Let $k$ be a field with algebraic closure $\overline{k}$. Let $f\colon X\to k$ be a smooth projective variety(geometrically connected) over $k$. 


Is the base change map $$\phi\_i\colon \mathrm{CH}^i(X)\to\mathrm{CH}^{i}(X\_{\overline{k}})$$ always injective?


(If $i=1$, $\mathrm{CH}^1(X)=\mathrm{Pic}(X)$, the Hochschild-Serre spectral sequence gives $$0\to H^1(\mathrm{Gal}(\overline{k}/k),f\_{\overline{k},\*}\mathbb{G}\_m)\to\mathrm{Pic}(X)\to\mathrm{Pic}\_{X\_\overline{k}/\overline{k}}(k)\to H^2(\mathrm{Gal}(\overline{k}/k),f\_{\overline{k},\*}\mathbb{G}\_m)$$
The condition implies $f\_{\overline{k},\*}\mathbb{G}\_m=\mathbb{G}\_m$, so $H^1(\mathrm{Gal}(\overline{k}/k),f\_{\overline{k},\*}\mathbb{G}\_m)=0$, and $\mathrm{Pic}\_{X\_{\overline{k}}/\overline{k}}(k)\to\mathrm{Pic}\_{X\_{\overline{k}}/\overline{k}}(\overline{k})$ is injective, we know $\phi\_1$ is injective. 


If $X$ is quasi-projective, then $\phi\_1$ is not injective, for example $X=\mathbb{P}^1\_{\mathbb{R}}-\{\pm i\}$, then $\mathcal{O}(1)$ is a nontrivial element in $\mathrm{ker}(\phi\_1)$.) 



Chow group and base change
No, this is not true in general.


A counterexample occurs already for Severi-Brauer varieties. Since the Chow group $\text{CH}(\mathbf{P}^n)$ is torsion free, it's enough to show there are Severi-Brauer varieties with torsion in their Chow groups. This was (I think) first observed in:



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> Merkurjev, A. S. Certain K-cohomology groups of Severi-Brauer varieties. K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 319–331, Proc. Sympos. Pure Math., 58, Part 2, Amer. Math. Soc., Providence, RI, 1995.
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Merkurjev uses the BGQ spectral sequence to show that certain differentials that compute the Chow groups are nonzero. Since it's also known from the Grothendieck-Riemann-Roch without denominators that the image of these differentials is contained in the torsion subgroup, this proves the claim.


More precise results in this direction appeared in:



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> Karpenko, Nikita A. Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), no. 4, 305–330. (Available at the authors website <https://sites.ualberta.ca/~karpenko/publ/ch2.pdf>).
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Karpenko uses the gamma filtration on the Grothendieck ring of Severi-Brauer varieties to explicitly construct torsion elements in $\text{gr}\_\gamma^2K(-)$ for certain Severi-Brauer varieties. He then shows this is equal to $\text{CH}^2(-)$ for Severi-Brauer varieties of certain indecomposable algebras.


More recently, Karpenko has provided a complete description of the Chow group of certain Severi-Brauer varieties in:



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> Karpenko, Nikita A.(3-AB-MS)
>  Chow ring of generically twisted varieties of complete flags. (English summary) 
>  Adv. Math. 306 (2017), 789–806.
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A number of these have a lot of torsion in their Chow groups (see Examples 3.17-3.22).


