The Stacks project

Lemma 50.3.3. Let $p : X \to S$ be a proper morphism of schemes with $S$ locally Noetherian. Then $Rp_*\Omega ^\bullet _{X/S}$ is an object of $D_{\textit{Coh}}(\mathcal{O}_ S)$.

Proof. In this case by Morphisms, Lemma 29.32.12 the modules $\Omega ^ i_{X/S}$ are coherent. Hence we can use exactly the same argument as in the proof of Lemma 50.3.2 using Cohomology of Schemes, Proposition 30.19.1. $\square$

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