= Exactness of associated graded modules for induced filtrations
{title2=$0\to\operatorname{gr}N\to\operatorname{gr}M\to\operatorname{gr}(M/N)\to0$}
Give $N$ the <subspace filtration> and $M/N$ the <quotient filtration>. If the degree-$i$ class of $m\in F_iM$ maps to zero, then $m=n+m'$ with $n\in N$ and $m'\in F_{i-1}M$. Thus $n\in N\cap F_iM$ and the class comes from $\operatorname{gr}N$. Inclusion is <injective> in every degree and the quotient map is <surjective>, proving the displayed <short exact sequence> of <associated graded modules>. Exactness can fail for arbitrarily chosen incompatible filtrations.
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