TheoremStatement: 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 equals the maximal regular-sequence length

Statement

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

Let $R$ be Noetherian, $M \neq 0$ finite, and $I$ an ideal contained in the Jacobson radical with $I M \neq M$ . Then every maximal $M$ -regular sequence in $I$ has length ${depth}_{I} (M)$ , and this common integer is the first degree in which ${Ext}_{R}^{*} (R / I, M)$ is nonzero.

Facts & Assumptions

Given: The Axiom of Choice and the hypotheses in the statement.

Proof

technique · direct

1.1given

By lem-maximal-regular-sequences-have-common-length-ext, the length of every maximal regular sequence is the first nonzero Ext degree.

2.1step 1.1∎

The Ext characterization identifies that same degree with ${depth}_{I} (M)$ , proving both assertions.

Depends on

Used by

Nothing in the library uses this result yet.

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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)