Lemma 82.22.4. In Situation 82.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 82.22.1. Let $\alpha $ be a $(k + 1)$-cycle on $X$. Then $i_*i^*\alpha = c_1(\mathcal{L}) \cap \alpha $ in $\mathop{\mathrm{CH}}\nolimits _ k(X)$. In particular, if $D$ is an effective Cartier divisor, then $D \cdot \alpha = c_1(\mathcal{O}_ X(D)) \cap \alpha $.
Proof. Write $\alpha = \sum n_ j[W_ j]$ where $i_ j : W_ j \to X$ are integral closed subspaces with $\dim _\delta (W_ j) = k$. Since $D$ is the vanishing locus of $s$ we see that $D \cap W_ j$ is the vanishing locus of the restriction $s|_{W_ j}$. Hence for each $j$ such that $W_ j \not\subset D$ we have $c_1(\mathcal{L}) \cap [W_ j] = [D \cap W_ j]_ k$ by Lemma 82.18.4. So we have
\[ c_1(\mathcal{L}) \cap \alpha = \sum \nolimits _{W_ j \not\subset D} n_ j[D \cap W_ j]_ k + \sum \nolimits _{W_ j \subset D} n_ j i_{j, *}(c_1(\mathcal{L})|_{W_ j}) \cap [W_ j]) \]
in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ by Definition 82.18.1. The right hand side matches (termwise) the pushforward of the class $i^*\alpha $ on $D$ from Definition 82.22.1. Hence we win. $\square$
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