= Dimension of a Weyl algebra module for the order filtration
{title2=$d(M)=\dim_{\mathrm{Krull}}(\operatorname{gr}R/\operatorname{Ann}_{\operatorname{gr}R}\operatorname{gr}M)$}
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 $\operatorname{gr}R=\mathbb C[x_1,\ldots,x_n,\xi_1,\ldots,\xi_n]$. 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 $aF_jM\subseteq F_{j+d-1}M$ 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.
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