The Stacks project

Lemma 36.13.2. Let $X$ be a Noetherian scheme and let $j : U \to X$ be an open immersion. The functor $D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ is essentially surjective.

Proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. By Proposition 36.11.2 we can represent $K$ by a bounded complex $\mathcal{F}^\bullet $ of coherent $\mathcal{O}_ U$-modules. Say $\mathcal{F}^ i = 0$ for $i \not\in [a, b]$ for some $a \leq b$. Since $j$ is quasi-compact and separated, the terms of the bounded complex $j_*\mathcal{F}^\bullet $ are quasi-coherent modules on $X$, see Schemes, Lemma 26.24.1. We inductively pick a coherent submodule $\mathcal{G}^ i \subset j_*\mathcal{F}^ i$ as follows. For $i = a$ we pick any coherent submodule $\mathcal{G}^ a \subset j_*\mathcal{F}^ a$ whose restriction to $U$ is $\mathcal{F}^ a$. This is possible by Properties, Lemma 28.22.2. For $i > a$ we first pick any coherent submodule $\mathcal{H}^ i \subset j_*\mathcal{F}^ i$ whose restriction to $U$ is $\mathcal{F}^ i$ and then we set $\mathcal{G}^ i = \mathop{\mathrm{Im}}(\mathcal{H}^ i \oplus \mathcal{G}^{i - 1} \to j_*\mathcal{F}^ i)$. It is clear that $\mathcal{G}^\bullet \subset j_*\mathcal{F}^\bullet $ is a bounded complex of coherent $\mathcal{O}_ X$-modules whose restriction to $U$ is $\mathcal{F}^\bullet $ as desired. $\square$

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