Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The three Depth Lemma inequalities

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let (R,m) be a Noetherian local ring and 0→A→B→C→0 a short exact sequence of finite R-modules. With a=depth⁡R(A), b=depth⁡R(B), and c=depth⁡R(C), b≥min⁡{a,c},a≥min⁡{b,c+1},c≥min⁡{a−1,b}. The last inequality is vacuous when a=0.

Facts & Assumptions

Given: The Axiom of Choice, displayed exact sequence, and depth notation.

Proof

technique · direct
1.1given

The middle, left, and right lower bounds are respectively lem-depth-lemma-lower-bound-middle, lem-depth-lemma-lower-bound-left, and lem-depth-lemma-lower-bound-right.

2.1step 1.1algebra∎

Substituting a,b,c into those three bounds gives the three displayed inequalities, including the one-degree shifts and the stated a=0 boundary.

Depends on

Used by

Dependency tree · two levels

6 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