The Stacks project

Lemma 42.50.6. In Situation 42.50.1 we have

\[ P_ p(Z \to X, E) \cap i_*\alpha = P_ p(E|_ Z) \cap \alpha , \quad \text{resp.}\quad c_ p(Z \to X, E) \cap i_*\alpha = c_ p(E|_ Z) \cap \alpha \]

in $\mathop{\mathrm{CH}}\nolimits _*(Z)$ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(Z)$.

Proof. We only prove the second equality and we omit the proof of the first. Since $c_ p(Z \to X, E)$ is a bivariant class and since the base change of $Z \to X$ by $Z \to X$ is $\text{id} : Z \to Z$ we have $c_ p(Z \to X, E) \cap i_*\alpha = c_ p(Z \to X, E) \cap \alpha $. By Lemma 42.50.4 the restriction of $c_ p(Z \to X, E)$ to $Z$ (!) is the localized Chern class for $\text{id} : Z \to Z$ and $E|_ Z$. Thus the result follows from ( with $X = Z$. $\square$

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