Proposition 36.7.5 (08DB)—The Stacks project

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Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.7: The coherator
Proposition 36.7.5 (cite)

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Proposition 36.7.5. Let $X$ be a quasi-compact scheme with affine diagonal. Then the functor (36.3.0.1)

$D(\mathit{QCoh}(\mathcal{O}_ X)) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X)$

is an equivalence with quasi-inverse given by $RQ_ X$ .

Proof. Let $U \subset X$ be an affine open. Then the morphism $U \to X$ is affine by Morphisms, Lemma 29.11.11. Thus the assumption of Lemma 36.7.4 holds by Lemma 36.7.1 and we win. $\square$

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