Timeline for How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 26, 2010 at 5:42 | vote | accept | Shizhuo Zhang | ||
| Feb 25, 2010 at 20:13 | comment | added | Greg Stevenson | @Shizhuo: In the case of Balmer's approach one will recover the scheme corresponding to the tensor product one picks. If I understand correctly to use Rosenberg's approach one needs an abelian category? In this case one will get the scheme corresponding to the t-structure one picks. Either the choice of t-structure or tensor product fixes the geometry associated to the category. | |
| Feb 25, 2010 at 16:23 | answer | added | Zoran Skoda | timeline score: 8 | |
| Feb 25, 2010 at 11:55 | comment | added | Shizhuo Zhang | @Greg: Thank you for reminding me the work of Bridgeland. Yes, in general one can not recover the scheme back without extra structure as you mentioned. Then I have a question: if we do not have extra structure,and we still follow the reconstruction process in sense of Rosenberg or Balmers, what will we get? Different schemes but the same underlying topological space? | |
| Feb 25, 2010 at 11:40 | comment | added | Shizhuo Zhang | @Bryan: Yes, I asked him yesterday and he explained relations of theses two reconstruction theorems to me. From representation POV and algebraic geometry POV | |
| Feb 25, 2010 at 11:39 | comment | added | Shizhuo Zhang | @Matthew: Thank you for very enlighten comments, because I just took a glance at Balmer's paper and not very clear what roles tensor structure play. I am really happy to see the relation with Bondal-Orlov's results. I will check out the notes you mentioned. | |
| Feb 25, 2010 at 3:34 | history | edited | Shizhuo Zhang | CC BY-SA 2.5 |
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| Feb 25, 2010 at 2:25 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Feb 25, 2010 at 0:30 | answer | added | David Ben-Zvi | timeline score: 26 | |
| Feb 25, 2010 at 0:29 | comment | added | B. Bischof | If you still have your notes from last semester in the 800 class, Rosenberg gave us his perspective on Tannaka-Krein. | |
| Feb 24, 2010 at 23:36 | comment | added | Matthew Ballard | There is a lemma in Thomason's "Classification of triangulated subcategories" that says a category is tensor-closed if and only if it closed under tensoring by powers of an ample line bundle. Thus, Bondal-Orlov's reconstruction theorem is a consequence of Balmer's result. A proof of Bondal-Orlov's result along these lines is in Rouquier's notes - see paper 38 on his webpage. One caveat: Bondal-Orlov use only the graded structure of the derived category. With this argument, you need the triangles. | |
| Feb 24, 2010 at 23:16 | comment | added | David Roberts♦ | Continuing what Theo commented above, the group $G$ (in Tannaka-Krein) is reconstructible as the automorphisms of the functor $f$. This is more akin to Grothendieck's Galois theory, where the fundamental group is (re)constructed from the functor $C \to Set$ satisfying some nice properties. | |
| Feb 24, 2010 at 20:25 | comment | added | Greg Stevenson | I thought I'd point out that yes one needs extra structure such as the tensor product in the triangulated case. For instance consider two smooth projective birational Calabi-Yau 3-folds. They are derived equivalent (by Bridgeland) so one needs some extra data or it won't be possible to recover both schemes. I also thought I'd comment that (in my opinion) there should be some good notion of homological fibre functor which comes up when extending Balmer's theory to the compactly generated setting. Knowing these would determine the spectrum, but they should know other interesting things too! | |
| Feb 24, 2010 at 16:29 | comment | added | Theo Johnson-Freyd | Nice question. I don't know enough scheme theory to be able to start on an answer. One comment, however: reconstruction theorems generally require a bit more than just the (abstract) category "of representations". More precisely, the statement of Tannaka-Krein is that given a category C with nice properties and a functor f:C->Vect with nice properties, there is a unique (and constructible, from the data) group G up to isomorphism so that (C,f) is equivalent to (G-modules,forget). You can also relax the properties a bit and replace "group" by "(quasitriangular) (quasi)Hopf algebra", etc. | |
| Feb 24, 2010 at 12:18 | history | asked | Shizhuo Zhang | CC BY-SA 2.5 |