Lemma 37.36.4 (0DQ2)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.36: Étale neighbourhoods and branches
Lemma 37.36.4 (cite)

previous lemma

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Lemma 37.36.4. 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 $s$ .

Proof. Follows immediately from More on Algebra, Lemma 15.106.8 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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