 Dimension and multiplicity in a filtered exact sequence

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.