The Stacks project

Lemma 36.7.3. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Then

  1. $Q_ X : \textit{Mod}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ X)$ is the functor which sends $\mathcal{F}$ to the quasi-coherent $\mathcal{O}_ X$-module associated to the $A$-module $\Gamma (X, \mathcal{F})$,

  2. $RQ_ X : D(\mathcal{O}_ X) \to D(\mathit{QCoh}(\mathcal{O}_ X))$ is the functor which sends $E$ to the complex of quasi-coherent $\mathcal{O}_ X$-modules associated to the object $R\Gamma (X, E)$ of $D(A)$,

  3. restricted to $D_\mathit{QCoh}(\mathcal{O}_ X)$ the functor $RQ_ X$ defines a quasi-inverse to (36.3.0.1).

Proof. The functor $Q_ X$ is the functor

\[ \mathcal{F} \mapsto \widetilde{\Gamma (X, \mathcal{F})} \]

by Schemes, Lemma 26.7.1. This immediately implies (1) and (2). The third assertion follows from (the proof of) Lemma 36.3.5. $\square$

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