Timeline for The second Tate-Shafarevich group of a permutation module is trivial
Current License: CC BY-SA 4.0
16 events
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| Jul 2, 2023 at 11:23 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Title corrected
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| Jul 1, 2023 at 13:06 | vote | accept | Tuvasbien | ||
| Jul 1, 2023 at 12:01 | comment | added | Mikhail Borovoi | @LSpice: I have typed an answer. It is not just two comments! | |
| Jul 1, 2023 at 7:01 | answer | added | Mikhail Borovoi | timeline score: 6 | |
| Jun 30, 2023 at 20:07 | comment | added | Mikhail Borovoi | @LSpice: Yes, they are. I will type an answer when I have time, hopefully tomorrow. | |
| Jun 30, 2023 at 19:21 | comment | added | LSpice | @MikhailBorovoi, aren't 1 and 2 together an answer? | |
| Jun 30, 2023 at 19:03 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
typo
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| Jun 30, 2023 at 18:58 | comment | added | Mikhail Borovoi | Idea of proof: (1) You may assume that $M={\mathbb Z}$. (2) We have $$H^2(G,{\mathbb Z})\cong H^1(G,{\mathbb Q}/{\mathbb Z})={\rm Hom}(G, {\mathbb Q}/{\mathbb Z}).$$ (3) You use the Chebotarev density theorem: for any cyclic subgroup $H\subseteq G$ there exists $v$ and $w$ such that ${\rm Gal}(L_w/K_v)=H$. The lemma follows. | |
| Jun 30, 2023 at 18:29 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
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| Jun 30, 2023 at 18:27 | comment | added | Mikhail Borovoi | See Lemma 1.9 of J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80. | |
| Jun 30, 2023 at 18:05 | comment | added | Tuvasbien | I've added more context to my question. | |
| Jun 30, 2023 at 18:04 | history | edited | Tuvasbien | CC BY-SA 4.0 |
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| Jun 30, 2023 at 17:34 | history | edited | Mikhail Borovoi |
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| Jun 30, 2023 at 17:29 | comment | added | Mikhail Borovoi | For the MO community, please state clearly the desired assertion. What is a permutation module? What do you mean by $\rm Sha$? $\rm Sha^1$ or $\rm Sha^2$? Is the extension $L/K$ finite or infinite? After that I will type a proof or give a reference. | |
| Jun 30, 2023 at 15:19 | history | edited | Tuvasbien |
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| Jun 30, 2023 at 15:13 | history | asked | Tuvasbien | CC BY-SA 4.0 |