The Stacks project

Lemma 32.9.3. Let $f : X \to S$ be a morphism of schemes. Assume:

  1. The morphism $f$ is of locally of finite type.

  2. The scheme $X$ is quasi-compact and quasi-separated, and

  3. 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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