The Stacks project

Lemma 34.9.2. 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,

  2. each $f_ i$ is flat and for every affine open $U \subset T$ there exist quasi-compact opens $U_ i \subset T_ i$ which are almost all empty, such that $U = \bigcup f_ i(U_ i)$,

  3. each $f_ i$ is flat and there exists an affine open covering $T = \bigcup _{\alpha \in A} U_\alpha $ and for each $\alpha \in A$ there exist $i_{\alpha , 1}, \ldots , i_{\alpha , n(\alpha )} \in I$ and quasi-compact opens $U_{\alpha , j} \subset T_{i_{\alpha , j}}$ such that $U_\alpha = \bigcup _{j = 1, \ldots , n(\alpha )} f_{i_{\alpha , j}}(U_{\alpha , j})$.

If $T$ is quasi-separated, these are also equivalent to

  1. each $f_ i$ is flat, and for every $t \in T$ there exist $i_1, \ldots , i_ n \in I$ and quasi-compact opens $U_ j \subset T_{i_ j}$ such that $\bigcup _{j = 1, \ldots , n} f_{i_ j}(U_ j)$ is a (not necessarily open) neighbourhood of $t$ in $T$.

Proof. We omit the proof of the equivalence of (1), (2), and (3). From now on assume $T$ is quasi-separated. We prove (4) implies (2). Let $U \subset T$ be an affine open. To prove (2) it suffices to show that for every $t \in U$ there exist finitely many quasi-compact opens $U_ j \subset T_{i_ j}$ such that $f_{i_ j}(U_ j) \subset U$ and such that $\bigcup f_{i_ j}(U_ j)$ is a neighbourhood of $t$ in $U$. By assumption there do exist finitely many quasi-compact opens $U'_ j \subset T_{i_ j}$ such that such that $\bigcup f_{i_ j}(U'_ j)$ is a neighbourhood of $t$ in $T$. Since $T$ is quasi-separated we see that $U_ j = U'_ j \cap f_ j^{-1}(U)$ is quasi-compact open as desired. Since it is clear that (2) implies (4) the proof is finished. $\square$

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