分类空间的$\mathbb{A}^1$ -连通分支与torsor的纯性
$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors
摘要：
本文研究了约化群$G$的$G$ -扭子的同构类集与光滑概形相联系的预层的Nisnevich层化$\mathcal{H}^1_{\acute{e}t}(G)$.我们表明，如果$G$ -torsors仿射直线的延伸，那么$\mathcal{H}^1_{\acute{e}t}(G)$是同伦不变量的，并表明，层是unramified当且仅当Nisnevich局部纯度持有$G$ -torsors。我们还确定了层$\mathcal{H}^1_{\acute{e}t}(G)$与层的$\mathbb{A}^1$ -连接组件的分类空间${\rm B}_{\acute{e}t}G$。这建立了同伦不变性的层的组成部分，如所示的莫雷尔。此外，它提供了一个计算层的$\mathbb{A}^1$连接组件的unramified $G$ -torsors在功能领域时，尼斯涅维奇-本地纯度持有$G$ -torsors。
英文摘要：
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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