Timeline for Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S May 13, 2020 at 21:00 | history | bounty ended | CommunityBot | ||
| S May 13, 2020 at 21:00 | history | notice removed | CommunityBot | ||
| S May 5, 2020 at 18:36 | history | bounty started | Ekaterina Bogdanova | ||
| S May 5, 2020 at 18:36 | history | notice added | Ekaterina Bogdanova | Draw attention | |
| May 1, 2020 at 14:50 | comment | added | Jason Starr | You are correct, and I am incorrect. I misread the question. I thought you were just asking about surjectivity of the group homomorphism from $\textbf{GL}(R((h))$ to $\textbf{PGL}(R((h))$. Now I see that is not what you are asking. Sorry for the confusion. | |
| Apr 30, 2020 at 10:39 | comment | added | Ekaterina Bogdanova | Dear @JasonStarr, I am confused. I've looked it up in the Serre`s book and have not find information concerning this question. To this point I am not sure that I formulated it clear enough. My question is the following. For any ring $R$ an element in $PGL(R((h)))$ defines a line bundle $L$ on $Spec(R((h)))$. I wonder whether it is true that there exists an open covering $Spec R = \cup Spec R_{f_i}$ such that pullbacks of $L$ to $Spec(R_{f_i}((h)))$ are trivial. I don't see why your ring provides a counterexample, so I will be grateful if you explain it in more details. | |
| Apr 29, 2020 at 17:01 | comment | added | Jason Starr | I do not have a copy of the book with me, but it should be prior to the long exact sequence of non-Abelian cohomology. For instance, if $R$ equals $\mathbb{C}[x,y]/\langle y^2-x^2(x-1)\rangle$, the $n$-torsion elements in the Picard group give counterexamples. | |
| Apr 29, 2020 at 7:16 | comment | added | Ekaterina Bogdanova | @JasonStarr thanks a lot! But could you be a little more specific? In which chapter can I find this discussion? | |
| Apr 28, 2020 at 22:56 | comment | added | Jason Starr | Welcome new contributor. No, that morphism of Zariski sheaves is not surjective. There is a discussion of this in Serre's "Galois cohomology". | |
| Apr 28, 2020 at 22:05 | review | First posts | |||
| Apr 28, 2020 at 22:24 | |||||
| Apr 28, 2020 at 22:04 | history | asked | Ekaterina Bogdanova | CC BY-SA 4.0 |