Timeline for Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$.
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 13, 2019 at 8:42 | comment | added | Laurent Moret-Bailly | @KarlPeter: Every henselian valued field satisfies (WH): see Proposition 3.1.4 in content.algebraicgeometry.nl/2014-5/2014-5-025.pdf. You can also find a summary of this kind of condition in math.u-psud.fr/~cesnavicius/topology-torsors.pdf. | |
| Nov 12, 2019 at 18:25 | comment | added | user267839 | sorry for digging out this ancient thread, but I would very glad if you could loose some words on explanation in which way your condition (WH) can be understood as "weak" henselian property. my main reference on characterizations of equivlent formulations of "classical" henselian property is Milne's book "Etale Cohomology" Thm 4.2 (on page 32). how are these related to (WH) and why we understand (WH) as a "weaker" form of henselian property? | |
| Aug 13, 2010 at 15:59 | vote | accept | BCnrd | ||
| Aug 13, 2010 at 15:59 | history | bounty ended | BCnrd | ||
| Aug 9, 2010 at 21:05 | comment | added | BCnrd | Dear Laurent: OK, I have checked this and it all works out nicely (assuming separatedness of $X$, which I don't know if you need for your more general functor stuff). The only "loose end" is the issue of behavior with respect to extension $k'/k$ of complete fields, though this can be seen via the rigid-analytic method so I am fine with that (though would be neat if it could be extracted by the kind of approach you have suggested). | |
| Aug 9, 2010 at 15:11 | comment | added | BCnrd | Thanks. I suppose "pointed etale neighborhoods" means every $x \in X(k)$ admits an etale affine neighborhood with a $k$-point over $X$. I thought long before that this fact wouldn't avoid the interference of rising towers of Galois extensions. Not sure why I thought that. I'll check the rest...except I wonder about the property that $X(k)$ is closed in $X(k')$ for an ext'n of complete fields $k'/k$; clear via the rigid-analytic method, less so otherwise. Also, seems sep'td needed for $U(k) \rightarrow X(k)$ to be "loc. injective" for affine etale chart $U \rightarrow X$, which is useful. | |
| Aug 9, 2010 at 13:45 | history | answered | Laurent Moret-Bailly | CC BY-SA 2.5 |