TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-24 (gpt-6-sol)

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Local dimension is the minimal number of generators of an ideal with maximal radical

Statement

Assume the Axiom of Choice. Let $(R, m)$ be a finite-dimensional Noetherian local ring of dimension $d < \infty$ . Then $d$ is the least integer $n$ for which there exists an $n$ -generated ideal $J \subseteq R$ with $\sqrt{J} = m$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and a finite-dimensional Noetherian local ring $(R, m)$ with $d = \dim R < \infty$ .

[L1]

Under Choice, systems of parameters exist, and by definition a system of parameters gives a $d$ -generated ideal with radical $m$ (Every finite-dimensional Noetherian local ring has a system of parameters, Systems of parameters and parameter ideals).

[L2]

If an ideal generated by $n \geq 1$ elements has radical $m$ , then the maximal ideal is minimal over it; under Choice the height theorem bounds $d$ by $n$ (Krull's height theorem). If $n = 0$ , $J = (0)$ and $\sqrt{(0)} = m$ forces $d = 0$ .

[L3]

Under Choice, the local converse to the height theorem produces $d$ generators whose radical is the maximal ideal (Converse to Krull's height theorem in localised form, Parameter ideals are exactly the m-primary d-generated ideals).

Proof

technique · direct

1.1L1given

By [L1], there exists a $d$ -generated ideal with radical $m$ . So the least such number is at most $d$ .

1.2L2given

Conversely, let $J$ be an $n$ -generated ideal with $\sqrt{J} = m$ . Then $m$ is minimal over $J$ , so [L2] gives $d = ht (m) \leq n$ . Hence every such generating number is at least $d$ .

2.1L3step 1.1step 1.2∎

Steps 1.1 and 1.2 show that the least number is exactly $d$ . The same conclusion may also be read from [L3] as the local radical form of the converse height theorem.

Depends on

Used by

Dependency tree · two levels

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Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., §21 (standard reference, not scraped)
The Stacks Project, Section 10.60: Dimension (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)