The Stacks project

Proof. By Lemma 32.8.6 we see that $f_ i$ is separated for some $i \geq 0$. Replacing $0$ by $i$ we may assume that $f_0$ is separated. Observe that $f_0$ is quasi-compact, see Schemes, Lemma 26.21.14. By Lemma 32.12.1 we can choose a diagram

where $X_0' \to \mathbf{P}^ n_{Y_0}$ is an immersion, and $\pi : X_0' \to X_0$ is proper and surjective. Introduce $X' = X_0' \times _{Y_0} Y$ and $X_ i' = X_0' \times _{Y_0} Y_ i$. By Morphisms, Lemmas 29.41.4 and 29.41.5 we see that $X' \to Y$ is proper. Hence $X' \to \mathbf{P}^ n_ Y$ is a closed immersion (Morphisms, Lemma 29.41.7). By Morphisms, Lemma 29.41.9 it suffices to prove that $X'_ i \to Y_ i$ is proper for some $i$. By Lemma 32.8.5 we find that $X'_ i \to \mathbf{P}^ n_{Y_ i}$ is a closed immersion for $i$ large enough. Then $X'_ i \to Y_ i$ is proper and we win. $\square$

Proof. By Proposition 32.9.6 we can find a closed immersion $X \to Y$ with $Y$ separated and of finite presentation over $S$. By Lemma 32.12.1 we can find a diagram

where $Y' \to \mathbf{P}^ n_ S$ is an immersion, and $\pi : Y' \to Y$ is proper and surjective. By Lemma 32.9.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i \to Y$ a closed immersion of finite presentation. Denote $X'_ i \subset Y'$, resp. $X' \subset Y'$ the scheme theoretic inverse image of $X_ i \subset Y$, resp. $X \subset Y$. Then $\mathop{\mathrm{lim}}\nolimits X'_ i = X'$. Since $X' \to S$ is proper (Morphisms, Lemmas 29.41.4), we see that $X' \to \mathbf{P}^ n_ S$ is a closed immersion (Morphisms, Lemma 29.41.7). Hence for $i$ large enough we find that $X'_ i \to \mathbf{P}^ n_ S$ is a closed immersion by Lemma 32.4.20. Thus $X'_ i$ is proper over $S$. For such $i$ the morphism $X_ i \to S$ is proper by Morphisms, Lemma 29.41.9. $\square$

Proof. By Lemma 32.13.2 we can write $X = \mathop{\mathrm{lim}}\nolimits _{k \in K} X_ k$ with $X_ k \to S$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 32.5.4) we can write $S = \mathop{\mathrm{lim}}\nolimits _{j \in J} S_ j$ with $S_ j$ of finite type over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to S_ j$ of finite presentation with $X_ k \cong S \times _{S_ j} X_{k, j}$ as schemes over $S$, see Lemma 32.10.1. After increasing $j$ we may assume $X_{k, j} \to S_ j$ is proper, see Lemma 32.13.1. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to S_ j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $S_{j'} \to S_ j$ whose base change to $S$ gives the morphism $X_{k'} \to X_ k$ (follows again from Lemma 32.10.1). These morphisms form the transition morphisms of the system. Some details omitted. $\square$

Proof. Choose a closed immersion $Y \to Y'$ with $Y'$ proper and of finite presentation over $X$, see Lemma 32.13.2. Then choose an $i$ and a proper morphism $Y'_ i \to X_ i$ such that $Y' = X \times _{X_ i} Y'_ i$. This is possible by Lemmas 32.10.1 and 32.13.1. Then after replacing $i$ by a larger index we have that $Y \to Y'_ i$ is a closed immersion, see Lemma 32.4.16. $\square$

Proof. We may replace $X_0$ by the scheme theoretic support of $\mathcal{F}_0$. By Morphisms, Lemma 29.5.3 this guarantees that $X_ i$ is the support of $\mathcal{F}_ i$ and $X$ is the support of $\mathcal{F}$. Then, if $Z \subset X$ denotes the scheme theoretic support of $\mathcal{F}$, we see that $Z \to X$ is a universal homeomorphism. We conclude that $X \to Y$ is proper as this is true for $Z \to Y$ by assumption, see Morphisms, Lemma 29.41.9. By Lemma 32.13.1 we see that $X_ i \to Y$ is proper for some $i$. Then it follows that the scheme theoretic support $Z_ i$ of $\mathcal{F}_ i$ is proper over $Y$ by Morphisms, Lemmas 29.41.6 and 29.41.4. $\square$