The Stacks project

Lemma 68.9.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Every quasi-coherent $\mathcal{O}_ X$-module is a filtered colimit of finitely presented $\mathcal{O}_ X$-modules.

Proof. We may view as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 63.16.2 and Properties of Spaces, Definition 64.3.1. Thus we may apply Proposition 68.8.1 and write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $\mathbf{Z}$. Thus $X_ i$ is a Noetherian algebraic space, see Morphisms of Spaces, Lemma 65.28.6. The morphism $X \to X_ i$ is affine, see Lemma 68.4.1. Conclusion by Cohomology of Spaces, Lemma 67.15.2. $\square$


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Your email address will not be published.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 07V9. Beware of the difference between the letter 'O' and the digit '0'.