The Stacks project

Lemma 69.9.7. 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 69.9.5 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 69.9.1. 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 68.5.3. 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 69.9.6). 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 69.9.6 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 082C. Beware of the difference between the letter 'O' and the digit '0'.