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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.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 19D06 $$Q$$- and plus-constructions 55P47 Infinite loop spaces
##### Keywords:
motivic homotopy theory; algebraic cobordism
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