ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passverified 2026-09-24 (gpt-6-sol)

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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]

In a Noetherian ring the nilradical is nilpotent (The nilradical of a Noetherian ring is nilpotent).

Verification

technique · direct

1.1L2givenalgebra

In a zero-dimensional local ring every prime ideal is maximal, hence the only prime is $m$ . Its nilradical is therefore $m$ , and [L2] gives $m^{N} = 0$ for some $N \geq 1$ . Hence for every $n \geq N - 1$ , $M / m^{n + 1} M = M$ .

2.1L1step 1.1algebra

The finite module $M$ has finite length over the zero-dimensional Noetherian local ring: the filtration by powers of the nilpotent maximal ideal has finitely many quotients, each a finite-dimensional vector space over $R / m$ . Thus $ℓ_{R} (M)$ is defined by [L1]. Step 1.1 shows that the Hilbert-Samuel function is eventually the constant polynomial $ℓ_{R} (M)$ , uniquely so because two polynomials agreeing at all large integers are equal.

3.1step 2.1∎

This is exactly the zero-dimensional case.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)