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

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

2. If $\kappa (x)/\kappa (s)$ is a purely inseparable1 extension of fields, then number of branches of $X$ at $x$ is equal to the number of branches of $S$ at $s$.

Proof. Follows immediately from More on Algebra, Lemma 15.105.8 and the definitions. $\square$

[1] In fact, it would suffice if $\kappa (x)$ is geometrically irreducible over $\kappa (s)$. If we ever need this we will add a detailed proof.

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