Timeline for Group scheme with an isotrivial maximal torus
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 24, 2022 at 20:22 | vote | accept | prochet | ||
| Mar 24, 2022 at 17:14 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title
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| Mar 24, 2022 at 13:46 | answer | added | Jason Starr | timeline score: 6 | |
| Mar 21, 2022 at 9:30 | comment | added | Jason Starr | Okay, I think I have a counterexample where the base is an open affine over a “classical” Enriques surface (as opposed to “singular” or “supersingular”, all of which are smooth Enriques surfaces). I will try to write more details soon. | |
| Mar 21, 2022 at 7:27 | history | edited | prochet | CC BY-SA 4.0 |
added 7 characters in body
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| Mar 20, 2022 at 21:35 | comment | added | prochet | I'm really over an affine base. Several things are much different whe the base scheme is projective, there is much less torsors, but I'm curious about your counterexample. | |
| Mar 20, 2022 at 11:31 | comment | added | Jason Starr | It seems easier to find counterexamples if you allow a projective base scheme rather than the affine scheme $\text{Spec}\ A$. Are you interested in counterexamples over a projective base scheme? | |
| Mar 19, 2022 at 20:30 | comment | added | prochet | But why such a torsor would admit a maximal torus after a finite surjective (or finite flat) map? | |
| Mar 19, 2022 at 11:20 | comment | added | Jason Starr | For definiteness, consider torsors for a split, simple algebraic group $G$ of type $E_8$. Each of these gives rise to an inner form of $G$. Now consider the case that $A$ is the coordinate ring of a simply connected surface over an algebraically closed field $k$. By Serre's "Conjecture II" (solved by a large number of contributors in this case), every $G$-torsor is Zariski locally trivial (this also uses the Grothendieck-Serre Conjecture, which is also solved in this case). So you are asking whether every $G$-torsor is globally trivial. I believe that is false. | |
| Mar 19, 2022 at 9:19 | history | asked | prochet | CC BY-SA 4.0 |