On the infinite loop spaces of algebraic cobordism and the motivic sphere

Bachmann, Tom ; Elmanto, Elden ; Hoyois, Marc ; Khan, Adeel A. ; Sosnilo, Vladimir ; Yakerson, Maria

Épijournal de Géométrie Algébrique, Tome 5 (2021)

Résumé

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.

Détail

DOI : 10.46298/epiga.2021.volume5.6581

Classification : 14F42, 19D06, 55P47