Lemma 36.4.2. Let $f : X \to S$ be a quasi-separated and quasi-compact morphism of schemes. Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules each of which is right acyclic for $f_*$. Then $f_*\mathcal{F}^\bullet $ represents $Rf_*\mathcal{F}^\bullet $ in $D(\mathcal{O}_ S)$.
Proof. There is always a canonical map $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet $. Our task is to show that this is an isomorphism on cohomology sheaves. As the statement is invariant under shifts it suffices to show that $H^0(f_*(\mathcal{F}^\bullet )) \to R^0f_*\mathcal{F}^\bullet $ is an isomorphism. The statement is local on $S$ hence we may assume $S$ affine. By Lemma 36.4.1 we have $R^0f_*\mathcal{F}^\bullet = R^0f_*\tau _{\geq -n}\mathcal{F}^\bullet $ for all sufficiently large $n$. Thus we may assume $\mathcal{F}^\bullet $ bounded below. As each $\mathcal{F}^ n$ is right $f_*$-acyclic by assumption we see that $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet $ is a quasi-isomorphism by Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7). $\square$
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