Lemma 69.11.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume

1. $f$ is locally of finite type and quasi-affine, and

2. $Y$ is quasi-compact and quasi-separated.

Then there exists a morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ over $Y$.

Proof. By Morphisms of Spaces, Lemma 66.21.6 we can find a factorization $X \to Z \to Y$ where $X \to Z$ is a quasi-compact open immersion and $Z \to Y$ is affine. Write $Z = \mathop{\mathrm{lim}}\nolimits Z_ i$ with $Z_ i$ affine and of finite presentation over $Y$ (Lemma 69.11.1). For some $0 \in I$ we can find a quasi-compact open $U_0 \subset Z_0$ such that $X$ is isomorphic to the inverse image of $U_0$ in $Z$ (Lemma 69.5.7). Let $U_ i$ be the inverse image of $U_0$ in $Z_ i$, so $U = \mathop{\mathrm{lim}}\nolimits U_ i$. By Lemma 69.5.12 we see that $X \to U_ i$ is a closed immersion for some $i$ large enough. Setting $X' = U_ i$ finishes the proof. $\square$

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