Chow groups of projective space

Affine space. Applying homotopy invariance [L1] successively in the $n$ coordinates gives $A_d({\mathbb{A}}_k^n)\cong A_{d-n}(\mathrm{Spec}k)$, which is $\mathbb{Z}$ for $d=n$ and $0$ otherwise; the top class is $[{\mathbb{A}}^n]$ and for $n=1$ every closed point is the principal divisor of its monic irreducible polynomial in $k[t]$, which is linear precisely for a rational point.

Generation of the Chow groups of projective space. For $n=0$, projective space is $\mathrm{Spec}k$ and the assertion is immediate. Assume $n\ge 1$. Let $H={\mathbb{P}}^{n-1}\subseteq {\mathbb{P}}^n$ be a coordinate hyperplane with complement ${\mathbb{A}}^n$. Localization [L1] gives the exact sequence $A_d({\mathbb{P}}^{n-1})\stackrel{i_{\ast }}{\to }{A}_d({\mathbb{P}}^n)\stackrel{j^{\ast }}{\to }{A}_d({\mathbb{A}}^n)\to 0$. For $d<n$ the group $A_d({\mathbb{A}}^n)$ vanishes by step 1.1, so $i_{\ast }$ is surjective, and induction on $n$ proves that $A_d({\mathbb{P}}^n)$ is generated by the class $[{\mathbb{P}}^d]$ of a $d$-dimensional linear subspace: in the hyperplane the class of a $d$-dimensional linear subspace generates by induction, and its pushforward is the class of the corresponding linear subspace of ${\mathbb{P}}^n$; the induction starts at $n=d$, where ${\mathbb{P}}^d$ has no hyperplane below it. For $d=n$ the same exact sequence reads $0=A_n({\mathbb{P}}^{n-1})\to A_n({\mathbb{P}}^n)\to A_n({\mathbb{A}}^n)\cong \mathbb{Z}\to 0$ by step 1.1, so $A_n({\mathbb{P}}^n)=\mathbb{Z}[{\mathbb{P}}^n]$ is generated by the top class, and the cycle group $Z_n({\mathbb{P}}^n)=\mathbb{Z}[{\mathbb{P}}^n]$ admits no nonzero rational equivalences because none of its subvarieties has dimension $n+1$. For $d<0$ or $d>n$ there are no integral closed subschemes of dimension $d$, so $A_d({\mathbb{P}}^n)=0$.

Independence. Fix $d$ with $0\le d\le n$ and let $c_1=c_1(\mathcal{O}(1))\cap -$; by [L2] the iterated cap $c_1^d$ maps $A_m({\mathbb{P}}^n)$ to $A_{m-d}({\mathbb{P}}^n)$, is additive, and is defined by cutting with $d$ coordinate hyperplanes in general position. On the linear subspace ${\mathbb{P}}^d$, the $d$ coordinate hyperplanes cut it in a single $k$-rational point, so $c_1^d[{\mathbb{P}}^d]=[{\mathbb{P}}^0]$, and the degree homomorphism $\mathrm{deg}:A_0({\mathbb{P}}^n)\to \mathbb{Z}$ of [L3] sends $[{\mathbb{P}}^0]$ to $1$; hence $\mathrm{deg}\circ c_1^d$ is a homomorphism $A_d({\mathbb{P}}^n)\to \mathbb{Z}$ sending the generator $[{\mathbb{P}}^d]$ of step 2.1 to $1$. A cyclic group admitting a homomorphism onto $\mathbb{Z}$ with generator mapping to $1$ is infinite cyclic, so $A_d({\mathbb{P}}^n)=\mathbb{Z}\cdot [{\mathbb{P}}^d]$.

The degree isomorphism and closed points. By steps 2.1 and 3.1, $A_0({\mathbb{P}}^n)$ is generated by $[{\mathbb{P}}^0]$ for a $k$-rational point; for a closed point $x$ with residue field $\kappa (x)$, proper pushforward to $\mathrm{Spec}k$ is multiplication by $[\kappa (x):k]$ by [L3], so $[x]=[\kappa (x):k][{\mathbb{P}}^0]$ and $\mathrm{deg}$ is the stated isomorphism; a $0$-cycle supported on $k$-rational points has class $({\sum }_x{m}_x)[{\mathbb{P}}^0]$. The computation over $\mathbb{C}$ is the classical cellular decomposition under the same identification.