# Relations between $\mathcal D$-modules and Exterior Differential Systems?

As a followup of this answer, I wonder relations between $$\mathcal D$$-modules and Exterior Differential Systems, both of which could be used to describe systems of PDEs. If I understand correctly, $$\mathcal D$$-modules are about linear PDEs while exterior differential systems are about general PDEs. However, I am looking for an explicit and direct description of relations between these two, at least under certain linearity assumptions. This question is motivated by the Spencer resolution. To fix notations, let $$X$$ be a smooth complex variety of dimension $$n$$. As a (left) $$\mathcal D_X$$-module, $$\mathcal O_X$$ admits a locally free resolution $$\mathcal D_X\otimes_{\mathcal O_X}\bigwedge^*\Theta_X\to\mathcal O_X$$ called Spencer resolution, where $$\Theta_X:=\operatorname{\mathcal Der}_{\mathcal O_X}(\mathcal O_X)$$ is the tangent sheaf. This resolution is quite closely related to the de Rham complex therefore the question. I am not familiar with either, and I guess that this should be something already understood, if not well documented.

• Instead of looking at the Spencer complex, I’d suggest looking at the de Rham resolution of the canonical module. This is a complex of right $D$-modules. Jun 19, 2020 at 17:59