---
abstract: 'In this expository paper, we give a complete proof of van den Essen’s theorem that the de Rham cohomology spaces of a holonomic ${\mathcal{D}}$-module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen’s papers as well as his unpublished thesis, and until now has not been available in a self-contained document.'
address: |
    School of Mathematics\
    University of Minnesota\
    127 Vincent Hall\
    206 Church St. SE\
    Minneapolis, MN 55455
author:
- Nicholas Switala
title: 'Van den Essen’s theorem on the de Rham cohomology of a holonomic ${\mathcal{D}}$-module over a formal power series ring'
---

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Introduction {#intro}
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Let $k$ be a field of characteristic zero, let $R = k[x_1, \ldots, x_n]$ be a polynomial ring over $k$, and let ${\mathcal{D}}= {\mathcal{D}}(R,k)$ be the ring of $k$-linear differential operators on $R$ (the *Weyl algebra*). To any finitely generated left ${\mathcal{D}}$-module $M$, we can associate its *dimension* $d(M)$: if $M$ is nonzero, this dimension is an integer between $n$ and $2n$. The left ${\mathcal{D}}$-modules of minimal dimension (those for which $d(M) = n$) are called *holonomic*. A basic result 