contain many more results on kernels and cokernels of differential operators that are not strictly necessary for the proof of this theorem, and we encourage the interested reader to investigate further.

The structure of this paper is as follows. In section \[prelim\], we collect preliminary material on formal power series rings, ${\mathcal{D}}$-modules, de Rham cohomology, and Gabber’s theorem. In section \[kernels\], we give the proof that the kernel of $\partial_n$ acting on a holonomic ${\mathcal{D}}$-module is again holonomic (with no further conditions on the module). In section \[regularity\], we define the $x_n$-regularity condition for a holonomic ${\mathcal{D}}$-module and prove some technical results concerning the consequences of this condition. In section \[cokernels\], we give the proof that (possibly after a linear change of coordinates) the cokernel of $\partial_n$ acting on a holonomic ${\mathcal{D}}$-module is again holonomic, and then complete the proof of Theorem \[mainthm\].

Preliminaries {#prelim}
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Throughout the paper, $k$ denotes a field of characteristic zero, $R$ denotes the formal power series ring $k[[x_1, \ldots, x_n]]$, and $R_{n-1}$ denotes the subring $k[[x_1, \ldots, x_{n-1}]]$. The rings $R$ and $R_{n-1}$ are commutative, Noetherian, regular local rings. We denote by $\mathfrak{m}$ the unique maximal ideal $(x_1, \ldots, x_n)$ of $R$ (similarly, $\mathfrak{m}_{n-1}$ 