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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?

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  • $\begingroup$ 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). $\endgroup$
    – user40276
    Jun 19, 2016 at 1:28
  • $\begingroup$ 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. $\endgroup$
    – user40276
    Jun 19, 2016 at 1:29
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    $\begingroup$ 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? $\endgroup$
    – Alex Mennen
    Jun 19, 2016 at 1:52
  • $\begingroup$ 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. $\endgroup$
    – user40276
    Jun 19, 2016 at 2:08
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    $\begingroup$ 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)! :) $\endgroup$
    – user40276
    Jun 20, 2016 at 21:27

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