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

Abstract

We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $1$ 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.

13 pages. v5: published version; v4: final version, to appear in Épijournal Géom. Algébrique; v3: minor corrections; v2: added details in the moving lemma over finite fields

References in corpus (2)

• Modules over algebraic cobordism
• The unit map of the algebraic special linear cobordism spectrum

Cited by in corpus (3)

• Modules over algebraic cobordism
• On the Hilbert function of Artinian local complete intersections of codimension three
• Zariski-local framed ${\mathbb{A}}^1$-homotopy theory