Let \(\pi: X\to S\) be a smooth \(k\)-morphism of smooth \(k\)-schemes. We define the relative de Rham cohomology sheaf on \(S\) is defined as the hypercohomology sheaf
\[ \mathcal{H}_{dR}^q(X/S):= R^q\pi_*\Omega_{X/S}^\bullet \]
which is the cohomology of the derived complex \(R\pi_*\Omega_{X/S}^\bullet\in D^b(S)\), i.e.
\[\mathcal{H}_{dR}(X/S) = H(R\pi_*\Omega_{X/S}^\bullet)\in D^b(S).\]
Here \(\Omega_{X/S}^\bullet\) is the relative de Rham complex of sheaves on \(X\).
Typically the de Rham cohomology need to be computed with a Cech resolution.
If \(Y\) is separated and \(\pi: X\to Y\) is affine, then by Serre’s theorem, each sheaf \(\Omega^i_{X/Y}\) is acyclic for \(\pi_*\), so \(\Omega^\bullet_{X/Y}\) actually becomes an acyclic resolution, no need to resolve it again. This means
\[ \mathcal{H}^\bullet_{dR}(X/Y) = H(R\pi_*\Omega_{X/Y}^\bullet) = H(\pi_*\Omega_{X/Y}^\bullet). \]
For the general separated morphism \(\pi:X\to Y\), for simplicity let’s assume \(Y\) to be affine. Then we need an affine cover of \(U_i\to X\) such that \(U_i\to Y\) is affine (which is automatic in our case). This means each \((i_J)_* i_J^* \Omega_{X/Y}^p\) is acyclic for \(\pi_*\). Here \(i_J : U_J \hookrightarrow X\) is the inclusion of the open set \(U_J = \cap_{j\in J} U_j\).
We obtain a Cech complex (which is a \(\pi_*\)-acyclic resolution) for each \(\Omega_{X/Y}^p\),
\[ E^{pq}_0 = \check{C}^q(U_\bullet, \Omega_{X/Y}^p) = \bigoplus_{|J|=q+1} (i_J)_* i_J^* \Omega_{X/Y}^p, \]
and the cohomology of this total complex
\[T^n = \bigoplus_{p+q=n} \check{C}^q(U_\bullet, \Omega_{X/Y}^p),\] \[ \mathcal{H}_{dR}^\bullet(X/Y) = H(T^\bullet) \]
is the relative de Rham cohomology, which is also the spectral sequence associated to the double complex \(E^{pq}_0\), whose first page is the vertical cohomologies, and is clearly
\[ E^{pq}_1 = H^q(R\pi_*\Omega_{X/Y}^p) \Rightarrow \mathcal{H}^{p+q}_{dR}(X/Y). \]
The smoothness implies the exact sequence \(0\to \pi^*(\Omega_{S/k}^1)\to \Omega_{X/k}^1\to \Omega_{X/S}^1\to 0\). Which says the relative \(1\)-differentials are exactly those total \(1\)-differentials on \(X\) quotient by those \(1\)-differentials on \(S\). The \(\pi^*\) functor is natural and necessary here because \(\Omega_{S}^1\) is not a sheaf on \(X\).
