Lemma 58.32.2 (0G1G)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.32: Tricks in positive characteristic
Lemma 58.32.2 (cite)

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Lemma 58.32.2. Let $k$ be a field of characteristic $p > 0$ . Let $X \to \mathbf{A}^ n_ k$ be an étale morphism with $X$ affine. Then there exists a finite étale morphism $X \to \mathbf{A}^ n_ k$ .

Proof. Write $X = \mathop{\mathrm{Spec}}(C)$ . Set $A = 0$ and denote $I = k[x_1, \ldots , x_ n]$ . By assumption there exists some étale $k$ -algebra map $\varphi : k[x_1, \ldots , x_ n] \to C$ . Denote $\tau : C/\varphi (I)C \to 0$ the unique surjection. We may choose $\varphi$ and $\tau$ such that $N(C, \varphi , \tau )$ is minimal. By Lemma 58.32.1 we get $N(C, \varphi , \tau ) = 0$ . Hence $\varphi$ is finite étale. $\square$

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