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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