Lemma 37.48.6. 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 37.48.5. 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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