Lemma 59.52.6. Let f : X \to Y be a quasi-compact and quasi-sepated morphism of schemes. Let K_ i \in D(X_{\acute{e}tale}), i \in I be a family of objects. Assume given a \in \mathbf{Z} such that H^ n(K_ i) = 0 for n < a and i \in I. Then Rf_*(\bigoplus _ i K_ i) = \bigoplus _ i Rf_*K_ i.
Proof. We have to show that R^ pf_*(\bigoplus _ i K_ i) = \bigoplus _ i R^ pf_*K_ i for all p \in \mathbf{Z}. Choose complexes \mathcal{F}_ i^\bullet representing K_ i such that \mathcal{F}_ i^ n = 0 for n < a. The direct sum of the complexes \mathcal{F}_ i^\bullet represents the object \bigoplus K_ i by Injectives, Lemma 19.13.4. Since \bigoplus \mathcal{F}^\bullet is the filtered colimit of the finite direct sums, the result follows from Lemma 59.52.5. \square
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