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The Stacks project

Lemma 34.9.6. 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 34.9.5. \square


Comments (2)

Comment #3202 by Dario Weißmann on

Typo: 'a fpqc covering'

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

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