Localization and nilpotent spaces in A¹-homotopy theory

2019/09/11 by Asok, Aravind, Fasel, Jean, Hopkins, Michael J. · 5 citations
#14F42 19D45 20G15 55P60 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1909.05185

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 A¹-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in \mathbb A¹-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 A¹-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 A¹-homotopy theory: if -1 is a sum of squares in the base field, \mathbb Aⁿ ∖ 0 is rationally equivalent to a suitable motivic Eilenberg--Mac Lane space, and the special linear group decomposes as a product of motivic spheres.

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• Rational and p-local Motivic Homotopy Theory
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• Homotopy idempotent functors on classifying spaces
• Equivariant motivic homotopy theory
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