Lemma 82.23.3 (0ERB)—The Stacks project

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Part 5: Topics in Geometry
Chapter 82: Chow Groups of Spaces
Section 82.23: Gysin homomorphisms
Lemma 82.23.3 (cite)

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Lemma 82.23.3. In Situation 82.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 82.22.1. Let $\mathcal{N}$ be an invertible $\mathcal{O}_ X$ -module. Then $i^*(c_1(\mathcal{N}) \cap \alpha ) = c_1(i^*\mathcal{N}) \cap i^*\alpha$ in $\mathop{\mathrm{CH}}\nolimits _{k - 2}(D)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(Z)$ .

Proof. With exactly the same proof as in Lemma 82.23.2 this follows from Lemmas 82.19.4, 82.21.3, and 82.23.1. $\square$

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