Lemma 34.9.12 (03LB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.9: The fpqc topology
Lemma 34.9.12 (cite)

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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

each $f_ i$ is flat, and
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 February 19, 2018 at 15:32

Typo: 'a fpqc covering'

Comment #3307 by Johan on May 17, 2018 at 16:57

Fixed here.

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