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It is known that for smooth projective varieties $X,Y$ over $k=\bar k,$ $$CH^d(X_{k(Y)})=\varinjlim_{U\subset Y\ open}CH^d(X\times_k U)$$ I was wondering whether there was such an equality with algebraic equivalence (instead of rational equivalence).

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  • $\begingroup$ If you assume this equality then you can easily run into a certain contradiction. I will try to recall this argument here if nobody else would give an answer. $\endgroup$
    – Mikhail Bondarko
    Commented Dec 19, 2015 at 14:17

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It is false: see Remark 2.3.9 2) here.

(Appeared in AKT 1 (2016), 379-440.)

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