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The Stacks project

Lemma 59.43.2. Let f : X \to Y be a morphism of schemes. Suppose

  1. V \to Y is an étale morphism of schemes,

  2. \{ U_ i \to X \times _ Y V\} is an étale covering, and

  3. v \in V is a point.

Assume that for any such data there exists an étale neighbourhood (V', v') \to (V, v), a disjoint union decomposition X \times _ Y V' = \coprod W'_ i, and morphisms W'_ i \to U_ i over X \times _ Y V. Then property (B) holds.

Proof. Omitted. \square

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