# Full exceptional set on flag variety

Define a full exceptional set in a triangulated category to be a partially ordered set of objects $\Delta_i$ which generate the category and such that $\text{Ext}^\bullet(\Delta_i, \Delta_j) = 0$ unless $i \geq j$ and $\text{Ext}^\bullet(\Delta_i, \Delta_i) = k$.

Is there a known (nonequivariant) full exceptional set in coherent sheaves on the full flag variety? Or partial flag varieties that are not $\mathbb{P}^n$?

• Do you really want the set of objects to be partially ordered? I'd expect "linearly ordered". Also, you defined "exceptional set" in the first sentence, but then your question is about "quasiexceptional sets". It would probably help to explain the difference. Feb 15, 2014 at 0:49
• @Harrison Oh hey! I think there are known examples of such exceptional sets (of equivariant bundles) found by Kuznetsov, but it is not known that they are full i.e. generate $D^b(G/B)$. I don't think removing the "equivariant" assumption helps with proving that some set is full, so I guess the answer is no. The proof of fullness for $P^n$ uses a "box resolution of diagonal" argument, which works for quadrics and maybe Grassmannians, but not for other $G/P$. Feb 15, 2014 at 6:22
• The collection you want is probably here in this paper by Kuznetsov and Polishchuk arxiv.org/abs/1110.5607 Feb 15, 2014 at 6:23
• I think if you have an exceptional collection (modulo some technical hypotheses that I maybe forgetting) it can be made equivariant. There are some notes on Seidel's webpage that explain this(Categorical Dynamics and something,something) Feb 15, 2014 at 6:25
• The answer is in §3 of On the derived categories of coherent sheaves on some homogeneous spaces by M. Kapranov, Invent. Math. 92 (1988), no. 3, 479-508.
– abx
Feb 15, 2014 at 6:32

First, if a semisimple simply connected group $G$ acts on a variety $X$ then any exceptional object in $D^b(coh(X))$ has a $G$-equivariant structure (proved by Elagin and by Polishchuk).

Second, indeed, as abx mentioned Kapranov has constructed a full exceptional collection on all (partial) flag varieties of type $A$ and on quadrics. Moreover, using a result of Samokhin one can construct a full exceptional collection on $X$ for $X$ having a fibration structure over $Y$ if there is a collection of objects on $X$ which restrict to a full exceptional collection on all fibers of $X$ over $Y$ and if $Y$ itself has a full exceptional collection. This allows to construct a full exceptional collection on some partial flag varieties of type $B$, $C$ and $D$ (including full flags). Using a recent result of Manivel and Faenzi one can also construct a full exceptional collection on some partial flags of type $E_6$. Finally a full exceptional collection is known on all flag varieties of type $G_2$.

And last, as Daniel Pomerleano said, there is a general construction of exceptional collections on flag varieties which produces a collection of maximal possible length on all flag varieties for groups of type $B$, $C$, and $D$ which is not known to be full. And unfortunately, it is not true that an exceptional collection generating $K_0$ is full --- there are counterexamples by Boehning-Graf vn Bothmer-Katzarkov-Sosna and by Gorchinsky-Orlov.