The Stacks project

Lemma 75.8.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ such that the support of $H^ i(E)$ is proper over $Y$ for all $i$. Then $Rf_*E$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$.

Proof. Consider the spectral sequence

\[ R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E \]

see Derived Categories, Lemma 13.21.3. By assumption and Lemma 75.7.10 the sheaves $R^ pf_*H^ q(E)$ are coherent. Hence $R^{p + q}f_*E$ is coherent, i.e., $E \in D_{\textit{Coh}}(\mathcal{O}_ Y)$. Boundedness from below is trivial. Boundedness from above follows from Cohomology of Spaces, Lemma 69.8.1 or from Lemma 75.6.1. $\square$

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