Timeline for Trivializing principal bundles on a curve over a finite field
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Apr 21, 2013 at 17:03 | answer | added | Justin Campbell | timeline score: 1 | |
| Apr 9, 2013 at 23:22 | history | bounty ended | Justin Campbell | ||
| Apr 2, 2013 at 22:43 | history | bounty started | Justin Campbell | ||
| Apr 1, 2013 at 12:05 | comment | added | Jason Starr | @Xuhan: My reading of the OP's question is to determine when a given torsor is Zariski locally trivial. The Grothendieck-Serre conjecture states that, over a "sufficiently regular" base scheme, Zariski local triviality should be equivalent to existence of a rational point. Nisnevich proved this conjecture for one-dimensional schemes. Thus, Zariski local triviality of a given torsor is equivalent to existence of a rational point. If you have a different question you would like answered, I suggest you post it. | |
| Mar 31, 2013 at 21:32 | comment | added | user29283 | @Jason: Sorry, I remain puzzled. When you say "rational point" you are thinking about the generic fiber, yes? I think the confusion may be that I am trying to think of how one would spread out the trivial torsor at the generic fiber to a nontrivial torsor over the entirety of $X$ (for some $G$), and I don't see how Serre's book helps with that. Does it? | |
| Mar 31, 2013 at 10:43 | comment | added | Jason Starr | @Xuhuan: By Nisnevich, the question is equivalent to existence of a rational point. That is precisely the question considered by Serre in the last chapters of his book. | |
| Mar 31, 2013 at 3:58 | comment | added | user29283 | @Jason:Ah, OK, so by Nisnevich, the question posed is exactly asking about the non-triviality of $H^1_{\rm{Zar}}(X,G)$. But I don't think Serre's book addresses this sort of thing (or am I mistaken?), and there are so few Zariski-exact sequences of interesting algebraic groups that it isn't clear (to me) how one gets a handle on this $H^1$. Do you know any connected reductive $G$ over $k = \mathbf{F}_q$ for which $H^1_{\rm{Zar}}(X,G)$ is nontrivial? | |
| Mar 30, 2013 at 21:29 | comment | added | Jason Starr | @Xuhan. Regarding Zariski local triviality versus rational points, this follows from Nisnevich's work on the Grothendieck-Serre conjecture. | |
| Mar 30, 2013 at 21:10 | comment | added | user29283 | @Jason: That chapter doesn't seem to provide a non-German reference for the prof of Harder's theorem, and also doesn't seem to address whether the map $H^1(X,G) \rightarrow H^1(K,G)$ is nontrivial (which is what it seems the OP is asking about, weaker than triviality of $H^1(K,G)$). What sort of relevant information can be gleaned from that part of Serre's book? For example, knowing $H^1(K,G) \ne 1$ for some $G$ arising from the constant field doesn't seem to help to "spread out" a nontrivial $G$-torsor over $K$ to a $G$-torsor over the entirety of $X$ (e.g., even for $G$ a torus). | |
| Mar 30, 2013 at 20:04 | comment | added | Jason Starr | Serre's Galois Cohomology textbook. Read the chapter on "Conjecture II". | |
| Mar 30, 2013 at 18:27 | comment | added | user29283 | Any smooth connected unipotent group over a perfect field is filtered by $\mathbf{G}_a$'s, so the unipotent case is OK by additive Hilbert 90 for the function field $K$ of $X$. Since Br($X$) vanishes (CFT for $K$), the map $H^1(X,GL_n) \rightarrow H^1(X,PGL_n)$ is surjective and identifies the target with the quotient of the set of (isom. classes of) rank-$n$ vector bundles modulo line-bundle twisting, so $H^1(X,PGL_n)$ is infinite for $n > 1$, and $H^1(K,PGL_n)={\rm{Br}}(K)[n]$ is too (CFT). But $H^1(X,PGL_n)\rightarrow H^1(K,PGL_n)$ vanishes since $H^1(K,GL_n)=1$, so $PGL_n$ is OK too. | |
| Mar 30, 2013 at 17:44 | history | asked | Justin Campbell | CC BY-SA 3.0 |