## 50.12 The spectral sequence for a smooth morphism

Consider a commutative diagram of schemes

$\xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[ld]^ q \\ & S }$

where $f$ is a smooth morphism. Then we obtain a locally split short exact sequence

$0 \to f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0$

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

$\Omega ^ p_{X/S} = F^0\Omega ^ p_{X/S} \supset F^1\Omega ^ p_{X/S} \supset F^2\Omega ^ p_{X/S} \supset \ldots \supset F^{p + 1}\Omega ^ p_{X/S} = 0$

whose successive quotients are

$\text{gr}^ k\Omega ^ p_{X/S} = F^ k\Omega ^ p_{X/S}/F^{k + 1}\Omega ^ p_{X/S} = f^*\Omega ^ k_{Y/S} \otimes _{\mathcal{O}_ X} \Omega ^{p - k}_{X/Y} = f^{-1}\Omega ^ k_{Y/S} \otimes _{f^{-1}\mathcal{O}_ Y} \Omega ^{p - k}_{X/Y}$

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

$F^ k\Omega ^\bullet _{X/S} = \mathop{\mathrm{Im}}\left(\wedge : \text{Tot}( f^{-1}\sigma _{\geq k}\Omega ^\bullet _{Y/S} \otimes _{p^{-1}\mathcal{O}_ S} \Omega ^\bullet _{X/S}) \longrightarrow \Omega ^\bullet _{X/S}\right)$

is the image of a map of complexes on $X$. The filtered complex

$\Omega ^\bullet _{X/S} = F^0\Omega ^\bullet _{X/S} \supset F^1\Omega ^\bullet _{X/S} \supset F^2\Omega ^\bullet _{X/S} \supset \ldots$

has the following associated graded parts

$\text{gr}^ k\Omega ^\bullet _{X/S} = f^{-1}\Omega ^ k_{Y/S}[-k] \otimes _{f^{-1}\mathcal{O}_ Y} \Omega ^\bullet _{X/Y}$

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

$E_1^{p, q} = H^ q(\Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y})$

converging to $R^{p + q}f_*\Omega ^\bullet _{X/S}$.

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

$Rf_*\text{gr}^ k\Omega ^\bullet _{X/S} = \Omega ^ k_{Y/S}[-k] \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y}$

and thus we conclude. $\square$

Remark 50.12.2. In Lemma 50.12.1 consider the cohomology sheaves

$\mathcal{H}^ q_{dR}(X/Y) = H^ q(Rf_*\Omega ^\bullet _{X/Y})$

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)

$E_1^{p, q} = \Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y)$

and the differentials in the spectral sequence are maps

$d_1^{p, q} : \Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \longrightarrow \Omega ^{p + 1}_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y)$

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

$\mathcal{H}^ q_{dR}(X/Y) \to \Omega ^1_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \to \Omega ^2_{Y/S} \otimes _{\mathcal{O}_ Y} \mathcal{H}^ q_{dR}(X/Y) \to \ldots$

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.

Comment #8831 by Josh Lam on

Is there a reference for the claim (in Remark 0FMN) that the de Rham cohomology sheaves $\mathcal{H}^q_{dR}(X/Y)$ are locally free? Thanks!

Comment #9254 by on

Deligne, P. Théorème de Lefschetz et critères de dégénérescence de suites spectrales.(French)Inst. Hautes Études Sci. Publ. Math.(1968), no.35, 259–278.

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).