Timeline for The direct limit of invertible functions on a variety
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 19, 2021 at 10:55 | vote | accept | oleout | ||
| Jul 19, 2021 at 9:05 | review | Close votes | |||
| Jul 24, 2021 at 3:02 | |||||
| Jul 19, 2021 at 8:29 | answer | added | Martin Brandenburg | timeline score: 2 | |
| Jul 18, 2021 at 9:09 | comment | added | oleout | Alright, I will have a look at it, thanks. | |
| Jul 18, 2021 at 8:39 | comment | added | user127776 | Your last question is asking about $H_{et}^i(k,\mathbb{G}_m)$. It is torsion for $i\geq1$. Even if you replace $k$ by $X$ it is true for $i>1$. In the case of fields it can be shown it is also torsion for $i=1$. The easiest way that I can explain this is by using motivic cohomology and the fact $\mathbb{G}_m[-1]$ is the weight one etale motivic complex which rationally agrees with the motivic cohomology. But I am sure there is an elementary proof too, see page 88 here: jmilne.org/math/CourseNotes/LEC.pdf | |
| Jul 18, 2021 at 7:51 | comment | added | oleout | @user127776 most of what I encountered has the proper condition imposed on $X$, so $\bar{k}[X]^*$ is simply $\bar{k}^*$. I've never thought about the structure of $k[X]^*$ itself, could you explain it further? Also, the quotient $k[X]^*/k^*$ is finitely generated, I didn't know about it being free. | |
| Jul 18, 2021 at 7:46 | comment | added | user127776 | Are you aware that $k[X]^*$ ($k$ doesn't need to be algebraically closed) has a very nice group structure? In fact $k[X]^*/k^*$ is free abelian group of finite rank. | |
| Jul 18, 2021 at 7:12 | history | asked | oleout | CC BY-SA 4.0 |