I am reading (HTT, D-modules, Perverse Sheaf and Representation Theory). In (HTT, 1.5.7, page 32), it is claimed that an equivalence of categories between bounded derived category with quasi-coherent cohomologies and bounded derived category of quasi-coherent sheaves of $D_X$ modules

$$D_{qc}^b(mathrm{Mod}(D_X))sim D^b(mathrm{Mod}_{qc}(D_X)).$$

It says that “It will not be used in what then follows and the proof is omitted.” The refference it cited is (Borel, Algebraic D-modules VI.2.10. page 222), whose proof is really short (almost 1 page). However, it seems there are some problems.

Firstly, some notations do not make sense. In (Borel), $mathcal{A}$ is a quasi-coherent sheaf of $mathcal{O}_X$-algebras (Not necessarily be commutative. In our case, $D_X$ is non-commutative). Let us assume $X$ is a “good” scheme. It is denoted by $mu (mathcal{A})$ the category of quasi-coherent $mathcal{A}$-modules, where quasi-coherent means quasi-coherent over $mathcal{O}_X$. Our goal is to prove the fully faithfulness

$$mathrm{Hom}_{D(mu (A))}(F^cdot, G^cdot)=mathrm{Hom}_{D_{qc}(mathrm{Mod}(mathcal{A}))}(F^cdot, G^cdot),~F^cdot,G^cdotin D^b(mu (mathcal{A})),$$

In the second set $mathrm{Hom}_{D_{qc}(mathrm{Mod}(mathcal{A}))}(F^cdot, G^cdot)$ a morphism is represented by a third complex $H^cdot$ which may not in $mu(mathcal{A})$, and a quasi-isomorphism. In (Borel), the following notation of $mathrm{Hom}^cdot$-complex seems uncorrect

$$mathrm{Hom}^cdot_{D(mu (A))}(F^cdot, G^cdot) .$$

I will assume that this is a typo in the following.

Secondly, I do not understand how it was able to reduce to the affine case in (Borel). I do know the adjointness of pushforward and pullback

$$mathrm{Hom}_{mathcal{A}}(F^cdot,i^*(tilde{J}))=mathrm{Hom}_{mathcal{A}|_U}(F^cdot|_U,tilde{J}).$$

But I am not sure if the following is true

$$mathrm{Hom}_{D(mathcal{A})}(F^cdot,i^*(tilde{J}))=mathrm{Hom}_{D(mathcal{A}|_U)}(F^cdot|_U,tilde{J})$$

Thirdly, I have another possible explanation using $mathrm{Hom}^cdot$-complex and $Rmathrm{Hom}$ by using an injective resolution of $G^cdot$ in $K^+(mathrm{Mod}_{qc}(mathcal{A})))$, say $I^cdot in mathrm{Ob}(K^+(mathrm{Mod}_{qc}(mathcal{A})))$. Then

$$H^0Rmathrm{Hom}_{mu(mathcal{A})}(F^cdot,G^cdot)=H^0mathrm{Hom}^{cdot}_{mu(mathcal{A})}(F^cdot,I^cdot)cong mathrm{Hom}_{D^+(mu(mathcal{A}))}(F^cdot,G^cdot)$$

This approach finally suffers from I do not know if an injective object in $mu(mathcal{A})$ is injective in $mathrm{Mod}(mathcal{A})$. In (HTT, 1.4.14, page 29) it was particularly added a comment “We do not claim that $I$ is injective in $mathrm{Mod}(D_X)$“. In (Thomason and Trobaugh, Higher Algebraic K-theory of schemes and of derived categories, Appendix B.4, page 410), it is explained that in a general scheme, the right derived funtor in $mathrm{Mod}(mathcal{O}_X)$ and the right derived funtor in $mathrm{Mod}_{qc}(mathcal{O}_X)$ could have different effects on quasi-coherent objects (SGA 6, II App. I 0.2). While in Noetherian case, they are the same, essentially because the injective objects in $mathrm{Mod}(mathcal{O}_X)$ are also injective in $mathrm{Mod}_{qc}(mathcal{O}_X)$ on a Noetherian scheme (Hartshorne, Residues and Duality, 7.18, 7.19, page 133).

I noticed that in (Bökstedt-Neeman, Homotopy limit and derived category, corollary 5.5), it is true for a quasi-compact and separated scheme that the natrual inclusion

$$D(mathrm{QCoh}(mathcal{O}_X)) to D_{qc}(mathrm{Mod}(mathcal{O}_X))$$

is an equivalence of category

But their proof is pretty long and I am not familiar with them. I do not know what is essential so far and whether it extends to our case for D-modules.

Also see (Stacks project, 36.3) for the affine case, whose proof is not so short. These kind of results are unbounded.

Bernstein himself in his notes (Bernstein, Algebraic theory of D-moduls, page 14, lemma) claimed a similar without proof. He simply said “properties of coherent D-modules imply the natural morphism

$$D_{coh}(mathrm{Mod}(D_X))to D(mathrm{Mod}_{coh}(D_X))$$

is an equivalence of categories”. But it seems (Bökstedt-Neeman)’s long argument implies that this should not be simple.

I am a beginner, so I am not sure if this question is important. However, it seems that we could restrict ourselves in just one of them safely and put this question aside. Bernstein himself in his notes (Bernstein, Algebraic theory of D-moduls, page 22, remark) said that “I do not know whether the natural morphism $D(Hol(D_X))to D_{hol}(D_X)$ is an equivalence of categories. In a sense, I do not care.”