= Dimension of a filtered module
{title2=$d(M)=\deg\bigl(\dim_k F_jM\bigr)$}
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>.
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