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Lemma 70.9.7. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be an integral quasi-coherent $\mathcal{O}_ X$-algebra. Then

  1. $\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\mathcal{O}_ X$-subalgebras, and

  2. $\mathcal{A}$ is a directed colimit of finite and finitely presented $\mathcal{O}_ X$-algebras.

Proof. By Lemma 70.9.5 we have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ 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 10.36.5 on affine schemes ├ętale over $X$). This proves (1).

To prove (2), write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a colimit of finitely presented $\mathcal{O}_ X$-modules using Lemma 70.9.1. 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 $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$. Moreover, the quasi-coherent $\mathcal{O}_ X$-algebras $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$ are finite (see above). Write $\mathcal{J}_ i = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_{ik}$ 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 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). 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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