Timeline for Can $\mathcal O_X$ be recognized abstract-nonsensically?
Current License: CC BY-SA 3.0
21 events
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Dec 14, 2016 at 10:42 | comment | added | HeinrichD | @Joël: See ncatlab.org/nlab/show/Gabriel-Rosenberg+theorem | |
Dec 12, 2016 at 5:45 | comment | added | Yosemite Sam | @Joël That's Gabriel's theorem, which works for a very general class of schemes and also algebraic spaces. (but fails badly for stacks: Spec (C x C) and BZ/2Z is a counterexample). | |
Dec 11, 2016 at 20:40 | vote | accept | მამუკა ჯიბლაძე | ||
Dec 11, 2016 at 18:54 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
added stuff about not being a generator
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Dec 11, 2016 at 18:44 | comment | added | მამუკა ჯიბლაძე | @Joël I completely agree it has become confusing, and probably I will split the question, leaving here only the portion relevant for Ben Webster's fantastic answer. I just want to clarify the latter for myself better, but so far it is closest to what I wanted to know. Seems like there is a property of objects in general abelian categories which characterizes precisely (sheaves of sections of) line bundles at least for categories of coherent sheaves over schemes locally of finite type. | |
Dec 11, 2016 at 18:39 | comment | added | Joël | Interesting question, interesting answers, but the whole field has become quite confusing now, with many partial answers to many diiferent interpretations of the question. I think re-asking one or several independent questions may help bring some clarity. For instance, one of the question I'd like to see solved is: if $X$ and $Y$ are two schemes, is true that the category of quasi-coherent $O_X$-modules determines $X$? As the OP notes, this is true when $X$ and $Y$ are affine, and as Ben's answer shows, it at least determines $X$ as a topological space when $X$ is locally of finite type. | |
Dec 11, 2016 at 2:53 | answer | added | Ben Webster♦ | timeline score: 6 | |
Dec 10, 2016 at 16:54 | comment | added | მამუკა ჯიბლაძე | @HeinrichD Well that's why I mentioned endoadjunctions - they carry a canonical tensor structure (given by composition) which is always there. I just wonder what it is in this case - can it be (monoidally equivalent to) the category of sheaves of abelian groups with their tensor product? | |
Dec 10, 2016 at 10:30 | comment | added | HeinrichD | No, this enrichment does not come for free, and also notice that the enrichment in $\mathcal{O}_X$-modules is exactly equivalent to the tensor product, which you wanted to avoid. | |
Dec 10, 2016 at 6:08 | comment | added | მამუკა ჯიბლაძე | @GeorgLehner Interesting idea! Maybe one can discover this enrichment purely categorically. I was thinking about the category of all adjoint pairs of endofunctors of the category of $\mathcal O_X$-modules... | |
Dec 9, 2016 at 23:28 | comment | added | Georg Lehner | We can view the category of $O_X$-modules as enriched over the category of sheaves on $X$, and I believe it should be the case that the sheaf of Sh(X)-natural endotransformation of the identity functor on $O_X$-modules is isomorphic to $O_X$ analogously. Not sure how helpful that is though. | |
Dec 9, 2016 at 17:59 | comment | added | Yosemite Sam | The categorical center of Mod(R), which by definition is endomorphisms of the identity functor, is the center of R. Thus, when R is commutative you have an intrinsic way of characterizing R, however not as an object inside Mod(R), which is odd. This of course does not work globally. | |
Dec 9, 2016 at 16:29 | answer | added | HeinrichD | timeline score: 1 | |
Dec 9, 2016 at 15:35 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
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Dec 9, 2016 at 9:31 | comment | added | მამუკა ჯიბლაძე | But what does it actually mean that I said? :D | |
Dec 9, 2016 at 9:31 | comment | added | მამუკა ჯიბლაძე | Oh I think I understand. It is true locally, not globally? | |
Dec 9, 2016 at 9:30 | comment | added | მამუკა ჯიბლაძე | @HeinrichD Really? Sounds important. Could you please elaborate? | |
Dec 9, 2016 at 9:29 | comment | added | HeinrichD | " the subcategory generated by it is the category of quasicoherent sheaves" is not correct. | |
Dec 9, 2016 at 8:36 | answer | added | Will Sawin | timeline score: 12 | |
Dec 9, 2016 at 5:53 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
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Dec 9, 2016 at 5:37 | history | asked | მამუკა ჯიბლაძე | CC BY-SA 3.0 |