The Stacks project

Lemma 35.10.6. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a fpqc covering. Suppose that for each $i$ we have an open subset $W_ i \subset T_ i$ such that for all $i, j \in I$ we have $\text{pr}_0^{-1}(W_ i) = \text{pr}_1^{-1}(W_ j)$ as open subsets of $T_ i \times _ T T_ j$. Then there exists a unique open subset $W \subset T$ such that $W_ i = f_ i^{-1}(W)$ for each $i$.

Proof. Apply Lemma 35.10.1 to the map $\coprod _{i \in I} T_ i \to T$. It implies there exists a subset $W \subset T$ such that $W_ i = f_ i^{-1}(W)$ for each $i$, namely $W = \bigcup f_ i(W_ i)$. To see that $W$ is open we may work Zariski locally on $T$. Hence we may assume that $T$ is affine. Using the definition of a fpqc covering, this reduces us to the case where $\{ f_ i : T_ i \to T\} $ is a standard fpqc covering. In this case we may apply Morphisms, Lemma 29.25.12 to the morphism $\coprod T_ i \to T$ to conclude that $W$ is open. $\square$


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