False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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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FALSE: any sequence of morphisms is a chain complex

Statement

Any sequence of morphisms in an abelian category is a chain complex.

Facts & Assumptions

Given: The sequence $Z \overset{1_{Z}}{\to} Z \overset{1_{Z}}{\to} Z$ in $A b$ .

[L1]

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

[L2]

A chain complex requires consecutive composites to be zero (Chain complex in an abelian category).

Refutation

technique · direct

1.1L1given

By [L1], the displayed sequence is a sequence of morphisms in an abelian category.

2.1L2step 1.1algebra∎

Its consecutive composite is $1_{Z} \circ 1_{Z} = 1_{Z}$ , which is not the zero endomorphism of $Z$ . Therefore [L2] shows that this sequence is not a chain complex.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)