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Localization and nilpotent spaces in ${\mathbb {A}}^1$-homotopy theory

Part of: Homotopy theory Linear algebraic groups and related topics (Co)homology theory Higher algebraic $K$-theory

Published online by Cambridge University Press:  27 May 2022

Aravind Asok ,
Jean Fasel  and
Michael J. Hopkins
Aravind Asok
Affiliation:
Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, USA asok@usc.edu
Jean Fasel
Affiliation:
Institut Fourier - UMR 5582, Université Grenoble Alpes, 100, rue des Mathématiques, F-38402 Saint Martin d'Hères, France jean.fasel@gmail.com
Michael J. Hopkins
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA mjh@math.harvard.edu
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Abstract

For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb {A}}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${\mathbb {A}}^1$-homotopy theory, paying attention to future applications for vector bundles. We show that $R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${\mathbb {A}}^1$-nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${\mathbb {A}}^1$-homotopy theory: if $-1$ is a sum of squares in the base field, ${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.

Keywords

motivic homotopynilpotent spaces

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory 19D45: Higher symbols, Milnor $K$-theory 20G15: Linear algebraic groups over arbitrary fields 55P60: Localization and completion
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 3 , March 2022 , pp. 654 - 720
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

AA was partially supported by National Science Foundation Awards DMS-1254892 and DMS-1802060. MJH was partially supported by National Science Foundation Awards DMS-0906194, DMS-1510417 and DMS-1810917.

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