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.

Zero-dimensional finite local modules are Cohen--Macaulay

Statement

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

Every nonzero finite module $M$ of dimension $0$ over a Noetherian local ring is Cohen--Macaulay.

Facts & Assumptions

Given: The Axiom of Choice; $M \neq 0$ and $\dim_{R} (M) = 0$ .

Proof

technique · direct

1.1given

Depth is nonnegative, while the AC-qualified cor-depth-of-a-finite-local-module-at-most-its-dimension gives ${depth}_{R} (M) \leq 0$ .

2.1step 1.1algebra∎

Thus ${depth}_{R} (M) = 0 = \dim_{R} (M)$ , which is precisely the definition of Cohen--Macaulayness.

Depends on

Used by

Dense relative-dimension strata in flat finitely presented fibres Lemma

Dependency tree · two levels

7 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)