The Stacks project

Lemma 18.29.3. 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

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

By Lemma 18.28.14 we may, after replacing $U$ by the members of a covering, assume there exists a factorization

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

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. $\square$

Comments (2)

Comment #4403 by Manuel Hoff on

In the whole proof, should be replaced by .

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