Lemma 35.35.13. Let X \to S be a surjective, quasi-compact, flat morphism of schemes. Let (V, \varphi ) be a descent datum relative to X/S. Suppose that for all v \in V there exists an open subscheme v \in W \subset V such that \varphi (W \times _ S X) \subset X \times _ S W and such that the descent datum (W, \varphi |_{W \times _ S X}) is effective. Then (V, \varphi ) is effective.
Proof. Let V = \bigcup W_ i be an open covering with \varphi (W_ i \times _ S X) \subset X \times _ S W_ i and such that the descent datum (W_ i, \varphi |_{W_ i \times _ S X}) is effective. Let U_ i \to S be a scheme and let \alpha _ i : (X \times _ S U_ i, can) \to (W_ i, \varphi |_{W_ i \times _ S X}) be an isomorphism of descent data. For each pair of indices (i, j) consider the open \alpha _ i^{-1}(W_ i \cap W_ j) \subset X \times _ S U_ i. Because everything is compatible with descent data and since \{ X \to S\} is an fpqc covering, we may apply Lemma 35.13.6 to find an open U_{ij} \subset U_ j such that \alpha _ i^{-1}(W_ i \cap W_ j) = X \times _ S U_{ij}. Now the identity morphism on W_ i \cap W_ j is compatible with descent data, hence comes from a unique morphism \varphi _{ij} : U_{ij} \to U_{ji} over S (see Remark 35.35.12). Then (U_ i, U_{ij}, \varphi _{ij}) is a glueing data as in Schemes, Section 26.14 (proof omitted). Thus we may assume there is a scheme U over S such that U_ i \subset U is open, U_{ij} = U_ i \cap U_ j and \varphi _{ij} = \text{id}_{U_ i \cap U_ j}, see Schemes, Lemma 26.14.1. Pulling back to X we can use the \alpha _ i to get the desired isomorphism \alpha : X \times _ S U \to V. \square
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