Timeline for What information does the completion of a stalk at its maximal ideal give us in the holomorphic, analytic, or algebraic cases?
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8 events
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| Sep 12, 2010 at 15:26 | comment | added | BCnrd | Dear Laurent: the affirmative answer in the relative case of points in the special fiber of a finite type scheme over an excellent local ring (i.e., combining Artin approximation with a "formal isomorphism" to make an isomorphism between henselizations and then between etale neighborhoods) is used in the proof of the etale-local structure of ordinary double pts (defining "ordinary double pt" via condition on completed local rings on geom. fibers, say). In the example you mention that relates to the special case $x$ and $y$ closed, even though the argument works in general. | |
| Sep 12, 2010 at 15:03 | comment | added | Laurent Moret-Bailly | Probably yes, by Artin approximation applied to a morphism $\widehat{X}\to Y$ inducing an isomorphism on completions. | |
| Sep 12, 2010 at 14:41 | comment | added | Laurent Moret-Bailly | Torsten's answer leaves aside a related question I had never thought about: Assume $X$ and $Y$ are schemes of finite type over a field $k$, and $x\in X$, $y\in Y$ are points such that the completions are isomorphic. Are the henselizations isomorphic? | |
| Sep 12, 2010 at 12:38 | vote | accept | Harry Gindi | ||
| Sep 12, 2010 at 12:33 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added two examples
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| Sep 12, 2010 at 11:24 | comment | added | Harry Gindi | Dear Torsten, while this is certainly informative and a very good answer, could you touch a little bit more on what kinds of geometric data are available by looking at the completion/henselization? | |
| Sep 12, 2010 at 11:21 | vote | accept | Harry Gindi | ||
| Sep 12, 2010 at 11:23 | |||||
| Sep 12, 2010 at 10:00 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |