Lemma 26.21.14. Let f : X \to Y and g : Y \to Z be morphisms of schemes. If g \circ f is quasi-compact and g is quasi-separated then f is quasi-compact.
Proof. This is true because f equals the composition (1, f) : X \to X \times _ Z Y \to Y. The first map is quasi-compact by Lemma 26.21.11 because it is a section of the quasi-separated morphism X \times _ Z Y \to X (a base change of g, see Lemma 26.21.12). The second map is quasi-compact as it is the base change of g \circ f, see Lemma 26.19.3. And compositions of quasi-compact morphisms are quasi-compact, see Lemma 26.19.4. \square
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