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.


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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