Dimension of a Weyl algebra module for the order filtration ($d(M)={\dim }_{\mathrm{Krull}}(\mathrm{gr}R/{\mathrm{Ann}}_{\mathrm{gr}R}\mathrm{gr}M)$)

For a nonzero finitely generated module over a complex Weyl algebra, choose a good filtration for the order filtration and take the Krull dimension of its support of a module over $\mathrm{gr}R=\mathbb{C}[x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n]$. This is the dimension of its order characteristic variety of a Weyl algebra module. One may set $d(0)=-\infty$.

For independence from the good module filtration, any two such filtrations of a module are bounded shifts of one another, by their finite shifted-generating descriptions. If a homogeneous symbol $\sigma (a)$ of degree $d$ annihilates one associated graded module, then $a{F}_{j}M\subseteq F_{j+d-1}M$ and its $q$-th power lowers the expected degree by $q$. For shift bound $c$ and $q>2c$, comparison shows that $\sigma (a{)}^q$ annihilates the other associated graded module. Reverse the roles to obtain equality of the radicals of ideals of their annihilators, hence equality of their support of a module and Krull dimension. This argument holds the algebra filtration fixed.

The algebraic set defined by the annihilator of a module of the associated graded module for a good filtration for the order filtration is the order characteristic variety. It lies in affine space with coordinates $x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n$ and is invariant under scaling the ${\xi }_i$. Its dimension is the dimension of a Weyl algebra module for the order filtration. The order filtration of a Weyl algebra assigns degrees zero to the $x_i$ and one to the ${\xi }_i$; a characteristic variety formed with the Bernstein filtration need not be the same algebraic set.