Lemma 76.33.2 (0ADU)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.33: Étale localization of morphisms
Lemma 76.33.2 (cite)

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Lemma 76.33.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ with image $y \in |Y|$. Assume that

$f$ is locally of finite type,
$f$ is separated, and
$f$ is quasi-finite at $x$.

Then there exists an étale morphism $(U, u) \to (Y, y)$ of pointed algebraic spaces and a decomposition

\[ U \times _ Y X = W \amalg V \]

into open and closed subspaces such that the morphism $V \to U$ is finite and there exists a point $v \in |V|$ which maps to $x$ in $|X|$ and $u$ in $|U|$.

Proof. Pick a scheme $U$, a point $u \in U$, and an étale morphism $U \to Y$ mapping $u$ to $y$. There exists a point $x' \in |U \times _ Y X|$ mapping to $x$ in $|X|$ and $u$ in $|U|$ (Properties of Spaces, Lemma 66.4.3). To finish, apply Lemma 76.33.1 to the morphism $U \times _ Y X \to U$ and the point $x'$. It applies because $U$ is a scheme and hence $u$ comes from the monomorphism $\mathop{\mathrm{Spec}}(\kappa (u)) \to U$. $\square$

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