cokernel of $x_n$ acting on $M$ are holonomic ${\mathcal{D}}_{n-1}$-modules.

We remark that in the polynomial ring case, $x_n$ and $\partial_n$ play essentially symmetric roles, and so the question for $\partial_n$ is no more difficult than the question for $x_n$ (and has the same unconditional affirmative answer). The symmetry between $x_n$ and $\partial_n$ does not persist in the formal power series case, which is why the question for $\partial_n$ is significantly more difficult.

We next discuss the *de Rham complex* of a ${\mathcal{D}}$-module $M$. This is a complex of length $n$, denoted $M \otimes \Omega_R^{\bullet}$, whose objects are $R$-modules but whose differentials are merely $k$-linear. It is defined as follows: for $0 \leq i \leq n$, $M \otimes \Omega^i_R$ is a direct sum of $n \choose i$ copies of $M$, indexed by $i$-tuples $1 \leq j_1 < \cdots < j_i \leq n$. The summand corresponding to such an $i$-tuple will be written $M \, dx_{j_1} \wedge \cdots \wedge dx_{j_i}$.

The $k$-linear differentials $d^i: M \otimes \Omega_R^i \rightarrow M \otimes \Omega_R^{i+1}$ are defined by $$d^i(m \,dx_{j_1} \wedge \cdots \wedge dx_{j_i}) = \sum_{s=1}^n \partial_s(m)\, dx_s \wedge dx_{j_1} \wedge \cdots \wedge dx_{j_i},$$ with the usual exterior algebra conventions for rearranging the wedge terms, and 