Lemma 50.3.2 (0FLX)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 50: de Rham Cohomology
Section 50.3: de Rham cohomology
Lemma 50.3.2 (cite)

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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. $\square$

Comments (2)

Comment #6516 by zhang on August 25, 2021 at 23:43

The two $R^a p_{\ast} \Omega^q_{X/S}$ 's should be " $R^b p_{\ast} \Omega^{a}_{X/S}$ "; or change " $E_1^{a,b}$ " to " $E_1^{q,a}$ "

Comment #6574 by Johan on September 14, 2021 at 14:22

Thanks and fixed here.

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