The Stacks project

Lemma 80.10.11. In Situation 80.10.6 there exists an fpqc covering $\{ X_ i \to X\} _{i \in I}$ refining the family $\{ U \to X, Y \to X\} $.

Proof. For the definition and general properties of fpqc coverings we refer to Topologies, Section 34.9. In particular, we can first choose an étale covering $\{ X_ i \to X\} $ with $X_ i$ affine and by base changing $Y$, $Z$, and $U$ to each $X_ i$ we reduce to the case where $X$ is affine. In this case $U$ is quasi-compact and hence a finite union $U = U_1 \cup \ldots \cup U_ n$ of affine opens. Then $Z$ is quasi-compact hence also $f^{-1}Z$ is quasi-compact. Thus we can choose an affine scheme $W$ and an étale morphism $h : W \to Y$ such that $h^{-1}f^{-1}Z \to f^{-1}Z$ is surjective. Say $W = \mathop{\mathrm{Spec}}(B)$ and $h^{-1}f^{-1}Z = V(J)$ where $J \subset B$ is an ideal of finite type. By Pro-étale Cohomology, Lemma 61.5.1 there exists a localization $B \to B'$ such that points of $\mathop{\mathrm{Spec}}(B')$ correspond exactly to points of $W = \mathop{\mathrm{Spec}}(B)$ specializing to $h^{-1}f^{-1}Z = V(J)$. It follows that the composition $\mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(B) = W \to Y \to X$ is flat as by assumption $f : Y \to X$ is flat at all the points of $f^{-1}Z$. Then $\{ \mathop{\mathrm{Spec}}(B') \to X, U_1 \to X, \ldots , U_ n \to X\} $ is an fpqc covering by Topologies, Lemma 34.9.2. $\square$


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