Lemma 28.22.1. Let $j : U \to X$ be a quasi-compact open immersion of schemes.

Any quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$.

Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}|_ U$ be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf $\mathcal{H}$ of $\mathcal{F}$ such that $\mathcal{H}|_ U = \mathcal{G}$ as subsheaves of $\mathcal{F}|_ U$.

Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $U$. Let $\varphi : \mathcal{G} \to \mathcal{F}|_ U$ be a morphism of $\mathcal{O}_ U$-modules. There exists a quasi-coherent sheaf $\mathcal{H}$ of $\mathcal{O}_ X$-modules and a map $\psi : \mathcal{H} \to \mathcal{F}$ such that $\mathcal{H}|_ U = \mathcal{G}$ and that $\psi |_ U = \varphi $.

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