ExampleConstruction: Literature-sourcedVerification: 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.

The affine dimension formula on a plane curve domain

Example

Let $A = k [x, y] / (y - x^{2}) .$ Then $A ≅ k [x]$ is a one-dimensional affine domain. For the prime ideals $(0)$ and $m = (\overset{ˉ}{x}, \overset{ˉ}{y})$ , the affine dimension formula reads $ht (p) + \dim (A / p) = 1.$

Facts & Assumptions

Given: A field $k$ , the domain $A = k [x, y] / (y - x^{2})$ , and the primes $(0)$ and $m = (\overset{ˉ}{x}, \overset{ˉ}{y})$ .

[L1]

Under $A ≅ k [x]$ , the ring is a PID: nonzero prime ideals are maximal, while $(0)$ is prime (For every field $F$ , $F [x]$ is a principal ideal domain).

[L2]

The evaluation quotient $A / (\overset{ˉ}{x}, \overset{ˉ}{y}) ≅ k$ is a field, while $A / (0) ≅ k [x]$ .

Verification

technique · direct computation

1.1L1givenalgebra

Substituting $y = x^{2}$ gives $A ≅ k [x]$ . By [L1], any nonzero prime of this PID is maximal, and the chain $(0) ⊊ (x)$ shows that its dimension is exactly one. The zero prime has height zero and quotient $A$ of dimension one.

2.1L1L2step 1.1

By [L2], $A / m ≅ k$ has dimension zero. The chain $(0) ⊊ m$ and [L1] give $ht (m) = 1$ . Thus $ht (m) + \dim (A / m) = 1 + 0 = 1$ ; at $(0)$ the corresponding sum is $0 + 1 = 1$ .

3.1step 1.1step 2.1∎

Both displayed instances hold by direct computation in $k [x]$ , without invoking the general affine dimension formula.

Depends on

For every field $F$, $F[x]$ is a principal ideal domain

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)