ideal $J$, and hence the number $d(M)$, is independent of the choice of good filtration on $M$. By *Bernstein’s theorem*, if $M$ is a (nonzero) finitely generated ${\mathcal{D}}$-module, we have $n \leq d(M) \leq 2n$. In the case where $d(M) = n$ we say that $M$ is *holonomic*.

Some basic facts about holonomic modules are the following: submodules and quotients of holonomic ${\mathcal{D}}$-modules are holonomic, an extension of a holonomic ${\mathcal{D}}$-module by another holonomic ${\mathcal{D}}$-module is holonomic, holonomic ${\mathcal{D}}$-modules are of finite length over ${\mathcal{D}}$, and holonomic ${\mathcal{D}}$-modules are cyclic (generated over ${\mathcal{D}}$ by a single element). We will use these basic facts below without comment.

If $M$ is a ${\mathcal{D}}$-module, the operator $\partial_n \in {\mathcal{D}}$ acts on $M$ via a ${\mathcal{D}}_{n-1}$-linear map, and so its kernel and cokernel are ${\mathcal{D}}_{n-1}$-modules. The main question we will be concerned with in this paper is the following: if $M$ is holonomic, are the kernel and cokernel of $\partial_n$ acting on $M$ holonomic ${\mathcal{D}}_{n-1}$-modules?

Exactly the same question can be asked about the operator $x_n \in {\mathcal{D}}$. This question is easier, and we have the following unconditional affirmative answer:

[@bjork Thm. 3.4.2, Prop. 3.4.4]\[bjorkxn\] Let $M$ be a holonomic ${\mathcal{D}}$-module. The kernel and 