Lemma 68.11.4 (0BBP)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 68: Decent Algebraic Spaces
Section 68.11: Residue fields and henselian local rings
Lemma 68.11.4 (cite)

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Lemma 68.11.4. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ . For every point $x \in |X|$ there exists an étale morphism

$(U, u) \longrightarrow (X, x)$

where $U$ is an affine scheme, $u$ is the only point of $U$ lying over $x$ , and the induced homomorphism $\kappa (x) \to \kappa (u)$ is an isomorphism.

Proof. We may assume that $X$ is quasi-compact by replacing $X$ with a quasi-compact open containing $x$ . Recall that $x$ can be represented by a quasi-compact (mono)morphism from the spectrum a field (by definition of decent spaces). Thus the lemma follows from Lemma 68.8.3. $\square$

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