Let $\zeta=c_1(\mathcal O_{\mathbb P(E)}(1))$. The <projective bundle formula for Chow groups> is the map
$$
\Theta_E:\bigoplus_{i=0}^{e}A_{k-e+i}(X)\longrightarrow A_k(\mathbb P(E)),
\qquad
(\alpha_0,\ldots,\alpha_e)\longmapsto
\sum_{i=0}^{e}\zeta^i\cap p^*\alpha_i.
$$
Flat pullback raises dimension by $e$, and intersection with $\zeta^i$ lowers it by $i$, so every summand has dimension $k$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 162 / 2 / i / Solution

For a rank-$e+1$ vector bundle $E\to X$ and $p:\mathbb P(E)\to X$, powers of $\zeta=c_1(\mathcal O(1))$ give isomorphisms
$$
\bigoplus_{i=0}^e A_{k-e+i}(X)\xrightarrow{\sim}A_k(\mathbb P(E)).
$$


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