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TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Associated primes of Cohen--Macaulay modules

Statement

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

If 0≠M is a finite Cohen--Macaulay module of dimension d over a Noetherian local ring R, then every p∈Ass⁡R(M) is minimal in Supp⁡R(M) and satisfies dim⁡(R/p)=d.

Facts & Assumptions

Given: the Axiom of Choice and the local Cohen--Macaulay hypotheses stated above.

[L1]

Under Choice, every associated prime has full quotient dimension (Associated primes of a Cohen--Macaulay module have full dimension).

[L2]

Under the hypotheses of [L1], every associated prime is minimal in the support (Cohen--Macaulay modules have no embedded associated primes).

Proof

technique · direct
1.1L1L2given

Full quotient dimension is [L1], and minimality is [L2].

2.1step 1.1algebra∎

Combining those conclusions gives both assertions simultaneously.

Depends on

Used by

Dependency tree · two levels

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Sources