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 $\mathrm{ker}(\phi_1)$.)

isa base change to the algebraic closure. $\endgroup$ – R. van Dobben de Bruyn Jan 2 '18 at 19:26