The Stacks project

Lemma 31.5.11. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $j : U \to X$ be an open subscheme such that for $x \in X \setminus U$ we have $\text{depth}(\mathcal{F}_ x) \geq 2$. Then

\[ \mathcal{F} \longrightarrow j_*(\mathcal{F}|_ U) \]

is an isomorphism and consequently $\Gamma (X, \mathcal{F}) \to \Gamma (U, \mathcal{F})$ is an isomorphism too.

Proof. We claim Lemma 31.2.11 applies to the map displayed in the lemma. Let $x \in X$. If $x \in U$, then the map is an isomorphism on stalks as $j_*(\mathcal{F}|_ U)|_ U = \mathcal{F}|_ U$. If $x \in X \setminus U$, then $x \not\in \text{Ass}(j_*(\mathcal{F}|_ U))$ (Lemmas 31.5.9 and 31.5.3). Since we've assumed $\text{depth}(\mathcal{F}_ x) \geq 2$ this finishes the proof. $\square$

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