The Stacks project

Lemma 68.9.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then we can write $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ i$ where each $\mathcal{F}_ i$ is an $\mathcal{O}_ X$-module of finite presentation and all transition maps $\mathcal{F}_ i \to \mathcal{F}_{i'}$ surjective.

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ as a filtered colimit of finitely presented $\mathcal{O}_ X$-modules (Lemma 68.9.1). We claim that $\mathcal{G}_ i \to \mathcal{F}$ is surjective for some $i$. Namely, choose an ├ętale surjection $U \to X$ where $U$ is an affine scheme. Choose finitely many sections $s_ k \in \mathcal{F}(U)$ generating $\mathcal{F}|_ U$. Since $U$ is affine we see that $s_ k$ is in the image of $\mathcal{G}_ i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_ i$ be the kernel of the map $\mathcal{G}_ i \to \mathcal{F}$. Write $\mathcal{K} = \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma 68.9.2). Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i/\mathcal{K}_ a$ is a solution to the problem posed by the lemma. $\square$


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Your email address will not be published.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 086Y. Beware of the difference between the letter 'O' and the digit '0'.