hcju/stacksqa huggingface dataset:
Choose an affine scheme $V_0$ and a surjective \'etale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to V_0 \times_{Y_0} X_0$. Diagram $$ \xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 } $$ The vertical arrows are surjective and \'etale by construction. We can base change this diagram to $B_i$ or $B$ to get $$ \vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } } $$ Note that $U_i, V_i, U, V$ are affine schemes, the vertical morphisms are surjective \'etale, and the limit of the morphisms $U_i \to V_i$ is $U \to V$. In this situation $X_i \to Y_i$ has relative dimension $\leq d$ if and only if $U_i \to V_i$ has relative dimension $\leq d$ (as defined in Morphisms, Definition \ref{morphisms-definition-relative-dimension-d}). To see the equivalence, use that the definition for morphisms of algebraic spaces involves Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre} which uses \'etale localization. The same is true for $X \to Y$ and $U \to V$. Since $f_0$ is locally of finite type, so is the morphism $U_0 \to V_0$. Hence the lemma follows from the more general Limits, Lemma \ref{limits-lemma-limit-dimension}.
Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = X_i \times_{X_0} U_0$. Set $R_0 = U_0 \times_{X_0} U_0$ and $R_i = R_0 \times_{X_0} X_i$. Denote $s_i, t_i : R_i \to U_i$ and $s, t : R \to U$ the two projections. In the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} we have seen that there exists a presentation $X = U/R$ with $U = \lim U_i$ and $R = \lim R_i$. Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is quasi-separated). Moreover, it is also true that $R \times_{s, U, t} R = \colim R_i \times_{s_i, U_i, t_i} R_i$. Thus we know that $\QCoh(\mathcal{O}_U) = \colim \QCoh(\mathcal{O}_{U_i})$, $\QCoh(\mathcal{O}_R) = \colim \QCoh(\mathcal{O}_{R_i})$, and $\QCoh(\mathcal{O}_{R \times_{s, U, t} R}) = \colim \QCoh(\mathcal{O}_{R_i \times_{s_i, U_i, t_i} R_i})$ by Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}. We have $\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)$ and $\QCoh(\mathcal{O}_{X_i}) = \QCoh(U_i, R_i, s_i, t_i, c_i)$, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent}. Thus the result follows formally.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Since invertible modules are of finite presentation we can find an $i$ and modules $\mathcal{L}_i$ and $\mathcal{N}_i$ of finite presentation over $X_i$ such that $f_i^*\mathcal{L}_i \cong \mathcal{L}$ and $f_i^*\mathcal{N}_i \cong \mathcal{L}^{\otimes -1}$, see Lemma \ref{lemma-descend-modules-finite-presentation}. Since pullback commutes with tensor product we see that $f_i^*(\mathcal{L}_i \otimes_{\mathcal{O}_{X_i}} \mathcal{N}_i)$ is isomorphic to $\mathcal{O}_X$. Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that $f_{i'i}^*\mathcal{L}_i \otimes_{\mathcal{O}_{X_{i'}}} f_{i'i}^*\mathcal{N}_i$ is isomorphic to $\mathcal{O}_{X_{i'}}$ for some $i' \geq i$. It follows that $f_{i'i}^*\mathcal{L}_i$ is invertible (Modules on Sites, Lemma \ref{sites-modules-lemma-invertible}) and the proof is complete.
We may view as an algebraic space over $\Spec(\mathbf{Z})$, see Spaces, Definition \ref{spaces-definition-base-change} and Properties of Spaces, Definition \ref{spaces-properties-definition-separated}. Thus we may apply Proposition \ref{proposition-approximate} and write $X = \lim X_i$ with $X_i$ of finite presentation over $\mathbf{Z}$. Thus $X_i$ is a Noetherian algebraic space, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}. The morphism $X \to X_i$ is affine, see Lemma \ref{lemma-directed-inverse-system-has-limit}. Conclusion by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-direct-colimit-finite-presentation}.
If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are finite type quasi-coherent $\mathcal{O}_X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another finite type quasi-coherent $\mathcal{O}_X$-submodule which contains both of them. In this way we see that the system is directed. To show that $\mathcal{F}$ is the colimit of this system, write $\mathcal{F} = \colim_i \mathcal{F}_i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma \ref{lemma-colimit-finitely-presented}. Then the images $\mathcal{G}_i = \Im(\mathcal{F}_i \to \mathcal{F})$ are finite type quasi-coherent subsheaves of $\mathcal{F}$. Since $\mathcal{F}$ is the colimit of these the result follows.
