The Stacks project

Lemma 74.7.1. Let $S$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering of algebraic spaces over $S$. Suppose that for each $i$ we have an open subspace $W_ i \subset T_ i$ such that for all $i, j \in I$ we have $\text{pr}_0^{-1}(W_ i) = \text{pr}_1^{-1}(W_ j)$ as open subspaces of $T_ i \times _ T T_ j$. Then there exists a unique open subspace $W \subset T$ such that $W_ i = f_ i^{-1}(W)$ for each $i$.

Proof. By Topologies on Spaces, Lemma 73.9.5 we may assume each $T_ i$ is a scheme. Choose a scheme $U$ and a surjective ├ętale morphism $U \to T$. Then $\{ T_ i \times _ T U \to U\} $ is an fpqc covering of $U$ and $T_ i \times _ T U$ is a scheme for each $i$. Hence we see that the collection of opens $W_ i \times _ T U$ comes from a unique open subscheme $W' \subset U$ by Descent, Lemma 35.13.6. As $U \to X$ is open we can define $W \subset X$ the Zariski open which is the image of $W'$, see Properties of Spaces, Section 66.4. We omit the verification that this works, i.e., that $W_ i$ is the inverse image of $W$ for each $i$. $\square$

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