The Stacks project

Lemma 35.32.13. 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 35.10.6 to find an open $V_{ij} \subset V_ j$ such that $\alpha _ i^{-1}(W_ i \cap W_ j) = X \times _ S V_{ij}$. 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 35.32.12). 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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