in the theory of ${\mathcal{D}}$-modules, due to Bernstein, states that the de Rham cohomology spaces of a holonomic left ${\mathcal{D}}$-module $M$ are finite-dimensional over $k$. These spaces are the cohomology objects of a complex defined using the usual exterior derivative formulas with respect to the action of the partial derivatives $\partial_1, \ldots, \partial_n \in {\mathcal{D}}$ on $M$. The key idea in the proof of this finiteness is that the kernel and cokernel of $\partial_n$ acting on $M$ are holonomic ${\mathcal{D}}_{n-1}$-modules, where ${\mathcal{D}}_{n-1} = {\mathcal{D}}(k[x_1, \ldots, x_{n-1}],k)$; with this statement in hand, the finiteness of the de Rham cohomology follows by a routine induction.

Now consider the case where $R$ is a formal power series ring $k[[x_1, \ldots, x_n]]$, again over a field of characteristic zero. We again have the ring ${\mathcal{D}}= {\mathcal{D}}(R,k)$ of $k$-linear differential operators on $R$, and a well-defined notion of dimension for finitely generated left ${\mathcal{D}}$-modules (hence a notion of holonomy for left ${\mathcal{D}}$-modules). In this case, the analogue of Bernstein’s result is due to van den Essen. If $M$ is a holonomic left ${\mathcal{D}}$-module, its de Rham cohomology spaces are again finite-dimensional over $k$, just as in the polynomial case; in contrast to this case, however, 