$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors

Elden Elmanto; Girish Kulkarni; Matthias Wendt

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arXiv · 2104.06273

$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors

Elden Elmanto·Girish Kulkarni·Matthias Wendt

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Abstract

In this paper, we study the Nisnevich sheafification $\mathcal{H}^1_{\acute{e}t}(G)$ 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 $\mathcal{H}^1_{\acute{e}t}(G)$ 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 $\mathcal{H}^1_{\acute{e}t}(G)$ with the sheaf of $\mathbb{A}^1$-connected components of the classifying space ${\rm B}_{\acute{e}t}G$. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of $\mathbb{A}^1$-connected components in terms of unramified $G$-torsors over function fields whenever Nisnevich-local purity holds for $G$-torsors.

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BibTeX RIS

Elden Elmanto, Girish Kulkarni, Matthias Wendt. 2021-04-13. $\mathbb{A}^1$-connected components of classifying spaces and purity for torsors. https://arxiv.org/abs/2104.06273

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