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(most of this question is re-asking Schemification (schematization?) of locally ringed spaces, which did not get answered)

Given a locally ringed space $X$, say that a schemification of $X$ is a scheme $Y$ with a map $X\rightarrow Y$ that is initial among maps from $X$ to schemes. What are necessary and sufficient conditions for a locally ringed space to have a schemification? Is there an explicit construction producing a schemification of any locally ringed space that has one, or at least an explicit construction of a schemification for some fairly large class of locally ringed spaces?

Alex Mennen
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  • 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 http://arxiv.org/abs/1103.2139 for the latter). – user40276 Jun 19 '16 at 01:28
  • 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. – user40276 Jun 19 '16 at 01:29
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    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? – Alex Mennen Jun 19 '16 at 01:52
  • 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. – user40276 Jun 19 '16 at 02:08
  • I believe this answer may be useful http://mathoverflow.net/questions/216060/about-the-relation-between-the-categories-textsch-textlrs-and-text in order to obtain your desired criterium. – user40276 Jun 19 '16 at 02:17
  • Sorry, forget the stuff about Demazure book. I've got the names of the categories wrong, so it will not work that representability stuff. – user40276 Jun 19 '16 at 03:35
  • 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.] – Alex Mennen Jun 19 '16 at 03:42
  • I didn't see anything in answer to that question that directly relates to mine. – Alex Mennen Jun 19 '16 at 03:43
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    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)! :) – user40276 Jun 20 '16 at 21:27
  • I would say this question is a duplicate of the linked question, right? – Martin Brandenburg Jan 30 '20 at 04:18

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