Solution
= Solution
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.