Lemma 50.3.2. Let $p : X \to S$ be a morphism of schemes. If $p$ is quasi-compact and quasi-separated, then $Rp_*\Omega ^\bullet _{X/S}$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ S)$.

**Proof.**
There is a spectral sequence with first page $E_1^{a, b} = R^ bp_*\Omega ^ a_{X/S}$ converging to the cohomology of $Rp_*\Omega ^\bullet _{X/S}$ (see Derived Categories, Lemma 13.21.3). Hence by Homology, Lemma 12.25.3 it suffices to show that $R^ bp_*\Omega ^ a_{X/S}$ is quasi-coherent. This follows from Cohomology of Schemes, Lemma 30.4.5.
$\square$

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