Timeline for Is every projective space bundle locally trivial in the Zariski topology?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2010 at 8:00 | comment | added | BCnrd | Vivek, Serre describes the bijection onto $(n+1)$-torsion in the Brauer groups as well as the inverse map; look up "Brauer-Severi variety" in the index at the back of the book (I think). As for actually computing the Brauer group, this is subtle beyond the 1-dim'l case handled by Tsen's theorem. As an indication of the difficulties, in dimension 3 nontrivial 2-torsion Brauer classes can fail to be represented by quaternion division algebras (in contrast with number fields, local fields, and function fields of curves over finite fields). There's quite a bit of literature for surfaces. | |
| Dec 31, 2010 at 7:18 | comment | added | Vivek Shende | Thanks, Brian. I guess probably this is answered in those references, but does one generally expect to be able to compute the Br(k(Y)) and/or that map? | |
| Dec 31, 2010 at 7:00 | comment | added | BCnrd | Oops, as usual I made a typo and wrote PGL$_n$ rather than PGL$_{n+1}$, and likewise should use $(n+1)$-torsion in the Brauer group. | |
| Dec 31, 2010 at 6:57 | vote | accept | Vivek Shende | ||
| Dec 31, 2010 at 6:40 | comment | added | BCnrd | Vivek, since the aut. scheme of $\mathbf{P}^n$ is ${\rm{PGL}}_n$, and any smooth surjective map has etale-local sections, your question is about etale PGL$_n$-torsors. Using the central extension $1 \rightarrow {\mathbf{G}}_m \rightarrow {\rm{GL}}_n \rightarrow {\rm{PGL}}_n \rightarrow 1$, at the generic pt of $Y$ we get a connecting map of pointed sets ${\rm{H}}^1(k(Y), {\rm{PGL}}_n) \rightarrow {\rm{H}}^2(k(Y),\mathbf{G}_m) = {\rm{Br}}(k(Y))$ which is a bijection onto ${\rm{Br}}(k(Y))[n]$ (see Serre's book on Galois cohom). See Grothendieck's Brauer exposes for the finer theory over $Y$. | |
| Dec 31, 2010 at 5:48 | answer | added | Torsten Ekedahl | timeline score: 14 | |
| Dec 31, 2010 at 5:33 | history | edited | Vivek Shende | CC BY-SA 2.5 |
edited title
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| Dec 31, 2010 at 5:28 | history | asked | Vivek Shende | CC BY-SA 2.5 |