Timeline for Computing $H^1$ with coefficients in a torsion-free abelian group
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 11, 2021 at 16:02 | vote | accept | oleout | ||
| Jul 11, 2021 at 8:45 | comment | added | Daniel Loughran | The spectral sequence still tells you something in the non-projective case. See Lemma 6.3 of Sansuc - Groupe de Brauer et arithmetiques des groupes algebriques lineaires sur un corps de nombres | |
| Jul 11, 2021 at 8:31 | comment | added | oleout | I'll have a look at the thesis as you mentioned, thank you. The neat part of the "projective" assumption is that the varieties would satisfy $\bar{k}[X]^* = \bar{k}^*$, and so the Hochschild-Serre spectral sequence reduces to $$\mathrm{Br}(k) \rightarrow \mathrm{Br}_1(X) \rightarrow H^1(k,\mathrm{Pic}(X_{\bar{k}})) \rightarrow H^3(k,\bar{k}^*) = 0.$$ Thus one has the identification as you mentioned. However, removing the properness condition would instead need us to understand the groups $H^i(k,\bar{k}[X]^*)$ for $i =2,3$, at least. This is related to some other questions I asked on this site. | |
| Jul 11, 2021 at 7:59 | history | answered | Daniel Loughran | CC BY-SA 4.0 |