Lemma 37.41.6 (02LP)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.41: Étale localization of quasi-finite morphisms
Lemma 37.41.6 (cite)

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Lemma 37.41.6. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$ . Assume that

$f$ is locally of finite type,
$f$ is separated, and
$X_ s$ has at most finitely many isolated points.

Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

$U \times _ S X = W \amalg V$

into open and closed subschemes such that the morphism $V \to U$ is finite, and the fibre $W_ u$ of the morphism $W \to U$ contains no isolated points. In particular, if $f^{-1}(s)$ is a finite set, then $W_ u = \emptyset$ .

Proof. This is clear from Lemma 37.41.4 by choosing $x_1, \ldots , x_ n$ the complete set of isolated points of $X_ s$ and setting $V = \bigcup V_ i$ . $\square$

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