The Stacks project

Lemma 34.9.5. Let $T$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with target $T$. Assume that

  1. each $f_ i$ is flat, and

  2. there exists an fpqc covering $\{ g_ j : S_ j \to T\} _{j \in J}$ such that each $\{ S_ j \times _ T T_ i \to S_ j\} _{i \in I}$ is an fpqc covering.

Then $\{ f_ i : T_ i \to T\} _{i \in I}$ is an fpqc covering of $T$.

Proof. We will use Lemma 34.9.2 without further mention. Let $U \subset T$ be an affine open. By (2) we can find quasi-compact opens $V_ j \subset S_ j$ for $j \in J$, almost all empty, such that $U = \bigcup g_ j(V_ j)$. Then for each $j$ we can choose quasi-compact opens $W_{ij} \subset S_ j \times _ T T_ i$ for $i \in I$, almost all empty, with $V_ j = \bigcup _ i \text{pr}_1(W_{ij})$. Thus $\{ S_ j \times _ T T_ i \to T\} $ is an fpqc covering. Since this covering refines $\{ f_ i : T_ i \to T\} $ we conclude by Lemma 34.9.4. $\square$

Comments (2)

Comment #3202 by Dario WeiƟmann on

Typo: 'a fpqc covering'

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  • 8 comment(s) on Section 34.9: The fpqc topology

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