Lemma 37.37.1. 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 37.37.0.1)

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.

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