Lemma 37.36.3. 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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