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 right lower bound in the Depth Lemma

Statement

Assume the Axiom of Choice (The Axiom of Choice). Under the hypotheses of the Depth Lemma, ${depth}_{R} (C) \geq \min {{depth}_{R} (A) - 1, {depth}_{R} (B)} .$ When ${depth}_{R} (A) = 0$ , the right side is $- 1$ and the assertion is vacuous.

Facts & Assumptions

Given: The Axiom of Choice; a short exact sequence $0 \to A \to B \to C \to 0$ of finite modules over a Noetherian local ring $(R, m)$ ; write $a, b, c$ for their depths.

Proof

technique · direct

1.1given

If $a = 0$ there is nothing to prove. Otherwise put $s = \min {a - 1, b}$ . For every $i < s$ , the Ext characterization gives ${Ext}_{R}^{i} (R / m, B) = 0$ and ${Ext}_{R}^{i + 1} (R / m, A) = 0$ .

2.1step 1.1algebra∎

For each fixed $i < s$ , a finite partial free resolution of $R / m$ through degree $i + 2$ gives the exact fragment ${Ext}_{R}^{i} (R / m, B) ⟶ {Ext}_{R}^{i} (R / m, C) ⟶ {Ext}_{R}^{i + 1} (R / m, A) .$ The outer terms vanish by step 1.1. Thus the middle term vanishes for every $i < s$ , and $c \geq s$ . The construction is finite for each $i$ and needs no choice principle.

Depends on

Dependency tree · 2 levels

Sources

Depth and Cohen--Macaulay modules source treatment