Lemma 27.22.12. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_ X$-algebras such that all transition maps $\mathcal{A}_{i'} \to \mathcal{A}_ i$ are surjective.

Proof. By Lemma 27.22.8 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 $\mathcal{J} = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_ i$ as a filtered colimit of finite type $\mathcal{O}_ X$-submodules $\mathcal{E}_ i$ (Lemma 27.22.7). 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 surjections, and $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$. 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 affine open covering $X = U_1 \cup \ldots \cup U_ m$. Take generators $f_{j, 1}, \ldots , f_{j, N_ j} \in \Gamma (U_ i, \mathcal{F})$. As $\mathcal{A}(U_ j)$ is a finite $\mathcal{O}_ X(U_ j)$-algebra we see that for each $k$ there exists a monic polynomial $P_{j, k} \in \mathcal{O}(U_ j)[T]$ such that $P_{j, k}(f_{j, k})$ is zero in $\mathcal{A}(U_ j)$. Since $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ by construction, we have $P_{j, k}(f_{j, k}) = 0$ in $\mathcal{A}_ i(U_ j)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_ i$ are finite. $\square$

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