Motivic infinite loop spaces. (English) Zbl 07422194

Elmanto, Elden; Hoyois, Marc; Khan, Adeel A.; Sosnilo, Vladimir; Yakerson, Maria

Camb. J. Math. 9, No. 2, 431-549 (2021).

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)

Keywords:

motivic stable homotopy theory; infinite loop space theory; recognition principle; cotangent complex