Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module.

Now it is easy to see, that there is a bijection between $\mathcal D$-module structures on $\mathcal M$ and $\Omega$ dg-module structures on $\Omega \otimes_{\mathcal O_X} \mathcal M$.

For example given a $\mathcal D$-module structure on $\mathcal M$ we can define the corresponding differential by imposing the rule $d(m)(X)=X.m$ on $\Omega\otimes_{\mathcal O_X} \mathcal M$. (X denotes a vectorfield and $m$ a local section of $\mathcal M$.

A similar statement holds if one takes instead $\mathcal D=U(\mathfrak g)$ the universal envelope of a Lie-algebra and $\Omega=\bigwedge \mathfrak g^*$ the standard complex.

Now in both of there cases $\mathcal D$ and $\Omega$ are Koszul-dual, meaning that there are equivalences between carefully defined versions of their "derived categories". Yet formulations of Koszul duality I am aware of, seem not to extend the above correspondence. Roughly speaking they are $\mathcal M \mapsto Hom(\Omega, \mathcal M)$ instead of $\mathcal M \mapsto \Omega \otimes \mathcal M$.

So my questions are:

What does the above observation have to do with Kosul duality?

Is there a formulation of Koszul duality extending the above correspondence?

Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module.

Now it is easy to see, that there is a bijection between $\mathcal D$-module structures on $\mathcal M$ and $\Omega$ dg-module structures on $\Omega \otimes_{\mathcal O_X} \mathcal M$.

For example given a $\mathcal D$-module structure on $\mathcal M$ we can define the corresponding differential by imposing the rule $d(m)(X)=X.m$ on $\Omega\otimes_{\mathcal O_X} \mathcal M$. (X denotes a vectorfield and $m$ a local section of $\mathcal M$.

A similar statement holds if one takes instead $\mathcal D=U(\mathfrak g)$ the universal envelope of a Lie-algebra and $\Omega=\bigwedge \mathfrak g^*$ the standard complex.

Now in both of there cases $\mathcal D$ and $\Omega$ are Koszul-dual, meaning that there are equivalences between carefully defined versions of their "derived categories". Yet formulations of Koszul duality I am aware of, seem not to extend the above correspondence. Roughly speaking they are $\mathcal M \mapsto Hom(\Omega, \mathcal M)$ instead of $\mathcal M \mapsto \Omega \otimes \mathcal M$.

So my questions are:

What does the above observation have to do with Kosul duality?

Is there a formulation of Koszul duality extending the above correspondence?