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 vanishing>
$$
\operatorname{ord}_V(r)=\operatorname{length}_R(R/(a))-\operatorname{length}_R(R/(b)).
$$
This is independent of the representation. The associated principal $k$-cycle is
$$
[\operatorname{div}(r)]=\sum_{\substack{V\subseteq W\\\dim V=k}}\operatorname{ord}_V(r)[V]\in Z_k(X),
$$
where only finitely many terms are nonzero.

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

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-162/1/i/solution