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On the infinite loop spaces of algebraic cobordism and the motivic sphere. (English. French summary) Zbl 07402132
Summary: We obtain geometric models for the infinite loop spaces of the motivic spectra MGL, MSL, and 1 over a field. They are motivically equivalent to \(\mathbf{Z}\times\text{Hilb}^{\text{lci}}_\infty (\mathbf{A}^\infty)^+\), \(\mathbf{Z}\times\text {Hilb}^{\text{or}}_\infty(\mathbf{A}^\infty)^+\), and \(\mathbf{Z} \times\text{Hilb}^{\text{fr}}_\infty(\mathbf{A}^\infty)^+\), respectively, where \(\text{Hilb}^{\text{lci}}_d(\mathbf{A}^n)\) (resp. \(\text{Hilb}^{\text{or}}_d(\mathbf{A}^n)\), \(\text{Hilb}^{\text{fr}}_d(\mathbf{A}^n))\) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree \(d\) in \(\mathbf{A}^n\), and \(+\) is Quillen’s plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.

14F42 Motivic cohomology; motivic homotopy theory
19D06 \(Q\)- and plus-constructions
55P47 Infinite loop spaces
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