\mathbbA¹-connected components of classifying spaces and purity for torsors

2021/04/13 by Elmanto, Elden, Kulkarni, Girish, Wendt, Matthias
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.06273

Abstract

In this paper, we study the Nisnevich sheafification H¹_\acuteet(G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G-torsors, for a reductive group G. We show that if G-torsors on affine lines are extended, then H¹_\acuteet(G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G-torsors. We also identify the sheaf H¹_\acuteet(G) with the sheaf of \mathbbA¹-connected components of the classifying space \rm B_\acuteetG. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of \mathbbA¹-connected components in terms of unramified G-torsors over function fields whenever Nisnevich-local purity holds for G-torsors.

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