ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

Relative height in a quotient of k[x,y,z]

Example

Let $A = k [x, y, z]$ , let $p = (x)$ , and let $q = (x, y)$ . Then in $A / p ≅ k [y, z]$ the prime $q / p$ has height $1$ .

Facts & Assumptions

Given: A field $k$ , the polynomial ring $A = k [x, y, z]$ , and the primes $p = (x) \subset q = (x, y)$ .

[L1]

Height in a quotient is the relative chain length between the two primes upstairs (Height in a quotient measures chains between two primes).

[L2]

The affine-domain dimension formula gives heights in polynomial rings over a field (Height plus quotient dimension equals ambient dimension in an affine domain, A polynomial ring in n variables over a field has dimension n).

Verification

technique · direct computation

1.1

The only strict prime chain from $p$ to $q$ is $(x) ⊊ (x, y)$ , so [L1] gives $ht (q / p) = 1$ .

L1given

2.1

By [L2], $A$ has dimension $3$ , $A / q ≅ k [z]$ has dimension $1$ , and $A / p ≅ k [y, z]$ has dimension $2$ . Thus $ht (q) = 2$ and $ht (p) = 1$ , so the same relative height is $1$ .

L2step 1.1

3.1

So the quotient-chain computation and the affine-dimension computation agree.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)