Tom Bachmann ; Elden Elmanto ; Marc Hoyois ; Adeel A. Khan ; Vladimir Sosnilo et al.
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On the infinite loop spaces of algebraic cobordism and the motivic
sphere
On the infinite loop spaces of algebraic cobordism and the motivic
sphere
Authors: Tom Bachmann ; Elden Elmanto ; Marc Hoyois ; Adeel A. Khan ; Vladimir Sosnilo ; Maria Yakerson
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Tom Bachmann;Elden Elmanto;Marc Hoyois;Adeel A. Khan;Vladimir Sosnilo;Maria Yakerson
We obtain geometric models for the infinite loop spaces of the motivic
spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are
motivically equivalent to $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where
$\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp.
$\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$,
$\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci
points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$,
and $+$ is Quillen's plus construction. Moreover, we show that the plus
construction is redundant in positive characteristic.