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I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely generated.

What is the standard name for this conjecture? In private communication people referred to it as Beilinson's conjecture. I assume that it should have been formulated before Beilinson. What is the best paper to cite for this conjecture? Is it known in any non-trivial case?

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    $\begingroup$ This might be related to Bass's conjecture. I am not really an expert, though. $\endgroup$ – R. van Dobben de Bruyn Dec 16 '17 at 1:37
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    $\begingroup$ For any smooth variety $X$, you have a filtration $\{F^iK_0(X)\}$ of the $K_0$ and natural identification of $F^i/F^{i+1}\otimes \mathbb{Q}$ with $CH^i(X)\otimes\mathbb{Q}$, by Grothendieck Rieman-Roch without denominators. So, Bass conjecture for $K_0$ is equivalent to what you ask. I do not know how much is known for either. $\endgroup$ – Mohan Dec 16 '17 at 2:34
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    $\begingroup$ @Mohan I think that Bass conjecture is not precisely the same conjecture as one I am asking about. The reason is that varieties over number fields are not schemes of finite type over $\mathbf Z$. And I am not sure how spreading out techniques work for $K$-theory/Chow groups. $\endgroup$ – gdb Dec 16 '17 at 2:59
  • $\begingroup$ @gdb my thoughts as well. Although I think the same is conjectured over finite fields, where we also don't know anything. In that case it really does seem to follow from Bass. $\endgroup$ – R. van Dobben de Bruyn Dec 16 '17 at 3:03
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    $\begingroup$ It is stated as Swinnerton-Dyer conjecture in Beilinson's Height pairings between algebraic cycles (conjecture 5.0). $\endgroup$ – Watson Sep 30 '18 at 11:29
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The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own.

If $\Sigma$ is a sufficiently large finite set of primes, then $X$ will admit a smooth model $\mathfrak{X}$ over $\mathcal{O}_{K, \Sigma}$. Since $\mathfrak{X}$ is a finite-type $\mathbf{Z}$-scheme, Bass' conjecture implies that all Chow groups of $\mathfrak{X}$ are finitely generated. [Edit: as user "guest" points out, this last step only works after $\otimes \mathbf{Q}$; to get finite generation with $\mathbf{Z}$ coefficients you need something slightly stronger, the "motivic Bass conjecture" which is the conjecture that finite-type $\mathbf{Z}$-schemes have finitely-generated motivic cohomology.]

So it suffices to check that the natural map $CH^i(\mathfrak{X}) \to CH^i(X)$ is surjective, which is easy, because any codimension $i$ cycle on $X$ has a scheme-theoretic closure which is a codimension $i$ cycle on $\mathfrak{X}$.

This doesn't work for motivic cohomology in other degrees, incidentally (already $H^1(\operatorname{Spec} \mathbf{Q}, \mathbf{Q}(1)) = \mathbf{Q}^\times \otimes \mathbf{Q}$ has countably infinite dimension).

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  • $\begingroup$ I think you mean finitely generated, not finite. $\endgroup$ – TKe Dec 16 '17 at 9:03
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    $\begingroup$ Since the comparison of K-theory and Chow groups is only after tensoring with $\mathbb Q$, it is not clear why the finite-generation of K-groups (Bass conjecture) implies that the Chow groups of a finite type $\mathbb Z$-scheme is finitely generated; clearly, it does imply that the ranks are finite. Could you please say more about the finiteness of the torsion subgroups of the Chow groups? Thanks $\endgroup$ – guest Dec 16 '17 at 18:51
  • $\begingroup$ @guest You are quite right, I have edited my answer to correct this. $\endgroup$ – David Loeffler Dec 16 '17 at 20:50
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See Conjecture 5.0 (attributed to Swnnerton-Dyer) in the paper "Height pairing between algebraic cycles" by Beilinson. The paper by Swinnerton-Dyer that Beilinson refers to is "The conjectures of Birch-Swinnerton-Dyer and of Tate". Please take a look in these papers.

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Just adding to Lucifer's answer: This is Conjecture 5 in Beilinson's paper https://mathscinet.ams.org/mathscinet-getitem?mr=902590

For a smooth projective variety over a number field and a fixed codimension, Tate's conjecture predicts that the rank of algebraic cycles modulo homological equivalence is given by the order of the pole of an appropriate L-function; Beilinson's conjecture predicts that the rank of homologically equivalent cycles (modulo rational equivalence) is the order of the zero of an appropriate L-function; special cases were stated by Swinnerton-Dyer, Tate, Bloch before. See 6.2 and 6.5 of Nekovar's article below for the precise statements.

Nice introductions to Beilinson's conjectures are

1) Nekovar's article

http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf

2) Ramakrishnan's article https://mathscinet.ams.org/mathscinet-getitem?mr=991982

3) The Book: available at Professor Schneider's webpage https://ivv5hpp.uni-muenster.de/u/pschnei/publ/beilinson-volume/

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