Last step in proof of comparison theorem of etale and singular cohomology

Question

I am trying to understand the argument in the very last sentence in the proof of Lemma 4.5 in SGA 4 Exposé XI, page 47 here: https://www.normalesup.org/~forgogozo/SGA4/tomes/tome3.pdf

The argument that I am trying to understand is the following (english translation) "since $H^1(X_{ét}, \mathbb{Z}/n\mathbb{Z})\to H^1(X_{cl}, \mathbb{Z}/n\mathbb{Z})$ is bijective, we immediately obtain the vanishing in 4.5 for $q=1$".
Lemma 4.5 basically states the vanishing of $R^q \varepsilon_*\mathbb Z/n \mathbb Z$ for $q>0$, so what we are trying to deduce is $R^1 \varepsilon_*\mathbb Z/n \mathbb Z=0$.

My idea was to use the exact sequence of low degree of the Leray spectral sequence \begin{align*} H^p(X_{ét},R^q\epsilon_{*}\mathbb Z/n \mathbb Z) \Longrightarrow H^{p+q}(X_{cl},\mathbb Z/n \mathbb Z) \end{align*}

induced by the morphism of sites $\epsilon \colon X_{cl} \to X_{ét}$. This yields \begin{align} 0 \to H^1(X_{ét}, \epsilon_*\mathbb Z/n \mathbb Z) \to H^1(X_{cl}, \mathbb Z/n \mathbb Z) \to H^0(X_{ét}, R^1 \varepsilon_*\mathbb Z/n \mathbb Z) \to H^2(X_{ét}, \epsilon_*\mathbb Z/n \mathbb Z) \to H^2(X_{cl}, \mathbb Z/n \mathbb Z) \end{align} Now, knowing that the first map is an isomorphism doesn't seem to be enough to conclude that $H^0(X_{ét},R^1 \varepsilon_*\mathbb Z/n \mathbb Z)$ vanishes, so I am stuck here. Help with this argument, or an alternative argument, would be much appreciated.

Answer 1

I think the intended argument is as follows: Since $H^1(X_{\mathrm{\acute{e}t}},\mathbb Z/n\mathbb Z)\rightarrow H^1(X_{\mathrm{cl}},\mathbb Z/n\mathbb Z)$ is bijective for all $X$ (under the assumptions of Théorème 4.3, as then both sides parametrise $\mathbb Z/n\mathbb Z$-principal étale coverings), it suffices to show:

Lemma. For all $\xi\in H^1(X_{\mathrm{\acute{e}t}},\mathbb Z/n\mathbb Z)$ and all geometric points $x$ of $X$ there exists an étale neighbourhood $X'\rightarrow X$ of $x$ such that $\xi$ vanishes in $H^1(X'_{\mathrm{\acute{e}t}},\mathbb Z/n\mathbb Z)$.

This is a completely general fact about cohomology; see [Stacks Project, Tag 01FW] for a reference. In your situation, this can also be seen geometrically: $\xi$ parametrises a $\mathbb Z/n\mathbb Z$-principal étale covering, and any such covering is étale-locally trivial.