The Stacks project

Proof. By Lemma 28.22.11 we have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ where $\mathcal{A}_ i \subset \mathcal{A}$ runs through the quasi-coherent $\mathcal{O}_ X$-algebras of finite type. Any finite type quasi-coherent $\mathcal{O}_ X$-subalgebra of $\mathcal{A}$ is finite (apply Algebra, Lemma 10.36.5 to $\mathcal{A}_ i(U) \subset \mathcal{A}(U)$ for affine opens $U$ in $X$). 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 28.22.7. For each $i$, let $\mathcal{J}_ i$ be the kernel of the map

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

commute. This follows from Modules, Lemma 17.22.8. This induces a map

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 28.22.12). Finally, we have

Namely, the first equality was shown in the proof of Lemma 28.22.12 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$