Timeline for Are schemes pushouts of neighbourhoods and formal neighbourhoods?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 28, 2011 at 2:42 | comment | added | S. Carnahan♦ | I agree with Ryan. There are no formal neighborhoods here. | |
| Nov 27, 2011 at 20:11 | history | edited | Charles Staats | CC BY-SA 3.0 |
improved latex
|
| Nov 27, 2011 at 16:54 | answer | added | Laurent Moret-Bailly | timeline score: 3 | |
| Nov 26, 2011 at 6:32 | comment | added | Ryan Reich | The title of this question is misleading, since "formal neighborhood" could either refer to the formal scheme which is the completion of a scheme at a closed subscheme, or to the map $\mathrm{Spec}(\mathbb{C}[[t]]) \to \mathrm{Spec}(\mathbb{C}[t])$ (or its generalization when the target is a smooth curve). Though it would be interesting to record the answer to the question of whether a scheme is glued, in the sense of being a colimit in the appropriate category, from an open set and a formal neighborhood (in this sense) of its closed complement. | |
| Nov 25, 2011 at 17:59 | comment | added | Martin Brandenburg | Thanks! I was a bit puzzled because Jonathan Wise showed before (mathoverflow.net/questions/65506/…) that infinite products are very rare in the category of schemes; but apparently there are more examples for relative schemes, i.e. infinite fiber products. | |
| Nov 25, 2011 at 17:22 | comment | added | user2035 | Yes, that is correct. You may assume $X=\mathrm{Spec}(A)$ affine and restrict to the cofinal subsystems of neighborhoods of the form $\mathrm{Spec}(A_f)$, so $\prod_{x\in U}U=\prod_{f\notin x}\mathrm{Spec}(A_f)=\mathrm{Spec}(\varinjlim A_f)=\mathrm{Spec}(\mathcal O_{X,x})$. | |
| Nov 25, 2011 at 17:06 | comment | added | Martin Brandenburg | So you claim that $\mathrm{Spec}(O_{X,x}) = \prod_{x \in U} U$ in the category of schemes over $X$? | |
| Nov 25, 2011 at 17:00 | comment | added | user2035 | $\mathrm{Spec}(\mathcal O_{X,x})$ is the scheme-theoretic intersection of all open neighborhoods of $x$. This is not quite a description of the morphism $\mathrm{Spec}(\mathcal O_{X,x})\to X$, but at least this description may be checked in a single open affine chart. | |
| Nov 25, 2011 at 16:55 | answer | added | Martin Brandenburg | timeline score: 8 | |
| Nov 25, 2011 at 16:33 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
added 4 characters in body
|
| Nov 25, 2011 at 15:00 | answer | added | name | timeline score: 1 | |
| Nov 25, 2011 at 14:55 | history | asked | Sasha | CC BY-SA 3.0 |