Definition 50.7.1. Let X \to S be a morphism of schemes. The Hodge filtration on H^ n_{dR}(X/S) is the filtration with terms
where \sigma _{\geq p}\Omega ^\bullet _{X/S} is as in Homology, Section 12.15.
Let X \to S be a morphism of schemes. The Hodge filtration on H^ n_{dR}(X/S) is the filtration induced by the Hodge-to-de Rham spectral sequence (Homology, Definition 12.24.5). To avoid misunderstanding, we explicitly define it as follows.
Definition 50.7.1. Let X \to S be a morphism of schemes. The Hodge filtration on H^ n_{dR}(X/S) is the filtration with terms
where \sigma _{\geq p}\Omega ^\bullet _{X/S} is as in Homology, Section 12.15.
Of course \sigma _{\geq p}\Omega ^\bullet _{X/S} is a subcomplex of the relative de Rham complex and we obtain a filtration
of the relative de Rham complex with \text{gr}^ p(\Omega ^\bullet _{X/S}) = \Omega ^ p_{X/S}[-p]. The spectral sequence constructed in Cohomology, Lemma 20.29.1 for \Omega ^\bullet _{X/S} viewed as a filtered complex of sheaves is the same as the Hodge-to-de Rham spectral sequence constructed in Section 50.6 by Cohomology, Example 20.29.4. Further the wedge product (50.4.0.1) sends \text{Tot}(\sigma _{\geq i}\Omega ^\bullet _{X/S} \otimes _{p^{-1}\mathcal{O}_ S} \sigma _{\geq j}\Omega ^\bullet _{X/S}) into \sigma _{\geq i + j}\Omega ^\bullet _{X/S}. Hence we get commutative diagrams
In particular we find that
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Comment #8284 by Dan B on
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