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.
Keywords
Cite
@article{arxiv.1711.05248,
title = {Motivic infinite loop spaces},
author = {Elden Elmanto and Marc Hoyois and Adeel A. Khan and Vladimir Sosnilo and Maria Yakerson},
journal= {arXiv preprint arXiv:1711.05248},
year = {2021}
}
Comments
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