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

TL;DR

This paper investigates the properties of sheaves of $G$-torsors in algebraic geometry, establishing their homotopy invariance and unramified nature under certain conditions, and relates them to $A^1$-connected components of classifying spaces.

Contribution

It proves homotopy invariance of the sheaf of $G$-torsors and links it to $A^1$-connected components, confirming a conjecture by Morel and providing explicit computations.

Findings

Sheaf $^1_{ ext{ét}}(G)$ is homotopy invariant under torsor extension conditions.

Unramified sheaf property is equivalent to Nisnevich-local purity for $G$-torsors.

Identification of sheaf with $A^1$-connected components of classifying space.

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.