People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in derived categories of coherent sheaves (over algebraic varieties; possibly the first example corresponds to the Beilnson's description of the derived category of coherent sheaves on the projective space of dimension $n$). Are there any examples of this notion in some stable homotopy categories (in the sense of abstract model categories; one can consider the category of modules over a ring spectrum here)?
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asked Nov 21, 2012 at 3:39
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$\begingroup$ 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. $\endgroup$– Eric WofseyCommented Nov 21, 2012 at 9:04
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$\begingroup$ I updated the question. $\endgroup$– Mikhail BondarkoCommented Nov 21, 2012 at 9:42
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$\begingroup$ 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 $\endgroup$– Fernando MuroCommented Nov 21, 2012 at 10:51
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$\begingroup$ 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. $\endgroup$– Eric WofseyCommented Nov 21, 2012 at 10:55
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2$\begingroup$ 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 $i>j$. $\endgroup$– SashaCommented Nov 21, 2012 at 11:09
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1 Answer
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A natural topological example of a [higher] category with an exceptional collection is constructible complexes on a stratified space, locally constant along contractible strata. This holds with any coefficients - e.g. you can look at sheaves of S-modules or E-modules for an $E_\infty$-ring spectrum if you prefer. The exceptionality encodes a. the contractibility of strata, b. the absence of Exts "in the wrong direction" for extensions of constant sheaves off the strata.
I would presume you could also find examples of "derived Fano schemes" with exceptional collections - i.e. nontrivial analogs of the many (toric Fano e.g.) schemes admitting exceptional collections, but I don't know of such.
answered Nov 21, 2012 at 20:50
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$\begingroup$ nice! I took the 'topological' from the title to mean 'homotopical' or 'spectral', didn't think about plain topological. $\endgroup$ Commented Nov 22, 2012 at 23:07