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The Stacks project

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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