TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (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.

Localization of Cohen--Macaulay modules

Statement

Let $R$ be Noetherian and $M$ finite and globally Cohen--Macaulay. For every multiplicative set $S$ , each nonzero localization of $S^{- 1} M$ at a prime of $S^{- 1} R$ is Cohen--Macaulay. In particular, assuming the Axiom of Choice (The Axiom of Choice), if $R$ is local and $M$ is Cohen--Macaulay, then $M_{p}$ is Cohen--Macaulay for every $p \in Supp (M)$ ; primes outside the support give the zero module and are excluded by the local definition.

Facts & Assumptions

Given: primes of $S^{- 1} R$ correspond to primes of $R$ disjoint from $S$ , and iterated localization agrees with localization at the corresponding prime. The Axiom of Choice is assumed for the local special case only.

Proof

technique · direct

1.1given

At a corresponding support prime $p$ , the definition of global Cohen--Macaulayness says $M_{p}$ is Cohen--Macaulay (Maximal and global Cohen--Macaulay modules).

2.1step 1.1given∎

The relevant localization of $S^{- 1} M$ is canonically $M_{p}$ , so it is Cohen--Macaulay. Under Choice, the local special case follows from Cohen--Macaulayness localizes; primes outside the support give the zero module.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)