Lemma 37.35.3. Let $X \to S$ be a morphism of schemes and $x \in X$ a point with image $s$. Then

the number of branches of the fibre $X_ s$ at $x$ is equal to the supremum of the number of irreducible components of the fibre $U_ s$ passing through $u$ taken over elementary étale neighbourhoods $(U, u) \to (X, x)$,

the number of geometric branches of the fibre $X_ s$ at $x$ is equal to the supremum of the number of irreducible components of the fibre $U_ s$ passing through $u$ taken over étale neighbourhoods $(U, u) \to (X, x)$,

the fibre $X_ s$ is unibranch at $x$ if and only if for every elementary étale neighbourhood $(U, u) \to (X, x)$ there is exactly one irreducible component of the fibre $U_ s$ passing through $u$, and

$X$ is geometrically unibranch at $x$ if and only if for every étale neighbourhood $(U, u) \to (X, x)$ there is exactly one irreducible component of $U_ s$ passing through $u$.

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