Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (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.

Depth is bounded by the quotient dimension at every associated prime

Statement

Assume the Axiom of Choice.

Let (R,m) be Noetherian local, let M≠0 be finite, and let p∈Ass⁡R(M). Then depth⁡(M)≤dim⁡(R/p).

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice); the data in the statement.

Proof

technique · direct
1.1givenchoose

Induct on d=depth⁡(M). The case d=0 is immediate. If d>0, choose an M-regular x∈m. Then x∉p. Choose q minimal over p+(x). The principal ideal theorem and p⊊q give dim⁡(R/q)≤dim⁡(R/p)−1.

2.1step 1.1algebra∎

The power-quotient lemma puts q in Ass⁡(M/xnM) for some n≥1. Since xn is regular, the quotient has depth d−1. Induction gives d−1≤dim⁡(R/q)≤dim⁡(R/p)−1, as required.

Depends on

Used by

Dependency tree · two levels

18 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