have the relation $$[\partial_i, f] = \partial_i f - f \partial_i = \partial_i(f) \in R \subset {\mathcal{D}},$$ where $[,]$ denotes the commutator of two elements of ${\mathcal{D}}$ and the operator $\partial_i(f) \in {\mathcal{D}}$ is *multiplication* by $\partial_i(f) \in R$. Consequently, the associated graded object $\operatorname{gr}{\mathcal{D}}= \oplus_j {\mathcal{D}}_j/{\mathcal{D}}_{j-1}$ with respect to the order filtration is isomorphic to the polynomial ring $R[\zeta_1, \ldots, \zeta_n]$, where $\zeta_i$ is the image of $\partial_i$ in ${\mathcal{D}}_1/{\mathcal{D}}_0 \subset \operatorname{gr}{\mathcal{D}}$. (In particular, $\operatorname{gr}{\mathcal{D}}$ is commutative.) For all $i$, $\zeta_i$ is called the *principal symbol* of $\partial_i$, and we write $\zeta_i = \sigma(\partial_i)$. More generally, if $\delta \in {\mathcal{D}}_j \setminus {\mathcal{D}}_{j-1}$, its principal symbol $\sigma(\delta)$ is its class in $\operatorname{gr}^j {\mathcal{D}}= {\mathcal{D}}_j/{\mathcal{D}}_{j-1} \subset \operatorname{gr}{\mathcal{D}}$.

If $M$ is a finitely generated ${\mathcal{D}}$-module, there exists an increasing, exhaustive filtration $\{M_j\}$ of $M$ such that $M$ becomes a filtered ${\mathcal{D}}$-module with respect to the order filtration on ${\mathcal{D}}$ (so ${\mathcal{D}}_i \cdot M_j \subset M_{i+j}$ for all $i$ and $j$) *and* $\operatorname{gr}M = \oplus_j M_j/M_{j-1}$ is a finitely generated $\operatorname{gr}{\mathcal{D}}$-module. We call such a filtration *good*. Let $J$ be the radical of $\operatorname{Ann}_{\operatorname{gr}{\mathcal{D}}} \operatorname{gr}M$ (the *characteristic ideal* of $M$) and set $d(M) = \dim \, (\operatorname{gr}{\mathcal{D}})/J$ where $\dim$ denotes Krull dimension. The 