Characteristic variety of a Weyl algebra module ($\mathrm{Supp}(\mathrm{gr}M)$)

A good Bernstein filtration on a finitely generated module produces a finite graded module over $\mathrm{gr}{A}_n\cong \mathbb{C}[x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n]$. Its support is the characteristic variety for this filtration. Its algebraic dimension is the Bernstein growth dimension of a Weyl algebra module, which is independent of the choice of good filtration.

Fix the Bernstein filtration on the Weyl algebra. Two good filtrations of a module are bounded shifts of one another: $F_{i}M\subseteq G_{i+c}M$ and conversely. If an operator of degree $d$ has symbol annihilating ${\mathrm{gr}}_{F}M$, its $r$th power lowers the expected filtration degree by $r$. For $r>2c$, comparison of filtrations forces the same power of its symbol to annihilate ${\mathrm{gr}}_{G}M$. Reversing the roles gives equality of the radical annihilator ideals and hence equality of support of a module. The algebra filtration is held fixed; changing between order and Bernstein filtrations need not preserve the same variety.