Lemma 42.37.6. 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 42.28.1. 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 42.37.2 we have

$\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0$

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

$\xymatrix{ P_ D = \mathbf{P}(\mathcal{E}|_ D) \ar[r]_-{i'} \ar[d]_{\pi _ D} & P \ar[d]^\pi \\ D \ar[r]^ i & X }$

Note that $\mathcal{O}_{P_ D}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Apply the gysin map $(i')^*$ (Lemma 42.29.2) to the displayed equation above. Applying Lemmas 42.29.4 and 42.28.9 we obtain

$\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ D}(1))^ i \cap \pi _ D^*(i^*\alpha _{r - i}) = 0$

in the chow group of $P_ D$. By the characterization of Lemma 42.37.2 we conclude. $\square$

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