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.

Regular quotients and Cohen--Macaulayness

Statement

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

Let $(R, m)$ be Noetherian local, let $0 \neq M$ be finite, and let $x_{1}, \dots, x_{r}$ be both $M$ -regular and an initial segment of a system of parameters for $M$ . Then $M$ is Cohen--Macaulay if and only if $M / (x_{1}, \dots, x_{r}) M$ is Cohen--Macaulay.

Facts & Assumptions

Given: The Axiom of Choice; each successive quotient is nonzero, and the remaining parameter tuple shows that each $x_{i}$ is a parameter element on the preceding quotient.

Proof

technique · direct

1.1given

Apply cor-regular-quotient-cohen-macaulay-equivalence first to $M$ and $x_{1}$ , then to each successive quotient and $x_{i}$ .

2.1step 1.1algebra∎

Chaining the resulting equivalences proves the assertion. The case $r = 0$ is the identity statement.

Depends on

Used by

Dependency tree · two levels

5 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)