Timeline for What justifies the following isomorphism in Cassels' proof of the Cassels–Tate pairing?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 22 at 21:25 | comment | added | Will Sawin | It is possible that this is simply a mistake and Cassels meant to say "injective". | |
| Apr 22 at 17:02 | comment | added | Aphelli | Even if $\Gamma_2$ was normal in $\Gamma$, you would expect the action of $\Gamma/\Gamma_2$ to be nontrivial in $H^{\ast}(\Gamma/\Gamma_2,-)$, unless you had a very good reason to believe otherwise. Indeed, in finite characteristic prime to $|\Gamma/\Gamma_2|$, then $H^{\ast}(\Gamma,-)=H^0(\Gamma/\Gamma_2,H^{\ast}(\Gamma_2,-))$. | |
| Apr 22 at 16:57 | comment | added | Aphelli | I can’t see any reason why $\Gamma_2$ should be normal (eg if the action is surjective on $\mathrm{Aut}(A)$). Consider the case where $GL_2(\mathbb{F}_p)$ acts tautologically on $\mathbb{F}_p^2$. Then the cohomology vanishes (for $p>2$) because the center acts trivially on the cohomology. But if you restrict a $p$-Sylow, the cohomology doesn’t vanish (it has constant cardinality independently of the degree, controlled by the cardinality of $\hat{H}^0$), so it can’t be formal. Maybe there’s something specific to number fields, but it looks likelier given the situation that it’s a mistake. | |
| Apr 22 at 1:04 | history | edited | LSpice | CC BY-SA 4.0 |
Tidying
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| Apr 21 at 22:30 | history | asked | Snacc | CC BY-SA 4.0 |