The Stacks project

Lemma 34.9.12. 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. every affine scheme $Z$ and morphism $h : Z \to T$ there exists a standard fpqc covering $\{ Z_ j \to Z\} _{j = 1, \ldots , n}$ which refines the family $\{ T_ i \times _ T Z \to Z\} _{i \in I}$.

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

Proof. Let $T = \bigcup U_\alpha $ be an affine open covering. For each $\alpha $ the pullback family $\{ T_ i \times _ T U_\alpha \to U_\alpha \} $ can be refined by a standard fpqc covering, hence is an fpqc covering by Lemma 34.9.5. As $\{ U_\alpha \to T\} $ is an fpqc covering we conclude that $\{ T_ i \to T\} $ is an fpqc covering by Lemma 34.9.6. $\square$

Comments (2)

Comment #3203 by Dario Weißmann on

Typo: 'a fpqc covering'

There are also:

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

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