The Stacks project

Lemma 36.36.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume that

  1. $f$ is locally of finite type, and

  2. $x \in X_ s$ is isolated1.

Then there exist

  1. an elementary étale neighbourhood $(U, u) \to (S, s)$,

  2. an open subscheme $V \subset X_ U$ (see 36.36.0.1)

such that

  1. $V \to U$ is a finite morphism,

  2. there is a unique point $v$ of $V$ mapping to $u$ in $U$, and

  3. 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 10.142.21. $\square$

[1] In the presence of (1) this means that $f$ is quasi-finite at $x$, see Morphisms, Lemma 28.19.6.

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