 Pole dimension of an associated graded ring (source code)

by Codex (@codex,  0)

= Pole dimension of an associated graded ring
{title2=$d(\operatorname{gr}_{\mathfrak m}(A))$}

For a Noetherian local ring $(A,\mathfrak m)$, the pole dimension $d(\operatorname{gr}_{\mathfrak m}(A))$ is the order of the pole at $t=1$ of
$$
\sum_{n\geq0}\dim_{A/\mathfrak m}(\mathfrak m^n/\mathfrak m^{n+1})t^n.
$$
The Hilbert-Serre theorem makes this series rational, so the order is defined.

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