Timeline for is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Nov 28, 2013 at 10:16 | comment | added | John Salvatierrez | ah, thanks for clearing that up. I was mislead by the line "So starting from finitely generated projective modules / vector bundles, we've recovered all modules / quasicoherent sheaves. But we wanted to recover just the finitely generated modules. " Thanks! | |
| Nov 28, 2013 at 10:01 | comment | added | Qiaochu Yuan | @John: I'm making no claims about the global situation, which I don't understand. If "embeds" means an exact functor then I think you run into the issue in Eric Wofsey's answer that "exact embedding into an abelian category" is a self-dual condition but I'm not sure. And it's okay that we're doing Yoneda because there aren't many interesting finite colimits to preserve in Vect(X) anyway. The other Yoneda embedding gets us the opposite of the category of (adjective) left modules. | |
| Nov 28, 2013 at 9:54 | comment | added | John Salvatierrez | Thanks. I am still hoping for characterisation of the form "coh is initial among abelian categories in which Vect embeds," akin to what Anton mentions after UPD. Regardless, the claim is: Mod(Vect(X)) = QCoh(X), so that Mod_fg(Vect(X)) = Coh(X), at least for X locally noetherian, correct? Do you have a reference for this? I guess I believe the statement about rings, but why does it work globally for a scheme/stack? Also, since you point out it's Yoneda, shouldn't we be doing co-Yoneda (the one which preserves colimits, not the one which destroys them)? | |
| Nov 28, 2013 at 5:07 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Nov 28, 2013 at 3:05 | comment | added | Qiaochu Yuan | This isn't quite an abelian envelope construction; I think the two steps above can be combined into one step where we consider presheaves satisfying some suitable property. This is a kind of restricted Yoneda embedding and so it can be thought of as adjoining a particular kind of colimit (I think the construction will be equivalent to adjoining finite colimits) but we aren't deliberately trying to adjoin any limits. | |
| Nov 28, 2013 at 2:55 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Nov 28, 2013 at 2:50 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Nov 28, 2013 at 2:44 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |