Lemma 41.18.2. Let f : X \to S be a morphism of schemes. Let x_1, \ldots , x_ n \in X be points having the same image s in S. Assume f is separated and f is étale at each x_ i. Then there exists an étale neighbourhood (U, u) \to (S, s) and a finite disjoint union decomposition
of schemes such that
V_{i, j} \to U is an isomorphism,
the fibre W_ u contains no point mapping to any x_ i.
In particular, if f^{-1}(\{ s\} ) = \{ x_1, \ldots , x_ n\} , then the fibre W_ u is empty.
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