CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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A complete local domain is finite over a regular power-series ring

Statement

Assume the Axiom of Choice.

Let $(A, m)$ be a complete equicharacteristic Noetherian local domain of dimension $d$ . Then there exists a coefficient field $k \subseteq A$ and an injective local homomorphism $k ⟦ X_{1}, \dots, X_{d} ⟧ ↪ A$ whose image is a regular complete local subring over which $A$ is module-finite.

Facts & Assumptions

Given: A complete equicharacteristic Noetherian local domain $(A, m)$ of dimension $d$ and the Axiom of Choice.

[L1]

The parameter power-series map makes $A$ finite over its image (Parameters make a complete local domain finite over the image of a power-series map).

[L2]

The same map is injective (The parameter power-series map is injective by dimension).

[L3]

A system of parameters is the $d$ -tuple that determines the relevant map (Systems of parameters and parameter ideals).

Proof

technique · choose parameters, then identify the source with its image

1.1L3givenchoose

Choose a coefficient field $k \subseteq A$ and a system of parameters $x_{1}, \dots, x_{d}$ . By [L3], these parameters determine the continuous map $ϕ : k ⟦ X_{1}, \dots, X_{d} ⟧ \to A, X_{i} \mapsto x_{i} .$

2.1L1L2step 1.1

By [L1], $A$ is finite over $ϕ (k ⟦ X_{1}, \dots, X_{d} ⟧)$ , and by [L2] the map $ϕ$ is injective. Therefore we may identify the source with a subring $A_{0} \subseteq A$ over which $A$ is module-finite. Standard formal-power-series theory makes $A_{0} ≅ k ⟦ X_{1}, \dots, X_{d} ⟧$ a regular complete local ring.

3.1step 2.1∎

Hence $A$ is finite over a regular power-series subring in $d$ variables over a coefficient field.

Depends on

Used by

Dependency tree · two levels

12 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

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Example 22.31 (standard reference, not scraped)
The Stacks Project, Section 10.160: The Cohen structure theorem (standard reference, not scraped)