Write $\mathcal{F} = \colim \mathcal{G}_i$ as a filtered colimit of finitely presented $\mathcal{O}_X$-modules (Lemma \ref{lemma-colimit-finitely-presented}). We claim that $\mathcal{G}_i \to \mathcal{F}$ is surjective for some $i$. Namely, choose an \'etale surjection $U \to X$ where $U$ is an affine scheme. Choose finitely many sections $s_k \in \mathcal{F}(U)$ generating $\mathcal{F}|_U$. Since $U$ is affine we see that $s_k$ is in the image of $\mathcal{G}_i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_i$ be the kernel of the map $\mathcal{G}_i \to \mathcal{F}$. Write $\mathcal{K} = \colim \mathcal{K}_a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma \ref{lemma-directed-colimit-finite-type}). Then $\mathcal{F} = \colim \mathcal{G}_i/\mathcal{K}_a$ is a solution to the problem posed by the lemma.
First we write $\mathcal{A} = \colim_i \mathcal{F}_i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma \ref{lemma-colimit-finitely-presented}. For each $i$ let $\mathcal{B}_i = \text{Sym}(\mathcal{F}_i)$ be the symmetric algebra on $\mathcal{F}_i$ over $\mathcal{O}_X$. Write $\mathcal{I}_i = \Ker(\mathcal{B}_i \to \mathcal{A})$. Write $\mathcal{I}_i = \colim_j \mathcal{F}_{i, j}$ where $\mathcal{F}_{i, j}$ is a finite type quasi-coherent submodule of $\mathcal{I}_i$, see Lemma \ref{lemma-directed-colimit-finite-type}. Set $\mathcal{I}_{i, j} \subset \mathcal{I}_i$ equal to the $\mathcal{B}_i$-ideal generated by $\mathcal{F}_{i, j}$. Set $\mathcal{A}_{i, j} = \mathcal{B}_i/\mathcal{I}_{i, j}$. Then $\mathcal{A}_{i, j}$ is a quasi-coherent finitely presented $\mathcal{O}_X$-algebra. Define $(i, j) \leq (i', j')$ if $i \leq i'$ and the map $\mathcal{B}_i \to \mathcal{B}_{i'}$ maps the ideal $\mathcal{I}_{i, j}$ into the ideal $\mathcal{I}_{i', j'}$. Then it is clear that $\mathcal{A} = \colim_{i, j} \mathcal{A}_{i, j}$.
By Lemma \ref{lemma-finite-directed-colimit-surjective-maps} there exists a finitely presented $\mathcal{O}_X$-module $\mathcal{F}$ and a surjection $\mathcal{F} \to \mathcal{A}$. Using the algebra structure we obtain a surjection $$ \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}) \longrightarrow \mathcal{A} $$ Denote $\mathcal{J}$ the kernel. Write $\mathcal{J} = \colim \mathcal{E}_i$ as a filtered colimit of finite type $\mathcal{O}_X$-submodules $\mathcal{E}_i$ (Lemma \ref{lemma-directed-colimit-finite-type}). Set $$ \mathcal{A}_i = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})/(\mathcal{E}_i) $$ where $(\mathcal{E}_i)$ indicates the ideal sheaf generated by the image of $\mathcal{E}_i \to \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})$. Then each $\mathcal{A}_i$ is a finitely presented $\mathcal{O}_X$-algebra, the transition maps are surjective, and $\mathcal{A} = \colim \mathcal{A}_i$. To finish the proof we still have to show that $\mathcal{A}_i$ is a finite $\mathcal{O}_X$-algebra for $i$ sufficiently large. To do this we choose an \'etale surjective map $U \to X$ where $U$ is an affine scheme. Take generators $f_1, \ldots, f_m \in \Gamma(U, \mathcal{F})$. As $\mathcal{A}(U)$ is a finite $\mathcal{O}_X(U)$-algebra we see that for each $j$ there exists a monic polynomial $P_j \in \mathcal{O}(U)[T]$ such that $P_j(f_j)$ is zero in $\mathcal{A}(U)$. Since $\mathcal{A} = \colim \mathcal{A}_i$ by construction, we have $P_j(f_j) = 0$ in $\mathcal{A}_i(U)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_i$ are finite.
