Timeline for Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 4, 2013 at 3:15 | answer | added | David Anderson | timeline score: 2 | |
| Jul 22, 2012 at 10:01 | answer | added | Muhammad | timeline score: 1 | |
| May 25, 2010 at 2:00 | answer | added | Victor Protsak | timeline score: 10 | |
| May 24, 2010 at 16:25 | answer | added | Hailong Dao | timeline score: 16 | |
| May 24, 2010 at 10:44 | answer | added | Georges Elencwajg | timeline score: 8 | |
| May 24, 2010 at 10:37 | vote | accept | Pete L. Clark | ||
| May 24, 2010 at 10:35 | answer | added | Steve D | timeline score: 19 | |
| May 24, 2010 at 10:21 | comment | added | Pete L. Clark | Thanks for the clarification. I was thinking of this ring as $\mathbb{C}[x,y,z]/(xy-z^2)$ and got as far as verifying that it is not a UFD (by a theorem of Samuel) and that it is normal. But I don't know anything about Kang's theorem. Anyway, this certainly sounds like the answer: would you please leave it as such and receive your due reward? :) | |
| May 24, 2010 at 10:07 | comment | added | Steve D | To clarify, $\mathbb{C}[x^2,y^2,xy]$ is the fixed subalgebra of $\mathbb{C}[x,y]$ being acted on by $C_2$, where the generator sends $x$ to $-x$ and $y$ to $-y$. This subalgebra is a Noetherian normal domain. Its Picard group is trivial (a theorem of Kang), but its divisor class group is order 2 (a theorem of Nakajima). A reference would be ch. 3 of Benson's "Polynomial Invariants of Finite Groups". | |
| May 24, 2010 at 9:55 | comment | added | Steve D | Why doesn't something like $\mathbb{C}[x^2,y^2,xy]$ work? | |
| May 24, 2010 at 9:32 | history | asked | Pete L. Clark | CC BY-SA 2.5 |