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
\xymatrix{ X \ar[d]_ p \ar[r]_ f & Y \ar[d]^ q \\ S \ar[r]^ h & T }
of schemes there is a canonical relative cup product
\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}
in D(Y, q^{-1}\mathcal{O}_ T) which is associative and which on cohomology reproduces the cup product discussed above.
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
\xi ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[n]
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
\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] }in D(Y, q^{-1}\mathcal{O}_ T).
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