TheoremStatement: Literature-sourcedProof: 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.

An exact functor commutes with homology

Statement

Let $F : A \to B$ be an exact functor between abelian categories. For every chain complex $C_{∙}$ and every $n \in Z$ , there is a canonical isomorphism $F (H_{n} (C)) \tilde{\to} H_{n} (F (C)) .$ These isomorphisms are natural in chain maps.

Facts & Assumptions

Given: An exact functor $F : A \to B$ , a chain complex $C_{∙}$ , and an integer $n$ .

[L1]

An exact functor preserves kernels and cokernels (An additive functor is exact exactly when it preserves kernels and cokernels).

[L2]

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

[L3]

$H_{n} (C)$ is the cokernel of the canonical map $β_{n} : B_{n} (C) \to Z_{n} (C)$ (Homology object of a chain complex).

Proof

technique · direct

1.1

By [L2], $F (C)$ is a chain complex. Because $F$ preserves kernels and cokernels by [L1], it carries $\ker (d_{n}^{C})$ to $\ker (d_{n}^{F (C)})$ and $im (d_{n + 1}^{C})$ to $im (d_{n + 1}^{F (C)})$ . Thus it identifies $F (Z_{n} (C))$ with $Z_{n} (F (C))$ and $F (B_{n} (C))$ with $B_{n} (F (C))$ .

L1L2L3

2.1

Applying $F$ to the cokernel description in [L3] and using [L1], the object $F (H_{n} (C))$ is the cokernel of the image of $β_{n}$ under $F$ . Under the identifications of step 1.1, that cokernel is exactly $H_{n} (F (C))$ . This gives the canonical isomorphism $F (H_{n} (C)) ≅ H_{n} (F (C))$ .

L1L3step 1.1

3.1

Naturality follows because the cycle maps, boundary maps, and quotient maps used in steps 1.1 and 2.1 are all functorial in the chain map input.

L2step 1.1step 2.1∎

Depends on

Used by

An exact functor preserves quasi-isomorphisms and reflects them when it is conservative Corollary
FALSE: an additive functor commutes with homology False statement

Dependency tree · two levels

14 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, Lemma 12.7.2 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra, Lemma 2.5.2 (standard reference, not scraped)