ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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In dimension zero the Hilbert-Samuel polynomial is constant and equals the module length

Example

Let $(R, m)$ be a zero-dimensional Noetherian local ring and let $M$ be a finite $R$ -module. Then some power of $m$ annihilates $M$ , so for all sufficiently large $n$ ,

M / m^{n + 1} M ≅ M .

Hence the Hilbert-Samuel polynomial is the constant polynomial

P_{m, M} (n) = ℓ_{R} (M) .

Facts & Assumptions

Given: A zero-dimensional Noetherian local ring $(R, m)$ and a finite $R$ -module $M$ .

[L1]

The length $ℓ_{R} (M)$ is defined for finite-length modules (Composition series and length of a module).

[L2]

The Hilbert-Samuel polynomial exists (The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form).

Verification

technique · direct

1.1

In a zero-dimensional Noetherian local ring, the maximal ideal is nilpotent on every finite module, so there is $N$ with $m^{N} M = 0$ . Hence for every $n \geq N - 1$ , $M / m^{n + 1} M = M .$

givenalgebra

1.2

Therefore the Hilbert-Samuel function is eventually constant equal to $ℓ_{R} (M)$ , which is defined by [L1]. So the eventual polynomial provided by [L2] is the constant polynomial $ℓ_{R} (M)$ .

L1L2

2.1

This is exactly the zero-dimensional case.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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, Section 10.59: Noetherian local rings (standard reference, not scraped)