Lemma 37.33.4. Let $X \to S$ be a smooth morphism of schemes. Let $x \in X$ with image $s \in S$. Then

The number of geometric branches of $X$ at $x$ is equal to the number of geometric branches of $S$ at $s$.

If $\kappa (x)/\kappa (s)$ is a purely inseparable

^{1}extension of fields, then number of branches of $X$ at $x$ is equal to the number of branches of $S$ at $s$.

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