CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-09 (gpt-6-astra)

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.

Completion preserves Cohen--Macaulayness in both directions

Statement

Assume the Axiom of Choice. If $(R, m)$ is Noetherian local and $0 \neq M$ is finite, then $M$ is Cohen--Macaulay over $R$ if and only if $\hat{M} = M \otimes_{R} \hat{R}$ is Cohen--Macaulay over $\hat{R}$ .

Facts & Assumptions

Given: the stated Noetherian local ring, nonzero finite module, and Axiom of Choice (The Axiom of Choice).

Proof

technique · direct

1.1given

By Completion reflects depth, ${depth}_{\hat{R}} (\hat{M}) = {depth}_{R} (M)$ . Under the stated AC assumption, Completion preserves dimension and Hilbert-Samuel data gives $\dim_{\hat{R}} (\hat{M}) = \dim_{R} (M)$ , where dimensions mean support dimensions. AC is used through this supplier's dimension assertion and through the completion flatness supplier in the next step.

2.1step 1.1algebra∎

By Jacobson-adic completion is faithfully flat, completion is faithfully flat, so $\hat{M} \neq 0$ ; it is finite because the images of any finite generating set of $M$ generate it over $\hat{R}$ . Thus Cohen--Macaulay local modules and rings applies on both sides. The two equalities in step 1.1 show that depth equals support dimension on one side exactly when it does on the other, proving both directions.

Depends on

Used by

Completion preserves Cohen--Macaulayness Theorem

Dependency tree · two levels

18 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)