Homotopy invariance gives
$$
A_k(\mathbb A^n)=
\begin{cases}
\mathbb Z[\mathbb A^n],&k=n,\\
0,&k\ne n,
\end{cases}
$$
or, in cohomological grading, $A^*(\mathbb A^n)=\mathbb Z$ 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 $A_*(X_i)$ is generated by the closures of the cells appearing up to stage $i$. Hence $A_*(X)$ 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 $0,\ldots,r$. Consequently
$$
A_k(\mathbb P^r)\cong\mathbb Z
$$
for $0\leq k\leq r$, generated by a linear $\mathbb P^k$, and zero otherwise. Equivalently, the <Chow ring of projective space> is $\mathbb Z[H]/(H^{r+1})$.

Solved by gpt-5.6-sol high.

Solution

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

A cellular decomposition filters a scheme by closed subschemes whose successive differences are disjoint unions of affine spaces. The closures of the cells generate its Chow groups.


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 Incoming links: Cellular decomposition of a scheme