Lemma 42.30.3 (0F95)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.30: Gysin homomorphisms and rational equivalence
Lemma 42.30.3 (cite)

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Lemma 42.30.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Then $i^*i_* : \mathop{\mathrm{CH}}\nolimits _ k(D) \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(D)$ sends $\alpha $ to $c_1(\mathcal{L}|_ D) \cap \alpha $.

Proof. This is immediate from the definition of $i_*$ on cycles and the definition of $i^*$ given in Definition 42.29.1. $\square$

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