The Stacks project

Lemma 27.22.13. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. 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 direct colimit of finite and finitely presented quasi-coherent $\mathcal{O}_ X$-algebras.

Proof. By Lemma 27.22.11 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 10.35.5 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 27.22.6. 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 Modules, Lemma 17.11.6. 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 27.22.12). 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 27.22.12 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$


Comments (2)

Comment #1924 by Guignard on

Typo in the statement of Lemma : "coherent" instead of "cohernet".


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