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The integral Chow groups are infinite dimensional. Can we say something about their dimension, for example, how many elements are required to generate them? Or their vector space dimension after tensoring with $\mathbb{Q}$. What is known in this direction? I recall hearing some statements whose proof uses the theory of Chow varieties.

Also, is there a good reference for Chow varieties. I am looking for something which will assume familiarity with algebraic geometry at the level of Hartshorne and which uses that language. Since I am reading Vistoli's notes on stacks, reference to some construction using the language of stacks might also be helpful.

EDIT: Removed the remarks on countable generation.

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    $\begingroup$ A countably generated abelian group is a quotient of a direct sum of countably many abelian groups, and hence is countable. $\endgroup$
    – Emerton
    May 19, 2011 at 12:23
  • $\begingroup$ Thanks for pointing that out. It was really stupid of me. $\endgroup$
    – Rex
    May 19, 2011 at 13:15

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Chow groups are not in general countable. For example, $CH^1(X) = Pic(X)$ is uncountable for a smooth projective curve $X$ of positive genus over any uncountable algebraically closed field.

The theory of Chow varieties allows one to prove the countability of Chow groups modulo algebraic equivalence (this follows easily from the definitions). A nice reference for Chow varieties is Chapter I of the book "Rational curves on algebraic varieties" by Kollar.

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