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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