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

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. [ABSTRACT FROM AUTHOR]