Good filtration of a module

For a filtered algebra $A$ with finite-dimensional filtration pieces, a good filtration of a module on a finitely generated module is compatible with multiplication and has finitely generated associated graded module over the associated graded ring. Taking bounded-degree translates of a finite set of module generators supplies such a filtration of a module when the associated graded ring is Noetherian.

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 $\mathbb{C}$.

For a good filtration of a module over a fixed finite-dimensional filtration of a ring whose associated graded algebra is commutative and finitely generated, the cumulative Hilbert function is eventually a quasipolynomial by the Hilbert-Serre theorem. Its leading asymptotic degree is the dimension of the nonzero filtered module. Monotonicity forces the leading coefficients in its residue classes to agree. If the associated graded algebra is a standard graded algebra, the Hilbert function is eventually polynomial. Two good module filtrations bound one another after fixed shifts, so they have the same growth degree. Set $d(0)=-\infty$. For the Bernstein filtration this is the Bernstein growth dimension of a Weyl algebra module.

With the fixed algebra-filtration convention in dimension of a filtered module, the multiplicity is factorial times the common leading coefficient of the cumulative Hilbert function for $M\ne 0$, and is zero for $M=0$. It is a positive integer with a standard graded algebra normalization, and can be a positive rational number for a weighted grading. Fixed shifts between good filtrations of a module preserve its leading Hilbert coefficient. Rescaling the algebra filtration itself can change multiplicity, so its normalization must be specified.

The subspace filtration and quotient filtration in a short exact sequence are good when the associated graded algebra is Noetherian. Their cumulative Hilbert functions add in every degree. Positive leading coefficients make the degree of the sum the larger degree; when both degrees agree, factorial times the leading coefficients add. Thus equal-dimensional nonzero factors consume positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral.