Lemma 35.13.3 (0EUB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.13: Fpqc coverings are universal effective epimorphisms
Lemma 35.13.3 (cite)

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Lemma 35.13.3. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a family of morphisms of schemes.

If the family is universal effective epimorphism in the category of schemes, then $\coprod f_ i$ is surjective.
If $X$ and $X_ i$ are affine and the family is a universal effective epimorphism in the category of affine schemes, then $\coprod f_ i$ is surjective.

Proof. Omitted. Hint: perform base change by $\mathop{\mathrm{Spec}}(\kappa (x)) \to X$ to see that any $x \in X$ has to be in the image. $\square$

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