DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

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Hilbert-Samuel multiplicity as the factorial-scaled leading coefficient

Definition

Let $(R, m)$ be a Noetherian local ring, let $M$ be a finite $R$ -module, and let $I$ be an ideal of definition for $M$ .

If $M = 0$ , define

e_{I} (M) : = 0.

If $M \neq 0$ , let $P_{I, M}$ be the eventual Hilbert-Samuel polynomial from The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form, and let $d = \deg P_{I, M}$ . Because $I \subseteq m$ and $M \neq 0$ , Nakayama's lemma makes $M / I^{n + 1} M$ nonzero for every $n$ , so $P_{I, M}$ is not the zero polynomial and $d$ is defined.

The Hilbert-Samuel multiplicity of $M$ with respect to $I$ is

e_{I} (M) : = d! \cdot (leading coefficient of P_{I, M}) .

Equivalently, when $M \neq 0$ and

P_{I, M} (n) = \frac{e_{I} (M)}{d!} n^{d} + lower-degree terms,

then $e_{I} (M)$ is the integer scaling the top term.

Depends on

The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stacks Project, Definition 10.59.6 and Lemma 10.59.7 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §21 (standard reference, not scraped)