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!

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    $\begingroup$ For an etale morphism $f:X\to Y$ the schematic fiber over a $k$-point $x:Spec\, k\to Y$ is the spectrum of an etale algebra over $k$, that is, of a direct sum of separable extensions of $k$ which all must be equal to $k$ by the assumption that $k$ is separably closed. The base change of this algebra to $\bar{k}$ is non-zero by the surjectivity assumption, hence the algebra itself is non-zero and isomorphic to $k^{\oplus d},d>0$ thus giving $d$ $k$-points of $X$ that are mapped to $x$. $\endgroup$ – SashaP Jul 1 '19 at 10:41
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    $\begingroup$ The fiber of $\phi$ over a $k$-point is Spec of an etale $k$-algebra. Since $k$ is separably closed, that $k$-algebra is a product of copies of $k$. Since the base change to $\overline{k}$ is nonempty, that product is not an empty product (the $k$-algebra is nonzero). $\endgroup$ – Jason Starr Jul 1 '19 at 10:42
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    $\begingroup$ You are not comfortable with algebraic geometry, yet are reading SGA 7? That seems like... not an easy thing to do. I'm upvoting for sheer courage. $\endgroup$ – RP_ Jul 1 '19 at 12:12
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    $\begingroup$ @RP_ Surprisingly enough, this chapter one of the last exposé of SGA is about linear algebra! I was just missing some properties of étale morphism to understand it fully. Thanks for your support :) $\endgroup$ – Stabilo Jul 1 '19 at 13:09
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    $\begingroup$ It would be a cool idea to assign SGA 7 to students of a linear algebra course. :) $\endgroup$ – RP_ Jul 1 '19 at 13:59

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