Timeline for Exceptional collections of objects in topological triangulated categories?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21, 2012 at 20:50 | answer | added | David Ben-Zvi | timeline score: 4 | |
| Nov 21, 2012 at 12:38 | comment | added | Sasha | @Eric: I would rather expect them to be modules over $S$ itself. | |
| Nov 21, 2012 at 12:24 | comment | added | Eric Wofsey | @Sasha: Am I correct in thinking that the $E_i$ should be something like modules over some augmented $S$-algebra? | |
| Nov 21, 2012 at 11:50 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
By a 'stable homotopy category' I mean any 'topological' triangulated category.
|
| Nov 21, 2012 at 11:09 | comment | added | Sasha | So, I guess in the topological situation you first have to specify the base ring object. Then an exceptional object over a ring object $S$ should be an object $E$ on which $S$ acts and such that $Map(E,E)\cong S$. A collection $E_1,…,E_n$ of $S$-exceptional objects should be called exceptional if $Map(E_i,E_j)$ is contractible for $i>j$. | |
| Nov 21, 2012 at 11:02 | comment | added | Sasha | @Eric: An object $E$ in a $k$-linear triangulated category is exceptional if $Ext^\bullet(E,E) = k$. A collection $E_1,\dots,E_n$ of exceptional objects is exceptional if $Ext^\bullet(E_i,E_j) = 0$ for $i > j$. | |
| Nov 21, 2012 at 10:55 | comment | added | Eric Wofsey | The condition that Ext vanishes in all degrees except 0 is kind of bizarre from a topological point of view--it says that the spectrum of maps between two objects is an Eilenberg-MacLane spectrum concentrated in degree 0. This is not something I can imagine coming up in a "topological" setting unless the mapping spectra were all actually 0. | |
| Nov 21, 2012 at 10:51 | comment | added | Fernando Muro | I find this question very interesting, as most of your questions. I've wondered the same in the past, but never thought of it seriously. I don't remember to have seen anything like that in purely topological triangulated categories (algebraic ones, like those arising in algebraic geometry, are also topological by trivial reasons). I have the feeling that it is a typical phenomenon of algebraic contexts. This may be suported by the fact that, under some circumstances, the triangulated category you start with ends up being equivalent to the derived category of the endomorphism algebra of the col | |
| Nov 21, 2012 at 9:42 | comment | added | Mikhail Bondarko | I updated the question. | |
| Nov 21, 2012 at 9:42 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
added 217 characters in body
|
| Nov 21, 2012 at 9:04 | comment | added | Eric Wofsey | Can you say more about what "exceptional" means and the algebraic examples you have in mind? There are certainly notions of "orthogonal" subcategories that come up in the theory of Bousfield localization. | |
| Nov 21, 2012 at 3:39 | history | asked | Mikhail Bondarko | CC BY-SA 3.0 |