The Stacks project

Lemma 69.12.1. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ quasi-compact and quasi-separated. Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of algebraic spaces $X_ i$ proper and of finite presentation over $Y$ and with transition morphisms and morphisms $X \to X_ i$ closed immersions.

Proof. By Proposition 69.11.7 we can find a closed immersion $X \to X'$ with $X'$ separated and of finite presentation over $Y$. By Lemma 69.11.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i \to X'$ a closed immersion of finite presentation. We claim that for all $i$ large enough the morphism $X_ i \to Y$ is proper which finishes the proof.

To prove this we may assume that $Y$ is an affine scheme, see Morphisms of Spaces, Lemma 66.40.2. Next, we use the weak version of Chow's lemma, see Cohomology of Spaces, Lemma 68.18.1, to find a diagram

\[ \xymatrix{ X' \ar[rd] & X'' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ Y \ar[dl] \\ & Y & } \]

where $X'' \to \mathbf{P}^ n_ Y$ is an immersion, and $\pi : X'' \to X'$ is proper and surjective. Denote $X'_ i \subset X''$, resp. $\pi ^{-1}(X)$ the scheme theoretic inverse image of $X_ i \subset X'$, resp. $X \subset X'$. Then $\mathop{\mathrm{lim}}\nolimits X'_ i = \pi ^{-1}(X)$. Since $\pi ^{-1}(X) \to Y$ is proper (Morphisms of Spaces, Lemmas 66.40.4), we see that $\pi ^{-1}(X) \to \mathbf{P}^ n_ Y$ is a closed immersion (Morphisms of Spaces, Lemmas 66.40.6 and 66.12.3). Hence for $i$ large enough we find that $X'_ i \to \mathbf{P}^ n_ Y$ is a closed immersion by Lemma 69.5.16. Thus $X'_ i$ is proper over $Y$. For such $i$ the morphism $X_ i \to Y$ is proper by Morphisms of Spaces, Lemma 66.40.7. $\square$

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