Motivic infinite loop spaces

TL;DR

This paper establishes a recognition principle for motivic infinite P1-loop spaces over perfect fields by developing a theory of framed motivic spaces, linking motivic spectra with geometric objects like Hilbert schemes.

Contribution

It introduces a theory of framed motivic spaces as a motivic analogue of E-infinity-spaces, providing a new framework for understanding motivic spectra.

Findings

Grouplike framed motivic spaces are equivalent to certain motivic spectra.

Derived representability results for suspension spectra of smooth varieties.

Established connections between motivic spectra and Hilbert schemes of points.

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.