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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