= Multiplicity of a filtered module
{title2=$m(M)=d(M)!\,[j^{d(M)}]\dim_k F_jM$}
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 $M\ne0$, 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.
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