sequence relating the de Rham cohomology of a ${\mathcal{D}}$-module $M$ with the de Rham cohomology of the kernel and cokernel of $\partial_n$ acting on $M$ (which are ${\mathcal{D}}_{n-1}$-modules):

[@bjork Prop. 2.4.13]\[derhamles\] Let $M$ be a ${\mathcal{D}}$-module. Let $M_*$ (resp. $\overline{M}$) be the kernel (resp. cokernel) of $\partial_n$ acting on $M$. Then there is a long exact sequence $$\cdots \rightarrow H^{i-2}_{dR}(\overline{M}) \rightarrow H^i_{dR}(M_*) \rightarrow H^i_{dR}(M) \rightarrow H^{i-1}_{dR}(\overline{M}) \rightarrow \cdots$$ of $k$-spaces, where $H^j_{dR}(M_*)$ and $H^j_{dR}(\overline{M})$ are de Rham cohomology spaces of the ${\mathcal{D}}_{n-1}$-modules $M_*$ and $\overline{M}$, defined using only $\partial_1, \ldots, \partial_{n-1}$.

Finally, we will need Gabber’s theorem on involutivity of characteristic ideals, originally proved in [@gabber]. We need to introduce the Poisson bracket (see [@gabber] or [@hotta App. D]). Its definition makes sense, and Gabber’s theorem holds, for more general filtered rings, but we content ourselves here with stating everything for the ring ${\mathcal{D}}$ and its order filtration $\{{\mathcal{D}}_j\}$. Suppose that $\delta \in \operatorname{gr}^i {\mathcal{D}}$ and $\delta' \in \operatorname{gr}^j {\mathcal{D}}$ for some $i$ and $j$. Then we can write $\delta = \sigma(d)$ and $\delta' = \sigma(d')$ for some $d \in {\mathcal{D}}_i$ and $d' \in {\mathcal{D}}_j$, where $\sigma$ denotes the principal symbol. Since $\operatorname{gr}{\mathcal{D}}$ is commutative, the commutator $[d,d']$ belongs to 