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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
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