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 $\mathbb L^\ast=MU^{2\ast}$ 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 $M$, $m=\dim\_k M$.