Lemma 50.4.1. Let $p : X \to S$ be a morphism of schemes. The cup product on $H^*_{dR}(X/S)$ is associative and graded commutative.
50.4 Cup product
Consider the maps $\Omega ^ p_{X/S} \times \Omega ^ q_{X/S} \to \Omega ^{p + q}_{X/S}$ given by $(\omega , \eta ) \longmapsto \omega \wedge \eta $. Using the formula for $\text{d}$ given in Section 50.2 and the Leibniz rule for $\text{d} : \mathcal{O}_ X \to \Omega _{X/S}$ we see that $\text{d}(\omega \wedge \eta ) = \text{d}(\omega ) \wedge \eta + (-1)^{\deg (\omega )} \omega \wedge \text{d}(\eta )$. This means that $\wedge $ defines a morphism
50.4.0.1
\begin{equation} \label{derham-equation-wedge} \wedge : \text{Tot}( \Omega ^\bullet _{X/S} \otimes _{p^{-1}\mathcal{O}_ S} \Omega ^\bullet _{X/S}) \longrightarrow \Omega ^\bullet _{X/S} \end{equation}
of complexes of $p^{-1}\mathcal{O}_ S$-modules.
Combining the cup product of Cohomology, Section 20.31 with (50.4.0.1) we find a $H^0(S, \mathcal{O}_ S)$-bilinear cup product map
\[ \cup : H^ i_{dR}(X/S) \times H^ j_{dR}(X/S) \longrightarrow H^{i + j}_{dR}(X/S) \]
For example, if $\omega \in \Gamma (X, \Omega ^ i_{X/S})$ and $\eta \in \Gamma (X, \Omega ^ j_{X/S})$ are closed, then the cup product of the de Rham cohomology classes of $\omega $ and $\eta $ is the de Rham cohomology class of $\omega \wedge \eta $, see discussion in Cohomology, Section 20.31.
Given a commutative diagram
\[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S } \]
of schemes, the pullback maps $f^* : R\Gamma (X, \Omega ^\bullet _{X/S}) \to R\Gamma (X', \Omega ^\bullet _{X'/S'})$ and $f^* : H^ i_{dR}(X/S) \longrightarrow H^ i_{dR}(X'/S')$ are compatible with the cup product defined above.
Proof. This follows from Cohomology, Lemmas 20.31.5 and 20.31.6 and the fact that $\wedge $ is associative and graded commutative. $\square$
Remark 50.4.2. Let $p : X \to S$ be a morphism of schemes. Then we can think of $\Omega ^\bullet _{X/S}$ as a sheaf of differential graded $p^{-1}\mathcal{O}_ S$-algebras, see Differential Graded Sheaves, Definition 24.12.1. In particular, the discussion in Differential Graded Sheaves, Section 24.32 applies. For example, this means that for any commutative diagram of schemes there is a canonical relative cup product in $D(Y, q^{-1}\mathcal{O}_ T)$ which is associative and which on cohomology reproduces the cup product discussed above.
\[ \xymatrix{ X \ar[d]_ p \ar[r]_ f & Y \ar[d]^ q \\ S \ar[r]^ h & T } \]
\[ \mu : Rf_*\Omega ^\bullet _{X/S} \otimes _{q^{-1}\mathcal{O}_ T}^\mathbf {L} Rf_*\Omega ^\bullet _{X/S} \longrightarrow Rf_*\Omega ^\bullet _{X/S} \]
Remark 50.4.3. Let $f : X \to S$ be a morphism of schemes. Let $\xi \in H_{dR}^ n(X/S)$. According to the discussion Differential Graded Sheaves, Section 24.32 there exists a canonical morphism in $D(f^{-1}\mathcal{O}_ S)$ uniquely characterized by (1) and (2) of the following list of properties:
$\xi '$ can be lifted to a map in the derived category of right differential graded $\Omega ^\bullet _{X/S}$-modules, and
$\xi '(1) = \xi $ in $H^0(X, \Omega ^\bullet _{X/S}[n]) = H^ n_{dR}(X/S)$,
the map $\xi '$ sends $\eta \in H^ m_{dR}(X/S)$ to $\xi \cup \eta $ in $H^{n + m}_{dR}(X/S)$,
the construction of $\xi '$ commutes with restrictions to opens: for $U \subset X$ open the restriction $\xi '|_ U$ is the map corresponding to the image $\xi |_ U \in H^ n_{dR}(U/S)$,
for any diagram as in Remark 50.4.2 we obtain a commutative diagram
in $D(Y, q^{-1}\mathcal{O}_ T)$.
\[ \xi ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[n] \]
\[ \xymatrix{ Rf_*\Omega ^\bullet _{X/S} \otimes _{q^{-1}\mathcal{O}_ T}^\mathbf {L} Rf_*\Omega ^\bullet _{X/S} \ar[d]_{\xi ' \otimes \text{id}} \ar[r]_-\mu & Rf_*\Omega ^\bullet _{X/S} \ar[d]^{\xi '} \\ Rf_*\Omega ^\bullet _{X/S}[n] \otimes _{q^{-1}\mathcal{O}_ T}^\mathbf {L} Rf_*\Omega ^\bullet _{X/S} \ar[r]^-\mu & Rf_*\Omega ^\bullet _{X/S}[n] } \]
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