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.

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).