The Stacks project

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 succesive 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.

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