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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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