Quotients of MGL, Their Slices and Their Geometric Parts

Let $x_1, x_2,ldots$ 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 $S$ \cite{VoevICM}. Relying on Hopkins-Morel-Hoyois isomorphism \cite{Hoyois} of the 0th slice $s_0\MGL_S$ for Voevodsky's slice tower with $\MGL_S/(x_1, x_2,ldots)$ (after inverting all residue characteristics of $S$), Spitzweck \cite{Spitzweck10} computes the remaining slices of $\MGL_S$ as $s_n\MGL_S=\Sigma^n_TH\Z\otimes \L^{-n}$ (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 $\N$, 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 $\Z[{x_j:j\not\in I}]$. In case $S=\Spec k$, $k$ a field of characteristic zero, we apply this to show that for $\sE$ a localization of a quotient of $\MGL$ as above, there is a natural isomorphism for the theory with support \[ \Omega_*(X)\otimes_{\L^{-*}}\sE^{-2*,-*}(k)\to \sE^{2m-2*, m-*}_X(M) \] for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=\dim_kM$.

2010 Mathematics Subject Classification:

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