Tom Bachmann ; Elden Elmanto ; Marc Hoyois ; Adeel A. Khan ; Vladimir Sosnilo et al. - On the infinite loop spaces of algebraic cobordism and the motivic sphere

epiga:6581 - Épijournal de Géométrie Algébrique, May 19, 2021, Volume 5 - https://doi.org/10.46298/epiga.2021.volume5.6581
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

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.

Volume: Volume 5
Published on: May 19, 2021
Accepted on: May 19, 2021
Submitted on: June 19, 2020
Keywords: Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - K-Theory and Homology