Lemma 75.7.10. 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 locally Noetherian. Let \mathcal{F} be a coherent \mathcal{O}_ X-module with support proper over Y. Then R^ pf_*\mathcal{F} is a coherent \mathcal{O}_ Y-module for all p \geq 0.
Proof. By Lemma 75.7.7 there exists a closed immersion i : Z \to X with g = f \circ i : Z \to Y proper and \mathcal{F} = i_*\mathcal{G} for some coherent module \mathcal{G} on Z. We see that R^ pg_*\mathcal{G} is coherent on S by Cohomology of Spaces, Lemma 69.20.2. On the other hand, R^ qi_*\mathcal{G} = 0 for q > 0 (Cohomology of Spaces, Lemma 69.12.9). By Cohomology on Sites, Lemma 21.14.7 we get R^ pf_*\mathcal{F} = R^ pg_*\mathcal{G} and the lemma follows. \square
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