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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