False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30

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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

1.1L1L3givenalgebra

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 $H_{1} ≅ Z / 2$ .

2.1L2step 1.1∎

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.

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)