Timeline for Chow group and base change
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2018 at 2:20 | vote | accept | CommunityBot | ||
| Jan 3, 2018 at 2:20 | comment | added | user39380 | @JasonStarr Sorry I misunderstood your example, thank you! | |
| Jan 2, 2018 at 22:59 | answer | added | Eoin | timeline score: 5 | |
| Jan 2, 2018 at 20:28 | comment | added | Jason Starr | Typo correction: The appropriate Chow group is $\text{CH}_0(X)$, not $\text{CH}^1(X)$. | |
| Jan 2, 2018 at 19:26 | comment | added | R. van Dobben de Bruyn | @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. | |
| Jan 2, 2018 at 19:16 | comment | added | Jason Starr | 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 $k$. | |
| Jan 2, 2018 at 14:54 | comment | added | Jason Starr | 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. | |
| Jan 2, 2018 at 14:48 | history | edited | user39380 | CC BY-SA 3.0 |
added 25 characters in body
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| Jan 2, 2018 at 14:43 | history | asked | user39380 | CC BY-SA 3.0 |