TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Completion preserves dimension and Hilbert-Samuel data

Statement

Assume the Axiom of Choice.

Let $(R, m)$ be a Noetherian local ring, let $M \neq 0$ be a finitely generated $R$ -module, and let $\hat{R}$ , $\hat{M}$ denote the $m$ -adic completions.

For every $n \geq 0$ , $\hat{M} / m^{n + 1} \hat{M} ≅ M / m^{n + 1} M .$ In particular the Hilbert-Samuel functions of $M$ and $\hat{M}$ agree.
The Hilbert-Samuel multiplicity of $M$ equals that of $\hat{M}$ .
The support dimensions of $M$ and $\hat{M}$ are equal.

Facts & Assumptions

Given: A Noetherian local ring $(R, m)$ and a nonzero finitely generated $R$ -module $M$ .

[L1]

Completion of a Noetherian local ring is again local with maximal ideal $m \hat{R}$ and the same residue field (Completion of a Noetherian local ring is local with the same residue field).

[L2]

Completion commutes with finite quotients and with powers of the defining ideal (Completion commutes with finite quotients and induced submodules).

[L3]

Hilbert-Samuel multiplicity is read from the leading coefficient of the eventual Hilbert-Samuel polynomial (Hilbert-Samuel multiplicity as the factorial-scaled leading coefficient).

[L4]

For a nonzero finite module, the degree of the Hilbert-Samuel polynomial equals the support dimension (The degree of the Hilbert-Samuel polynomial equals the dimension of the support).

Proof

technique · direct

1.1L1L2

By [L2], for every $n \geq 0$ , $\hat{M} / m^{n + 1} \hat{M} ≅ M / m^{n + 1} M .$ Since [L1] identifies the residue fields of $R$ and $\hat{R}$ , the two sides have the same finite length. Hence the Hilbert-Samuel functions agree term by term.

2.1L3step 1.1

Equality of the Hilbert-Samuel functions implies equality of their eventual polynomials. Therefore the Hilbert-Samuel multiplicities, which are read from the leading coefficients of those polynomials by [L3], are equal.

2.2L4step 1.1

By [L4], the degree of that common eventual polynomial is the support dimension of $M$ , and the same degree computed over $\hat{R}$ is the support dimension of $\hat{M}$ . Hence those dimensions are equal.

3.1step 1.1step 2.1step 2.2∎

This proves all three claims.

Depends on

Used by

Completion preserves Cohen--Macaulayness in both directions Corollary
A nonfree maximal Cohen--Macaulay module Example
A parameter sequence regular in a hypersurface Example
completion regularity invariance Example
formal power series ring regular Example
completion preserves regular local rings Theorem

Dependency tree · two levels

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Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise 22.14(2) (standard reference, not scraped)
The Stacks Project, Section 10.97 (standard reference, not scraped)