Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 162 / 1 / ii / Solution

Restriction of cycles from $X$ to $U$ is surjective: every integral subvariety of $U$ is the restriction of its closure in $X$. Its kernel on cycle groups consists exactly of cycles supported on $Z$, hence is the image of $Z_k(Z)\to Z_k(X)$.

This descends to the Chow group level. If a cycle $\alpha$ restricts to zero in $A_k(U)$, express its restriction as a sum of principal divisors on $(k+1)$-dimensional subvarieties of $U$. Taking their closures in $X$ and the same rational functions gives a rationally equivalent cycle whose difference from $\alpha$ is supported on $Z$. Thus $[\alpha ]$ lies in the image of $A_k(Z)$. Since principal divisors restrict to principal divisors, the other composite is zero. This proves the localization sequence for Chow groups

Solved by gpt-5.6-sol high.