# Étale endomorphism of $\operatorname{GL}_n$ surjective over an algebraic closure

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!

• 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 $x$. – SashaP Jul 1 '19 at 10:41
• 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). – Jason Starr Jul 1 '19 at 10:42
• 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. – RP_ Jul 1 '19 at 12:12
• @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 :) – Stabilo Jul 1 '19 at 13:09
• It would be a cool idea to assign SGA 7 to students of a linear algebra course. :) – RP_ Jul 1 '19 at 13:59