By Lemma \ref{lemma-algebra-directed-colimit-finite-type} we have $\mathcal{A} = \colim \mathcal{A}_i$ where $\mathcal{A}_i \subset \mathcal{A}$ runs through the quasi-coherent $\mathcal{O}_X$-sub algebras of finite type. Any finite type quasi-coherent $\mathcal{O}_X$-subalgebra of $\mathcal{A}$ is finite (use Algebra, Lemma \ref{algebra-lemma-characterize-finite-in-terms-of-integral} on affine schemes \'etale over $X$). This proves (1). \medskip\noindent To prove (2), write $\mathcal{A} = \colim \mathcal{F}_i$ as a colimit of finitely presented $\mathcal{O}_X$-modules using Lemma \ref{lemma-colimit-finitely-presented}. For each $i$, let $\mathcal{J}_i$ be the kernel of the map $$ \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i) \longrightarrow \mathcal{A} $$ For $i' \geq i$ there is an induced map $\mathcal{J}_i \to \mathcal{J}_{i'}$ and we have $\mathcal{A} = \colim \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i$. Moreover, the quasi-coherent $\mathcal{O}_X$-algebras $\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i$ are finite (see above). Write $\mathcal{J}_i = \colim \mathcal{E}_{ik}$ as a colimit of finitely presented $\mathcal{O}_X$-modules. Given $i' \geq i$ and $k$ there exists a $k'$ such that we have a map $\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}$ making $$ \xymatrix{ \mathcal{J}_i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] } $$ commute. This follows from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit}. This induces a map $$ \mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'} $$ where $(\mathcal{E}_{ik})$ denotes the ideal generated by $\mathcal{E}_{ik}$. The quasi-coherent $\mathcal{O}_X$-algebras $\mathcal{A}_{ki}$ are of finite presentation and finite for $k$ large enough (see proof of Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented}). Finally, we have $$ \colim \mathcal{A}_{ik} = \colim \mathcal{A}_i = \mathcal{A} $$ Namely, the first equality was shown in the proof of Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented} and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_i$.
Denote $j : U \to X$ the inclusion morphism. As $X$ is quasi-separated and $U$ quasi-compact, the morphism $j$ is quasi-compact. Hence $j_*\mathcal{G} \subset j_*\mathcal{F}|_U$ are quasi-coherent modules on $X$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}). Let $\mathcal{H} = \Ker(j_*\mathcal{G} \oplus \mathcal{F} \to j_*\mathcal{F}|_U)$. Then $\mathcal{H}|_U = \mathcal{G}$. By Lemma \ref{lemma-directed-colimit-finite-type} we can find a finite type quasi-coherent submodule $\mathcal{H}' \subset \mathcal{H}$ such that $\mathcal{H}'|_U = \mathcal{H}|_U = \mathcal{G}$. Set $\mathcal{G}' = \Im(\mathcal{H}' \to \mathcal{F})$ to conclude.
