Lemma 50.8.3. Assume $X \to S$ and $Y \to S$ are smooth and quasi-compact and the morphisms $X \to X \times _ S X$ and $Y \to Y \times _ S Y$ are affine. Then the relative cup product

is an isomorphism in $D(\mathcal{O}_ S)$.

Lemma 50.8.3. Assume $X \to S$ and $Y \to S$ are smooth and quasi-compact and the morphisms $X \to X \times _ S X$ and $Y \to Y \times _ S Y$ are affine. Then the relative cup product

\[ Ra_*\Omega ^\bullet _{X/S} \otimes _{\mathcal{O}_ S}^\mathbf {L} Rb_*\Omega ^\bullet _{Y/S} \longrightarrow Rf_*\Omega ^\bullet _{X \times _ S Y/S} \]

is an isomorphism in $D(\mathcal{O}_ S)$.

**Proof.**
Immediate consequence of Lemma 50.8.2.
$\square$

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