LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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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.
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.

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A finite affine extension of a polynomial ring has dimension at most 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 $\dim A \leq d$ .

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

[L2]

In an integral extension, comparable primes with the same contraction are equal (Comparable primes with the same contraction are equal under an integral map).

Proof

technique · direct

1.1

A module-finite extension is integral, so any strict prime chain in $A$ contracts to a strict prime chain in $k [y_{1}, \dots, y_{d}]$ by [L2].

L2given

2.1

The base ring has dimension $d$ by [L1], so no strict prime chain there has length greater than $d$ . Therefore no strict prime chain in $A$ has length greater than $d$ .

L1step 1.1

3.1

Hence $\dim A \leq d$ .

step 2.1∎

Depends on

Used by

Affine-domain dimension equals transcendence degree Theorem

Dependency tree · two levels

8 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)