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I have two questions concerning the existence and uniqueness of enhancements in the following cases: i.) Let $A$ be a finite dimensional $k$ algebra of finite global dimension. Does the triangulated category $D^b(\mathrm{mod}A)$ of finitely generated $A$ modules admit (unique) enhancement?

ii.) let $X$ be a smooth projective scheme over $k$. Then $D^b(X)$ has a unique enhancement. What is about admissible subcategories $R\subset D^b(X)$...Do they have (unique) enhancements?

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    $\begingroup$ hi Aleksa, may I ask what kind of enhancement are you thinking of? $\endgroup$ Jan 3, 2014 at 22:37
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    $\begingroup$ @FernandoMuro, why would it matter? Surely Aleksa means some kind of stable linear (infty,1)-categorical enhancements. $\endgroup$
    – AAK
    Jan 3, 2014 at 23:15
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    $\begingroup$ Adeel, maybe it matters, maybe not, let Aleksa say. $\endgroup$ Jan 3, 2014 at 23:16
  • $\begingroup$ I imagine ii) should be true, but I'm not familiar enough with the techniques of Lunts-Orlov to be sure. $\endgroup$
    – AAK
    Jan 3, 2014 at 23:17
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    $\begingroup$ The enhancement in both cases should be a pretriangulated DG-categories just as assumend in the work of Lunts-Orlov. $\endgroup$
    – Aleksa
    Jan 4, 2014 at 9:34

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