Consider a commutative diagram of schemes
where $f$ is a smooth morphism. Then we obtain a locally split short exact sequence
by Morphisms, Lemma 29.34.16. Let us think of this as a descending filtration $F$ on $\Omega _{X/S}$ with $F^0\Omega _{X/S} = \Omega _{X/S}$, $F^1\Omega _{X/S} = f^*\Omega _{Y/S}$, and $F^2\Omega _{X/S} = 0$. Applying the functor $\wedge ^ p$ we obtain for every $p$ an induced filtration
whose successive quotients are
for $k = 0, \ldots , p$. In fact, the reader can check using the Leibniz rule that $F^ k\Omega ^\bullet _{X/S}$ is a subcomplex of $\Omega ^\bullet _{X/S}$. In this way $\Omega ^\bullet _{X/S}$ has the structure of a filtered complex. We can also see this by observing that
is the image of a map of complexes on $X$. The filtered complex
has the following associated graded parts
by what was said above.
Lemma 50.12.1. Let $f : X \to Y$ be a quasi-compact, quasi-separated, and smooth morphism of schemes over a base scheme $S$. There is a bounded spectral sequence with first page converging to $R^{p + q}f_*\Omega ^\bullet _{X/S}$.
\[ E_1^{p, q} = H^ q(\Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y}) \]
Proof. Consider $\Omega ^\bullet _{X/S}$ as a filtered complex with the filtration introduced above. The spectral sequence is the spectral sequence of Cohomology, Lemma 20.29.5. By Derived Categories of Schemes, Lemma 36.23.2 we have
and thus we conclude. $\square$
Remark 50.12.2. In Lemma 50.12.1 consider the cohomology sheaves If $f$ is proper in addition to being smooth and $S$ is a scheme over $\mathbf{Q}$ then $\mathcal{H}^ q_{dR}(X/Y)$ is finite locally free (insert future reference here). If we only assume $\mathcal{H}^ q_{dR}(X/Y)$ are flat $\mathcal{O}_ Y$-modules, then we obtain (tiny argument omitted) and the differentials in the spectral sequence are maps In particular, for $p = 0$ we obtain a map $d_1^{0, q} : \mathcal{H}^ q_{dR}(X/Y) \to \Omega ^1_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y)$ which turns out to be an integrable connection $\nabla $ (insert future reference here) and the complex with differentials given by $d_1^{\bullet , q}$ is the de Rham complex of $\nabla $. The connection $\nabla $ is known as the Gauss-Manin connection.
\[ \mathcal{H}^ q_{dR}(X/Y) = H^ q(Rf_*\Omega ^\bullet _{X/Y}) \]
\[ E_1^{p, q} = \Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \]
\[ d_1^{p, q} : \Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \longrightarrow \Omega ^{p + 1}_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \]
\[ \mathcal{H}^ q_{dR}(X/Y) \to \Omega ^1_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \to \Omega ^2_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \to \ldots \]
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Comments (2)
Comment #8831 by Josh Lam on
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