Lemma 89.6.3 (0BHR)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 89: Resolution of Surfaces Revisited
Section 89.6: Base change to the completion
Lemma 89.6.3 (cite)

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Lemma 89.6.3. Let $(A, \mathfrak m)$ be a local Noetherian ring. Let $X$ be an algebraic space over $A$ . Assume

$A$ is analytically unramified (Algebra, Definition 10.162.9),
$X$ is locally of finite type over $A$ ,
$X \to \mathop{\mathrm{Spec}}(A)$ is étale at every point of codimension $0$ in $X$ .

Then the normalization of $X$ is finite over $X$ .

Proof. Choose a scheme $U$ and a surjective étale morphism $U \to X$ . Then $U \to \mathop{\mathrm{Spec}}(A)$ satisfies the assumptions and hence the conclusions of Resolution of Surfaces, Lemma 54.11.5. $\square$

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