The Stacks project

Lemma 29.42.14. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms.

  1. If $g \circ f$ is finite and $g$ separated then $f$ is finite.

  2. If $g \circ f$ is integral and $g$ separated then $f$ is integral.

Proof. Assume $g \circ f$ is finite (resp. integral) and $g$ separated. The base change $X \times _ Z Y \to Y$ is finite (resp. integral) by Lemma 29.42.6. The morphism $X \to X \times _ Z Y$ is a closed immersion as $Y \to Z$ is separated, see Schemes, Lemma 26.21.11. A closed immersion is finite (resp. integral), see Lemma 29.42.12. The composition of finite (resp. integral) morphisms is finite (resp. integral), see Lemma 29.42.5. Thus we win. $\square$


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