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.

  • $\begingroup$ 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. $\endgroup$ Jun 19, 2020 at 17:59


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