I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which is the intersection of two hypersurface sections in $X$. Is the pushforward $CH_i(Z)\otimes \mathbb{Q}\rightarrow CH_i(X)\otimes \mathbb{Q}$ injective?
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2$\begingroup$ This is not true for surfaces $S \subseteq \mathbf P^3$ (embed them linearly in $\mathbf P^4$ if you insist on codimension $2$ complete intersections). The Picard rank can be more than $1$. $\endgroup$– R. van Dobben de BruynCommented Jan 3, 2021 at 5:35
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$\begingroup$ Not true either for a curve in $\Bbb{P}^3$, for the same reason. $\endgroup$– abxCommented Jan 3, 2021 at 8:16
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1$\begingroup$ @abx over finite fields and when tensored with $\mathbf Q$, it should be injective for curves, as $\operatorname{Pic}^0(C)$ is torsion and smooth projective curves have Picard rank $1$. $\endgroup$– R. van Dobben de BruynCommented Jan 3, 2021 at 21:19
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$\begingroup$ @R. van Dobben de Bruyn: Oops, you are right of course. I forgot the $\otimes \mathbb{Q}$. $\endgroup$– abxCommented Jan 4, 2021 at 7:01
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