CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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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.

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An acyclic noncontractible complex from a nonsplit extension

Statement refuted

Every acyclic complex is contractible.

Facts & Assumptions

Given: The three-term complex $0 \to Z \overset{2}{\to} Z \overset{m o d 2}{\to} Z / 2 \to 0.$

[L1]

Every contractible complex is acyclic (A contractible complex is acyclic).

[L2]

Being zero in the homotopy category is stronger than having zero homology (Zero homology does not make an object zero in the homotopy category).

[L3]

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

Counterexample

technique · direct

1.1L3givenalgebra

The complex is acyclic: multiplication by $2$ is injective, reduction mod $2$ is surjective, and its kernel is $2 Z$ , which is the image of the first map.

2.1L1L2step 1.1algebra∎

If the complex were contractible, its identity map would be null-homotopic. In degree $0$ , that would force the surjection $Z \to Z / 2$ to have a section, so the short exact sequence would split. It does not split, so the complex is not contractible. Hence the displayed complex refutes the statement, exactly as [L2] warns; [L1] remains true as the forward implication.

Depends on

Used by

A quasi-isomorphism with no homotopy inverse Counterexample

Dependency tree · two levels

13 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)