The Stacks project

Lemma 50.12.1. Let $f : X \to Y$ be a quasi-compact, quasi-separated, and smooth morphism of schemes over a base scheme $S$. There is a bounded spectral sequence with first page

\[ E_1^{p, q} = H^ q(\Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y}) \]

converging to $R^{p + q}f_*\Omega ^\bullet _{X/S}$.

Proof. Consider $\Omega ^\bullet _{X/S}$ as a filtered complex with the filtration introduced above. The spectral sequence is the spectral sequence of Cohomology, Lemma 20.29.5. By Derived Categories of Schemes, Lemma 36.23.2 we have

\[ Rf_*\text{gr}^ k\Omega ^\bullet _{X/S} = \Omega ^ k_{Y/S}[-k] \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y} \]

and thus we conclude. $\square$

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