Timeline for Surjectivity of frobenius
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2013 at 16:41 | vote | accept | Lan | ||
| Jul 19, 2013 at 19:41 | answer | added | almostuser | timeline score: 2 | |
| Jul 19, 2013 at 15:08 | comment | added | user36938 | I was probably mistaken to suggest it is ind-etale (otherwise Faltings would have proved that). | |
| Jul 19, 2013 at 14:33 | comment | added | Lan | Can you give some reference for the proof for ind-etale? | |
| Jul 19, 2013 at 14:25 | comment | added | user36938 | @Tom: Have you tried to show that if $A \rightarrow B$ is almost etale then the Frobenius on $B/pB$ is surjective if that of $A/pA$ is. (This is true for "ind-etale", so maybe a variant of the same argument will work for "almost etale".) | |
| Jul 19, 2013 at 13:38 | comment | added | Lan | Faltings has proved that $\bar{R}$ is almost etale over $R_{\infty}$. Weather it is releavent? | |
| Jul 19, 2013 at 13:31 | comment | added | user36938 | Ah, I think I see what to do: how about showing that in effect $R_{\infty}$ "eats up" all such wild ramification, meaning that $\overline{R}$ over $R_{\infty}$ is built from parts that are tame at the generic points in char. $p$, so you can use Abyhankar's Lemma to get that $\overline{R}$ is ind-etale over $R_{\infty}$ (so then the problem reduces to $R_{\infty}$, where everything is clear)? | |
| Jul 19, 2013 at 13:27 | comment | added | user36938 | In view of Abhyankar's Lemma, the main issue is to grapple with the connected finite etale covers of $R[1/p]$ that are wildly ramified over the generic points in char. $p$, and there will be plenty of these that don't arise from such covers of the punctured affine line (which just gives $R_{\infty}$ that you have mentioned). | |
| Jul 19, 2013 at 13:22 | comment | added | Lan | $\bar{R}$ containe a subring $R_{\infty}$ generted by $p$-power roots of $T$ over $R\otimes_{\mathbb{Z}_p}{\bar{\mathbb{Q}}_p}$ | |
| Jul 19, 2013 at 13:18 | comment | added | user36938 | Keerthi: That will introduce ramification in char. 0, so something like $t^p - ph(t) - r$ is more suitable, but not clear what to take for $h$ to avoid char-0 ramification. | |
| Jul 19, 2013 at 13:15 | comment | added | Keerthi Madapusi | You might want to consider $\overline{R}$-algebras of the form $\overline{R}[t]/(t^p-r)$, where $r$ is an element of $\overline{R}$. | |
| Jul 19, 2013 at 13:15 | comment | added | Lan | Yes, allow $T$'s root. | |
| Jul 19, 2013 at 13:08 | comment | added | user36938 | I imagined "typical Faltings style" is part of the issue. But we can hold ourselves to higher standards of clarity. :) OK, so we'll assume geometric integrality for $R[1/p]$ over ${\mathbf{Q}}_p$. But you don't really mean normalization in $R \otimes \overline{\mathbf{Q}}_p$ as you have written, correct? (For example, it seems you want to allow root extractions of $T$.) I think you mean the normalization in the "universal cover" of Spec($R[1/p]$); is that right? | |
| Jul 19, 2013 at 12:52 | comment | added | Lan | Any way it is the exact description in the paper- a typical falings style. We may just assume that $R\otimes_{\mathbb{Z}_p}\bar{\mathbb{Q}}_p$ is integral ring. | |
| Jul 19, 2013 at 12:44 | comment | added | Lan | I think "geometric integral" means that $R\otimes_{\mathbb{Z}_p}\bar{\mathbb{Q}}_p$ is integral ring. | |
| Jul 19, 2013 at 12:30 | comment | added | user36938 | This is better, but still unclear. The way you use "geometrically integral" is meaningless -- do you mean Spec($R$) has geometrically integral fibers over Spec($\mathbf{Z}_p$), or perhaps that $R[1/p]$ is geometrically integral over $\mathbf{Q}_p$? Also, the end of the 2nd paragraph rules out extracting roots of $T$, yet that does provide an etale extension over $\mathbf{Q}_p$. So you give inconsistent descriptions of what you want. Why not focus on $R$ a domain, since your initial $R$ of interest is a product of domains anyway, so the domain case should be all that matters. | |
| Jul 19, 2013 at 9:09 | history | edited | David Loeffler | CC BY-SA 3.0 |
Corrected spelling (again)
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| Jul 19, 2013 at 8:29 | comment | added | Lan | I have edited the qustion to make it more clear. | |
| Jul 19, 2013 at 8:28 | history | edited | Lan | CC BY-SA 3.0 |
added 447 characters in body
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| Jul 19, 2013 at 7:03 | comment | added | user36938 | Probably $R$ is meant to have connected spectrum (so it is a 2-dimensional regular excellent domain) and you only permit extensions that have connected Spec or else "maximal extension..." would make no sense. But you must intend more conditions left unstated, since otherwise $R[1/p]$ is such an extension and so $p$ would a unit in the "maximal" extension. Do you want the normalization of $R$ in the "universal cover" (direct limit of connected finite etale covers, controlled by geometric generic point) of Spec$(R[1/p])$? Please rewrite the question so we don't have to guess what you're asking. | |
| Jul 18, 2013 at 10:25 | history | edited | David Loeffler | CC BY-SA 3.0 |
corrected spelling and grammar
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| Jul 18, 2013 at 9:09 | history | asked | Lan | CC BY-SA 3.0 |