Theorem 115.21.5. Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\textit{Coh}_{X/B}$ is an algebraic stack over $S$.

**Proof.**
This theorem is a copy of Quot, Theorem 99.6.1. The reason we have this copy here is that with the material in this section we get a second proof (as discussed at the beginning of this section). Namely, we argue exactly as in the proof of Quot, Theorem 99.5.12 except that we substitute Lemma 115.21.4 for Quot, Lemma 99.5.11.
$\square$

## 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Your email address will not be published.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.

## Comments (0)