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Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$.

1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck category? I suspect not, just based on the fact that people don't usually talk about projective $D$-modules, only about locally projective ones.

2) Anyhow, Consider the dg-category $D(X)$ got by considering the dg-category of unbounded complexes in $M(X)$ and localizing by quasi-isomorphisms. Is it "presentable"? How to perform calculations, i.e. are there enough $k$-injective complexes or something like that?

Thank you

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    $\begingroup$ Grothendieck categories don't have enough projectives in general, rather they have enough injectives. So there might be something wrong with question 1. $\endgroup$
    – pbelmans
    Commented Jul 6, 2016 at 11:40
  • $\begingroup$ @pbelmans : You are right, I was confused; But this is better, now there is more chance for it being a Grothendieck category. $\endgroup$
    – Sasha
    Commented Jul 6, 2016 at 12:46
  • $\begingroup$ Regarding 1), I think the answer is affirmative, see Lemma A.0.5 1 in webusers.imj-prg.fr/~haohao.liu/DFM.pdf. $\endgroup$
    – Doug Liu
    Commented Jul 15, 2023 at 16:53

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