ExampleConstruction: AI-adaptedVerification: 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.

Coordinate ideals show the height bound is sharp

Example

In $A = k [x_{1}, \dots, x_{n}]$ , the coordinate ideal $I = (x_{1}, \dots, x_{n})$ is maximal, hence minimal over itself, and has height exactly $n$ .

Facts & Assumptions

Given: A field $k$ and the polynomial ring $A = k [x_{1}, \dots, x_{n}]$ .

[L1]

The polynomial ring $A$ has dimension $n$ , so every prime height in $A$ is at most $n$ (A polynomial ring in n variables over a field has dimension n).

[L2]

The quotient by $I$ is the field $k$ by direct evaluation.

Verification

technique · direct computation

1.1L2given

The quotient $A / I ≅ k$ is a field, so $I$ is maximal and therefore prime.

2.1L1L2step 1.1algebra

For $n \geq 1$ , the chain $(0) ⊊ (x_{1}) ⊊ \dots ⊊ (x_{1}, \dots, x_{n}) = I$ consists of primes, since each quotient is a polynomial domain over $k$ ; it gives $ht (I) \geq n$ . Fact [L1] gives the opposite bound. When $n = 0$ , $A = k$ and $I = (0)$ has height zero. Thus $ht (I) = n$ in all cases.

3.1step 2.1∎

Therefore the height bound in Krull's height theorem is sharp.

Depends on

A polynomial ring in n variables over a field has dimension n

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)