# Quotients of MGL, their slices and their geometric parts

### Marc Levine

### Girja Shanker Tripathi

## Abstract

Let $x_1, x_2,\dots$ be a system of homogeneous polynomial generators for the Lazard ring $\mathbb L^\ast=MU^{2\ast}$ and let $MGL_S$ 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 $s_0MGL_S$ for Voevodsky's slice tower with $MGL_S/(x_1, x_2,\dots)$ (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 $MGL_S$ as $s_nMGL_S=\sum^n_TH\mathbb Z \otimes \mathbb L^{-n}$ (again, after inverting all residue characteristics of $S$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MGL_S/(\{x_i:i \in I\})$ for $I$ an arbitrary subset of $\mathbb N$, as well as the ${mod } p$ version $MGL_S/(\{p, x_i:i \in I\})$ and localizations with respect to a system of homogeneous elements in $\mathbb Z[\{x_j:j \not\in I\}]$. In case $S=\operatorname{Spec} k$, $k$ 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 $M$, $m=\dim_k M$.