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mathbb{A}^1$ -connected components of classifying spaces and purity for torsors
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mathbb{A}^1$ -connected components of classifying spaces and purity for torsors
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Journal Article
mathbb{A}^1$ -connected components of classifying spaces and purity for torsors
2022
Overview
In this paper, we study the Nisnevich sheafification H^1_ét(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 H^1_ét(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 H^1_ét(G) with the sheaf of A^1 -connected components of the classifying space B_étG . This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of A^1 -connected components in terms of unramified G -torsors over function fields whenever Nisnevich-local purity holds for G -torsors.
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