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Define $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ to be the full subcategory of the derived category $D^{\text{#}}(Mod(\mathcal{D}_X))$ of complexes of $\mathcal{D}_X$-modules whose cohomology groups belong to $Mod_{\bullet}(\mathcal{D}_X)$, # = $+,-,b$, $X$ smooth algebraic variety over $\mathbb{C}$.

Is it true that $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ is equivalent to the category $D^{\text{#}}(Mod_{\bullet}(\mathcal{D}_X))$?

It is true that $Mod_{\bullet}(\mathcal{D}_X)$ are closed under kernels, cokernels and extensions because we are working over a Noetherian scheme. However, Kashiwara-Schapira Category and Sheaves states the equivalence under one more hypothesis. Apart from the answer, if you can quote any reference I would be glad.

EDIT: I forgot to say that $\bullet =$ quasi coherent modules or coherent modules (a quasi coherent $\mathcal{D}_X$ module is a $\mathcal{D}_X$ modules which is quasi coherent over $\mathcal{O}_X$, instead coherence is over $\mathcal{D}_X$)

Federico Barbacovi
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1 Answers1

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I post it as an answer because I found it on a book: Theorem 1.5.7 of D-Modules, perverse sheaves, and representation theory by Hotta, Takeuchi, and Tanisaki states that the natural functors give equivalence or categories

$$D^{b}(Mod_{\bullet}(\mathcal{D}_X)) \rightarrow D^{b}_{\bullet}(\mathcal{D}_X)$$

The book says that this result was due to Bernstein, and it can be found Algebraic D-Modules, Perspectives in Mathematics, Vol 2, 1987, by Borel, Grivel, Kaup, Haefliger, Malgrange, And Ehlers.

Federico Barbacovi
  • 1,315