Cambridge Journal of Mathematics

Volume 9 (2021)

Number 2

Motivic infinite loop spaces

Pages: 431 – 549



Elden Elmanto (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Marc Hoyois (Fakultät für Mathematik, Universität Regensburg, Germany)

Adeel A. Khan (Institute of Mathematics, Academia Sinica, Taipei, Taiwan)

Vladimir Sosnilo (Laboratory “Modern Algebra and Applications”, St. Petersburg State University, Saint Petersburg, Russia)

Maria Yakerson (Institute for Mathematical Research, ETH Zürich, Switzerland)


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.


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

2010 Mathematics Subject Classification

Primary 14F42, 55P47. Secondary 14C05.

In Memory of Vladimir Voevodsky.

E.E. was supported by a Institut Mittag-Leffler postdoctoral fellowship and NSF grant DMS-1508040.

M.H. was partially supported by NSF grants DMS-1508096 and DMS-1761718.

A.A.K. was supported by a Institut Mittag-Leffler postdoctoral fellowship.

V.S. was supported by the grant of the Government of the Russian Federation for the state support of scientific research carried out under the supervision of leading scientists, agreement 14.W03.31.0030 dated 15.02.2018.

M.Y. was supported by SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”.

Published 7 October 2021