Timeline for Categorical construction of the category of schemes?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 31, 2010 at 14:45 | comment | added | BCnrd | @Martin: when one asks for a "characterization" of a property it is desired to give something not trivially equivalent to the initial definition (cf. Peter's comment). Could choose broader intent, but of limited use (e.g., is your "characterization" useful to represent Hilbert functors?). Was a huge problem in 1960's to give a characterization of functors represented by schemes. Grothendieck discovered necessary conditions (fpqc descent, algebraization, etc.), essential in Artin's solution via alg. spaces (e.g., new representability proofs for Hilb, Pic, etc., beyond Grothendieck's results). | |
| May 31, 2010 at 14:01 | comment | added | Peter LeFanu Lumsdaine | This is a nice answer to part 1 of the question, but I think BCnrd is pointing out that this doesn't answer part 2, since this answer takes "the Zariski topology" as part of the data, presumably defined in the classical way, hence in the language of commutative rings. | |
| May 31, 2010 at 13:05 | comment | added | Martin Brandenburg | ? "This isn't really a "characterization" of functors represented by schemes: as you know, it's just the usual definition of a scheme recast in more categorical language." Yeah, and that was the question. And it does provide a construction without locally ringed spaces. | |
| May 31, 2010 at 12:42 | comment | added | BCnrd | This isn't really a "characterization" of functors represented by schemes: as you know, it's just the usual definition of a scheme recast in more categorical language. The answer to question #2 is, sadly, "no" (as far as anyone knows). The raison d'etre for the theory of algebraic spaces is that there a characterization really is possible (assuming mild finiteness hypotheses, say); see my comments on David's answer above. | |
| May 31, 2010 at 5:08 | history | answered | Martin Brandenburg | CC BY-SA 2.5 |