Lemma 36.11.4. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ such that the support of $H^ i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^+_{\textit{Coh}}(\mathcal{O}_ S)$.

**Proof.**
The proof is the same as the proof of Lemma 36.11.3. You can also deduce it from Lemma 36.11.3 by considering what the exact functor $Rf_*$ does to the distinguished triangles $\tau _{\leq a}E \to E \to \tau _{\geq a + 1}E \to \tau _{\leq a}E[1]$.
$\square$

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