Hello,
I read that if $A\subset D$ is the heart of a bounded t structure, D a triangulated category, then D=D^b(A).
Have not been able to find a proper reference for this. Can anyone confirm this result?
Thanks!
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You're never going to find a reference because it's false. Just take $D$ to be the stable homotopy category of Postnikov pieces, i.e. spectra with finitely many non-trivial homotopy groups. Take $D_{\geq 0}$ (resp. $D_{\leq 0}$) the full subcategory of spectra with non-trivial homotopy groups concentrated in non-negative (resp. non-positive) degrees. This is a $t$-structure whose heart $D_{\geq 0}\cap D_{\leq 0}$ is equivalent to the category $Ab$ of abelian groups. There's even a functor $D^b(Ab)\rightarrow D$ preserving the $t$-structure (take the canonical $t$-structure on the left) and inducing an equivalence between the hearts, but $D$ cannot be equivalent to $D^b(Ab)$. Actually $D$ has no algebraic model. |
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To add to what Fernando Muro said, there are examples where this is true. Beilinson showed in his paper "On the derived category of perverse sheaves" (Springer LNM 1289 pp. 27-41) that it is true when you are talking about the middle perverse t-structure in the constructible derived category of sheaves on schemes. If you look through Chapter 3 of BBD ("Faisceaux pervers") you will see how subtle it is even to define a functor from $D^b(A)$ to D, and you can also see from Beilinson's solo article how different that task is from the task of showing that such a map is an equivalence. That it works at all depends quite a bit on the fact that the perverse t-structure is used. |
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