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.

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