CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.
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Cohen--Macaulay modules have no embedded associated primes

Statement

Under the hypotheses of lem-associated-primes-of-cohen-macaulay-module-have-full-dimension, every associated prime of $M$ is minimal in ${Supp}_{R} (M)$ . Thus $M$ has no embedded associated primes.

Facts & Assumptions

Given: $p \in {Ass}_{R} (M)$ .

Proof

technique · direct

1.1given

Suppose $q \in Supp (M)$ and $q ⊊ p$ . A chain of length $d$ above $p$ , preceded by the strict inclusion $q ⊊ p$ , gives $\dim (R / q) > d$ .

2.1step 1.1algebra∎

But $V (q) \subseteq Supp (M)$ implies $\dim (R / q) \leq \dim M = d$ , a contradiction. Hence no such $q$ exists.

Depends on

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

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)