Lemma 32.9.3. Let f : X \to S be a morphism of schemes. Assume:
The morphism f is of locally of finite type.
The scheme X is quasi-compact and quasi-separated, and
The scheme S is quasi-separated.
Then there exists a morphism of finite presentation f' : X' \to S and a closed immersion X \to X' of schemes over S.
Proof.
By Lemma 32.9.1 above there exists a morphism Y \to S of finite presentation and an immersion i : X \to Y of schemes over S. For every point x \in X, there exists an affine open V_ x \subset Y such that i^{-1}(V_ x) \to V_ x is a closed immersion. Since X is quasi-compact we can find finitely may affine opens V_1, \ldots , V_ n \subset Y such that i(X) \subset V_1 \cup \ldots \cup V_ n and i^{-1}(V_ j) \to V_ j is a closed immersion. In other words such that i : X \to X' = V_1 \cup \ldots \cup V_ n is a closed immersion of schemes over S. Since S is quasi-separated and Y is quasi-separated over S we deduce that Y is quasi-separated, see Schemes, Lemma 26.21.12. Hence the open immersion X' = V_1 \cup \ldots \cup V_ n \to Y is quasi-compact. This implies that X' \to Y is of finite presentation, see Morphisms, Lemma 29.21.6. We conclude since then X' \to Y \to S is a composition of morphisms of finite presentation, and hence of finite presentation (see Morphisms, Lemma 29.21.3).
\square
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