On the infinite loop spaces of algebraic cobordism and the motivic sphere.(English. French summary)Zbl 1478.14043

The authors give explicit geometric descriptions of the motivic infinite loop spaces associated with the motivic spectra of algebraic cobordism $$\mathrm{MGL}$$, special linear algebraic cobordism $$\mathrm{MSL}$$, and the motivic sphere spectrum $$\mathbf{1}$$ in Morel-Voevodsky’s unstable motivic homotopy category over a base field $$k$$. By former work of the last named five authors it was already known that the objects in question are equivalent to the motivic localisations of the group completion of the moduli stack $$\mathcal{FS}yn$$ of finite syntomic $$k$$-schemes, its framed version $$\mathcal{FS}yn^\mathrm{fr}$$, and its oriented version $$\mathcal{FS}yn^\mathrm{or}$$, respectively [E. Elmanto et. al., Camb. J. Math. 9, No. 2, 431–549 (2021; Zbl 07422194)]. The current paper replaces the group completion by a generalisation of Quillen’s plus construction (Corollary 4.2, Proposition 4.8). Furthermore, it is shown that the plus construction can be ommited in positive charactersitic (Proposition 5.6).

MSC:

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

Zbl 07422194
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