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.

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