Localization and nilpotent spaces in -homotopy theory

Abstract

AbstractFor 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.