The Stacks project

Lemma 37.47.6. Let $S$ be a quasi-compact and quasi-separated scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exists a surjective finite morphism $S' \to S$ of finite presentation and an open covering $S' = \bigcup U'_\alpha $ such that for each $\alpha $ the morphism $U'_\alpha \to S$ factors through $S_ i \to S$ for some $i$.

Proof. Let $Y \to X$ be the integral surjective morphism found in Lemma 37.47.5. Choose a finite affine open covering $Y = \bigcup V_ j$ such that $V_ j \to X$ factors through $S_{i(j)}$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_\lambda $ with $Y_\lambda \to X$ finite and of finite presentation, see Limits, Lemma 32.7.3. For large enough $\lambda $ we can find affine opens $V_{\lambda , j} \subset Y_\lambda $ whose inverse image in $Y$ recovers $V_ j$, see Limits, Lemma 32.4.11. For even larger $\lambda $ the morphisms $V_ j \to S_{i(j)}$ over $X$ come from morphisms $V_{\lambda , j} \to S_{i(j)}$ over $X$, see Limits, Proposition 32.6.1. Setting $S' = Y_\lambda $ for this $\lambda $ finishes the proof. $\square$

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