Localization and nilpotent spaces in A^1-homotopy theory

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.