CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (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.

A finite local module has depth at most its dimension

Statement

Assume the Axiom of Choice.

If $(R, m)$ is Noetherian local and $0 \neq M$ is a finite $R$ -module, then ${depth}_{R} (M) \leq \dim_{R} (M) : = \dim {Supp}_{R} (M) .$

Facts & Assumptions

Given: The Axiom of Choice, and $m \in {Supp}_{R} (M)$ because $M \neq 0$ is finite.

Proof

technique · direct

1.1given

Under the stated AC, apply Depth at a prime is bounded by local support dimension at $p = m$ .

2.1step 1.1algebra∎

Localization at the maximal ideal changes neither $R$ , $M$ , nor the support dimension, giving the displayed inequality.

Depends on

Used by

Dependency tree · two levels

10 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)