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Is $D^b(Mod_{qc}(D_X)) \to D^b_{qc}(D_X)$ an equivalence of categories for singular $X$? Where $Mod_{qc}(D_X)$ is the category of quasi-coherent modules over $D_X$ and $D^b_{qc}(D_X)$ is the category consisting of objects $X$ such that $H^j (X)$ is quasi-coherent for all $j$. This is known to be an equivalence for $X$ smooth, by Ryoshi Hotta Kiyoshi Takeuchi Toshiyuki Tanisaki D-Modules, Perverse Sheaves, and Representation Theory Theorem 1.5.7. Thanks in advance.

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  • $\begingroup$ Maybe this will be helpful: the LHS has enough injectives even when $X$ is singular. $\endgroup$ – FunctionOfX Feb 12 at 16:20

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