Remark 42.63.4. The upshot of Lemmas 42.63.2 and 42.63.3 is the following. Let $(S, \delta )$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(X)$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \in A^ q(Y \to Z)$. Then

$\alpha \times (c' \cap \beta ) = c' \cap (\alpha \times \beta )$

in $\mathop{\mathrm{CH}}\nolimits _*(X \times _ S Y)$ for any $\beta \in \mathop{\mathrm{CH}}\nolimits _*(Z)$. Namely, this follows by taking $c = c_\alpha \in A^*(X \to S)$ the bivariant class corresponding to $\alpha$, see proof of Lemma 42.63.2.

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