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Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?

This statement appears to be easy if $K$ is stably rational over $k$. What about other extensions; any references on this matter?

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    $\begingroup$ I assume that you aware that you are asking which projective $k$-varieties have an integral decomposition-of-the-diagonal. $\endgroup$
    – Jason Starr
    Commented Jul 18, 2020 at 18:06
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    $\begingroup$ I believe this is in general open. When K is purely transcendental over k, one can have injectivity maybe? But surjectivity does not seem immediate to me in general. Though I think there are some works in this direction, by e.g. Vishik, Karpenko, Merkurjev, Zainoulline, S. Gille, R. Fino, S. Baek, etc. and I think you may find some partial answers from some of their papers. Like A. Vishik, in Geometric methods in the algebraic theory of quadratic forms, 25–101, Lecture Notes in Math., 1835, Springer, Berlin, 2004? This long paper seems to consider the function field case. $\endgroup$
    – Jinhyun Park
    Commented Jul 19, 2020 at 2:42
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    $\begingroup$ Yes, this question may be attacked by means of decomposing the diagonal; yet I am not sure that looking at smooth projectives is optimal here. On the other case, the injectivity question appears to be much easier; it suffices to have a degree 1 zero-cycle on any (and thus, on all) smooth X such that $k(X)=K$. $\endgroup$
    – Mikhail Bondarko
    Commented Jul 19, 2020 at 10:19

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