is the unique maximal ideal of $R_{n-1}$). Since $R$ is local, any element of $R$ with a nonzero constant term is a unit.

\[xnregf\] A formal power series $f \in R$ is said to be *$x_n$-regular* if $f(0,0,\ldots, 0, x_n) \neq 0$ in $k[[x_n]]$, that is, if a term $c_{0,\ldots, 0, i}x_n^i$ with $c_{0,\ldots, 0, i} \in k \setminus \{0\}$ occurs in $f$.

The following theorem clarifies the significance of the $x_n$-regularity hypothesis:

[@lang Thm. IV.9.2]\[weierprep\] Suppose that $f \in R$ is $x_n$-regular. There exists a unique expression $f = u(x_n^d + b_{n-1}x_n^{d-1} + \cdots + b_0)$ where $u \in R$ is a unit and each $b_i \in \mathfrak{m}_{n-1}$.

\[wpfingen\] The Weierstrass preparation theorem has the following consequence: if $f \in R$ is $x_n$-regular, then $R/fR$ is finitely generated (by the classes of $x_n^i$ with $0 \leq i \leq d-1$) as a module over $R_{n-1}$. It follows that *any* finitely generated $R/fR$-module is in fact a finitely generated $R_{n-1}$-module. In the sequel, our appeals to the “Weierstrass preparation theorem” will in fact be appeals to this consequence.

We now review some definitions and properties of ${\mathcal{D}}$-modules and de Rham complexes. Our basic reference for what follows is [@bjork]. The ring 