Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$ the algebra of differential operators over it.

The overall vague question is what kind of algebraic object is $D$ and what kind of category is the category of its modules? Here are some points whose answers could together be considered an answer to this.

When (meaning for what kind of $D$-modules $M,N$ left or right) is there a $D$-module structure on $Hom_R(M,N)$? On $M\otimes_RN$? Does this make $D$-mod into an

abelian monoidal category with fiber functor to $R$-mod?If so isit closed monoidal?

When does a $D$-module $M$ admit a dual $M^*$?(in the sense of monoidal categories).

Is the abelian category of $D$-modules isomorphic to $D^{op}$-mod?(before deriving). If not how are they related? Maybe the correct thing to consider is the opposite co-opposite $D^{op}_{cop}$? SpecificallyWhy does the dualizing sheaf $\omega_X$ pop up in this context?Is the $R$-linear dual $Hom_R(D,R)$ a bialgebra? Is it related to the opposite of $D$?

How do I derive correctly the category of $D$-modules?Suppose $M$ and $N$ are $D$-modules and suppose $Hom_R(M,N)$ is the "correct" internal Hom in the abelian monoidal category of $D$-modules. If we hope to have a fiber functor between the derived categories $Hom_R(Q,N)$ (with $Q \to M$ a resolution as a $D$-module) should go to (something quasi isomorphic) to $Hom_R(P,N)$ (with $P \to M$ a resolution as an $R$-module). It doesn't seem to follow easily from the rest of the structure.

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