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We survey recent developments related to the problem of classifying vector bundles on algebraic varieties. We focus on the striking analogies between topology and algebraic geometry, and the way in which the Morel–Voevodsky motivic homotopy category can be used to exploit those analogies. 

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.