Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 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.

Depth depends only on the radical of the ideal

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let R be Noetherian, M finite, and I,J ideals contained in the Jacobson radical. If I=J, then depth⁡I(M)=depth⁡J(M), including the value ∞.

Facts & Assumptions

Given: The Axiom of Choice and the ring, module, and ideals in the statement.

[L1]

Equal radicals give equal first nonzero Ext degrees, including simultaneous infinity (Radical invariance of first nonzero Ext).

[L2]

Under the assumed Axiom of Choice, depth for a finite module and a Jacobson-radical ideal is its first nonzero Ext degree (Depth as the first nonzero Ext degree).

Proof

technique · direct
1.1L1given

Apply [L1] to I and J: the least nonvanishing degrees of Ext⁡R∗(R/I,M) and Ext⁡R∗(R/J,M) agree, with both infinite if neither family has a nonzero group.

2.1L2step 1.1given∎

Apply [L2] separately to I and J; both lie in the Jacobson radical by hypothesis. Their depths are the degrees equal in step 1.1, including the infinite case.

Depends on

Used by

Dependency tree · two levels

9 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