Lemma 42.50.7. In Situation 42.50.1 if $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ has support disjoint from $Z$, then $P_ p(Z \to X, E) \cap \alpha = 0$, resp. $c_ p(Z \to X, E) \cap \alpha = 0$.

**Proof.**
This is immediate from the construction of the localized Chern classes. It also follows from the fact that we can compute $c_ p(Z \to X, E) \cap \alpha $ by first restricting $c_ p(Z \to X, E)$ to the support of $\alpha $, and then using Lemma 42.50.4 to see that this restriction is zero.
$\square$

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