# Computing the constant D-module on an intersection

Let $M$ be a smooth variety, say over the complex numbers, and let $i:W \hookrightarrow M, j: Z \hookrightarrow M$ be smooth subvarieties. Let $i_{+},j_{+}$ denote (derived) pushforward of D-modules, $i^{*},j^{*}$ (derived) pullback.

Question, in general: How do I compute the (complex) of $D$-modules $i_{+}\mathcal{O}_{W} \otimes^{L} j_{+} \mathcal{O}_{Z} \simeq i_{+}i^{*} \mathcal{O}_{M} \otimes^{L} j_{+}j^{*} \mathcal{O}_{M}$?

If $X=W \cap Z$ is smooth and of the expected codimension, then base change and the projection formula imply that $i_{+}\mathcal{O}_{W} \otimes^{L} j_{+} \mathcal{O}_{Z} \simeq i_{+}i^{*} \mathcal{O}_{M} \otimes^{L} j_{+}j^{*} \mathcal{O}_{M} \simeq h_{+}h^{*}\mathcal{O}_{M} \simeq h_{+} \mathcal{O}_{X}$, where $h: X \rightarrow M$ is the inclusion. Thus we get the constant $D$-module supported along $X$.

Actually, I'm most interested in the case when $X$ is not necessarily smooth or of the expected codimension. Then the answer for the corresponding question for $\mathcal{O}$-modules is $i_{*}\mathcal{O}_{W} \otimes^{L} j_{*}\mathcal{O}_{Z}$, which is the structure sheaf of the derived intersection of $W$ and $Z$, and its cohomology sheaves are just $Tor_{k}(i_{*}\mathcal{O}_{W}, j_{*}\mathcal{O}_{Z})$, and (in theory and often in practice) I know how to compute these by writing down a locally free resolution of $i_{*}\mathcal{O}_{W}$ as an $\mathcal{O}_{M}$-module.

In theory, I also know that I can take a resolution of $i_{+}\mathcal{O}_{W}$ by $D$-modules that are locally free as $O$-modules. But in practice I don't know how to do this.

Question, more specific: Suppose I have a locally free resolution of the $\mathcal{O}_{M}$-module $i_{*}\mathcal{O}_{W}$. Can I use this to build a $D$-module resolution that is locally free over $\mathcal{O}$?

Question, very specific: Let $W, M=T^{*}W$, and $i: W \hookrightarrow T^{*}W$ is the zero section. Then I have the standard Koszul resolution of $i_{*}\mathcal{O}_{W}$. Is there a 'Koszul resolution' of the $D$-module $i_{+}\mathcal{O}_{W}$? More generally, you could replace $T^{*}W$ with any vector bundle.

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