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

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

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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

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Homotopy invariance givesor, in cohomological grading, $A^{\ast }({\mathbb{A}}^n)=\mathbb{Z}$ concentrated in degree zero.

Apply the localization sequence successively to $X_{i-1}\subseteq X_i$. Since the Chow group of each stratum is freely generated in the dimensions of its affine components, induction shows that $A_{\ast }(X_i)$ is generated by the closures of the cells appearing up to stage $i$. Hence $A_{\ast }(X)$ is generated by the classes $[V_{i,j}]$ of a cellular decomposition of a scheme.

The standard cells of ${\mathbb{P}}^r$ have one affine space in each dimension $0,\dots ,r$. Consequentlyfor $0\le k\le r$, generated by a linear ${\mathbb{P}}^k$, and zero otherwise. Equivalently, the Chow ring of projective space is $\mathbb{Z}[H]/(H^{r+1})$.

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The exceptional divisor is $E\cong {\mathbb{P}}^{r-1}$, so its Chow groups are generated by the linear subspaces $[{\Gamma }_k]$. The complement $X\setminus E$ is isomorphic through the blow-up map to ${\mathbb{P}}^r\setminus \{p\}$. The localization sequence for $\{p\}\subseteq {\mathbb{P}}^r$ shows that the latter's Chow groups are generated by the restrictions of linear spaces $H_j$ through $p$ for $1\le j\le r$; its zero-dimensional Chow group vanishes.

Under the complement isomorphism, $H_j\setminus \{p\}$ corresponds to the open part of the strict transform ${\tilde{H}}_j$. A second localization sequence, now for $E\subseteq X$, shows that $A_{\ast }(X)$ is generated by the lifts $[{\tilde{H}}_j]$ together with the images $[{\Gamma }_k]$ from $A_{\ast }(E)$, exactly as claimed.

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