## Motivic infinite loop spaces.(English)Zbl 07422194

Summary: We prove a recognition principle for motivic infinite $$\mathsf{P}^1$$-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of $$\mathcal{E}_\infty$$-spaces. A framed motivic space is a motivic space equipped with transfers along finite syntomic morphisms with trivialized cotangent complex in K-theory. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties, and in particular for the motivic sphere spectrum, in terms of Hilbert schemes of points in affine spaces.

### MSC:

 14F42 Motivic cohomology; motivic homotopy theory 55P47 Infinite loop spaces 14C05 Parametrization (Chow and Hilbert schemes) 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects)
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