Lemma 63.10.1. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a ring.

Let $K_ i \in D^+_{tors}(X_{\acute{e}tale}, \Lambda )$, $i \in I$ 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 $Rf_!(\bigoplus _ i K_ i) = \bigoplus _ i Rf_!K_ i$.

If $\Lambda $ is torsion, then the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ commutes with direct sums.

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