Timeline for Etale cohomology of and forms of algebraic groups
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 9, 2013 at 23:02 | vote | accept | Craig Westerland | ||
| Jul 9, 2013 at 2:43 | answer | added | Jacob Lurie | timeline score: 13 | |
| Jul 9, 2013 at 2:04 | comment | added | Daniel Litt | Just an obvious comment--if you take $X=\operatorname{Spec} \mathbb{R}\sqcup \operatorname{Spec} \mathbb{R}$ and $Y=\mathbb{C}$, viewed as $\mathbb{R}$-schemes. The homotopy type of the former is $B(\mathbb{Z}/2\mathbb{Z})\sqcup B(\mathbb{Z}/2\mathbb{Z})$ and the latter is that of a point. If you'd like an group example where the same phenomenon happens, consider $\mu_4$ and the constant group scheme $\mathbb{Z}/4\mathbb{Z}$. | |
| Jul 9, 2013 at 1:37 | comment | added | abz | Let $\Gamma=Gal(K/k)$. For a variety $X$ over $k$ and a finite abelian group $F$ say, there is a spectral sequence $H^{r}(\Gamma,H^{s}(X_{K},F))\implies H^{r+s}(X,F)$. In general, if you take a different form of $X$, you will get a different action of $\Gamma$ on $H^{s}(X_{K},F)$, and hence you will expect to get different cohomology. It shouldn't be too hard to work everything out in the particular case you mention (but I also am lazy). | |
| Jul 9, 2013 at 0:40 | history | asked | Craig Westerland | CC BY-SA 3.0 |