Timeline for Compute higher direct image for Gm under open embedding
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2014 at 23:40 | vote | accept | lime | ||
| Sep 19, 2014 at 20:13 | comment | added | user27920 | (1) We don't, it is relevant in the later parts. (2) There are general commutation theorems for etale cohomology and limits; learn them. (3) This is where we use the Dedekind property (to be sure the strict henselization is a local domain). | |
| Sep 19, 2014 at 18:31 | comment | added | lime | Thank you for your answer! Three more questions about this argument: 1) How did we use fact that $X$ - Dedekind to conclude that $j_*|_U$ is identity? 2) Surely, we can do the base change $\mathcal O^{sh}_p \to \mathcal O_p$ and compute group $H^1(j^{-1}(\mathcal O^{sh}_p), \mathbb G_m) = 0$. But why for any $V$ containing point $p$ we always have a map $\mathcal O^{sh}_p \to V$ (to conclude that whatever $H^1(J^{-1}(V), \mathbb G_m)$ is, its elements vanish in inverse limit) ? 3) Why $j^{-1}(\mathcal O^{sh}_p) \cong K$? | |
| Sep 19, 2014 at 5:57 | comment | added | user27920 | For a dense open immersion $j:U \hookrightarrow X$ into a Dedekind scheme $X$, ${\rm{R}}^i j_{\ast}|_U$ vanishes for $i > 0$ since $j_{\ast}|_U$ is the identity functor. Hence, these higher direct images are skyscraper with support at the points of $X-U$. The formation of this higher direct image commutes with strict henselization at such points, so we're reduced to the case $X = {\rm{Spec}}(R)$ for a strictly henselian dvr $R$ with fraction field $K$. There it is ${\rm{H}}^1(K, \mathbf{G}_m)$, which vanishes by Hilbert 90. | |
| Sep 19, 2014 at 2:52 | answer | added | Piotr Achinger | timeline score: 1 | |
| Sep 18, 2014 at 23:16 | review | First posts | |||
| Sep 19, 2014 at 0:37 | |||||
| Sep 18, 2014 at 23:09 | history | asked | lime | CC BY-SA 3.0 |