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

TL;DR

This paper constructs geometric models for the infinite loop spaces of certain motivic spectra related to algebraic cobordism, linking them to Hilbert schemes of points with specific structures, and shows the plus construction's redundancy in positive characteristic.

Contribution

It provides explicit geometric models for the infinite loop spaces of motivic spectra MGL, MSL, and the sphere, connecting them to Hilbert schemes of points and analyzing the plus construction.

Findings

Models are motivically equivalent to Hilbert schemes of points with structures.

The plus construction is unnecessary in positive characteristic.

Provides geometric descriptions for infinite loop spaces of key motivic spectra.

Abstract

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.