= Hilbert series multiplication exact sequence
{c}
{title2=$(1-t^e)P_M=P_{M/xM}-t^eP_{(0:_Mx)}$}
For a homogeneous element $x$ of degree $e>0$ acting on a finite <graded module>, set $N=(0:_Mx)$ and $C=M/xM$. The multiplication exact sequence $0\to N(-e)\to M(-e)\to M\to C\to0$ and additivity of component lengths give $(1-t^e)P_M=P_C-t^eP_N$. This is the inductive step in the <Hilbert-Serre theorem>; the <annihilator> term must be kept unless multiplication by $x$ is injective.
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