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

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    $\begingroup$ The base change homomorphism is not always injective. For a smooth, projective, rationally (chain) connected variety $Y$ over a field $\kappa$, let $k$ be $\kappa(Y)$, and let $X$ be the generic fiber of $$\text{pr}_1:Y\times_{\text{Spec}\ \kappa}Y \to Y.$$ Consider the cycle class of the diagonal, and consider the cycle class of $Y\times\{y_0\}$, where $y_0$ is a $\kappa$-point. These induce zero cycles on $X$ whose images in $\text{CH}^1(X_{\overline{k}})$ are equal. However, these cycles are equal in $\text{CH}^1(X)$ if and only if there is an integral decomposition of the diagonal. $\endgroup$
    – Jason Starr
    Jan 2, 2018 at 14:54
  • $\begingroup$ I do not quite understand the question. It is true that the scheme $X$ over $k$ is the base change of $Y$ by the field extension $k/\kappa$. However, there are birational modifications where that is no longer true, e.g., if we blow up $Y\times_{\text{Spec}\ \kappa} Y$ along a "typical" subvariety that dominates $Y$ with respect to $\text{pr}_1$. For an appropriate blowing up $\mathcal{X}\to Y\times_{\text{Spec}\ \kappa}Y$, for the generic fiber $\widetilde{X}$ over $\text{Spec}\ k$, the smallest "field of definition" of $X$ equals the field $k$. $\endgroup$
    – Jason Starr
    Jan 2, 2018 at 19:16
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    $\begingroup$ @Qixiao: Jason's example is about the base change from $K(Y)$ to $\overline{K(Y)}$ (not $\kappa$ to $K(Y)$), so that is a base change to the algebraic closure. $\endgroup$
    – R. van Dobben de Bruyn
    Jan 2, 2018 at 19:26
  • $\begingroup$ Typo correction: The appropriate Chow group is $\text{CH}_0(X)$, not $\text{CH}^1(X)$. $\endgroup$
    – Jason Starr
    Jan 2, 2018 at 20:28
  • $\begingroup$ @JasonStarr Sorry I misunderstood your example, thank you! $\endgroup$
    – user39380
    Jan 3, 2018 at 2:20

1 Answer 1

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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:

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.

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:

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).

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:

Karpenko, Nikita A.(3-AB-MS) Chow ring of generically twisted varieties of complete flags. (English summary) Adv. Math. 306 (2017), 789–806.

A number of these have a lot of torsion in their Chow groups (see Examples 3.17-3.22).

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