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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