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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. $\endgroup$ Commented Feb 27, 2018 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
    Commented Feb 27, 2018 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). $\endgroup$ Commented Feb 27, 2018 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
    Commented Feb 27, 2018 at 5:45

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The paper

R. Guralnick, D. B. Jaffe, W. Raskind, and R. Wiegand, On the Picard group: Torsion and the kernel induced by a faithfully flat map, J. Algebra, vol. 183, no. 2, pp. 420–455, 1996. DOI: 10.1006/jabr.1996.0228.

proves the following finite generation result for normal schemes over more general fields.

Definition. A field $k$ is absolutely finitely generated if $k$ is finitely generated (as a field) over its prime field.

Equivalently, $k$ is absolutely finitely generated if and only if $k \cong F(t_1,\ldots,t_n)$, where $F$ is a number field or a finite field.

Theorem [GJRW, Proposition 6.1]. Let $k$ be an absolutely finitely generated field and let $X$ be a normal scheme of finite type over $k$. Then the Picard group $\mathrm{Pic}(X)$ is finitely generated.

They also show:

Theorem [GJRW, Theorem 6.6]. Let $k$ be an absolutely finitely generated field of characteristic $p>0$ and let $X$ be a scheme of finite type over $k$. Then $$\mathrm{Pic}(X) \cong \left(\substack{\text{countably generated} \\ \text{free abelian group}} \right) \oplus (\text{bounded $p$-group}) \oplus (\text{finite group}) \rlap{\ .} $$

A different paper of Jaffe (which appeared on the arXiv a bit earlier)

D. B. Jaffe, Coherent functors, with application to torsion in the Picard group, Trans. Amer. Math. Soc., vol. 349, no. 2, pp. 481–527, 1997. DOI: 10.1090/S0002-9947-97-01616-4.

shows a more refined result over finite fields:

Theorem [J, Theorem 9.6]. Let $X$ be a scheme of finite type over $\mathbb{F}_p$. Then there exists a finite $p$-group $A$ such that $$\mathrm{Pic}(X) \cong \left(\substack{\text{finitely generated} \\ \text{abelian group}} \right) \oplus \bigoplus_{i=1}^{\infty} A \rlap{\ .} $$

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  • $\begingroup$ It seems likely that over a finite field normality is unnecessary; this is easy to see if $X$ is a curve. $\endgroup$
    – naf
    Commented Sep 9, 2023 at 10:31

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