Timeline for What is the relationship between connective and nonconnective derived algebraic geometry?
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| May 1, 2017 at 21:45 | comment | added | AAK | @TimCampion Well, there are a lot more affine non-connective spectral schemes than just of this form. Re: smoothness, the definition that works in the connective case is a locally fp morphism whose cotangent complex is of tor-amplitude $[0,0]$ (i.e. a retract of a free module of finite rank). With this definition, $V_X(F)$ would be smooth as a non-connective spectral scheme iff $F$ is of tor-amplitude $[0,0]$, so there are a lot of smooth connective spectral schemes which become non-smooth with this definition. | |
| Apr 30, 2017 at 22:19 | comment | added | Tim Campion | Thanks, this is really helpful! From your first point, I'm tempted to conclude "Oh, maybe nonconnective DAG is just geometry locally modeled on quasiaffine spectral schemes!" -- but maybe it's not that simple? Regarding your second point, I'm not really familiar with these ways of encoding data in the cotangent complex, but I'll ask -- is there even a definition of "smooth nonconnective spectral scheme"? | |
| Apr 30, 2017 at 17:47 | history | edited | AAK | CC BY-SA 3.0 |
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| Apr 30, 2017 at 17:42 | comment | added | AAK | I think the issue is more fundamental. I'm saying that the embedding of SAG into nonconnective SAG is not compatible with many basic notions that I would call "geometric", like affineness, open immersions, smooth and étale morphisms... It's a genuinely different geometry, which makes the generalization from classical AG to connective SAG look very tame by comparison. So if you want to import some theorems from classical AG into SAG, you might have to be content with proving their connective versions! | |
| Apr 30, 2017 at 16:46 | comment | added | Denis Nardin | Well, this is more impressionistic than anything else, but in the second example you are essentially saying that you cannot use the t-structure on R-modules to get information from the cotangent complex anymore. And that is fair, since the t-structure does not exists anymore. I don't know what should replace it though. Anyway I'll shut up now, since I don't know nearly enough about derived algebraic geometry... | |
| Apr 30, 2017 at 16:12 | comment | added | Denis Nardin | I don't know... To me your example suggests that the homotopy groups are the "wrong" invariant for modules over a nonconnective rings, rather than nonconnective rings be wrong in themselves. I wouldn't know what would be the "right" invariant though... | |
| Apr 30, 2017 at 14:50 | history | edited | AAK | CC BY-SA 3.0 |
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| Apr 30, 2017 at 14:43 | history | answered | AAK | CC BY-SA 3.0 |