#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 407-442

** Marc Levine, Girja Shanker Tripathi **
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:

Keywords and Phrases:

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