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CorollaryStatement: Literature-sourcedProof: Literature-sourcedverified 2026-09-24 (gpt-6-sol)
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Height plus quotient dimension equals ambient dimension in an affine domain

Statement

Assume the Axiom of Choice. Let k be a field, let A be a finite-type k-domain, and let p∈Spec⁡(A). Then ht⁡(p)+dim⁡(A/p)=dim⁡A.

Facts & Assumptions

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

[L1]

Under Choice, the affine-domain dimension formula identifies ht⁡(p)+trdeg⁡kFrac⁡(A/p)=trdeg⁡kFrac⁡(A). (The dimension formula for affine domains).

[L2]

For affine domains, dimension equals transcendence degree of the fraction field (Affine-domain dimension equals transcendence degree).

Proof

technique · direct
1.1L1L2given

Replace the two transcendence degrees in [L1] using [L2], once for A and once for the quotient domain A/p.

2.1step 1.1∎

The result is exactly ht⁡(p)+dim⁡(A/p)=dim⁡A.

Depends on

Used by

Dependency tree · two levels

15 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