PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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 additive functor applies degreewise to complexes and chain maps

Statement

Let $F : A \to B$ be an additive functor between abelian categories. Applying $F$ degreewise to a chain complex $C_{∙}$ and to a chain map $f : C_{∙} \to D_{∙}$ produces a chain complex $F (C)_{∙}$ and a chain map $F (f) : F (C)_{∙} \to F (D)_{∙}$ .

Facts & Assumptions

Given: An additive functor $F : A \to B$ between abelian categories.

[L1]

Additive functors preserve zero morphisms (An additive functor preserves zero morphisms).

[L2]

A chain complex satisfies $d_{n - 1} d_{n} = 0$ (Chain complex in an abelian category).

[L3]

A chain map commutes with differentials (Chain map).

Proof

technique · direct

1.1

If $C_{∙}$ is a chain complex, then $F (d_{n - 1}^{C}) F (d_{n}^{C}) = F (d_{n - 1}^{C} d_{n}^{C}) = F (0) = 0$ by [L1] and [L2]. Hence the objects $F (C_{n})$ with differentials $F (d_{n}^{C})$ form a chain complex.

L1L2givenalgebra

2.1

If $f : C \to D$ is a chain map, then [L3] gives $d_{n}^{D} f_{n} = f_{n - 1} d_{n}^{C}$ . Applying $F$ yields $F (d_{n}^{D}) F (f_{n}) = F (f_{n - 1}) F (d_{n}^{C}),$ so the family $F (f_{n})$ is a chain map $F (C) \to F (D)$ .

L3step 1.1algebra∎

Depends on

Used by

FALSE: an additive functor commutes with homology False statement
An exact functor commutes with homology Theorem

Dependency tree · two levels

7 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)
The Stacks Project, Section 12.7: Additive functors (standard reference, not scraped)