The Stacks project

Lemma 17.16.11. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be locally of finite presentation and flat. Then $\mathcal{F}$ is locally a direct summand of a finite free $\mathcal{O}_ X$-module.

Proof. After replacing $X$ by the members of an open covering, we may assume there exists a presentation

\[ \mathcal{O}_ X^{\oplus r} \to \mathcal{O}_ X^{\oplus n} \to \mathcal{F} \to 0 \]

Let $x \in X$. By Lemma 17.16.10 we can, after shrinking $X$ to an open neighbourhood of $x$, assume there exists a factorization

\[ \mathcal{O}_ X^{\oplus n} \to \mathcal{O}_ X^{\oplus n_1} \to \mathcal{F} \]

such that the composition $\mathcal{O}_ X^{\oplus r} \to \mathcal{O}_ X^{\oplus n} \to \mathcal{O}_ X^{\oplus n_1}$ annihilates the first summand of $\mathcal{O}_ X^{\oplus r}$. Repeating this argument $r - 1$ more times we obtain a factorization

\[ \mathcal{O}_ X^{\oplus n} \to \mathcal{O}_ X^{\oplus n_ r} \to \mathcal{F} \]

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


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Your email address will not be published.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 08BL. Beware of the difference between the letter 'O' and the digit '0'.