Lemma 42.38.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Then $c_ j(\mathcal{E}|_ D) \cap i^*\alpha = i^*(c_ j(\mathcal{E}) \cap \alpha )$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$.

**Proof.**
Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have

in the chow group of the projective bundle $(\pi : P \to X, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram

Note that $\mathcal{O}_{P_ D}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Apply the gysin map $(i')^*$ (Lemma 42.30.2) to the displayed equation above. Applying Lemmas 42.30.4 and 42.29.9 we obtain

in the chow group of $P_ D$. By the characterization of Lemma 42.38.2 we conclude. $\square$

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