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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 $D^b(\mathbb{P}^n_k)$.

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    $\begingroup$ 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 $\endgroup$
    – Anton Fonarev
    Commented Nov 9, 2013 at 20:04
  • $\begingroup$ @AntonFonarev: Thanks a lot for the reference. $\endgroup$
    – naf
    Commented Nov 10, 2013 at 6:46
  • $\begingroup$ 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. $\endgroup$
    – pbelmans
    Commented Apr 9, 2021 at 13:22
  • $\begingroup$ @pbelmans: Thanks, and yes, I'd be happy to accept this as an answer. $\endgroup$
    – naf
    Commented Apr 9, 2021 at 13:47

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Reposting my comment as a slightly extended answer.

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.

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