Title: Motivic infinite loop spaces,
Abstract: We prove a recognition principle for motivic infinite P1-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 E-infinity-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.