Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Transcendence degrees along affine prime quotients add correctly

Statement

Let k be a field, let A be a finite-type k-domain, and let p⊆q be prime ideals of A. Then

ht⁡(q/p)+trdeg⁡kFrac⁡(A/q)=trdeg⁡kFrac⁡(A/p).

Facts & Assumptions

Given: A field k, a finite-type k-domain A, and prime ideals p⊆q.

[L1]

The affine-domain dimension formula applies to the quotient domain A/p and the prime q/p (The dimension formula for affine domains).

Proof

technique · direct
1.1

The quotient A/p is a finite-type k-domain, and q/p is a prime ideal of it. Applying [L1] to that quotient domain gives ht⁡A/p(q/p)+trdeg⁡kFrac⁡(A/q)=trdeg⁡kFrac⁡(A/p).

L1given
2.1

This is exactly the displayed identity.

step 1.1∎

Depends on

Used by

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