The Stacks project

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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