Koszul complex of a $\mathcal{D}$-module I am trying to understand the construction of push forward in the derived category of $\mathcal{D}$-modules using the Koszul complex. My question will be about the latter notion.
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Let $\mathcal{D}_X$ be the sheaf of algebraic differential operators. Let $dR(\mathcal{D}_X)$ be the de Rham complex of $\mathcal{D}_X$ considered as a left $\mathcal{D}_X$-module. For a right $\mathcal{D}_X$-module $M$ consider its Koszul complex
$$Kos(M):=M\otimes_{\mathcal{O}_X}dR(\mathcal{D}_X)\otimes_{\mathcal{O}_X}K_X^{-1},$$
where $\mathcal{O}_X$ is the sheaf of regular functions, $K_X$ is the sheaf of top degree forms.
Questions. (1) What is the differential in $Kos(M)$?
(2) What are the structures on $Kos(M)$ and how they are compatible with the differential? In particular is $Kos(M)$ a complex of right $\mathcal{D}_X$-modules with the differential commuting with this structure?
Is there a good place to read about it?
 A: If I understand what you're trying to get at, this is not quite the correct thing. The correct "Koszul" complex (as you're calling it) is the complex
$$\begin{aligned}
(\operatorname{DR}_{X/pt}(M))^k
&:=\begin{cases}
\Omega^{n+k}_{X/\Bbb C}\otimes_{\mathcal{O}_X}M, &\text{if }-n\leq k \leq 0,\\
0, &\text{otherwise}
\end{cases}\\
d(\omega\otimes s) &= d\omega\otimes s + \sum_{i=1}^n (dz_i\wedge \omega)\otimes \partial_is,
\end{aligned}$$
where $\{z_i,\partial_i\}_{1\leq i\leq n}$ is a local coordinate system of $X$. This complex is not a $D_X$-module of any sort! Its hypercohomology $R\Gamma(X, \operatorname{DR}_{X/pt}(M))$ is the $D$-module direct image of $M$ via the map to a point. This hypercohomology is a complex of vector spaces, and it doesn't have any additional structure.
You can read more about this starting on page 45 of D-Modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi, and Tanasaki (HTT).
Edit
I didn't notice at first that your $M$ is a right $D_X$-module. Anyway, the complex that you call $dR(\mathcal{D}_X)\otimes_{\mathcal{O}_X} K_X^{-1}$ is (isomorphic to) what is usually called the Spencer complex, and it is an explicit $\mathcal{D}_X$-module resolution of the left $\mathcal{D}_X$-module $\mathcal{O}_X$. See Lemma 1.5.27 of HTT. It's common (though not universal) to denote this complex by $\operatorname{Sp}^\bullet_X$. Explicitly, if $\Theta_X$ denotes the tangent sheaf of $X$, then
$$\begin{aligned}
\operatorname{Sp}^{-k}_X
&:=\mathcal{D}_X\otimes_{\mathcal{O}_X}\mathsf{\Lambda}^{k}\Theta_X\\
d(P\otimes \partial_{i_1}\wedge\cdots \wedge \partial_{i_k}) &= \sum_j (-1)^{j+1} P\partial_{i_j}\otimes \partial_{i_1}\wedge\cdots \wedge\widehat{\partial_{i_j}}\wedge\cdots\wedge \partial_{i_k},
\end{aligned}$$
where $\{z_i,\partial_i\}_{1\leq i\leq n}$ is a local coordinate system of $X$. Then
$$\begin{aligned}
M\otimes_{\mathcal{D}_X}\operatorname{Sp}^{-k}_X
&\cong M\otimes_{\mathcal{O}_X}\mathsf{\Lambda}^{k}\Theta_X\\
d(s\otimes \partial_{i_1}\wedge\cdots \wedge \partial_{i_k}) &= \sum_j (-1)^{j+1} s\partial_{i_j}\otimes \partial_{i_1}\wedge\cdots \wedge\widehat{\partial_{i_j}}\wedge\cdots\wedge \partial_{i_k}.
\end{aligned}$$
