I am currently reading chapter 1, exposé XXII of SGA7 and I am stuck at the following argument, left without explanation. It can be formulated like this:

Let $k$ be a separably closed field and $\bar{k}$ be an algebraic closure of it. Let $\varphi$ be an étale endomorphism of the variety $\operatorname{GL}_n/k$. If $\varphi_{\bar{k}}:\operatorname{GL}_n(\bar{k})\to \operatorname{GL}_n(\bar{k})$ is surjective, then $\varphi_{k}:\operatorname{GL}_n(k)\to \operatorname{GL}_n(k)$ is surjective.

I should say that I am not so confortable with algebraic geometry, but I am interested in a gentle explanation of this fact.

Many thanks!

1more comment