Section 50.7 (0FM7): The Hodge filtration

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

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

In particular we find that

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

Comments (2)

Comment #8284 by Dan B on March 28, 2023 at 16:07

In the commutative diagram, the first $\sigma_{\geq j}$ in the top left term should be replaced by $\sigma_{\geq i}$ .

Comment #8916 by Stacks project on April 10, 2024 at 12:50

Thanks and fixed here.

The Stacks project

50.7 The Hodge filtration

Comments (2)

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