The Stacks project

Lemma 36.4.3. Let $X$ be a quasi-separated and quasi-compact scheme. Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules each of which is right acyclic for $\Gamma (X, -)$. Then $\Gamma (X, \mathcal{F}^\bullet )$ represents $R\Gamma (X, \mathcal{F}^\bullet )$ in $D(\Gamma (X, \mathcal{O}_ X)$.

Proof. Apply Lemma 36.4.2 to the canonical morphism $X \to \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$. Some details omitted. $\square$

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