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.

The middle lower bound in the Depth Lemma

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(R, m)$ be a Noetherian local ring and let $0 ⟶ A ⟶ B ⟶ C ⟶ 0$ be a short exact sequence of finite $R$ -modules. Then ${depth}_{R} (B) \geq \min {{depth}_{R} (A), {depth}_{R} (C)} .$

Facts & Assumptions

Given: The Axiom of Choice; the displayed short exact sequence; write $a, b, c$ for the depths of $A, B, C$ . Depth $+ \infty$ has the convention fixed in def-depth-with-respect-to-an-ideal.

Proof

technique · direct

1.1given

By cor-depth-as-first-nonzero-ext, ${Ext}_{R}^{i} (R / m, A)$ and ${Ext}_{R}^{i} (R / m, C)$ vanish for every $i < \min {a, c}$ .

2.1step 1.1algebra∎

Fix $i < \min {a, c}$ . A finite partial free resolution of $R / m$ through degree $i + 1$ exists because $R$ is Noetherian. Applying its finite free terms to the short exact sequence gives the exact cohomology fragment ${Ext}_{R}^{i} (R / m, A) ⟶ {Ext}_{R}^{i} (R / m, B) ⟶ {Ext}_{R}^{i} (R / m, C) .$ Its outer terms vanish by step 1.1, so the middle term vanishes. This argument is finite for each $i$ and uses no choice principle. The Ext characterization now gives $b \geq \min {a, c}$ .

Depends on

Dependency tree · 2 levels

Sources

Depth and Cohen--Macaulay modules source treatment