TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The local depth-zero associated-prime criterion

Statement

Let $(R, m)$ be a Noetherian local ring and let $M \neq 0$ be a finite $R$ -module. Then $depth (M) = 0 ⟺ m \in {Ass}_{R} (M) .$

Facts & Assumptions

Given: A Noetherian local ring and a nonzero finite module.

Proof

technique · direct

1.1

Nakayama gives $m M \neq M$ , so the ideal-relative criterion applies with $I = m$ . It says depth is zero exactly when $m \subseteq p$ for some associated prime $p$ .

given

2.1

Every associated prime is proper and every proper ideal of a local ring is contained in $m$ . Hence $m \subseteq p$ forces $p = m$ , proving both directions.

step 1.1algebra∎

Depends on

Depth zero and associated primes

Used by

Depth is additive for a flat local homomorphism Corollary
A non-Cohen--Macaulay local quotient Example
A parameter sequence that fails in a non-Cohen--Macaulay ring Example

Dependency tree · two levels

3 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)