \otimes \mathbb L^{-n} (again, after inverting all residue characteristics of SS). We apply Spitzweck's method to compute the slices of a quotient spectrum MGLS/({xi:iI})MGL_S/(\{x_i:i \in I\}) for II an arbitrary subset of N\mathbb N, as well as the modp{mod } p version MGLS/({p,xi:iI})MGL_S/(\{p, x_i:i \in I\}) and localizations with respect to a system of homogeneous elements in Z[{xj:j∉I}]\mathbb Z[\{x_j:j \not\in I\}]. In case S=SpeckS=\operatorname{Spec} k, kk a field of characteristic zero, we apply this to show that for E\mathcal E a localization of a quotient of MGLMGL as above, there is a natural isomorphism for the theory with support Ω(X)LE2,(k)E2m2,m(M) \Omega_\ast (X) \otimes _{\mathbb L^{-\ast}}\mathcal E^{-2\ast,-\ast}(k) \to \mathcal E^{2m-2\ast, m-\ast}(M) for XX a closed subscheme of a smooth quasi-projective kk-scheme MM, m=dimkMm=\dim_k M.