# Quotients of MGL, their slices and their geometric parts

### Marc Levine

### Girja Shanker Tripathi

## Abstract

Let $x_{1},x_{2},…$ be a system of homogeneous polynomial generators for the Lazard ring $L_{∗}=MU_{2∗}$ 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 $s_{0}MGL_{S}$ for Voevodsky's slice tower with $MGL_{S}/(x_{1},x_{2},…)$ (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 $s_{n}MGL_{S}=∑_{T}HZ⊗L_{−n}$ (again, after inverting all residue characteristics of $S$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MGL_{S}/({x_{i}:i∈I})$ for $I$ an arbitrary subset of $N$, as well as the $modp$ version $MGL_{S}/({p,x_{i}:i∈I})$ and localizations with respect to a system of homogeneous elements in $Z[{x_{j}:j∈I}]$. In case $S=Speck$, $k$ a field of characteristic zero, we apply this to show that for $E$ a localization of a quotient of $MGL$ as above, there is a natural isomorphism for the theory with support $Ω_{∗}(X)⊗_{L_{−∗}}E_{−2∗,−∗}(k)→E_{2m−2∗,m−∗}(M)$ for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=dim_{k}M$.