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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