The Stacks project

Lemma 3.9.12. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fppf covering of a scheme. There exists an fppf covering $\{ W_ j \to X\} _{j \in J}$ which is a refinement of $\{ X_ i \to X\} _{i \in I}$ such that $\text{size}(\coprod W_ j) \leq \text{size}(X)$.

Proof. Choose an affine open covering $X = \bigcup _{a \in A} U_ a$ with $|A| \leq \text{size}(X)$. For each $a$ we can choose a finite subset $I_ a \subset I$ and for $i \in I_ a$ a quasi-compact open $W_{a, i} \subset X_ i$ such that $U_ a = \bigcup _{i \in I_ a} f_ i(W_{a, i})$. Then $\text{size}(W_{a, i}) \leq \text{size}(X)$ by Lemma 3.9.7. We conclude that $\text{size}(\coprod _ a \coprod _{i \in I_ a} W_{i, a}) \leq \text{size}(X)$ by Lemma 3.9.5. $\square$

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