Good filtration for the order filtration ($M_j={\sum }_{\ell =1}^s{F}_{j-d_{\ell }}{A}_n{m}_{\ell }$)

For the order filtration of a Weyl algebra, a filtration of a module is good if it is increasing, exhaustive, zero in sufficiently negative degrees, compatible with multiplication, and has associated graded module finite over the associated graded ring. Equivalently it has the displayed finite shifted-generating description. Choose homogeneous generators of the associated graded module, lift them to $m_{\ell }$, and successively remove the top symbol of any element; the lower bound on the filtration of a module makes this process terminate. Conversely, the displayed description makes the symbols of the $m_{\ell }$ generate. Each $M_j$ is finite over the degree-zero polynomial ring $\mathbb{C}[x_1,\dots ,x_n]$; it need not be a finite-dimensional vector space over $\mathbb{C}$.