The total differential complex \(\Omega^\bullet_{X/k}\) admits a filtration given by counting the number of terms that essentially coming from \(S\),
\[ F^p \Omega^\bullet_{X/k} = \mathrm{Image}(\Omega_{X/k}^\bullet[-p]\otimes_{\mathcal{O}_X}\pi^*(\Omega_{S/k}^p) \to \Omega_{X/k}^\bullet) \]
i.e. \(F^p\) is the differentials where at least \(p\)-terms come from \(S\). The grading shift \([-p]\) is necessary to match up the degrees of differentials, you can think of it as coming from \(\Omega^p_{S/k}[-p]\). This is a descending filtration
\[ \Omega_{X/k}^\bullet = F^0\Omega_{X/k}^\bullet \supset F^1\Omega_{X/k}^\bullet \supset F^2\Omega_{X/k}^\bullet \supset \cdots \]
with graded terms
\[ \mathrm{Gr}^p\Omega_{X/k}^\bullet = F^p/F^{p+1} = \Omega_{X/S}^\bullet[-p]\otimes_{\mathcal{O}_X}\pi^*(\Omega_{S/k}^p),\]
for example
\[ \begin{align*} \mathrm{Gr}^0(\Omega_{X/k}^\bullet) & = \Omega_{X/k}^\bullet / F^1\Omega_{X/k}^\bullet \\ & = \Omega_{X/k}^\bullet / \{\text{Anything containing a term from } \pi^*\Omega_{S/k}^\bullet\} \\ & \cong \Omega_{X/S}^\bullet \end{align*} \]
Note that the graded pieces \(\mathrm{Gr}^p\) has cohomology starting at degree \(p\). This hints us to use the spectral sequence associated to the filtration of \(R\pi_*\Omega_{X/k}^\bullet\), \(E_0^{p\bullet} = \pi_*\mathrm{Gr}^p\Omega_{X/k}^\bullet[p]\), so
\[\begin{align*} E^{pq}_1 &= R^{q}(\pi_*\mathrm{Gr}^p[p]) \\ &= R^{p+q}(\pi_*\mathrm{Gr}^p) \\ &= R^{p+q}(\pi_* (\Omega_{X/S}^\bullet[-p]\otimes_{\mathcal{O}_X}\pi^*\Omega_{S/k}^p))\\ &= R^{p+q}(\pi_* \Omega_{X/S}^\bullet[-p]\otimes_{\mathcal{O}_S}\Omega_{S/k}^p)\\ &= R^{p+q}(\pi_* \Omega_{X/S}^\bullet[-p])\otimes_{\mathcal{O}_S}\Omega_{S/k}^p\\ &= R^q(\pi_*\Omega_{X/S}^\bullet)\otimes_{\mathcal{O}_S}\Omega_{S/k}^p\\ &= \mathcal{H}_{dR}^q(X/S)\otimes_{\mathcal{O}_S}\Omega_{S/k}^p. \end{align*}\]
Here we used the fact that the differential in \(\Omega_{X/S}^\bullet\) is \(\mathcal{O}_S\)-linear and that \(\Omega_{S/k}^p\) is locally free.
There is a product
\[ E^{pq}_r \times E^{p'q'}_r \to E^{p+p',q+q'}_r \]
\[ (e,e')\mapsto e\cdot e' \]
with
\[ \begin{align*} e\cdot e' &= (-1)^{(p+q)(p'+q')} e'\cdot e, \\ d_r(e\cdot e') &= d_r(e)\cdot e' + (-1)^{p+q}e\cdot d_r(e'). \end{align*} \]
Gauss-Manin connection is a natural connection associated with a family \(X\to S\) on the \(\mathcal{H}_{dR}^q(X/S)\) of relative de Rham cohomology sheaves. It can be obtained by the above spectral sequence associated to its filtrations induced by pulling back differentials from \(S\).
The differential \(d_1\) in \(E_1\) actually gives the Gauss-Manin connection, writing it out we have
\[ d^{pq}_1 : E^{pq}_1 = R^{p+q}\pi_*(\mathrm{Gr}^p) \to R^{p+1+q}\pi_*\mathrm{Gr}^{p+1} = E^{p+1, q}_1 \]
or
\[ \mathcal{H}_{dR}^q(X/S)\otimes_{\mathcal{O}_S}\Omega_{S/k}^p \to \mathcal{H}_{dR}^q(X/S)\otimes_{\mathcal{O}_S}\Omega_{S/k}^{p+1}. \]
And this is the connecting homomorphism of the exact sequence by applying the cohomology functors \(R^q\pi_*\circ [p]\)
\[ 0\to \mathrm{Gr}^{p+1}\to F^p/F^{p+2}\to \mathrm{Gr}^p\to 0 \]