Since $|f|(|X|)$ is quasi-compact we may replace $Y$ by a quasi-compact open subspace whose set of points contains $|f|(|X|)$. Hence we may assume $Y$ is quasi-compact as well. Write $X = \lim X_a$ and $Y = \lim Y_b$ as in Proposition \ref{proposition-approximate}, i.e., with $X_a$ and $Y_b$ of finite type over $\mathbf{Z}$ and with affine transition morphisms. By Proposition \ref{proposition-characterize-locally-finite-presentation} we find that for each $b$ there exists an $a$ and a morphism $f_{a, b} : X_a \to Y_b$ making the diagram $$ \xymatrix{ X \ar[d] \ar[r] & Y \ar[d] \\ X_a \ar[r] & Y_b } $$ commute. Moreover the same proposition implies that, given a second triple $(a', b', f_{a', b'})$, there exists an $a'' \geq a'$ such that the compositions $X_{a''} \to X_a \to X_b$ and $X_{a''} \to X_{a'} \to X_{b'} \to X_b$ are equal. Consider the set of triples $(a, b, f_{a, b})$ endowed with the preordering $$ (a, b, f_{a, b}) \geq (a', b', f_{a', b'}) \Leftrightarrow a \geq a',\ b' \geq b,\text{ and } f_{a', b'} \circ h_{a, a'} = g_{b', b} \circ f_{a, b} $$ where $h_{a, a'} : X_a \to X_{a'}$ and $g_{b', b} : Y_{b'} \to Y_b$ are the transition morphisms. The remarks above show that this system is directed. It follows formally from the equalities $X = \lim X_a$ and $Y = \lim Y_b$ that $$ X = \lim_{(a, b, f_{a, b})} X_a \times_{f_{a, b}, Y_b} Y. $$ where the limit is over our directed system above. The transition morphisms $X_a \times_{Y_b} Y \to X_{a'} \times_{Y_{b'}} Y$ are affine as the composition $$ X_a \times_{Y_b} Y \to X_a \times_{Y_{b'}} Y \to X_{a'} \times_{Y_{b'}} Y $$ where the first morphism is a closed immersion (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fibre-product-after-map}) and the second is a base change of an affine morphism (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-affine}) and the composition of affine morphisms is affine (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-affine}). The morphisms $f_{a, b}$ are of finite presentation (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation} and \ref{spaces-morphisms-lemma-finite-presentation-permanence}) and hence the base changes $X_a \times_{f_{a, b}, S_b} S \to S$ are of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-presentation}).
Consider the quasi-coherent $\mathcal{O}_Y$-module $\mathcal{A} = f_*\mathcal{O}_X$. By Lemma \ref{lemma-algebra-directed-colimit-finite-presentation} we can write $\mathcal{A} = \colim \mathcal{A}_i$ as a directed colimit of finitely presented $\mathcal{O}_Y$-algebras $\mathcal{A}_i$. Set $X_i = \underline{\Spec}_Y(\mathcal{A}_i)$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. By construction $X_i \to Y$ is affine and of finite presentation and $X = \lim X_i$.
Consider the finite quasi-coherent $\mathcal{O}_Y$-module $\mathcal{A} = f_*\mathcal{O}_X$. By Lemma \ref{lemma-integral-algebra-directed-colimit-finite} we can write $\mathcal{A} = \colim \mathcal{A}_i$ as a directed colimit of finite and finitely presented $\mathcal{O}_Y$-algebras $\mathcal{A}_i$. Set $X_i = \underline{\Spec}_Y(\mathcal{A}_i)$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. By construction $X_i \to Y$ is finite and of finite presentation and $X = \lim X_i$.
Consider the finite quasi-coherent $\mathcal{O}_Y$-module $\mathcal{A} = f_*\mathcal{O}_X$. By Lemma \ref{lemma-finite-algebra-directed-colimit-finite-finitely-presented} we can write $\mathcal{A} = \colim \mathcal{A}_i$ as a directed colimit of finite and finitely presented $\mathcal{O}_Y$-algebras $\mathcal{A}_i$ with surjective transition maps. Set $X_i = \underline{\Spec}_Y(\mathcal{A}_i)$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. By construction $X_i \to Y$ is finite and of finite presentation, the transition maps are closed immersions, and $X = \lim X_i$.
Let $\mathcal{I} \subset \mathcal{O}_Y$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subspace of $Y$. By Lemma \ref{lemma-directed-colimit-finite-type} we can write $\mathcal{I} = \colim \mathcal{I}_i$ as the filtered colimit of its finite type quasi-coherent submodules. Let $X_i$ be the closed subspace of $X$ cut out by $\mathcal{I}_i$. Then $X_i \to Y$ is a closed immersion of finite presentation, and $X = \lim X_i$. Some details omitted.
By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-characterize-quasi-affine} 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 $Z = \lim Z_i$ with $Z_i$ affine and of finite presentation over $Y$ (Lemma \ref{lemma-affine-morphism-is-limit}). 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 \ref{lemma-descend-opens}). Let $U_i$ be the inverse image of $U_0$ in $Z_i$, so $U = \lim U_i$. By Lemma \ref{lemma-finite-type-eventually-closed} we see that $X \to U_i$ is a closed immersion for some $i$ large enough. Setting $X' = U_i$ finishes the proof.
