Lemma 26.21.11 (01KT)—The Stacks project

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Part 2: Schemes
Chapter 26: Schemes
Section 26.21: Separation axioms
Lemma 26.21.11 (cite)

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Lemma 26.21.11. Let $f : X \to S$ be a morphism of schemes. Let $s : S \to X$ be a section of $f$ (in a formula $f \circ s = \text{id}_ S$ ). Then $s$ is an immersion. If $f$ is separated then $s$ is a closed immersion. If $f$ is quasi-separated, then $s$ is quasi-compact.

Proof. This is a special case of Lemma 26.21.10 applied to $g =s$ so the morphism $i = s : S \to S \times _ S X$ . $\square$

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