The Stacks project

50.7 The Hodge filtration

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

\[ F^ pH^ n_{dR}(X/S) = \mathop{\mathrm{Im}}\left(H^ n(X, \sigma _{\geq p}\Omega ^\bullet _{X/S}) \longrightarrow H^ n_{dR}(X/S)\right) \]

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

\[ \Omega ^\bullet _{X/S} = \sigma _{\geq 0}\Omega ^\bullet _{X/S} \supset \sigma _{\geq 1}\Omega ^\bullet _{X/S} \supset \sigma _{\geq 2}\Omega ^\bullet _{X/S} \supset \sigma _{\geq 3}\Omega ^\bullet _{X/S} \supset \ldots \]

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

\[ \xymatrix{ H^ n(X, \sigma _{\geq j}\Omega ^\bullet _{X/S})) \times H^ m(X, \sigma _{\geq j}\Omega ^\bullet _{X/S})) \ar[r] \ar[d] & H^{n + m}(X, \sigma _{\geq i + j}\Omega ^\bullet _{X/S})) \ar[d] \\ H^ n_{dR}(X/S) \times H^ m_{dR}(X/S) \ar[r]^\cup & H^{n + m}_{dR}(X/S) } \]

In particular we find that

\[ F^ iH^ n_{dR}(X/S) \cup F^ jH^ m_{dR}(X/S) \subset F^{i + j}H^{n + m}_{dR}(X/S) \]

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