The Stacks project

Proof. Since both $i_* \circ j^* \circ C$ and $p^*$ are bivariant classes (see Lemmas 42.33.2 and 42.33.4) we can use the equation

(suitably interpreted) to define $c(Z \to X, \mathcal{N})$ as a bivariant class. This works because $p^*$ is always bijective on chow groups by Lemma 42.36.3.

Let $X' \to X$, $Z' \to X'$, and $\mathcal{N}'$ be as in Lemma 42.54.1. Write $c = c(Z \to X, \mathcal{N})$ and $c' = c(Z' \to X', \mathcal{N}')$. The second statement of the lemma means that $c'$ is the restriction of $c$ as in Remark 42.33.5. Since we claim this is true for all $X'/X$ locally of finite type, a formal argument shows that it suffices to check that $c' \cap \alpha ' = c \cap \alpha '$ for $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _ k(X')$. To see this, note that we have a commutative diagram

which induces closed immersions:

To get $c \cap \alpha '$ we use the class $C \cap \alpha '$ defined using the morphism $W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'} \to \mathbf{P}^1_{X'}$ in Lemma 42.48.1. To get $c' \cap \alpha '$ on the other hand, we use the class $C' \cap \alpha '$ defined using the morphism $W' \to \mathbf{P}^1_{X'}$. By Lemma 42.48.3 the pushforward of $C' \cap \alpha '$ by the closed immersion $W'_\infty \to (W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'})_\infty $, is equal to $C \cap \alpha '$. Hence the same is true for the pullbacks to the opens

by Lemma 42.15.1. Since we have a commutative diagram

these classes pushforward to the same class on $N'$ which proves that we obtain the same element $c \cap \alpha ' = c' \cap \alpha '$ in $\mathop{\mathrm{CH}}\nolimits _*(Z')$. $\square$