The Stacks project

Remark 49.11.3. In Lemma 49.11.2 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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