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

How statement and proof provenance work

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A finite affine extension of a polynomial ring has dimension at least the number of variables

Statement

Let $k$ be a field, let $A$ be a finite-type $k$ -domain, and suppose $A$ is module-finite over a polynomial subring $k [y_{1}, \dots, y_{d}]$ . Then $\dim A \geq d$ .

Facts & Assumptions

Given: A field $k$ , a finite-type $k$ -domain $A$ , and a module-finite inclusion $k [y_{1}, \dots, y_{d}] \subseteq A$ .

[L1]

The coordinate-prime chain in $k [y_{1}, \dots, y_{d}]$ has length $d$ (A polynomial ring in n variables over a field has dimension n).

[L2]

A supplied finite prime chain lifts through an injective module-finite extension of domains by a finite-dimensional lying-over argument (Finite prime chains lift through module-finite domain extensions without Choice).

Proof

technique · direct

1.1L1given

The coordinate-prime chain $(0) ⊊ (y_{1}) ⊊ \dots ⊊ (y_{1}, \dots, y_{d})$ in $k [y_{1}, \dots, y_{d}]$ has length $d$ by [L1].

2.1L2step 1.1

The extension is an injective module-finite extension of domains, so [L2] lifts that chain to a strict prime chain in $A$ of the same length $d$ .

3.1step 2.1∎

Therefore $\dim A \geq d$ .

Depends on

Used by

Affine-domain dimension equals transcendence degree Theorem

Dependency tree · two levels

9 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, §§18, 21 (standard reference, not scraped)
The Stacks Project, Section 10.116: Dimension of finite type algebras over fields, reprise (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)