Lemma 46.7.5. If $U = \mathop{\mathrm{Spec}}(A)$ is an affine scheme, then the bounded below version
of the functor above is an equivalence.
Proof. Let $A : \textit{Mod}(\mathcal{O}) \to \textit{Adeq}(\mathcal{O})$ be the right adjoint to the inclusion functor constructed in Lemma 46.7.3. Since $A$ is left exact and since $\textit{Mod}(\mathcal{O})$ has enough injectives, $A$ has a right derived functor $RA : D^+_{\textit{Adeq}}(\mathcal{O}) \to D^+(\textit{Adeq}(\mathcal{O}))$. We claim that $RA$ is a quasi-inverse to (46.7.5.1). To see this the key fact is that if $\mathcal{F}$ is an adequate module, then the adjunction map $\mathcal{F} \to RA(\mathcal{F})$ is a quasi-isomorphism by Lemma 46.7.4.
Namely, to prove the lemma in full it suffices to show:
Given $\mathcal{F}^\bullet \in K^+(\textit{Adeq}(\mathcal{O}))$ the canonical map $\mathcal{F}^\bullet \to RA(\mathcal{F}^\bullet )$ is a quasi-isomorphism, and
given $\mathcal{G}^\bullet \in K^+(\textit{Mod}(\mathcal{O}))$ the canonical map $RA(\mathcal{G}^\bullet ) \to \mathcal{G}^\bullet $ is a quasi-isomorphism.
Both (1) and (2) follow from the key fact via a spectral sequence argument using one of the spectral sequences of Derived Categories, Lemma 13.21.3. Some details omitted. $\square$
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