Lemma 35.13.1 (02KI)—The Stacks project

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Part 2: Schemes
Chapter 35: Descent
Section 35.13: Fpqc coverings are universal effective epimorphisms
Lemma 35.13.1 (cite)

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Lemma 35.13.1. For a scheme $X$ denote $|X|$ the underlying set. Let $f : X \to S$ be a morphism of schemes. Then

\[ |X \times _ S X| \to |X| \times _{|S|} |X| \]

is surjective.

Proof. Follows immediately from the description of points on the fibre product in Schemes, Lemma 26.17.5. $\square$

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