Timeline for Monoid cohomology of $\mathbb{N}$ for a linear algebraic group
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 18, 2018 at 11:21 | comment | added | LSpice | Is monoid cohomology different from group cohomology? That is, is the restriction map $\mathrm H^1(\mathbb Z, \mathbb G(k_E^{\mathrm{sep}}) \to \mathrm H^1(\mathbb N, \mathbb G(k_E^{\mathrm{sep}}))$ an isomorphism, with inverse given by $\alpha \mapsto -n \mapsto \alpha(0)\mathbb G(\varphi^n)(\alpha(n))^{-1}$? (You've described triviality but not the equivalence relation, so I'm not quite sure. Also, is $\varphi$ required to be an isomorphism?) If so, then that seems to make it even more likely that you can use standard group-cohomology tricks. | |
| Jul 17, 2018 at 12:52 | comment | added | Estus | This does seem to be the correct approach. Thank you very much! | |
| Jul 17, 2018 at 12:28 | comment | added | LSpice | This feels (which I guess is your motivation) like Lang–Steinberg + Hensel's lemma. Does any approach like that seem plausible? | |
| Jul 17, 2018 at 12:21 | comment | added | LSpice | (I'm sorry, I see now that you already explicitly said in your post that $k_E^{sep}$ was the separable closure.) | |
| Jul 17, 2018 at 12:05 | history | edited | Estus | CC BY-SA 4.0 |
added 35 characters in body
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| Jul 17, 2018 at 11:57 | comment | added | Estus | I meant the seperable closure of $k_E$. For $H^1$ you can define a 1-cocycle to be a map $\alpha:\mathbb{N}\rightarrow\mathbb{G}(k_E^{sep})$, which satisfies $$\alpha(n+m)=\alpha(n)\cdot\mathbb{G}(\varphi^n)(\alpha(m))$$ and a subset of 1-coborders to be the maps $\alpha_A(n)=A^{-1}\cdot\mathbb{G}(\varphi^n)(A)$ for some $A\in\mathbb{G}(k_E^{sep})$. The vanishing of $H^1$ means, that the 1-coborders are exactly the 1-cocycles. | |
| Jul 17, 2018 at 11:53 | comment | added | LSpice | Also, does $k_E^{sep}$ mean $(k^{\mathrm{sep}})_E = k^{\mathrm{sep}}((X))$, the maximal unramified extension of $k_E$, or $(k_E)^{\mathrm{sep}}$, the separable closure of $k_E$? | |
| Jul 17, 2018 at 11:52 | comment | added | LSpice | I've never seen monoid (as opposed to group) cohomology before. Is your definition of the vanishing of $\mathrm H^1$ all that you are asking? (That is, would a positive answer to the reformulated question—which I can't provide—suffice?) | |
| Jul 17, 2018 at 11:29 | review | First posts | |||
| Jul 17, 2018 at 11:31 | |||||
| Jul 17, 2018 at 11:27 | history | asked | Estus | CC BY-SA 4.0 |