Timeline for Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?
Current License: CC BY-SA 2.5
6 events
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| Mar 13, 2010 at 0:30 | comment | added | David E Speyer | And, as you might guess, that inverse is the opposite derived reflection functor. A big hint here is to understand that Rep Q is hereditary. A consequence is that every object in the dervied category is a direct sum of shifts of objects from Rep Q. | |
| Mar 12, 2010 at 22:27 | comment | added | Ben Webster♦ | I think you may need to take some time to learn more about derived categories if you want to understand this statement. The derived reflection functor is a functor between two derived categories, and is an equivalence in the usual sense: it has an inverse up to isomorphism of functors. | |
| Mar 12, 2010 at 21:10 | comment | added | darij grinberg | Hmm, could you please be more precise about what "the derived reflection functors are equivalences of derived categories" means? I am understanding this as: "if A -> B -> C is a complex of representations of a quiver Q, and i is a sink of Q, then Ker (F_i^+ B -> F_i^+ C) / Im (F_i^+ A -> F_i^+ B) is isomorphic to Ker (B -> C) / Im (A -> B)". But I can't confirm this. I'd be glad to know whether I'm trying to prove the right thing at all. | |
| Mar 12, 2010 at 14:10 | comment | added | darij grinberg | That's a lot of stuff I want to understand one day. As for derived category, all I know is the definition. Will try to prove that the derived reflection functors are isomorphisms. | |
| Mar 12, 2010 at 14:04 | vote | accept | darij grinberg | ||
| Mar 12, 2010 at 13:54 | history | answered | David E Speyer | CC BY-SA 2.5 |