 Solution

Compactify the total space of $E$ as $\overline{E}=\mathbb{P}(E\oplus {\mathcal{O}}_X)$; its hyperplane at infinity is $\mathbb{P}(E)$ and its open complement is $E$. Under the projective bundle formula for $\overline{E}$, the pushforward from that hyperplane spans the summands containing positive powers of $\zeta$. The localization quotient therefore retains the zeroth summandwhose restriction to $E$ is precisely the vector-bundle pullback ${\pi }^{\ast }:A_k(X)\to A_{k+e+1}(E)$. It survives injectively. In fact this proves the stronger homotopy invariance of Chow groups: ${\pi }^{\ast }$ is an isomorphism.

Solved by gpt-5.6-sol high.