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.

Parameters and regular sequences in Cohen--Macaulay modules

Statement

Assume the Axiom of Choice. For a nonzero finite module $M$ over a Noetherian local ring, the following are equivalent:

$M$ is Cohen--Macaulay;
every system of parameters for $M$ is $M$ -regular;
some system of parameters for $M$ is $M$ -regular.

Facts & Assumptions

Given: The Axiom of Choice and a nonzero finite module $M$ over a Noetherian local ring.

[L1]

Under AC, every system of parameters of a Cohen--Macaulay module is regular (Every system of parameters is regular in a Cohen--Macaulay module).

[L2]

A regular system of parameters makes its module Cohen--Macaulay (One regular system of parameters implies Cohen--Macaulayness).

[L3]

Systems of parameters exist for nonzero finite modules over Noetherian local rings, and their length is $\dim_{R} (M)$ (For a finite module, the dimension is the least size of an ideal of definition, and such tuples are systems of parameters).

Proof

technique · direct

1.1givenL1L3

Under the stated AC, [L1] gives (1) $\Rightarrow$ (2). By [L3] there is at least one system of parameters, so (2) $\Rightarrow$ (3) is not merely vacuous.

2.1L2step 1.1∎

By [L2], (3) $\Rightarrow$ (1). Thus all three conditions are equivalent.

Depends on

Used by

Dependency tree · two levels

15 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)