TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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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

1.1L1L2L3

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 $B_{n} (F (C))$ .

2.1L1L3step 1.1

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 $F (H_{n} (C)) ≅ H_{n} (F (C))$ .

3.1L2step 1.1step 2.1∎

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.

An exact functor commutes with homology

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources