${\mathcal{D}}= {\mathcal{D}}(R,k)$ of $k$-linear differential operators, a subring of $\operatorname{End}_k(R)$, takes the form ${\mathcal{D}}= R\langle \partial_1, \ldots, \partial_n \rangle$, where $\partial_i = \frac{\partial}{\partial x_i}$. (This notation is meant to indicate that, after adjoining the new variables $\partial_i$ to $R$, we do *not* obtain a commutative ring.) As a left $R$-module, ${\mathcal{D}}$ is free on monomials in the $\partial_i$. If $R$ is any commutative ring and $A \subset R$ any commutative subring, there is a more general definition ([@EGAIV §16]) of the ring ${\mathcal{D}}(R,A)$ of $A$-linear differential operators on $R$. See [@EGAIV Thm. 16.11.2] for a proof that our definition coincides with this more general one in the formal power series case.

Unless expressly indicated otherwise, by a *${\mathcal{D}}$-module* we will always mean a *left* module over ${\mathcal{D}}$.

The ring ${\mathcal{D}}$ has an increasing, exhaustive filtration $\{{\mathcal{D}}_j\}$, called the *order filtration*, where ${\mathcal{D}}_j$ is the $R$-submodule consisting of those differential operators of order $\leq j$ (the order of an element of ${\mathcal{D}}$ is the maximum of the orders of its summands, and the order of a single summand $\rho \partial_1^{a_1} \cdots \partial_n^{a_n}$ with $\rho \in R$ is $\sum a_i$). Note that for all $f \in R$ and for all $i$, we 