The Stacks project

Proof. Consider the quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{A} = f_*\mathcal{O}_ X$. By Lemma 69.9.4 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finitely presented $\mathcal{O}_ Y$-algebras $\mathcal{A}_ i$. Set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$, see Morphisms of Spaces, Definition 66.20.8. By construction $X_ i \to Y$ is affine and of finite presentation and $X = \mathop{\mathrm{lim}}\nolimits X_ i$. $\square$

Proof. Consider the quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{A} = f_*\mathcal{O}_ X$. By Lemma 69.9.7 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finite and finitely presented $\mathcal{O}_ Y$-algebras $\mathcal{A}_ i$. Set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$, see Morphisms of Spaces, Definition 66.20.8. By construction $X_ i \to Y$ is finite and of finite presentation and $X = \mathop{\mathrm{lim}}\nolimits X_ i$. $\square$

Proof. Consider the finite quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{A} = f_*\mathcal{O}_ X$. By Lemma 69.9.6 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finite and finitely presented $\mathcal{O}_ Y$-algebras $\mathcal{A}_ i$ with surjective transition maps. Set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$, see Morphisms of Spaces, Definition 66.20.8. By construction $X_ i \to Y$ is finite and of finite presentation, the transition maps are closed immersions, and $X = \mathop{\mathrm{lim}}\nolimits X_ i$. $\square$

Proof. Let $\mathcal{I} \subset \mathcal{O}_ Y$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subspace of $Y$. By Lemma 69.9.2 we can write $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits \mathcal{I}_ i$ 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 $X = \mathop{\mathrm{lim}}\nolimits X_ i$. Some details omitted. $\square$

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. $\square$

Proof. By Proposition 69.8.1 we can write $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ 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

Note that $X_ i$ is of finite presentation over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Morphisms of Spaces, Lemma 66.28.7. Hence the base change $X_{i, Y} \to Y$ is of finite presentation by Morphisms of Spaces, Lemma 66.28.3. Observe that $\mathop{\mathrm{lim}}\nolimits X_{i, Y} = X \times Y$ and that $X \to X \times Y$ is a monomorphism. By Lemma 69.5.12 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 66.23.6) and a monomorphism, hence separated and locally quasi-finite (Morphisms of Spaces, Lemma 66.27.10). 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 64.5.8 and the result for morphisms of schemes More on Morphisms, Lemma 37.43.2. Thus Lemma 69.11.5 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 66.28.2). $\square$

Proof. By Lemma 69.11.6 there is a closed immersion $X \to Z$ with $Z/Y$ of finite presentation. Let $\mathcal{I} \subset \mathcal{O}_ Z$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subscheme of $Y$. By Lemma 69.9.2 we can write $\mathcal{I}$ as a directed colimit $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits _{a \in A} \mathcal{I}_ a$ of its quasi-coherent sheaves of ideals of finite type. Let $X_ a \subset Z$ be the closed subspace defined by $\mathcal{I}_ a$. These form an inverse system indexed by $A$. The transition morphisms $X_ a \to X_{a'}$ are affine because they are closed immersions. Each $X_ a$ is quasi-compact and quasi-separated since it is a closed subspace of $Z$ and $Z$ is quasi-compact and quasi-separated by our assumptions. We have $X = \mathop{\mathrm{lim}}\nolimits _ a X_ a$ as follows directly from the fact that $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits _{a \in A} \mathcal{I}_ a$. Each of the morphisms $X_ a \to Z$ is of finite presentation, see Morphisms, Lemma 29.21.7. Hence the morphisms $X_ a \to Y$ are of finite presentation. Thus it suffices to show that $X_ a \to Y$ is separated for some $a \in A$. This follows from Lemma 69.5.13 as we have assumed that $X \to Y$ is separated. $\square$