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 $S$.Take $S$ essentially smooth over a field $k$. Relying on Hopkins-Morel-Hoyois isomorphism 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 the characteristic of $k$), Spitzweck computes the remaining slices of $MG{L}_S$ as $s_{n}MG{L}_S={\Sigma }_T^{n}H\mathbb{Z}\otimes {\mathbb{L}}^{-n}$ (again, after inverting the characteristic of $k$). 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 mod $p$ 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 $S=\text{Spec}k$, $k$ 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 \[ Ω_*(X)\otimes_{\mathbb{L}^{-*}}\mathcal{E}^{-2*,-*}(k)\to \mathcal{E}^{2m-2*, m-*}_X(M) \] for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=$dim${}_{k}M$.

30 pages