Quotients of MGL, their slices and their geometric parts

Abstract

Let $x_1, x_2,\ldots$ be a system of homogeneous polynomial generators for the Lazard ring $\mathbb{L}^*=MU^{2*}$ and let $MGL_S$ 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 $s_0MGL_S$ for Voevodsky's slice tower with $MGL_S/(x_1, x_2,\ldots)$ (after inverting the characteristic of $k$), Spitzweck computes the remaining slices of $MGL_S$ as $s_nMGL_S=\Sigma^n_TH\mathbb{Z}\otimes \mathbb{L}^{-n}$ (again, after inverting the characteristic of $k$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MGL_S/(\{x_i:i\in I\})$ 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 $S=\text{Spec}\, k$, $k$ 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 \[ \Omega_*(X)\otimes_{\mathbb{L}^{-*}}\mathcal{E}^{-2*,-*}(k)\to \mathcal{E}^{2m-2*, m-*}_X(M) \] for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=$dim${}_kM$.