Timeline for Is $G \rightarrow G/P$ surjective on $K$-points over a local field?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 15, 2015 at 16:06 | comment | added | LSpice | @DanielLitt, yes, of course you're right. Thanks for clearing up my confusion. | |
| Jan 15, 2015 at 7:32 | comment | added | Daniel Litt | @LSpice: That's the case for maps which are quotients by free group actions (as in QuestionMark's example) but the situation is rather more complicated in general. The sense in which my example is "cohomological" is that the curve in question exhibits a Severi-Brauer curve whose associated Brauer class (in $H^2(K, \mu_2)$) is non-trivial. | |
| Jan 14, 2015 at 22:56 | comment | added | LSpice | @DanielLitt, probably I'm being foolish, but isn't the failure of surjectivity always measured by the non-vanishing of a first cohomology set (or at least the non-injectivity of a certain map on first cohomology sets)? | |
| Jan 14, 2015 at 8:13 | comment | added | Daniel Litt | @LSpice: In some sense, the two examples are of the same nature; QuestionMark's example is related to the non-vanishing of $H^1(K, \mu_n)$ whereas mine is related to the non-vanishing of $H^2(K, \mu_2)$. | |
| Jan 13, 2015 at 20:41 | comment | added | LSpice | @DanielLitt, there is probably some sort of personality test inherent in the question of whether your example or QuestionMark's is the first one to come to mind. :-) | |
| Oct 18, 2014 at 21:09 | comment | added | Daniel Litt | Even more explicit, how about the map $C\to \text{Spec}(\mathbb{R})$ where $C$ is the smooth projective conic defined by $x^2+y^2+z^2=0$? | |
| Oct 18, 2014 at 15:12 | vote | accept | Question Mark | ||
| Oct 18, 2014 at 12:34 | comment | added | user27920 | @abx: For a more geometric example consider a smooth projective geometrically connected curve $X_0$ over the finite residue field $k$ of $K$ such that $X_0(k)$ is empty, and let $X$ be a proper flat lift of $X_0$ over $O_K$. The generic fiber $X_K$ is smooth, proper, and geometrically connected curve over $K$ with no $K$-points. | |
| Oct 18, 2014 at 9:53 | answer | added | Peter McNamara | timeline score: 8 | |
| Oct 18, 2014 at 6:12 | comment | added | Question Mark | @abx: Nope, not so: consider $\mathrm{SL}_n(K) \rightarrow \mathrm{PGL}_n(K)$ (for $K$ of characteristic $0$) for one of many counterexamples. | |
| Oct 18, 2014 at 5:50 | comment | added | abx | Any smooth surjective morphism is surjective on $K$-points. | |
| Oct 18, 2014 at 5:00 | history | asked | Question Mark | CC BY-SA 3.0 |