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.

78 pages. v6: final version, to appear in the Cambridge Journal of Mathematics; v5: replace Nisnevich by Zariski; v4: include the case of characteristic 2; v3: fix a mistake in Appendix B and state explicitly the "BPQ" and "framed cobordism" descriptions of the motivic sphere spectrum; v2: generalized the main results to finite fields