LemmaStatement: 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 gives the stated depth inequality

Statement

Assume the Axiom of Choice.

Let $(R, m)$ be Noetherian local, let $M$ be finite, and let $p \in Spec R$ . Then ${depth}_{R_{p}} (M_{p}) + \dim (R / p) \geq {depth}_{R} (M),$ with the left side infinite when $M_{p} = 0$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice); the data in the statement.

Proof

technique · direct

1.1givenchoose

The claim is immediate if $M_{p} = 0$ or if $depth M \leq \dim (R / p)$ . Otherwise the associated-prime dimension bound shows that $p$ is contained in no member of $Ass (M)$ . Finite prime avoidance supplies an $M$ -regular $x \in p$ .

2.1step 1.1algebra∎

Localizing preserves regularity of $x$ , and quotienting by it lowers both $depth M$ and $depth M_{p}$ by one. Induction on $depth M$ applied to $M / x M$ proves the inequality after adding one.

Depends on

Used by

Dependency tree · two levels

16 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)