4
$\begingroup$

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 $\mathcal{D}_X$)

$\endgroup$
4
  • $\begingroup$ What is $Mod_\bullet$? $\endgroup$
    – Bugs Bunny
    Mar 29, 2018 at 9:45
  • $\begingroup$ @BugsBunny Forgive me, I added what is $\bullet$ $\endgroup$ Mar 29, 2018 at 10:32
  • 1
    $\begingroup$ Did you check mathoverflow.net/questions/236245/… $\endgroup$
    – Bugs Bunny
    Mar 29, 2018 at 11:14
  • $\begingroup$ IMHO, the answer is UNKNOWN. If there is a good reason for it to be YES, it is known thanks to Exercise on p.153 of Gelfand-Manin or 2.42 and 3.4 of Huybrechts... $\endgroup$
    – Bugs Bunny
    Mar 29, 2018 at 11:16

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ May I ask if the result is know when $X$ is singular? $\endgroup$ Feb 18, 2019 at 11:30
  • 1
    $\begingroup$ Unfortunately, I don't know. The book I referred to deal with D-modules on smooth varieties. $\endgroup$ Feb 18, 2019 at 16:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.