The Stacks project

Lemma 70.16.2. Let $S$ be a scheme. Let $f : X \to Y$ be an integral morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Let $V \subset Y$ be a quasi-compact open subspace such that $f^{-1}(V) \to V$ is finite and of finite presentation. Then $X$ can be written as a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ where $f_ i : X_ i \to Y$ are finite and of finite presentation such that $f^{-1}(V) \to f_ i^{-1}(V)$ is an isomorphism for all $i$.

Proof. This lemma is a slight refinement of Proposition 70.16.1. Consider the integral quasi-coherent $\mathcal{O}_ Y$-algebra $\mathcal{A} = f_*\mathcal{O}_ X$. In the next paragraph, we will write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finite and finitely presented $\mathcal{O}_ Y$-algebras $\mathcal{A}_ i$ such that $\mathcal{A}_ i|_ V = \mathcal{A}|_ V$. Having done this we set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$, see Morphisms of Spaces, Definition 67.20.8. By construction $X_ i \to Y$ is finite and of finite presentation, $X = \mathop{\mathrm{lim}}\nolimits X_ i$, and $f_ i^{-1}(V) = f^{-1}(V)$.

The proof of the assertion on algebras is similar to the proof of part (2) of Lemma 70.9.7. First, write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a colimit of finitely presented $\mathcal{O}_ Y$-modules using Lemma 70.9.1. Since $\mathcal{A}|_ V$ is a finite type $\mathcal{O}_ V$-module we may and do assume that $\mathcal{F}_ i|_ V \to \mathcal{A}|_ V$ is surjective for all $i$. 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 $\mathcal{J}_ i \to \mathcal{J}_{i'}$. We have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$. Moreover, the quasi-coherent $\mathcal{O}_ X$-algebras $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$ are finite (as finite type quasi-coherent subalgebras of the integral quasi-coherent $\mathcal{O}_ Y$-algebra $\mathcal{A}$ over $\mathcal{O}_ X$). The restriction of $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$ to $V$ is $\mathcal{A}|_ V$ by the surjectivity above. Hence $\mathcal{J}_ i|_ V$ is finitely generated as an ideal sheaf of $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)|_ V$ due to the fact that $\mathcal{A}|_ V$ is finitely presented as an $\mathcal{O}_ Y$-algebra. Write $\mathcal{J}_ i = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_{ik}$ as a colimit of finitely presented $\mathcal{O}_ X$-modules. We may and do assume that $\mathcal{E}_{ik}|_ V$ generates $\mathcal{J}_ i|_ V$ as a sheaf of ideal of $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)|_ V$ by the statement on finite generation above. 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 69.5.3. 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 70.9.6). Moreover we have $\mathcal{A}_{ik}|_ V = \mathcal{A}|_ V$ by construction. Finally, we have

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{A}_{ik} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i = \mathcal{A} \]

Namely, the first equality was shown in the proof of Lemma 70.9.6 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$

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