Lemma 37.41.1 (02LK)—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 37.41.1 (cite)

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Lemma 37.41.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ . Set $s = f(x)$ . Assume that

$f$ is locally of finite type, and
$x \in X_ s$ is isolated ¹.

Then there exist

an elementary étale neighbourhood $(U, u) \to (S, s)$ ,
an open subscheme $V \subset X_ U$ (see 37.41.0.1)

such that

$V \to U$ is a finite morphism,
there is a unique point $v$ of $V$ mapping to $u$ in $U$ , and
the point $v$ maps to $x$ under the morphism $X_ U \to X$ , inducing $\kappa (x) = \kappa (v)$ .

Moreover, for any elementary étale neighbourhood $(U', u') \to (U, u)$ setting $V' = U' \times _ U V \subset X_{U'}$ the triple $(U', u', V')$ satisfies the properties (i), (ii), and (iii) as well.

Proof. Let $Y \subset X$ , $W \subset S$ be affine opens such that $f(Y) \subset W$ and such that $x \in Y$ . Note that $x$ is also an isolated point of the fibre of the morphism $f|_ Y : Y \to W$ . If we can prove the theorem for $f|_ Y : Y \to W$ , then the theorem follows for $f$ . Hence we reduce to the case where $f$ is a morphism of affine schemes. This case is Algebra, Lemma 10.145.2. $\square$

[1] In the presence of (1) this means that

$f$ is quasi-finite at

$x$ , see Morphisms, Lemma 29.20.6.

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