Timeline for Why central isogeny of reductive group over general field F map maximal F split torus onto a maximal split F torus
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 22, 2019 at 20:53 | answer | added | yshuai Qin | timeline score: 0 | |
| Apr 22, 2019 at 1:02 | comment | added | LSpice | In your comments, you mentioned pull-backs, but I have been reminded by @anon's answer that your actual question addresses push-forwards. Accordingly, I'm not sure whether your comment about pull-backs was a separate question, or whether I missed your point. | |
| Apr 22, 2019 at 0:32 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Typo in the title corrected
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| Apr 22, 2019 at 0:28 | answer | added | anon | timeline score: 2 | |
| Apr 19, 2019 at 2:04 | comment | added | LSpice | Ah, I see. If you wish to work with pullbacks in the scheme-theoretic sense (not underlying reduced schemes), then the statement is not true. Let $k = \mathbb F_2((t))$, let $G$ and $G'$ be the group schemes underlying $\ker \mathrm N_{D/k}$ and $D^\times/k^\times$ where $D/k$ is the quaternionic division algebra, and let $f : G \to G'$ be the natural projection. Then the maximal split torus in $G'$ is trivial, but its pullback to $G$ is the non-smooth scheme $Z(G) = \mu_2$. | |
| Apr 19, 2019 at 0:51 | comment | added | yshuai Qin | @LSpice, I think the pull back (in scheme sense) is not a smooth subgroup scheme over a non perfect field $F$ in general. The argument works when F is perfect field. | |
| Apr 18, 2019 at 23:05 | comment | added | LSpice | This is not the place to get detailed proofs of standard results. One approach (probably not optimal) is to notice that the character lattice of $f^{-1}(T_{F^{\text{alg}}})$ (which is certainly a torus) has the trivial Galois action, so that the $F$-algebra it generates is an $F$-structure for $f^{-1}(T_{F^{\text{alg}}})$. | |
| Apr 18, 2019 at 22:11 | comment | added | yshuai Qin | @LSpice Thank you for the explanation. I only know how to show it if F is perfect field. The inverse image of a maximal torus defined over F is also maximal torus defined over F. But I don't know how to show it for reductive group over a non perfect field. Could you explain how to proved it in details. | |
| Apr 18, 2019 at 13:50 | review | Close votes | |||
| Apr 30, 2019 at 3:05 | |||||
| Apr 18, 2019 at 13:33 | comment | added | LSpice | Because it induces an isomorphism of rational-ised character lattices (with Galois action). This question is not research level, and can be found in the part of any of the standard books dealing with rationality questions. | |
| Apr 18, 2019 at 6:38 | history | edited | yshuai Qin | CC BY-SA 4.0 |
Correct e
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| Apr 18, 2019 at 3:10 | review | First posts | |||
| Apr 18, 2019 at 4:02 | |||||
| Apr 18, 2019 at 3:09 | history | asked | yshuai Qin | CC BY-SA 4.0 |