By Proposition \ref{proposition-approximate} we can write $X = \lim_i X_i$ with $X_i$ quasi-separated of finite type over $\mathbf{Z}$ and with transition morphisms $f_{ii'} : X_i \to X_{i'}$ affine. Consider the commutative diagram $$ \xymatrix{ X \ar[r] \ar[rd] & X_{i, Y} \ar[r] \ar[d] & X_i \ar[d] \\ & Y \ar[r] & \Spec(\mathbf{Z}) } $$ Note that $X_i$ is of finite presentation over $\Spec(\mathbf{Z})$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}. Hence the base change $X_{i, Y} \to Y$ is of finite presentation by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-presentation}. Observe that $\lim X_{i, Y} = X \times Y$ and that $X \to X \times Y$ is a monomorphism. By Lemma \ref{lemma-finite-type-eventually-closed} we see that $X \to X_{i, Y}$ is a monomorphism for $i$ large enough. Fix such an $i$. Note that $X \to X_{i, Y}$ is locally of finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}) and a monomorphism, hence separated and locally quasi-finite (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}). Hence $X \to X_{i, Y}$ is representable. Hence $X \to X_{i, Y}$ is quasi-affine because we can use the principle Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication} and the result for morphisms of schemes More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine}. Thus Lemma \ref{lemma-quasi-affine-closed-in-finite-presentation} gives a factorization $X \to X' \to X_{i, Y}$ with $X \to X'$ a closed immersion and $X' \to X_{i, Y}$ of finite presentation. Finally, $X' \to Y$ is of finite presentation as a composition of morphisms of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-presentation}).
By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to X'$ with $X'$ separated and of finite presentation over $Y$. By Lemma \ref{lemma-closed-is-limit-closed-and-finite-presentation} we can write $X = \lim X_i$ 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. \medskip\noindent To prove this we may assume that $Y$ is an affine scheme, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-proper-local}. Next, we use the weak version of Chow's lemma, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}, 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 $\lim X'_i = \pi^{-1}(X)$. Since $\pi^{-1}(X) \to Y$ is proper (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-proper}), we see that $\pi^{-1}(X) \to \mathbf{P}^n_Y$ is a closed immersion (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-universally-closed-permanence} and \ref{spaces-morphisms-lemma-immersion-when-closed}). Hence for $i$ large enough we find that $X'_i \to \mathbf{P}^n_Y$ is a closed immersion by Lemma \ref{lemma-eventually-closed-immersion}. Thus $X'_i$ is proper over $Y$. For such $i$ the morphism $X_i \to Y$ is proper by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}.
By Lemma \ref{lemma-proper-limit-of-proper-finite-presentation} we can write $X = \lim_{k \in K} X_k$ with $X_k \to Y$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition \ref{proposition-approximate}) we can write $Y = \lim_{j \in J} Y_j$ with $Y_j$ of finite presentation over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to Y_j$ of finite presentation with $X_k \cong Y \times_{Y_j} X_{k, j}$ as algebraic spaces over $Y$, see Lemma \ref{lemma-descend-finite-presentation}. After increasing $j$ we may assume $X_{k, j} \to Y_j$ is proper, see Lemma \ref{lemma-eventually-proper}. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to Y_j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $Y_{j'} \to Y_j$ whose base change to $Y$ gives the morphism $X_{k'} \to X_k$ (follows again from Lemma \ref{lemma-descend-finite-presentation}). These morphisms form the transition morphisms of the system. Some details omitted.
We may replace $X_0$ by the scheme theoretic support of $\mathcal{F}_0$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type} 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 \ref{morphisms-lemma-image-proper-is-proper}. By Lemma \ref{lemma-eventually-proper} 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 of Spaces, Lemmas \ref{spaces-morphisms-lemma-closed-immersion-proper} and \ref{spaces-morphisms-lemma-composition-proper}.
Say $U = \Spec(A)$. Write $A = \colim A_i$ as a filtered colimit of finite type $\mathbf{Z}$-subalgebras. For each $i$ the morphism $U \to U_i = \Spec(A_i)$ induces a morphism $$ U \longrightarrow X \times U_i $$ over $X$. In the limit the morphism $U \to X \times U$ is an immersion as $X$ is locally separated, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-semi-diagonal}. By Lemma \ref{lemma-finite-type-eventually-closed} we see that $U \to X \times U_i$ is an immersion for some $i$. Since $U_i$ is isomorphic to a closed subscheme of $\mathbf{A}^n_{\mathbf{Z}}$ the lemma follows.