False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30‡ rests on unproved material (inherited)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: an additive functor commutes with homology

Statement

Every additive functor between abelian categories commutes with homology.

Facts & Assumptions

Given: The additive functor $F : A b \to A b$ defined by $F (A) = A / 2 A$ , and the complex $0 \to Z \overset{\times 2}{\to} Z \to 0.$

[L1]

Additive functors apply degreewise to complexes and chain maps (An additive functor applies degreewise to complexes and chain maps).

[L2]

Exactness is the hypothesis that makes a functor commute with homology (An exact functor commutes with homology).

[L3]

Additivity means preservation of sums on hom-groups (Additive functor).

Refutation

technique · direct

1.1

The functor $F$ is additive in the sense of [L3], and by [L1] it sends the displayed complex to $0 \to Z / 2 \overset{0}{\to} Z / 2 \to 0,$ because multiplication by $2$ becomes zero after quotienting by $2 A$ . The original complex has $H_{1} = 0$ and $H_{0} ≅ Z / 2$ , while the new complex has $H_{1} ≅ Z / 2$ .

L1L3givenalgebra

2.1

Therefore $F (H_{1} (C)) = 0 but H_{1} (F (C)) ≅ Z / 2,$ so $F$ does not commute with homology. The contrast with [L2] shows that exactness is genuinely load-bearing.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

The Stacks Project, Section 12.7: Additive functors (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)