The Stacks project

Lemma 28.22.5. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{G}$ be an $\mathcal{O}_ U$-module.

  1. If $\mathcal{G}$ is quasi-coherent and of finite type, then there exists a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{G}'$ of finite type such that $\mathcal{G}'|_ U = \mathcal{G}$.

  2. If $\mathcal{G}$ is of finite presentation, then there exists an $\mathcal{O}_ X$-module $\mathcal{G}'$ of finite presentation such that $\mathcal{G}'|_ U = \mathcal{G}$.

Proof. Part (2) is the special case of Lemma 28.22.4 where $\mathcal{F} = 0$. For part (1) we first write $\mathcal{G} = \mathcal{F}|_ U$ for some quasi-coherent $\mathcal{O}_ X$-module by Lemma 28.22.1 and then we apply Lemma 28.22.2 with $\mathcal{G} = \mathcal{F}|_ U$. $\square$

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