Lemma 42.28.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 2}(X)$ we have
as elements of $\mathop{\mathrm{CH}}\nolimits _ k(X)$.
Proof. Write $\alpha = \sum m_ j[Z_ j]$ for some locally finite collection of integral closed subschemes $Z_ j \subset X$ with $\dim _\delta (Z_ j) = k + 2$. Consider the proper morphism $p : \coprod Z_ j \to X$. Set $\alpha ' = \sum m_ j[Z_ j]$ as a $(k + 2)$-cycle on $\coprod Z_ j$. By several applications of Lemma 42.26.4 we see that $c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap c_1(p^*\mathcal{N}) \cap \alpha ')$ and $c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{N}) \cap c_1(p^*\mathcal{L}) \cap \alpha ')$. Hence it suffices to prove the formula in case $X$ is integral and $\alpha = [X]$. In this case the result follows from Lemma 42.28.1 and the definitions. $\square$
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