Lemma 81.18.2. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{L}$, $\mathcal{N}$ be an invertible sheaves on $X$. Then

$c_1(\mathcal{L}) \cap \alpha + c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}) \cap \alpha$

in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ for every $\alpha \in Z_{k - 1}(X)$. Moreover, $c_1(\mathcal{O}_ X) \cap \alpha = 0$ for all $\alpha$.

Proof. The additivity follows directly from Spaces over Fields, Lemma 71.7.5 and the definitions. To see that $c_1(\mathcal{O}_ X) \cap \alpha = 0$ consider the section $1 \in \Gamma (X, \mathcal{O}_ X)$. This restricts to an everywhere nonzero section on any integral closed subspace $W \subset X$. Hence $c_1(\mathcal{O}_ X) \cap [W] = 0$ as desired. $\square$

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