Given $i: Z \subset X$, a closed immersion of smooth schemes over some field $k$. Is there an open subscheme $U$ of $X$ such that $Z \cap U$ is non-empty and such that the Gysin map of the Chow groups (CH is the total Chow ring here)
$$i^!: CH(U) \rightarrow CH(Z \cap U)$$
is surjective?
(Variants I would also be interested in, if these are helpful simplifications: $k$ algebraically closed or both groups tensored with $\mathbf Q$, $Z$ a divisor in $X$)
Thank you,
Jakob Scholbach
I don't think that this is true in general. Take $X=\mathbb{P}^2$, then $CH(X)$ is finitely generated. Moreover, finite generation holds for any open $U\subset X$ by the exact sequence $$CH(X-U)\to CH(X)\to CH(U)\to 0$$ [Fulton, Intersection theory I, 1.8.]
On the other when $Z$ is curve of degree $3$ or more, $CH(Z\cap U)$ is not finitely generated when $k$ is algebraically closed. To see this, observe that $CH(Z)$ contains the rational points of the Jacobian $J(Z)$, which is not finitely generated (it is uncountable if $k=\mathbb{C}$). Now use the above sequence, to conclude that $CH(Z\cap U)$ is also non finitely generated.
