Lemma 37.23.6. Let $S$ be a scheme. Let $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ be an fppf covering of $S$, see Topologies, Definition 34.7.1. Then there exists an fppf covering $\mathcal{V} = \{ T_ j \to S\} _{j \in J}$ which refines (see Sites, Definition 7.8.1) $\mathcal{U}$ such that each $T_ j \to S$ is locally quasi-finite.

Proof. For every $s \in S$ there exists an $i \in I$ such that $s$ is in the image of $S_ i \to S$. By Lemma 37.23.5 we can find a morphism $g_ s : T_ s \to S$ such that $s \in g_ s(T_ s)$ which is flat, locally of finite presentation and locally quasi-finite and such that $g_ s$ factors through $S_ i \to S$. Hence $\{ T_ s \to S\}$ is the desired covering of $S$ that refines $\mathcal{U}$. $\square$

Comment #417 by on

It's insanely nitpicky, but quasi-finite is written quasi finite in the proof. And $s\in g_s(T)_s$ should be $s\in g_s(T_s)$.

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