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 summand
$$
p^*A_k(X)\subseteq A_{k+e+1}(\overline E),
$$
whose restriction to $E$ is precisely the vector-bundle pullback $\pi^*:A_k(X)\to A_{k+e+1}(E)$. It survives injectively. In fact this proves the stronger <homotopy invariance of Chow groups>: $\pi^*$ is an isomorphism.

Solution

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

For a rank-$r$ vector bundle $E\to X$, flat pullback is an isomorphism $A_k(X)\cong A_{k+r}(E)$.


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 Incoming links: Homotopy invariance of Chow groups