Localization and nilpotent spaces in A^1-homotopy theory - Archive ouverte HAL
Article Dans Une Revue Compositio Mathematica Année : 2022

Localization and nilpotent spaces in A^1-homotopy theory

Aravind Asok
  • Fonction : Auteur
Jean Fasel
Michael Hopkins
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-03470522 , version 1 (08-12-2021)

Identifiants

Citer

Aravind Asok, Jean Fasel, Michael Hopkins. Localization and nilpotent spaces in A^1-homotopy theory. Compositio Mathematica, 2022, 158 (3), pp.654-720. ⟨10.1112/S0010437X22007321⟩. ⟨hal-03470522⟩
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