Lemma 30.9.11. Let $X$ be a scheme. Let $j : U \to X$ be the inclusion of an open. Let $T \subset X$ be a closed subset contained in $U$. If $\mathcal{F}$ is a coherent $\mathcal{O}_ U$-module with $\text{Supp}(\mathcal{F}) \subset T$, then $j_*\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module.

Proof. Consider the open covering $X = U \cup (X \setminus T)$. Then $j_*\mathcal{F}|_ U = \mathcal{F}$ is coherent and $j_*\mathcal{F}|_{X \setminus T} = 0$ is also coherent. Hence $j_*\mathcal{F}$ is coherent. $\square$

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