The Stacks project

Lemma 34.9.3. Let $T$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent

  1. $\{ f_ i : T_ i \to T\} _{i \in I}$ is an fpqc covering, and

  2. setting $T' = \coprod _{i \in I} T_ i$, and $f = \coprod _{i \in I} f_ i$ the family $\{ f : T' \to T\} $ is an fpqc covering.

Proof. Suppose that $U \subset T$ is an affine open. If (1) holds, then we find $i_1, \ldots , i_ n \in I$ and affine opens $U_ j \subset T_{i_ j}$ such that $U = \bigcup _{j = 1, \ldots , n} f_{i_ j}(U_ j)$. Then $U_1 \amalg \ldots \amalg U_ n \subset T'$ is a quasi-compact open surjecting onto $U$. Thus $\{ f : T' \to T\} $ is an fpqc covering by Lemma 34.9.2. Conversely, if (2) holds then there exists a quasi-compact open $U' \subset T'$ with $U = f(U')$. Then $U_ j = U' \cap T_ j$ is quasi-compact open in $T_ j$ and empty for almost all $j$. By Lemma 34.9.2 we see that (1) holds. $\square$

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