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For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups $$Ch^i(Y)\rightarrow Ch^i(X).$$ A particular example of this is of course an open immersion $U\rightarrow X$. In that case we in fact know that the induced morphism on Chow groups is surjective (and we also know the kernel). Hence I'm curious if this localization property is also true for finer topologies, i.e. how fine can we make our topology such that we still have surjectivity of restriction?

I think this fails already for the Nisnevich topology. Consider a scheme $X$ and the Nisnevich open $$X\coprod X\rightarrow X$$ then we don't get a surjection $$Ch^i(X)\rightarrow Ch^i(X\coprod X)\cong Ch^i(X)\times Ch^i(X), \quad [D]\mapsto ([D],[D]).$$ The main examples I would like to be able to apply this would be for a flat base change, i.e. if $X$ is defined over some field $k$, and we compare the chow groups $$Ch^i(X) \rightarrow Ch^i(X\times_{\text{Spec} k} \text{Spec } l)$$ for some (possibly infinite) field extension $l/k$.

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    $\begingroup$ Note that $X \amalg X$ is already a covering of $X$ in the (big) Zariski site. Somehow the objects that appear in the site and the families that form coverings are separate (unrelated) notions. This question is only about objects and does not see the Grothendieck (pre)topology. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Aug 15, 2020 at 21:52
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    $\begingroup$ For $X$ a smooth projective curve of genus $\geq 1$ defined over $\mathbb{Q}$, $CH^1(X)=\operatorname{Pic}(X) $ is finitely generated, while $CH^1(X\otimes _{\mathbb{Q}}\mathbb{C})$ is an extension of $\mathbb{Z} $ by a complex torus. $\endgroup$
    – abx
    Commented Aug 16, 2020 at 3:41

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