CorollaryStatement: Literature-sourcedProof: Literature-sourcedverified 2026-09-24 (gpt-6-sol)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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 $p \in Spec (A)$ . 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 $p \subset A$ .

[L1]

Under Choice, the affine-domain dimension formula identifies $ht (p) + {trdeg}_{k} Frac (A / p) = {trdeg}_{k} Frac (A) .$ (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 $ht (p) + \dim (A / p) = \dim A$ .

Height plus quotient dimension equals ambient dimension in an affine domain

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources