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I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely generated left $R'$-modules for some left noetherian ring $R'$.

Which examples are known? Do all of them support full exceptional collections (see https://en.wikipedia.org/wiki/Semiorthogonal_decomposition#Exceptional_collection)?

More generally, which examples of this sort are known if $P$ is a regular scheme that is proper over the spectrum of a (commutative) noetherian ring?

This question is close to Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts have enough injectives?

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  • $\begingroup$ I think what you ask for is existing of a tilting generator $T$ in $D^b(P)$; then $R = End(T)$ is the corresponding ring, in fact, finite-dimensional noncommutative $k$-algebra. $\endgroup$
    – Evgeny Shinder
    Commented Nov 24, 2021 at 20:06
  • $\begingroup$ Regarding exceptional collections, it not true that if $R$ is a finite dimensional $k$-algebra of finite global dimension (this corresponds to $P$ being smooth proper) has a full exceptional collection in $D^b(R-mod)$. $\endgroup$
    – Evgeny Shinder
    Commented Nov 24, 2021 at 20:08
  • $\begingroup$ Well, not all t-structures come from isomorphisms to the corresponding derived category. However, examples of tilting objects are certainly interesting to me. And may I ask you to explain "this corresponds to P being smooth proper"? I can prove something like that; yet possibly you know more nice statements on this matter. $\endgroup$
    – Mikhail Bondarko
    Commented Nov 25, 2021 at 8:36
  • $\begingroup$ A good reference is Orlov's paper on smooth and proper triangulated dg-categories: arxiv.org/abs/1402.7364 $\endgroup$
    – Evgeny Shinder
    Commented Nov 25, 2021 at 21:05

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