I have two questions regarding the Chow Ring $A_*(X)$ of an affine scheme $X$.
- Find an affine smooth variety $X/\mathbb{C}$ such that $A_*(X) \ncong \mathbb{Z}$: My idea: we know that $A_n(X)=\mathbb{Z}$ generated by its irreducible component $[X]$, so we just have to find a non trivial $A_k$ for $k<n$. If we let $E$ be a smooth projective elliptic curve in $\mathbb{P}^2$ then projecting away from a point gives an exact $$A_0(pt) \rightarrow A_0(E) \rightarrow A_0(U) \rightarrow 0$$ where $U=X$ is an affine curve and $A_0(pt)=\mathbb{Z}$, $A_0(E)\cong Pic(E)=Pic_0(E)\oplus\mathbb{Z}$ via the first Chern map, hence its cokerel is not trivial, $Pic_0(E)$ corresponding to the group structure endowed on $E$. Does that work?
- Do we even find any nonempty scheme $X/k$, finite type over some $k$, such that its entire chow ring vanishes?