Lemma 18.28.13. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be locally of finite presentation and flat. Then given an object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $\mathcal{F}|_{U_ i}$ is a direct summand of a finite free $\mathcal{O}_{U_ i}$-module.

**Proof.**
Choose an object $U$ of $\mathcal{C}$. After replacing $U$ by the members of a covering, we may assume there exists a presentation

By Lemma 18.28.12 we may, after replacing $U$ by the members of a covering, assume there exists a factorization

such that the composition $\mathcal{O}_ U^{\oplus r} \to \mathcal{O}_ U^{\oplus n} \to \mathcal{O}_ U^{\oplus n_ r}$ is zero. This means that the surjection $\mathcal{O}_ U^{\oplus n_ r} \to \mathcal{F}|_ U$ has a section and we win. $\square$

## 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Your email address will not be published.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 (2)

Comment #4403 by Manuel Hoff on

Comment #4505 by Johan on