The Stacks project

Lemma 58.32.3. Let $k$ be a field of characteristic $p > 0$. Let $Z \subset \mathbf{A}^ n_ k$ be a closed subscheme. Let $Y \to Z$ be finite étale. There exists a finite étale morphism $f : U \to \mathbf{A}^ n_ k$ such that there is an open and closed immersion $Y \to f^{-1}(Z)$ over $Z$.

Proof. Let us turn the problem into algebra. Write $\mathbf{A}^ n_ k = \mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n])$. Then $Z = \mathop{\mathrm{Spec}}(A)$ where $A = k[x_1, \ldots , x_ n]/I$ for some ideal $I \subset k[x_1, \ldots , x_ n]$. Write $Y = \mathop{\mathrm{Spec}}(B)$ so that $Y \to Z$ corresponds to the finite étale $k$-algebra map $A \to B$.

By Algebra, Lemma 10.143.10 there exists an étale ring map

\[ \varphi : k[x_1, \ldots , x_ n] \to C \]

and a surjective $A$-algebra map $\tau : C/\varphi (I)C \to B$. (We can even choose $C, \varphi , \tau $ such that $\tau $ is an isomorphism, but we won't use this). 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.

Let $f : U = \mathop{\mathrm{Spec}}(C) \to \mathbf{A}^ n_ k$ be the finite étale morphism corresponding to $\varphi $. The morphism $Y \to f^{-1}(Z) = \mathop{\mathrm{Spec}}(C/\varphi (I)C)$ induced by $\tau $ is a closed immersion as $\tau $ is surjective and open as it is an étale morphism by Morphisms, Lemma 29.36.18. This finishes the proof. $\square$

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