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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?

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    $\begingroup$ This is all correct. However, you should know that proving the theorem of the base in positive characteristic is definitely not easy. We can only access the Néron–Severi group through the $\ell$-adic and $p$-adic realisations, which does not immediately imply finite generation over $\mathbb Z$. Moreover, proving finiteness of torsion also requires work. $\endgroup$ – R. van Dobben de Bruyn Feb 27 '18 at 3:38
  • $\begingroup$ I had just slightly modified my question, removing smoothness of $X'$. It doesn't look so immediate anymore. As far as I know the Theorem of the Base is available only in the smooth projective case. $\endgroup$ – user119470 Feb 27 '18 at 3:40
  • $\begingroup$ Hmm, that complicates things. I see no reason why $\operatorname{CH}^1(X')$ is finitely generated in this case, so I don't know how to make this argument work. However, an elementary argument is given in the article by Bruno Kahn that I posted in response to this question (that looks very similar to your second question). $\endgroup$ – R. van Dobben de Bruyn Feb 27 '18 at 3:44
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    $\begingroup$ The Theorem of the Base is valid for arbitrary proper schemes over a field: see Theorem 5.1 in Exp. XIII of SGA6 for an impressive relative formulation (the proof of which uses resolution of singularities for proper surfaces over an algebraically closed field, available in all characteristics). $\endgroup$ – nfdc23 Feb 27 '18 at 5:45

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