Timeline for ring-valued points of locally ringed spaces
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jan 3, 2014 at 14:59 | comment | added | S. Carnahan♦ | @CharlesSiegel I think you are correct that the map $(pt,A) \to (pt,\hat{A})$ of locally ringed spaces does not exist in general. On the other hand, the error with covariance versus contravariance is easy to fix. | |
| Jan 23, 2010 at 20:32 | comment | added | Harry Gindi | This is of course why we need to impose the locality condition for covering functors of points. | |
| Jan 23, 2010 at 20:25 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
added 305 characters in body; deleted 7 characters in body
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| Jan 7, 2010 at 16:08 | comment | added | Charles Siegel | Ahh, of course. Don't know what I was thinking. I'm withdrawing my objection. | |
| Jan 7, 2010 at 15:46 | comment | added | Emerton | There's an additional restriction on morphisms of locally ringed spaces: the maps on stalks must be local maps. (This is not automatic; it must be imposed. Thus locally ringed spaces are not a full subcategory of ringed spaces.) This means that the map $A \to R$ must take the maximal ideal of $A$ into the nilradical of $R$, and hence if that maximal ideal is f.g. (e.g. principal), to a nilpotent ideal. Thus Martin's statement is correct (even for all Noetherian local rings). | |
| Jan 7, 2010 at 14:50 | comment | added | Charles Siegel | There's something funny here in your set up, I think. A morphism of LRS is a continuous map and a map of sheaves in the other direction, so that natural map $A\to \hat{A}$ would give a map $|\hat{A}|\to|A|$ (using $|A|$ to denote the LRS you defined), so if it induces a bijection, then should be $Hom(Spec R,\hat{A})\to Hom(Spec R,A)$, but I still don't think I believe your statement. My proposed counterexample is in an answer below. | |
| Dec 30, 2009 at 11:38 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |