Lemma 36.13.1 (08ED)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.13: Lifting complexes
Lemma 36.13.1 (cite)

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Lemma 36.13.1. Let $X$ be a scheme and let $j : U \to X$ be a quasi-compact open immersion. The functors

$D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ U) \quad \text{and}\quad D^+_\mathit{QCoh}(\mathcal{O}_ X) \to D^+_\mathit{QCoh}(\mathcal{O}_ U)$

are essentially surjective. If $X$ is quasi-compact, then the functors

$D^-_\mathit{QCoh}(\mathcal{O}_ X) \to D^-_\mathit{QCoh}(\mathcal{O}_ U) \quad \text{and}\quad D^ b_\mathit{QCoh}(\mathcal{O}_ X) \to D^ b_\mathit{QCoh}(\mathcal{O}_ U)$

are essentially surjective.

Proof. The argument preceding the lemma applies for the first case because $Rj_*$ maps $D_\mathit{QCoh}(\mathcal{O}_ U)$ into $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.4.1. It is clear that $Rj_*$ maps $D^+_\mathit{QCoh}(\mathcal{O}_ U)$ into $D^+_\mathit{QCoh}(\mathcal{O}_ X)$ which implies the statement on bounded below complexes. Finally, Lemma 36.4.1 guarantees that $Rj_*$ maps $D^-_\mathit{QCoh}(\mathcal{O}_ U)$ into $D^-_\mathit{QCoh}(\mathcal{O}_ X)$ if $X$ is quasi-compact. Combining these two we obtain the last statement. $\square$

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