Timeline for Polytopal domains in non-archimedean torus
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 31, 2019 at 7:35 | comment | added | Jérôme Poineau | Yes, opens in rigid geometry need not be open in Berkovich theory, so you should really be clear on which setting you use. | |
| Aug 30, 2019 at 21:50 | comment | added | Hang | Thank you very much for the clarification! Now I think I understand. The issue is we should be more careful for Berkovich topology, but this is ok for rigid geometry. | |
| Aug 30, 2019 at 21:03 | comment | added | Jérôme Poineau | OK. I had the Berkovich picture in mind where affinoid domains are closed but not open in general. Anyway, $\mathrm{val}^{-1}(\Delta') \subset \mathrm{val}^{-1}(\Delta')$ is an embedding of an affinoid domain, so it would indeed be an open immersion in rigid geometry. | |
| Aug 30, 2019 at 16:21 | comment | added | Hang | @JérômePoineau Let's say we are in the category of the classical rigid analytic space and let's even consider the special case $\Delta'=\{0\}$. It seems strange to me why $\mathrm{val}^{-1}(\Delta')$ is not open in $\mathrm{val}^{-1}(\Delta)$, because both are open in the whole $\mathbb G_m^n$, if we consider subspace topology (or maybe not). | |
| Aug 30, 2019 at 15:41 | comment | added | Jérôme Poineau | You should make clear which space and which topology you use: the Berkovich analytification of $\mathbb{G}_m^n$? Then, I do not see how $\textrm{val}^{-1}(\Delta')$ could be ever be open except when it is the whole thing. Do you have any example? | |
| Aug 30, 2019 at 14:11 | comment | added | Hang | @JérômePoineau Thanks! So, let put it in a simpler way: if $\Delta'\subset \Delta$ is not a closed face, then is $\mathrm{val}^{-1}(\Delta')$ an open subspace of $\mathrm{val}^{-1}(\Delta)$? | |
| Aug 30, 2019 at 6:31 | comment | added | Jérôme Poineau | You got the statement wrong: the open immersion is not the map between the Weierstrass domains, but the induced map between their reductions. | |
| Aug 30, 2019 at 3:53 | history | asked | Hang | CC BY-SA 4.0 |