The Stacks project

Lemma 37.48.2. Let $S$ be a scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exist

  1. a Zariski open covering $S = \bigcup U_ j$,

  2. surjective finite locally free morphisms $W_ j \to U_ j$,

  3. Zariski open coverings $W_ j = \bigcup _ k W_{j, k}$,

  4. surjective finite locally free morphisms $T_{j, k} \to W_{j, k}$

such that the fppf covering $\{ T_{j, k} \to S\} $ refines the given covering $\{ S_ i \to S\} $.

Proof. Let $\{ V_ a \to S\} _{a \in A}$ be the fppf covering found in Lemma 37.48.1. In other words, this covering refines $\{ S_ i \to S\} $ and each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ ├ętale and $V_ a \to S'_ a$ surjective finite locally free.

By Remark 37.40.3 there exists a Zariski open covering $S = \bigcup U_ j$, for each $j$ a finite locally free, surjective morphism $W_ j \to U_ j$, and for each $j$ a Zariski open covering $\{ W_{j, k} \to W_ j\} $ such that the family $\{ W_{j, k} \to S\} $ refines the ├ętale covering $\{ S'_ a \to S\} $, i.e., for each pair $j, k$ there exists an $a(j, k)$ and a factorization $W_{j, k} \to S'_ a \to S$ of the morphism $W_{j, k} \to S$. Set $T_{j, k} = W_{j, k} \times _{S'_ a} V_ a$ and everything is clear. $\square$

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