# Admissible subcategories of $D^b(\mathbb{P}^n)$

Recall that a triangulated subcategory $\mathcal{A}$ of a triangulated category $\mathcal{B}$ is called admissible if the inclusion functor has both left and right adjoints.

Is it true that all admissible subcategories of $D^b(\mathbb{P}^n_k)$ (the bounded derived category of coherent sheaves on $\mathbb{P}^n_k$, for a field $k$) are generated by exceptional collections?

Perhaps the right setting for the question is to consider admissible subcategories $\mathcal{A}$ of a $k$-linear triangulated category $\mathcal{B}$ generated by a (strong) exceptional collection rather than just $D^b(\mathbb{P}^n_k)$.

• This is definitely not true for a category generated by an exceptional collection (without strongness assumption though): an elementary geometric counterexample can be found in the paper arxiv.org/abs/1304.0903 Nov 9, 2013 at 20:04
• @AntonFonarev: Thanks a lot for the reference.
– naf
Nov 10, 2013 at 6:46
• For $n=1$ this is folklore, whilst for $n=2$ it was settled last year by Pirozhkov in arxiv.org/abs/2006.07643. For $n\geq 3$ it's open. If @naf thinks this should be an accepted answer I can turn this comment into an answer. Apr 9, 2021 at 13:22
• @pbelmans: Thanks, and yes, I'd be happy to accept this as an answer.
– naf
Apr 9, 2021 at 13:47

For $$n=1$$ this is folklore (it is I think a pleasant exercise using global dimension 1 and the description of coherent sheaves on curves), whilst for $$n=2$$ it was settled last year by Pirozhkov in his preprint Admissible subcategories of del Pezzo surfaces. He also gives the analogous statement for del Pezzo surfaces.
For $$n\geq 3$$ it's still open as far as I know.