The Stacks project

Lemma 51.15.2. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $\mathcal{F}' = j_*(\mathcal{F}|_ U)$ is coherent. Then $\mathcal{F} \to \mathcal{F}'$ is the unique map of coherent $\mathcal{O}_ X$-modules such that

  1. $\mathcal{F}|_ U \to \mathcal{F}'|_ U$ is an isomorphism,

  2. $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) \geq 2$ for $x \in X$, $x \not\in U$.

If $f : Y \to X$ is a flat morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}'$ is the corresponding map for $f^{-1}(U) \subset Y$.

Proof. We have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) \geq 2$ by Divisors, Lemma 31.6.6 part (3). The uniqueness of $\mathcal{F} \to \mathcal{F}'$ follows from Divisors, Lemma 31.5.11. The compatibility with flat pullbacks follows from flat base change, see Cohomology of Schemes, Lemma 30.5.2. $\square$

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