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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