Timeline for Which locally ringed spaces are schemifiable?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2020 at 4:18 | comment | added | Martin Brandenburg | I would say this question is a duplicate of the linked question, right? | |
| Jun 20, 2016 at 21:27 | comment | added | user40276 | I just referred the answer to that question, because schemefication seems to be directly related to existence of colimits (because of the cocompleteness of LRS and failure of cocompleteness in Sch). Therefore a general compilation of general facts about these categories may be useful. I'm not claiming that it will be useful (see the "may be useful" above)! :) | |
| Jun 19, 2016 at 3:43 | comment | added | Alex Mennen | I didn't see anything in answer to that question that directly relates to mine. | |
| Jun 19, 2016 at 3:42 | comment | added | Alex Mennen | I was looking for the largest full subcategory C of LocallyRingedSpaces such that the inclusion from Schemes to C has a left adjoint. A full subcategory C of LocallyRingedSpaces that includes Schemes and such that the inclusion from C to LocallyRingedSpaces has a left adjoint could also be interesting in a closely related way, but it looks like prop 4.1 says that a functor from LocallyRingedSpaces to another category has a left adjoint, which is backwards from the sort of thing I'm looking for. [Edit: oh, it looks like you just figured that out.] | |
| Jun 19, 2016 at 3:35 | comment | added | user40276 | Sorry, forget the stuff about Demazure book. I've got the names of the categories wrong, so it will not work that representability stuff. | |
| Jun 19, 2016 at 2:17 | comment | added | user40276 | I believe this answer may be useful mathoverflow.net/questions/216060/… in order to obtain your desired criterium. | |
| Jun 19, 2016 at 2:08 | comment | added | user40276 | I was reffering to prop 4.1 (at pag 15 of my edition). However I've just noticed that it's a forgetful functor that goes to ME (which is a category of presheaves of some rings) instead of Sch. So your condition, under prop 4.1, reduces to representability of this presheaf. | |
| Jun 19, 2016 at 1:52 | comment | added | Alex Mennen | No, there isn't a left adjoint of the forgetful functor from schemes to locally ringed spaces, because as the question I linked to pointed out, there exist locally ringed spaces that are not schemifiable. What are you referring to from Demazure & Gabriel? | |
| Jun 19, 2016 at 1:29 | comment | added | user40276 | It's analogous to a realization functor (actually I believe it's a realization functor according to nlab definition). You can write it as a coend. | |
| Jun 19, 2016 at 1:28 | comment | added | user40276 | I believe the construction of a left adjoint to the forgetful functor is in the first chapter of Demazure & Gabriel's book in algebraic geometry and algebraic groups. Actually it's possible to even get a left adjoint in the case of ringed spaces by composing with a "localization" of a ringed space (see arxiv.org/abs/1103.2139 for the latter). | |
| Jun 19, 2016 at 0:18 | history | asked | Alex Mennen | CC BY-SA 3.0 |