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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12 events
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| Sep 12, 2010 at 18:18 | comment | added | Harry Gindi | Dear BCnrd: Thanks, I will check it out. Also, yes, I am familiar with excellent rings. | |
| Sep 12, 2010 at 13:45 | comment | added | BCnrd | Dear Harry: Here is a geometric example showing merits of completion and henselization. Let $X$ be proper flat scheme over local noetherian $R$ with fiber dim. 1. Is it projective? Using ample line bundle on special fiber, infinitesimal def. theory and formal GAGA (for coherent sheaves) imply lifts to $X' := X_ {\widehat{R}}$, so $X'$ projective over $\widehat{R}$. If $R$ is excellent, Artin approx. implies such a lift then exists on $X'' := X_ {R^{\rm{h}}}$, so $X''$ is projective over $R^{\rm{h}}$. Thus, $X$ is projective over a (residually trivial) etale nbhd of $R$. Seems best can do. | |
| Sep 12, 2010 at 13:37 | comment | added | BCnrd | Dear Harry: Completions admit technique of working by successive approx., not available with henselization (until Artin approx. came along, but that often doesn't suffice). Learn deformation theory. Henselization and strict henselization are the "local rings" for the etale topology, and so are ubiquitous for local calculations on alg. spaces and DM stacks; local rings for their Zar. topologies are often useless. As Torsten notes, solving problem over str. hens. often allows to descend to actual etale nbhd; can't do with completion. Read Knutson's "Alg. Spaces" and sec. 3.6 of "Neron Models". | |
| Sep 12, 2010 at 13:26 | comment | added | BCnrd | Dear Harry: Do you know about excellent rings (Chapter 13 of Matsumura's "Commutative Algebra" -- not "Commutative Ring Theory" -- and section 7.7 or 7.8 in EGA IV$_3$)? Gives powerful technique to prove for "most" (not all!) local noetherian rings that properties hold if and only if they do for completion. As Torsten notes, if $X$ is an algebraic scheme over $\mathbf{C}$ and $x \in X(\mathbf{C})$ then $O_ {X,x}$ and $O_ {X^{\rm{an}},x}$ have the same completion. So $X$ is reduced, normal, Cohen-Macaulay, etc. iff $X^{\rm{an}}$ is. I doubt there's a proof without going through completion. | |
| Sep 12, 2010 at 12:38 | vote | accept | Harry Gindi | ||
| Sep 12, 2010 at 11:21 | vote | accept | Harry Gindi | ||
| Sep 12, 2010 at 11:23 | |||||
| Sep 12, 2010 at 10:00 | answer | added | Torsten Ekedahl | timeline score: 14 | |
| Sep 12, 2010 at 9:39 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Sep 12, 2010 at 9:34 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Sep 12, 2010 at 9:33 | comment | added | Harry Gindi | From what I can gather, it seems that the henselization has better finiteness properties than the completion but contains all of the important data in many important cases (It is my understanding that this is the geometric content of the Artin approximation theorem and the result of Popescu on General Néron Desingularization) | |
| Sep 12, 2010 at 8:35 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Sep 12, 2010 at 8:23 | history | asked | Harry Gindi | CC BY-SA 2.5 |