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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