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

At the generic point ${\eta }_V$ of $V$, the local ring $R={\mathcal{O}}_{W,{\eta }_V}$ is a one-dimensional Noetherian local domain. Write $r=a/b$ with nonzero $a,b\in R$ and define the order of vanishingThis is independent of the representation. The associated principal $k$-cycle iswhere only finitely many terms are nonzero.

The subgroup ${\mathrm{Rat}}_k(X)\subseteq Z_k(X)$ is generated by these cycles as $W$ ranges over integral $(k+1)$-dimensional subvarieties and $r$ over $R(W{)}^{\times }$. Two $k$-cycles are rationally equivalent when their difference lies in this subgroup, and the Chow group is

Solved by gpt-5.6-sol high.