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I am reading the paper $\mathbb{A}^1$-homotopy theory of schemes by Morel and Voevodsky from 1999. There is a proposition saying that

Let $G$ be a finite étale group scheme over $S$ of order prime to the characteristic of $S$. Then the object $B_\text{ét}G$ in the model category of simplicial étale sheaves is $\mathbb{A}_1$-local. Here $B_\text{ét}G$ is the étale sheafification of $BG$ constructed by applying bar construction sectionwise.

I am confused about the proof. First, we can assume that $S$ is $\operatorname{Spec} R$ where $R$ is a strict henselian local ring. Then to prove that $B_\text{ét}G$ is $\mathbb{A}_1$-local, we need to show that $BG(\mathbb{A}^1_S)$ is weakly equivalence to $*$. Since $\pi_0(BG(\mathbb{A}^1_S))$ is the set of $G$-torsors over $\mathbb{A}^1_S$, we need to show that such torsors are trivial.

Then they said that it is because of the homotopy invariance of the completion of $\pi_1^{ét}$ outside of characteristic, and referred to SGA4 without classifying which part they referred to. I wonder whether there is an explanation?

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    $\begingroup$ What does "characteristic of $G$" mean? $\endgroup$
    – LSpice
    Commented Dec 12, 2021 at 19:02
  • $\begingroup$ You don't have to show that BG(A¹) is weakly equivalent to the point, but that it's weakly equivalent to BG(S) (that is, the nerve of the groupoid with one object and $G$ automorphisms, since S is strictly henselian). $\endgroup$
    – Denis Nardin
    Commented Dec 13, 2021 at 13:38
  • $\begingroup$ @DenisNardin Sure. But in order to show that $BG(\mathbb{A}^1)$ is w.e. to $BG(S)$, we need to show that $\pi_0(BG(\mathbb{A}^1))$ is trivial since so is $BG(S)$ (in étale topology). $\endgroup$
    – XT Chen
    Commented Dec 13, 2021 at 14:18
  • $\begingroup$ @DenisNardin Sorry, there is a typo in my original post. I said we needed to show that $\pi_1(BG(\mathbb{A}^1))$ is trivial, but I should say we need to show that $\pi_0(BG(\mathbb{A}^1))$ is trivial. $\endgroup$
    – XT Chen
    Commented Dec 13, 2021 at 14:22
  • $\begingroup$ A $G$-torsor over $\mathbb{A}^1$ would be an étale $G$-covering of $\mathbb{A}^1$. The homotopy invariance of the étale fundamental group implies that completion away from the characteristic of the fundamental group of $\mathbb{A}^1_S$ (over strictly henselian $S$) is trivial. Therefore, $G$-torsors must be trivial if the order of $G$ is prime to the characteristic. $\endgroup$
    – Matthias Wendt
    Commented Dec 13, 2021 at 18:43

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