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LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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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.

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Associated primes of a Cohen--Macaulay module have full dimension

Statement

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

Let (R,m) be Noetherian local and let 0≠M be a finite Cohen--Macaulay R-module of dimension d. For every p∈Ass⁡R(M), dim⁡(R/p)=d.

Facts & Assumptions

Given: the Axiom of Choice; every associated prime belongs to Supp⁡(M).

[L1]

Under Choice, depth is bounded by the quotient dimension at every associated prime (Depth is bounded by the quotient dimension at every associated prime).

Proof

technique · direct
1.1L1given

Cohen--Macaulayness and [L1] give d=depth⁡R(M)≤dim⁡(R/p).

2.1step 1.1algebra∎

Since V(p)⊆Supp⁡(M), the reverse inequality dim⁡(R/p)≤d holds. The two inequalities give equality.

Depends on

Used by

Dependency tree · two levels

13 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