Timeline for Can $\mathcal O_X$ be recognized abstract-nonsensically?
Current License: CC BY-SA 3.0
12 events
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| Dec 9, 2016 at 15:19 | comment | added | მამუკა ჯიბლაძე | @DenisNardin As Will Sawin points out the question is vague about that point, but I had in mind properties of $\mathcal O_X$ as an object of an abelian category. | |
| Dec 9, 2016 at 14:14 | comment | added | Denis Nardin | @მამუკაჯიბლაძე I'm not sure if this is what you wanted, but if the scheme $X$ is quasi-compact and quasi-separated you can reconstruct it completely from the category of perfect complexes if you remember the symmetric monoidal structure (if I recall correctly this is in Balmer's Presheaves of triangulated categories and reconstruction of schemes) | |
| Dec 9, 2016 at 9:40 | comment | added | მამუკა ჯიბლაძე | @HeinrichD And you still cannot pin down $R$ among rank one projectives? Strange... | |
| Dec 9, 2016 at 9:39 | comment | added | HeinrichD | @მამუკაჯიბლაძე: No, the isomorphism is canonical. No choices are involved. | |
| Dec 9, 2016 at 9:38 | comment | added | HeinrichD | What Will Sawin has sketched in the last paragraph has been worked out for the category of quasi-coherent sheaves by Gabriel, Rosenberg and others. See again arxiv.org/abs/1310.5978 . For the whole category of modules there is a similar construction using idempotents (unpublished work). | |
| Dec 9, 2016 at 9:38 | comment | added | მამუკა ჯიბლაძე | @HeinrichD Presumably to identify these groups means to pick a(n equivalence class of a) symmetric monoidal structure? | |
| Dec 9, 2016 at 9:31 | comment | added | HeinrichD | Yes, the group of auto-equivalences of $\mathsf{Mod}(R)$ is $\mathsf{Pic}(R) \rtimes \mathrm{Aut}(R)$. Something similar holds for quasi-separated schemes. See arxiv.org/abs/1310.5978 | |
| Dec 9, 2016 at 9:27 | comment | added | მამუკა ჯიბლაძე | I think I get that part now. You might have several non-isomorphic projective generators with commutative endomorphism rings, the latter rings being necessarily isomorphic to each other. And there is no way to tell which one is "the" $R$, right? What you get is a sort of a "Pic-torsor", a canonical one, but without a distinguished element... Funny, is not it? | |
| Dec 9, 2016 at 9:22 | comment | added | მამუკა ჯიბლაძე | I see, thanks. I will think about modifying the question accordingly; although I still do not understand well enough the difference you pointed out, I acknowledge that there is a difference. | |
| Dec 9, 2016 at 9:20 | comment | added | Will Sawin | @მამუკაჯიბლაძე I completely agree with (2). I'm just saying that is not a way of detecting $R$ inside the category of $R$-modules. It's a way of constructing $R$, outside the category of $R$-modules, using the category of $R$-modules. | |
| Dec 9, 2016 at 9:17 | comment | added | მამუკა ჯიბლაძე | Thank you for the very informative answer. I still have to digest it, but here are some quick questions/comments. I agree I should be more accurate, but I don't understand already the part about $R$. (1) is not it correct that if an abelian category is equivalent to the category of modules over a ring, then this ring is isomorphic to the endomorphism ring of a projective generator? (2) if further one knows that the ring in question is commutative, is not it true that it is necessarily isomorphic to the ring of endotransformations of the identity functor? | |
| Dec 9, 2016 at 8:36 | history | answered | Will Sawin | CC BY-SA 3.0 |