always satisfied up to a linear change of variables in $R$. This is the technical assumption necessary for the cokernel of $\partial$ acting on a holonomic ${\mathcal{D}}$-module to again be holonomic.

[@essthesis p. 21]\[E\] Let $M$ be a ${\mathcal{D}}$-module, let $m \in M$, and suppose that $\tau \in {\mathcal{D}}$ is a $k$-linear derivation. We write $E_{\tau}(m)$ for the $R$-submodule $\sum_{i=0}^{\infty} R \cdot \tau^i(m)$ of $M$ generated by the family $\{\tau^i(m)\}$. If $N \subset M$ is an $R$-submodule, $E_{\tau}(N)$ is the $R$-submodule of $M$ generated by $E_{\tau}(n)$ for $n \in N$.

For a given $\tau$ and $m$, if $E_{\tau}(m)$ is a finitely generated $R$-module, then there exists $p$ such that $\tau^p(m) \in \sum_{i=0}^{p-1} R \cdot \tau^i(m)$. In this case, $E_{\tau}(m) = \sum_{i=0}^{p-1} R \cdot \tau^i(m)$.

[@essthesis Ch. II, Prop. 1.16]\[reglink\] Let $M$ be a ${\mathcal{D}}$-module, and suppose there exists a nonzero $f \in R$ such that the localization $M_f$ is a finitely generated $R_f$-module. Then for any $m \in M$, there exists $s \geq 0$ such that $E_{f^s \partial}(m)$ is a finitely generated $R$-module.

Let $m \in M$ be given. Since $R$ (and hence $R_f$) is Noetherian, any $R$-submodule of $M_f$ is also finitely generated over $R$. In particular, this is 