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Modules over algebraic cobordism

Part of:(Co)homology theoryFamilies, fibrations

Published online by Cambridge University Press:  17 December 2020

Elden Elmanto,
Marc Hoyois,
Adeel A. Khan,
Vladimir Sosnilo and
Maria Yakerson
Elden Elmanto
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138, USA; E-mail: elmanto@math.harvard.edu
Marc Hoyois
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138, USA; E-mail: elmanto@math.harvard.eduFakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany; E-mail: marc.hoyois@ur.de
Adeel A. Khan
Affiliation:
IHES, 35 route de Chartres, 91440 Bures-sur-Yvette, France; E-mail: khan@ihes.fr
Vladimir Sosnilo
Affiliation:
Laboratory “Modern Algebra and Applications”, St. Petersburg State University, 14th line, 29B, 199178 Saint Petersburg, Russia; E-mail: vsosnilo@gmail.com
Maria Yakerson
Affiliation:
Institute for Mathematical Research (FIM), ETH Zürich, Rämistr. 101, 8092 Zürich, Switzerland; E-mail: maria.yakerson@math.ethz.ch

Abstract

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We prove that the $\infty $-category of $\mathrm{MGL} $-modules over any scheme is equivalent to the $\infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$-loop spaces, we deduce that very effective $\mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers.

Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $ is the $\mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $ is the $\mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.

MSC classification

Primary:14F42: Motivic cohomology; motivic homotopy theory
Secondary:14D23: Stacks and moduli problems

Information

Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Pi , Volume 8 , 2020, e14
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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