The Stacks project

Lemma 28.18.2. Let $A$ be a ring and let $U \subset \mathop{\mathrm{Spec}}(A)$ be a quasi-compact open subscheme. For $\mathcal{F}$ quasi-coherent on $U$ the canonical map

\[ \widetilde{H^0(U, \mathcal{F})}|_ U \to \mathcal{F} \]

is an isomorphism.

Proof. Denote $j : U \to \mathop{\mathrm{Spec}}(A)$ the inclusion morphism. Then $H^0(U, \mathcal{F}) = H^0(\mathop{\mathrm{Spec}}(A), j_*\mathcal{F})$ and $j_*\mathcal{F}$ is quasi-coherent by Schemes, Lemma 26.24.1. Hence $j_*\mathcal{F} = \widetilde{H^0(U, \mathcal{F})}$ by Schemes, Lemma 26.7.5. Restricting back to $U$ we get the lemma. $\square$

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