CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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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.

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Unequal-depth equalities in a short exact sequence

Statement

Assume the Axiom of Choice (The Axiom of Choice). In the setting of thm-depth-lemma, the following implications hold: $\begin{array}{lll} a < c \Rightarrow b = a, & a > c + 1 \Rightarrow b = c, \\ a < b \Rightarrow c = a - 1, & a > b \Rightarrow c = b, \\ b < c \Rightarrow a = b, & b > c \Rightarrow a = c + 1. \end{array}$ Only implications whose strict inequality is meaningful for the depth conventions are asserted.

Facts & Assumptions

Given: The Axiom of Choice; write $a, b, c$ for the depths of $A, B, C$ in the short exact sequence.

Proof

technique · direct

1.1given

The first Depth Lemma inequality and either of the other two give $a < c \Rightarrow b = a$ and $a > c + 1 \Rightarrow b = c$ . Cyclically applying the same comparison to the remaining inequalities gives the other four displayed implications.

2.1step 1.1algebra∎

For example, if $a < c$ , then $b \geq a$ while $a \geq \min {b, c + 1}$ forces $a \geq b$ ; hence $b = a$ . If $a > c + 1$ , then $c \geq \min {a - 1, b}$ and $a - 1 > c$ force $c \geq b$ , while $b \geq c$ ; hence $b = c$ . If $a < b$ , then $c \geq a - 1$ , while $a \geq \min {b, c + 1}$ forces $a \geq c + 1$ ; hence $c = a - 1$ . The other three cases are the same comparisons.

Depends on

Used by

Nothing in the library uses this result yet.

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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)