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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?

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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 $$\begin{aligned} M\otimes_{\mathcal{D}_X}\operatorname{Sp}^{-k}_X &\cong M\otimes_{\mathcal{O}_X}\mathsf{\Lambda}^{k}\Theta_X\\ d(s\otimes \partial_{i_1}\wedge\cdots \wedge \partial_{i_k}) &= \sum_j (-1)^{j+1} s\partial_{i_j}\otimes \partial_{i_1}\wedge\cdots \wedge\widehat{\partial_{i_j}}\wedge\cdots\wedge \partial_{i_k}. \end{aligned}$$

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    $\begingroup$ What you wrote is the de Rham complex of $M$. This is not the same as the Koszul complex. $\endgroup$
    – asv
    Jan 4, 2022 at 18:52
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    $\begingroup$ @makt Then I'm not sure what you mean by the Koszul complex of a D-module. Can you point me to where you found it? $\endgroup$ Jan 4, 2022 at 20:10
  • $\begingroup$ See p.48 in these lecture notes people.math.harvard.edu/~gaitsgde/grad_2009/Dmod_brav.pdf $\endgroup$
    – asv
    Jan 5, 2022 at 7:29
  • $\begingroup$ @makt Aha! What those lecture notes call $Kos(D_X)$ is isomorphic to what is usually called the Spencer complex. I've added some stuff to my answer to address this. Hopefully it's what you needed! $\endgroup$ Jan 5, 2022 at 16:47
  • $\begingroup$ Thanks. I am wondering what is $\wedge^{-k} \Theta_X$. In other words why the exterior power is negative. $\endgroup$
    – asv
    Jan 5, 2022 at 17:02

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