The Stacks project

Lemma 37.48.1. Let $S$ be a scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exist

  1. an étale covering $\{ S'_ a \to S\} $,

  2. surjective finite locally free morphisms $V_ a \to S'_ a$,

such that the fppf covering $\{ V_ a \to S\} $ refines the given covering $\{ S_ i \to S\} $.

Proof. We may assume that each $S_ i \to S$ is locally quasi-finite, see Lemma 37.23.6.

Fix a point $s \in S$. Pick an $i \in I$ and a point $s_ i \in S_ i$ mapping to $s$. Choose an elementary étale neighbourhood $(S', s) \to (S, s)$ such that there exists an open

\[ S_ i \times _ S S' \supset V \]

which contains a unique point $v \in V$ mapping to $s \in S'$ and such that $V \to S'$ is finite, see Lemma 37.41.1. Then $V \to S'$ is finite locally free, because it is finite and because $S_ i \times _ S S' \to S'$ is flat and locally of finite presentation as a base change of the morphism $S_ i \to S$, see Morphisms, Lemmas 29.21.4, 29.25.8, and 29.48.2. Hence $V \to S'$ is open, and after shrinking $S'$ we may assume that $V \to S'$ is surjective finite locally free. Since we can do this for every point of $S$ we conclude that $\{ S_ i \to S\} $ can be refined by a covering of the form $\{ V_ a \to S\} _{a \in A}$ where each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ étale and $V_ a \to S'_ a$ surjective finite locally free. $\square$

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