ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A contracting homotopy for the two-term identity complex

Example

In $A b$ , consider the chain complex $0 \to Z \overset{1_{Z}}{\to} Z \to 0,$ with the left copy of $Z$ in degree $1$ and the right copy in degree $0$ . It is contractible: a contracting homotopy is given by $s_{0} = 1_{Z}$ and $s_{n} = 0$ for $n \neq 0$ .

Facts & Assumptions

Given: The two-term identity complex $C_{∙}$ in $A b$ .

[L1]

A chain homotopy from $1_{C}$ to $0$ is a degree- $1$ family $s$ with $1_{C} = d s + s d$ (A chain homotopy).

[L2]

A complex is contractible when its identity is null-homotopic (A contractible complex).

[L3]

$A b$ is an abelian category (Abelian groups form an abelian category).

Verification

technique · direct

1.1L1L3givenalgebra

Define $s_{0} = 1_{Z} : C_{0} \to C_{1}$ and $s_{n} = 0$ otherwise. In degree $1$ , $d_{2} s_{1} + s_{0} d_{1} = 0 + 1_{Z} = 1_{C_{1}},$ and in degree $0$ , $d_{1} s_{0} + s_{- 1} d_{0} = 1_{Z} + 0 = 1_{C_{0}} .$

2.1L1L2step 1.1algebra∎

Thus $1_{C} = d s + s d$ , so [L1] makes $1_{C}$ homotopic to $0$ . By [L2], the complex is contractible.

Depends on

Used by

Two homotopic maps with different components Example

Dependency tree · two levels

14 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)