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CorollaryStatement: Literature-sourcedProof: Literature-sourcedverified 2026-09-24 (gpt-6-sol)
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Maximal ideals of an affine domain have full height

Statement

Assume the Axiom of Choice. Let k be a field, let A be a finite-type k-domain, and let m be a maximal ideal of A. Then ht⁡(m)=dim⁡A.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), a field k, a finite-type k-domain A, and a maximal ideal m⊂A.

[L1]

The residue field A/m is a finite extension of k (A maximal ideal of an affine algebra has finite residue field over the base field).

[L2]

Under Choice, in an affine domain ht⁡(m)+dim⁡(A/m)=dim⁡A (Height plus quotient dimension equals ambient dimension in an affine domain).

Proof

technique · direct
1.1L1given

By [L1], the quotient A/m is a field. Hence dim⁡(A/m)=0.

2.1L2step 1.1

Applying [L2] now yields ht⁡(m)=dim⁡A.

3.1step 2.1∎

Thus every maximal ideal of an affine domain has full height.

Depends on

Used by

Dependency tree · two levels

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Sources