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question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? Assume if necessary $A^c$ idempotent complete.

Example I am interested is $A = D(Qcoh(X))$ where $X$ good scheme, so $A^c = D_{perf}(X)$. Is it possible to recover $D(Qcoh(X))$ from $D_{perf}(X)$ ?!

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    $\begingroup$ It might be worth noting that the answer to your question is yes in the setting of (cocomplete) stable $\infty$-categories: more precisely, $A$ is equivalent to the ind-completion of its compact objects. In particular, if $X$ is a nice scheme (e.g., quasi-projective), then the compact generation of the derived category quasi-coherent sheaves is known (I believe by Thomason-Neeman). $\endgroup$
    – Akhil Mathew
    Commented Aug 26, 2014 at 17:49
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    $\begingroup$ On the other hand, it's not true that the construction of $Ind$-objects commutes with passage to the homotopy category. Stated another way: filtered homotopy inverse limits don't correspond to inverse limits at the level of $\pi_0$ (because of the existence of $\lim^1$ phenomena). $\endgroup$
    – Akhil Mathew
    Commented Aug 26, 2014 at 17:51

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