1 results
Localization and nilpotent spaces in
${\mathbb {A}}^1$-homotopy theory
- Part of
-
- Journal:
- Compositio Mathematica / Volume 158 / Issue 3 / March 2022
- Published online by Cambridge University Press:
- 27 May 2022, pp. 654-720
- Print publication:
- March 2022
-
- Article
- Export citation
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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.
