Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 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.

Radical, localization, and regular-quotient properties of depth

Statement

Assume the Axiom of Choice (The Axiom of Choice).

For finite modules over Noetherian rings, depth has the following properties.

  1. Ideals with the same radical and contained in the Jacobson radical give the same depth.
  2. If (R,m) is local, then depth⁡(Mp)+dim⁡(R/p)≥depth⁡(M) for every prime p.
  3. If x∈I is M-regular and I is in the Jacobson radical, then depth⁡I(M/xM)=depth⁡I(M)−1.

Facts & Assumptions

Given: the Axiom of Choice and, in each part, the hypotheses stated there.

[L1]

Depth is invariant under radical for the ideals in part 1 (Depth depends only on the radical of the ideal).

[L2]

Under Choice, depth obeys the localization inequality of part 2, including the zero-localization convention (Localization gives the stated depth inequality).

[L3]

Quotienting by a regular element lowers depth by one as in part 3 (Depth drops by one after quotienting by a regular element).

Proof

technique · direct
1.1L1L2given

Part 1 follows from [L1], and part 2 follows from [L2] under the stated Choice assumption.

2.1L3step 1.1∎

Part 3 follows from [L3]. The three cited results prove the package.

Depends on

Used by

Dependency tree · two levels

12 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