The Stacks project

Proof. It suffices to prove this on generators for $\mathop{\mathrm{CH}}\nolimits _*(P)$. Thus it suffices to prove this for a cycle class of the form $\xi ^ i \cdot p^*\alpha$ where $0 \leq i \leq r - 1$ and $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(X)$. Note that $p_*(\xi ^ i \cdot p^*\alpha ) = 0$ if $i < r - 1$ and $p_*(\xi ^{r - 1} \cdot p^*\alpha ) = \alpha$. On the other hand, we have $\gamma (\xi ^ i \cdot p^*\alpha ) = c^ i \cup p^*\gamma (\alpha )$ and by the projection formula (Lemma 45.9.1) we have

Thus it suffices to show that $p_*c^ i = 0$ for $i < r - 1$ and $p_*c^{r - 1} = 1$. Equivalently, it suffices to prove that $\lambda _ P : H^{2d + 2r - 2}(P)(d + r - 1) \to F$ defined by the rules

satisfies the condition of axiom (A5). This follows from the computation of the class of the diagonal of $P$ in Lemma 45.6.2. $\square$