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.

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]

In the quotient $A / p ≅ k [y, z]$ , the prime $q / p$ is the principal ideal $(y)$ ; a prime strictly below $(y)$ must be zero by cancellation in the polynomial domain.

Verification

technique · direct computation

1.1L1L2givenalgebra

In $k [y, z]$ , let $Q ⊊ (y)$ be prime. If $Q$ contained a nonzero polynomial $f$ , write $f = y^{r} g$ with $r \geq 1$ and $y ∤ g$ . Primality would give $y \in Q$ or $g \in Q$ ; the latter is impossible because $g \notin (y)$ , while the former contradicts $Q ⊊ (y)$ . Hence $Q = (0)$ . By [L1], the only strict chain from $p$ to $q$ is $(x) ⊊ (x, y)$ and $ht (q / p) = 1$ .

2.1L1step 1.1

The quotient correspondence identifies the chain from step 1.1 with $(0) ⊊ (y)$ in $A / p$ , confirming its length is one.

3.1step 1.1step 2.1∎

Therefore the relative height in the stated quotient is one.

Depends on

Height in a quotient measures chains between two primes

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)