= Dimension and multiplicity in a filtered exact sequence
{title2=$d(M)=\max(d(N),d(M/N))$}
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.
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