Let $X$ be a smooth open sub-variety of a projective, not necessarily smooth, variety $X'$, defined over a finite field.

Is $\text{Pic}(X)$ a finitely generated abelian group?

I'm tempted to just say: call $Z$ the closed complement of $X$ in $X'$. We have a localization sequence:

$$\text{CH}^1(Z)\to\text{CH}^1(X')\to\text{CH}^1(X)\to 0$$

and we may replace $X$ with $X'$.

Specifically:

is the localization sequence for Chow groups available also when some of the varieties involved are not smooth?

For $P$ a projective, not necessarily smooth variety over a finite field, is $\text{CH}^1(P)$ finitely generated?

Thanks

**Previously**, my question was assuming $X'$ to be smooth, in addition to projectivity. In this case indeed $\text{CH}^1(X')$ is finitely generated. It is an extension of $\text{Pic}^0(X')$ by $\text{NS}(X')$. The latter is finitely generated by the Theorem of the Base, and the former is finite because it is a subgroup of $\underline{\text{Pic}}^0_{X'/k}(k)$.

**Remark** By the above argument, if $\text{CH}^1(P)$ is finitely generated whenever $P$ is **projective**, then so it is for all quasi-projective varieties over $k$. Is this true?

smoothprojective case. $\endgroup$