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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