Author: Eiko

Time: 2024-12-08 00:19:38 - 2024-12-08 00:19:38 (UTC)

In this article I want to summarize the connection between D-modules and differential equations.

D-modules and Differential Equations

${\mathrm{Hom}}_D(D/DP,\mathcal{O})$ corresponds to $Pf=0$

Let $P\in D_X$ be a differential operator, and let $M=D/DP$. Then

$$\begin{array}{rl}{\mathrm{Hom}}_{D_X}(M,{\mathcal{O}}_X)& ={\mathrm{Hom}}_{D_X}(D/DP,{\mathcal{O}}_X)\\ & =\{\phi \in {\mathrm{Hom}}_{D_X}(D,{\mathcal{O}}_X):\phi (P)=0\}\\ & =\{f=\phi (1)\in {\mathcal{O}}_X:\phi (P)=P\phi (1)=Pf=0\}\\ & =\ker (P:{\mathcal{O}}_X\to {\mathcal{O}}_X).\end{array}$$

we see that the dual module ${\mathrm{Hom}}_D(M,\mathcal{O})$ can be seen as the solution space of the differential equation $Pf=0$.