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

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

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FM7. Beware of the difference between the letter 'O' and the digit '0'.



View Section 50.7 as pdf