Lemma 37.41.5. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots , x_ n \in X_ s$. Assume that

1. $f$ is locally of finite type,

2. $f$ is separated, and

3. $x_1, \ldots , x_ n$ are pairwise distinct isolated points of $X_ s$.

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

$U \times _ S X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j}$

into open and closed subschemes such that the morphisms $V_{i, j} \to U$ are finite, the fibres of $V_{i, j} \to U$ over $u$ are singletons $\{ v_{i, j}\}$, each $v_{i, j}$ maps to $x_ i$, $\kappa (u) \subset \kappa (v_{i, j})$ is purely inseparable, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_ i$.

Proof. This is proved in exactly the same way as the proof of Lemma 37.41.4 except that it uses Lemma 37.41.3 instead of Lemma 37.41.2. $\square$

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