Quotients of MGL, their slices and their geometric parts

Abstract

Let $x_1,x_2,\dots$ be a system of homogeneous polynomial generators for the Lazard ring ${\mathbb{L}}^{\ast }=M{U}^{2\ast }$ and let $MG{L}_S$ denote Voevodsky's algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme $S$ [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 $s_{0}MG{L}_S$ for Voevodsky's slice tower with $MG{L}_S/(x_1,x_2,\dots )$ (after inverting all residue characteristics of $S$), M. Spitzweck [Homology Homotopy Appl. 12, No. 2, 335--351 (2010; Zbl 1209.14019)] computes the remaining slices of $MG{L}_S$ as $s_{n}MG{L}_S={\sum }_T^{n}H\mathbb{Z}\otimes {\mathbb{L}}^{-n}$ (again, after inverting all residue characteristics of $S$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MG{L}_S/(\{x_i:i\in I\})$ for $I$ an arbitrary subset of $\mathbb{N}$, as well as the $modp$ version $MG{L}_S/(\{p,x_i:i\in I\})$ and localizations with respect to a system of homogeneous elements in $\mathbb{Z}[\{x_j:j\notin I\}]$. In case $S=\mathrm{Spec}k$, $k$ a field of characteristic zero, we apply this to show that for $\mathcal{E}$ a localization of a quotient of $MGL$ as above, there is a natural isomorphism for the theory with support ${\Omega }_{\ast }(X){\otimes }_{{\mathbb{L}}^{-\ast }}{\mathcal{E}}^{-2\ast ,-\ast }(k)\to {\mathcal{E}}^{2m-2\ast ,m-\ast }(M)$ for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m={\dim }_{k}M$.